From SS-matrix theory to strings: scattering data and the commitment to non-arbitrariness 从 SS 矩阵理论到弦：散射数据和对非任意性的承诺
Robert van Leeuwen ^(1){ }^{1} 罗伯特-范-利乌文 ^(1){ }^{1}Institute for Theoretical Physics 理论物理研究所Vossius Center for History of the Humanities and the Sciences 沃西乌斯人文与科学史中心University of Amsterdam² 阿姆斯特丹大学²
Abstract 摘要
The early history of string theory is marked by a shift from strong interaction physics to quantum gravity. The first string models and associated theoretical framework were formulated in the late 1960s and early 1970s in the context of the SS-matrix program for the strong interactions. In the mid-1970s, the models were reinterpreted as a potential theory unifying the four fundamental forces. This paper provides a historical analysis of how string theory was developed out of SS-matrix physics, aiming to clarify how modern string theory, as a theory detached from experimental data, grew out of an SS-matrix program that was strongly dependent upon observable quantities. Surprisingly, the theoretical practice of physicists already turned away from experiment before string theory was recast as a potential unified quantum gravity theory. With the formulation of dual resonance models (the “hadronic string theory”), physicists were able to determine almost all of the models’ parameters on the basis of theoretical reasoning. It was this commitment to “nonarbitrariness”, i.e., a lack of free parameters in the theory, that initially drove string theorists away from experimental input, and not the practical inaccessibility of experimental data in the context of quantum gravity physics. This is an important observation when assessing the role of experimental data in string theory. 弦理论的早期历史以从强相互作用物理学转向量子引力为标志。最早的弦模型和相关理论框架是在20世纪60年代末和70年代初根据强相互作用的 SS 矩阵计划提出的。20 世纪 70 年代中期，这些模型被重新诠释为统一四种基本力的潜在理论。本文对弦理论如何从 SS 矩阵物理学发展而来进行了历史分析，旨在阐明现代弦理论作为一种脱离实验数据的理论，是如何从强烈依赖于可观测量的 SS 矩阵计划发展而来的。令人惊讶的是，在弦理论被重塑为潜在的统一量子引力理论之前，物理学家的理论实践就已经脱离了实验。随着双共振模型（"强子弦理论"）的提出，物理学家能够根据理论推理确定模型的几乎所有参数。正是这种对 "非任意性 "的承诺，即理论中缺乏自由参数，最初驱使弦理论学家远离实验输入，而不是量子引力物理学中实验数据的实际不可得性。在评估实验数据在弦理论中的作用时，这是一个重要的观察结果。
Keywords: history of string theory; S-matrix theory; duality; non-arbitrariness; bootstrap; history and philosophy of physics 关键词：弦理论史；S 矩阵理论；对偶性；非任意性；自举法；物理学史与物理学哲学
1. Introduction 1.导言
The early history of string theory is marked by a transition from nuclear physics to quantum gravity. The first string models and the associated theoretical framework were formulated in the late 1960s and early 1970s in an attempt to describe the properties of strongly interacting particles. In the mid1970s, proposals were made to drastically change the scale of the theory and to reinterpret it as a potential theory unifying the four fundamental forces. ^(3){ }^{3} It was in this guise that string theory experienced its major breakthrough as an important candidate for a unified theory of quantum gravity, due to an important result by Green and Schwarz (1984) that made it possible to formulate “finite” string theories encompassing Standard Model symmetries for the first time. ^(4){ }^{4} 弦理论的早期历史以从核物理过渡到量子引力为标志。最早的弦模型和相关理论框架是在 20 世纪 60 年代末和 70 年代初提出的，试图描述强相互作用粒子的特性。1970年代中期，有人建议大幅改变该理论的尺度，并将其重新解释为统一四种基本力的潜在理论。 ^(3){ }^{3} 正是在这一背景下，弦理论作为量子引力统一理论的重要候选理论取得了重大突破，这要归功于格林和施瓦茨（1984 年）的一项重要成果，它首次使人们有可能提出包含标准模型对称性的 "有限 "弦理论。 ^(4){ }^{4}
String theory’s shift from hadronic physics to quantum gravity also meant that the theory became disconnected from experimental data in any straightforward sense. In stark contrast to this, string theory originated in the so-called SS-matrix program for the strong interactions which was strongly dependent upon experimental results: the SS-matrix particle physics program was grounded in the attitude that the theory should be restricted to mathematical relations between observable scattering amplitudes only. ^(5){ }^{5} This somewhat puzzling shift is also noted by historian of science Dean Rickles (2014) in his A Brief History of String Theory. Rickles points out a 弦理论从强子物理转向量子引力，也意味着该理论在任何直接意义上都与实验数据脱节了。与此形成鲜明对比的是，弦理论起源于所谓的 SS 强相互作用矩阵计划，该计划强烈依赖于实验结果： SS 矩阵粒子物理学计划的基本态度是，理论应仅限于可观测散射振幅之间的数学关系。 ^(5){ }^{5} 科学史学者迪恩-里克斯（Dean Rickles，2014 年）在其《弦理论简史》中也提到了这一令人费解的转变。里克斯指出了
certain irony in how things have developed from SS-matrix theory since its primary virtue was that it meant that one was dealing entirely in observable quantities (namely, scattering amplitudes). Yet, string theory grew out of SS-matrix theory. Of course, most of the complaints with string theory, since its earliest days, have been levelled at its detachment from measurable quantities. (p. 16, italics in original) 从 SS -矩阵理论发展而来的东西具有某种讽刺意味，因为它的主要优点在于，它意味着人们完全是在处理可观测的量（即散射振幅）。然而，弦理论却是从 SS 矩阵理论发展而来的。当然，自弦理论诞生之日起，人们对它的大多数抱怨都是由于它脱离了可测量的量。(第 16 页，斜体为原文所加）
Yet, while noting the “irony” of the development from SS-matrix theory to quantum gravity string theory, Rickles’ account of string theory’s early history does mostly emphasize the break constituted by string theory’s reinterpretation as a potential unified theory. As he argues, theoretical notions in string theory underwent “several quite radical transformations”, and while there is a “clear continuity of structure linking these changes”, it is in some cases (and especially in the case of the shift to quantum gravity) better to think of the resulting theoretical structure as a different theory. Thus, while acknowledging that “certain philosophical residues (such as the distaste for arbitrariness in physics) from the SS-matrix program stuck to string theory”, Rickles hastens to make clear that string theory “soon became a very different structure”. ^(6){ }^{6} In his view, 然而，在指出从 SS 矩阵理论到量子引力弦理论的发展具有 "讽刺意味 "的同时，Rickles对弦论早期历史的描述主要强调了弦论被重新诠释为潜在统一理论所构成的断裂。正如他所言，弦理论的理论概念经历了 "几次相当彻底的转变"，虽然 "这些变化之间存在着明显的结构连续性"，但在某些情况下（尤其是在向量子引力转变的情况下），最好还是把由此产生的理论结构视为另一种理论。因此，虽然里克尔斯承认" SS -矩阵计划的某些哲学残余（如对物理学中任意性的反感）保留到了弦理论中"，但他还是赶紧明确指出，弦理论 "很快就变成了一种非常不同的结构"。 ^(6){ }^{6} 在他看来、
the switch that occurred when [string theory] changed from being a theory of strong interactions to a theory incorporating gravitational interactions and Yang-Mills fields [is] a clear case in which it makes sense to think of the resulting theory as a genuinely new theory, couched in a near-identical framework. There was no switch; rather, a distinct theory was constructed. (p. 17) 当[弦理论]从强相互作用理论转变为包含引力相互作用和杨-米尔斯场的理论时所发生的转换[是]一个明显的例子，在这个例子中，把由此产生的理论视为一个真正的新理论是有道理的，它被置于一个近乎相同的框架之中。没有转换，而是构建了一个独特的理论。(p. 17)
So for Rickles, with its reinterpretation as a candidate unified theory, string theory in a sense started anew. 因此，对里克斯来说，弦理论被重新诠释为候选统一理论，从某种意义上说是重新开始了。
This point of view is understandable: a unified theory of the fundamental interactions is of course different from a theory of hadronic particles in many ways, and with the transition to quantum gravity a whole range of new theoretical possibilities and interpretations opened up. More generally, one can of course speak of a “new” theory even when it is to a large extent building upon an older one. Yet, a highly problematic consequence of emphasizing too much the novelty of unified quantum gravity string theory is that it obscures certain motivations that were guiding string theory’s construction already as a hadronic theory, and that are crucial for a proper understanding of quantum gravity string theory’s relation to experimental data. The most important of these, I will argue, was the aspiration of particle theorists while developing dual resonance models (the “hadronic string theory”) to construct a theory with as few free parameters (to be determined on the basis of experiment) as possible. This is what Rickles designates above as a “distaste for arbitrariness” and what I will refer to as striving for “non-arbitrariness”. The commitment to the ideal of a theory without free parameters is not, as Rickles suggests, a passive “philosophical residue” from SS-matrix theory that stuck to string theory, but was instead crucial in driving the practice of theory construction away from the use of experimental data, already before string theory was recast as a candidate unified quantum gravity theory. As such, I claim, dual resonance models are the missing link between quantum gravity string theory, as a theory detached from experimental data, and SS matrix theory that was strongly dependent upon observable quantities. 这种观点是可以理解的：基本相互作用的统一理论当然在许多方面不同于强子粒子理论，而且随着向量子引力的过渡，一系列新的理论可能性和解释也随之出现。更一般地说，即使在很大程度上是建立在旧理论的基础上，我们当然也可以说它是一种 "新 "理论。然而，过分强调统一量子引力弦理论的新颖性会带来一个很大的问题，那就是它掩盖了指导弦理论作为强子理论构建的某些动机，而这些动机对于正确理解量子引力弦理论与实验数据的关系至关重要。我要论证的是，其中最重要的动机是粒子理论家在发展双共振模型（"强子弦理论"）时，希望构建一种自由参数（根据实验确定）尽可能少的理论。这就是里克尔斯在上文所说的 "厌恶任意性"，而我将称之为追求 "非任意性"。对没有自由参数的理论这一理想的承诺，并不像里克尔斯所说的那样，是 SS -矩阵理论被动的 "哲学残余"，而不是弦理论的 "哲学残余"，相反，在弦理论被重新塑造为统一量子引力理论候选者之前，它就已经在推动理论建构实践远离实验数据的使用方面发挥了至关重要的作用。因此，我认为，双共振模型是量子引力弦理论与 SS 矩阵理论之间缺失的一环，前者是一种脱离实验数据的理论，而后者则强烈依赖于可观测量。
This corrects the idea that string theory’s problematic relation with experimental data must be understood solely in the context of theory-driven quantum gravity research, as implied by the view of Dawid and Rickles. Surprisingly, the point where the theoretical practice of string theorists turned away from experiment was not the theory’s reinterpretation as a candidate unified theoryeven if this was the moment when experimental input became practically out of reach, given the extremely high energy scales of quantum gravity physics. Instead, the detachment of string theory from experimental input can already be located with the transition from SS-matrix theory to dual resonance models (that is, the original hadronic string models). This is a crucial observation when assessing the role of experimental data in the practice of string theorists: string theory did not initially become detached from experimental data because of the practical inaccessibility of experimental data on quantum gravity energy scales, but because of the involved physicists’ commitment to determine the theory’s parameters on the basis of theoretical reasoning, grounded in a set of principles. This non-arbitrariness was, however, appreciated much more in the context of unified quantum gravity, because in that case it promised an ultimate explanation for the features of the fundamental interactions. 这就纠正了达维德和里克斯的观点所暗示的，弦理论与实验数据之间的问题关系必须仅仅从理论驱动的量子引力研究的角度来理解的想法。令人惊讶的是，弦理论学家的理论实践脱离实验的起点并不是理论被重新诠释为候选统一理论--即使考虑到量子引力物理学的极高能量尺度，实验输入实际上已经变得遥不可及。相反，从 SS 矩阵理论过渡到双共振模型（即最初的强子弦模型）时，弦理论就已经脱离了实验。在评估实验数据在弦理论家实践中的作用时，这是一个至关重要的观察结果：弦理论最初脱离实验数据并不是因为量子引力能量尺度上的实验数据实际上无法获得，而是因为相关物理学家致力于根据理论推理确定理论参数，并以一系列原则为基础。然而，这种非任意性在统一量子引力的背景下更受赞赏，因为在这种情况下，它承诺对基本相互作用的特征做出终极解释。
This insight is important for the debates on string theory’s viability, which took off in the wake of string theory’s establishment as a candidate quantum gravity theory, and continue up to this day. Essentially, these debates are driven by the question whether string theorists’ claim that the theory should ultimately be able to lead to the correct unifying description of gravity and quantum theory is justified—and with that its dominant position within theoretical high-energy physics. ^(7){ }^{7} Historically, string theory’s critics have pointed out its detachment from experimental data, while proponents of string theory emphasized the merit of the theory’s internal consistency-a notion that is also of fundamental importance in the more recent philosophical defense of string theory by 在弦理论被确立为量子引力候选理论之后，关于弦理论可行性的争论就开始了，并一直持续到今天。从根本上说，这些争论的驱动力是弦理论学家关于弦理论最终应该能够导致引力和量子理论的正确统一描述的说法是否合理的问题，以及弦理论在理论高能物理中的主导地位是否合理的问题。 ^(7){ }^{7} 从历史上看，弦理论的批评者指出了弦理论与实验数据的脱节，而弦理论的支持者则强调了弦理论内部一致性的优点--这一概念在最近弦理论的哲学辩护中也至关重要。
philosopher of science Richard Dawid (2013). ^(8){ }^{8} However, it would be too simple to depict the debate as revolving solely around the question whether progress in theoretical high-energy physics can be made in the absence of empirical tests, as Dawid and some string proponents have suggested. ^(9){ }^{9} Indeed, it has become common practice in high-energy physics since at least the 1980s to search for unified theories, quantum gravity theories, and other physics beyond the Standard Model, with all practitioners agreeing that, with new experimental data practically out of reach, one simply only can rely on heuristics of theoretical judgment, even if experiment should in the end provide the verdict on a theory. Yet, in spite of this shared starting point, string theorists and their various critics diverge strongly in their judgment on string theory’s viability and on whether results in string theory can be considered “empirical”. In the words of historian of science Jeroen van Dongen (2021), string theorists “consider certain types of argument as epistemically relevant to, and valid expressions of empirical science”, while their critics disagree with that judgment. Such judgments are then ultimately “expressions of different cultures of rationality, rooted in different practices of theory” (p. 174). ^(10){ }^{10} In this paper, I present a historical analysis of the theoretical practices in which modern string theory originated. With that, I bring to the fore how the “non-arbitrariness” ideal of a theory without free parameters evolved from SS-matrix physics into string theory, was the crucial notion in string theorist’s turn away from experiment, and constitutes as such an important element of string theory’s “culture of rationality”. 科学哲学家理查德-达维德（Richard Dawid，2013 年）。 ^(8){ }^{8} 然而，如果像Dawid和一些弦支持者所说的那样，把这场争论仅仅描述为围绕着在缺乏经验检验的情况下理论高能物理能否取得进展这一问题，那就太简单了。 ^(9){ }^{9} 事实上，至少从20世纪80年代起，寻找统一理论、量子引力理论和其他超越标准模型的物理学理论就已经成为高能物理的普遍做法，所有实践者都同意，在新的实验数据几乎遥不可及的情况下，我们只能依靠理论判断的启发式方法，即使实验最终应该对理论做出裁决。然而，尽管有这个共同的出发点，弦理论家和他们的各种批评者在判断弦理论的可行性以及弦理论的结果是否可以被视为 "经验性的 "方面却存在着巨大分歧。用科学史家耶罗-范东根（Jeroen van Dongen，2021 年）的话说，弦理论家 "认为某些类型的论证在认识论上与实证科学相关，并且是实证科学的有效表达"，而他们的批评者则不同意这种判断。这种判断最终是 "不同理性文化的表达，植根于不同的理论实践"（第 174 页）。 ^(10){ }^{10} 在本文中，我对现代弦理论起源的理论实践进行了历史分析。通过这一分析，我揭示了无自由参数理论的 "非任意性 "理想是如何从 SS 矩阵物理学演变成弦理论的，它是弦理论家远离实验的关键概念，并因此构成了弦理论 "理性文化 "的一个重要元素。
In order to do so, I will divide the developments leading up to modern string theory into three phases, following a periodization that is also employed by Rickles (2014) and is reflected in the contributions in the volume The Birth of String Theory (Cappelli et al., 2012): 为此，我将把现代弦理论的发展分为三个阶段，这也是 Rickles（2014）所采用的分期方法，并反映在《弦理论的诞生》（Cappelli et al：）
Analytic SS-matrix theory for hadrons (Chapter 2). Hadronic SS-matrix theory was developed in the late 1950s and 1960s as an alternative to a field-theoretical approach to the strong interactions, indirectly motivated by Heisenberg’s SS-matrix program for quantum electrodynamics from the mid-1940s. The main aim of hadronic SS-matrix theory was to compute observable scattering amplitudes on the basis of a set of SS-matrix principles, while treating the dynamics of the scattering process as a black box. The virtue of non-arbitrariness was influential in hadronic SS-matrix theory through the notion of the “bootstrap” as advocated by Geoffrey Chew: the conjecture that imposing all SS-matrix principles on the theory’s equations could lead to a unique solution determining all features of strongly interacting particles. However, in practice the bootstrap ideal was unreachable, and SS-matrix theory was developed while experimentally obtained values were used as input. 强子的解析 SS 矩阵理论（第 2 章）。强子 SS 矩阵理论是在 20 世纪 50 年代末和 60 年代发展起来的，作为强相互作用场论方法的替代方案，其间接动机来自海森堡在 20 世纪 40 年代中期提出的量子电动力学 SS 矩阵计划。强子 SS 矩阵理论的主要目的是在一套 SS 矩阵原理的基础上计算可观测的散射振幅，同时将散射过程的动力学视为黑箱。非任意性的优点通过杰弗里-丘（Geoffrey Chew）倡导的 "自举"（bootstrap）概念在强子 SS 矩阵理论中产生了影响：将所有 SS 矩阵原理强加在理论方程上的猜想可能会导致确定强相互作用粒子所有特征的唯一解。然而，在实践中，bootstrap 的理想是无法实现的， SS - 矩阵理论是在使用实验得到的数值作为输入的情况下发展起来的。
2. Dual resonance models for hadrons (Chapter 3). Dual resonance models were a class of models that grew out of hadronic SS-matrix theory. In SS-matrix theory, the low-energy contributions to the scattering amplitude came from so-called “direct” resonances (usually pictured as a short-lived particle forming and decaying again), while the high-energy contributions were calculated using “exchanged” resonances (analogous to the exchange of force particles). In the late 1960s, an approximation that came to be known as “duality” suggested that either a sum of direct or of exchanged resonances would suffice in computing the value of the amplitude. Following the introduction of an amplitude function that satisfied this duality approximation by physicist Gabriele Veneziano (1968), a model-building enterprise took off that added duality as a principle of SS-matrix theory. These “dual resonance models” implemented non-arbitrariness, since all but one of the models’ parameters could be determined on the basis of theoretical reasoning, avoiding “arbitrary” input from experiment. Furthermore, the models admitted an interpretation in terms of string-like constituents, and the mathematical structure of string theory originated with them. 2.强子的双共振模型（第 3 章）。双共振模型是由强子 SS 矩阵理论发展而来的一类模型。在 SS -矩阵理论中，对散射振幅的低能贡献来自所谓的 "直接 "共振（通常描绘为一个短寿命粒子的形成和再次衰变），而高能贡献则通过 "交换 "共振（类似于力粒子的交换）来计算。20 世纪 60 年代末，一种后来被称为 "二元性 "的近似方法提出，直接共振或交换共振之和都足以计算振幅值。在物理学家加布里埃尔-威尼斯诺（Gabriele Veneziano，1968 年）提出了满足这种二重性近似的振幅函数之后，一种将二重性作为 SS 矩阵理论原则的模型建立事业开始兴起。这些 "二元共振模型 "实现了非任意性，因为除了一个参数之外，模型的所有参数都可以根据理论推理确定，避免了来自实验的 "任意 "输入。此外，这些模型还可以用类似弦的成分来解释，弦理论的数学结构就起源于这些模型。
3. Dual models/string theory as a candidate for a unified quantum gravity theory (1974-1984) (Chapter 3 and 4). In the mid-1970s, it was proposed to reinterpret dual models as a potential unified theory of all fundamental interactions, instead of a theory of hadrons. While work on hadronic dual resonance models diminished (also due to the empirical success of quantum chromodynamics as a theory for the strong interactions), a small group of physicists kept working on this unified theory proposal, eventually leading to string theory’s breakthrough as a major quantum gravity theory candidate in 1984. 3.双模型/弦理论作为统一量子引力理论的候选者（1974-1984 年）（第 3 章和第 4 章）。20 世纪 70 年代中期，有人提出把二元模型重新解释为所有基本相互作用的潜在统一理论，而不是强子理论。虽然有关强子双共振模型的工作有所减少（这也是由于量子色动力学作为强相互作用理论在经验上的成功），但一小群物理学家仍在继续研究这一统一理论的提议，最终导致弦理论在 1984 年作为主要量子引力理论候选者取得突破。
To avoid confusion further on, I here will briefly stipulate my use of the term “empiricist” in this analysis. A large part of this story will revolve around SS-matrix theory. The presupposition underlying the SS-matrix approach is that only mathematical relations between observable quantities (namely scattering amplitudes) are allowed. ^(11){ }^{11} That is, the theory relates the observable incoming and outgoing states, but is not concerned with a dynamical description of how the state changes over time during the scattering. In the following, I designate this aspect of SS-matrix theory as “empiricist”. My use of the term stems from the definition of constructive empiricism from philosopher Bas van Fraassen (1980). According to Van Fraassen, constructive empiricism entails that science aims to give us empirically adequate theories (i.e., theories that give correct predictions), acceptance of a theory involves only as belief that it is empirically adequate, and only those entities are to be accepted which are observable. In my use of the term as an analytical category in historiographical analysis, only the latter part of the definition is of importance: in SS-matrix theory, the starting point is to allow only those entities (scattering amplitudes) that are observable. This excludes the dynamics of scattering (most importantly, as described by field theory), which corresponds to physical states that are unobservable. I want to stress that throughout the paper I use “empiricism” in this limited sense. It is not my aim to engage in a debate on the level of ontology and belief, nor do I want to suggest that SS-matrix theorists were constructive empiricists in Van Fraassen’s sense. The reason I nevertheless emphasize the “empiricism” of SS-matrix theory is because it can inform us on how 为了避免进一步的混淆，我将在此简要说明我在分析中使用的 "经验主义者 "一词。这个故事的很大一部分将围绕 SS 矩阵理论展开。 SS 矩阵方法的前提是，只允许可观测量（即散射振幅）之间存在数学关系。 ^(11){ }^{11} 也就是说，该理论将可观测的入射和出射状态联系起来，但并不关注散射过程中状态如何随时间变化的动力学描述。在下文中，我将 SS 矩阵理论的这一方面称为 "经验主义"。我使用这个术语源于哲学家巴斯-范-弗拉森（Bas van Fraassen，1980 年）对建构经验主义的定义。根据范-弗拉森的观点，建构经验主义意味着科学旨在为我们提供经验上充分的理论（即能够给出正确预测的理论），接受一种理论只涉及相信它在经验上是充分的，并且只接受那些可观察到的实体。在我使用这个术语作为史学分析的一个分析范畴时，只有定义的后一部分是重要的：在 SS 矩阵理论中，出发点是只允许那些可观测的实体（散射振幅）。这就排除了散射的动力学（最重要的是场论所描述的），它对应于不可观测的物理状态。我想强调的是，在整篇论文中，我都是在这种有限的意义上使用 "经验主义"。我的目的不是参与本体论和信仰层面的辩论，也不想暗示 SS 矩阵理论家是范-弗拉森意义上的建设性经验主义者。不过，我之所以强调 SS 矩阵理论的 "经验主义"，是因为它能告诉我们如何
historical actors in practice dealt with the relation between theoretical structure and observable quantities during the formation of string theory. This is, I believe, of central importance to properly explain how SS-matrix theorists, with the construction of dual resonance models, turned away from experiment—and with that to identify the origins of string theory’s contested relation with experimental data. 在弦理论的形成过程中，历史行为者实际上处理了理论结构与可观测量之间的关系。我认为，这对于正确解释矩阵理论家如何通过构建双共振模型而远离实验--并由此确定弦理论与实验数据之间有争议的关系的起源--具有核心意义。
2. Analytic SS-matrix theory 2.解析 SS 矩阵理论
In the 1950s and 1960s, particle physics was oriented towards experiment, with an abundance of new data generated by new particle accelerators such as the Cosmotron at Brookhaven National Laboratory or CERN’s Proton Synchrotron. At that time it was unclear what the best theoretical framework was for particle interactions. The framework of quantum field theory, which formed the basis for the empirically successful theory of quantum electrodynamics, failed to work for both the strong interactions (governing the properties of atomic nuclei) and the weak interactions (responsible for radioactive decay). For the strong force, the main problem was the high value of the coupling constants determining the strength of the interaction, leading to a failure of the usual field theoretic approach of carrying out perturbative expansions in powers of the coupling constant. ^(12){ }^{12} 二十世纪五六十年代，粒子物理学以实验为导向，布鲁克海文国家实验室的宇宙加速器或欧洲核子研究中心的质子同步加速器等新型粒子加速器产生了大量新数据。当时还不清楚粒子相互作用的最佳理论框架是什么。量子场论框架是经验上成功的量子电动力学理论的基础，但它在强相互作用（控制原子核的性质）和弱相互作用（负责放射性衰变）方面都不起作用。对于强作用力，主要问题是决定相互作用强度的耦合常数值过高，导致以耦合常数的幂级数进行微扰展开的通常场论方法失效。 ^(12){ }^{12}
In order to make sense of accelerator data, approaches were pursued that aimed to describe experimental data from strongly interacting particles outside the framework of QFT. This led to the formation of the research program called analytic SS-matrix theory. The aim of SS-matrix theory was to obtain the entire scattering matrix of particle collisions on the basis of a set of fundamental principles, instead of calculating it from a dynamical theory, like field theory. The SS-matrix approach to elementary particle theory originated with Werner Heisenberg, who proposed an SS-matrix theory for quantum electrodynamics in 1943. It is insightful to start by briefly revisiting the original works of Heisenberg and collaborators and their motivations for it, because they greatly influenced the work on strong interaction SS-matrix theory, and thereby, albeit indirectly, also string theory. 为了弄清加速器数据的意义，人们寻求在 QFT 框架之外描述强相互作用粒子实验数据的方法。这导致了名为解析 SS 矩阵理论的研究计划的形成。 SS 矩阵理论的目的是在一套基本原理的基础上获得粒子碰撞的整个散射矩阵，而不是从动力学理论（如场论）中计算出来。基本粒子理论的 SS 矩阵方法起源于沃纳-海森堡（Werner Heisenberg），他于 1943 年提出了量子电动力学的 SS 矩阵理论。首先简要回顾一下海森堡及其合作者的原著和他们的动机是很有意义的，因为他们极大地影响了强相互作用 SS 矩阵理论的研究，从而也间接地影响了弦理论。
2.1 Heisenberg's SS-matrix program 2.1 海森堡的 SS 矩阵程序
Heisenberg’s original SS-matrix theory proposal was theoretically motivated to avoid the divergences encountered in field theory models of quantum electrodynamics. Heisenberg wanted to base his theory on finite, observable quantities only, avoiding reference to a Hamiltonian or to equations of motion. It was not that he denied the physical significance of these concepts, but he considered the route from a Hamiltonian or equations of motion to experimentally observable quantities to be too ill-defined and often leading to infinities. In quantum field theory, once the Hamiltonian is given the scattering matrix SS is determined, out of which the transition probabilities and cross sections can be calculated directly. Starting from the SS-matrix, Heisenberg wanted to extract from it all the general, model-independent features he foresaw would be part of a future, improved theory. He thought it plausible such a future theory would contain a fundamental length. ^(13){ }^{13} 海森堡最初提出 SS 矩阵理论的理论动机是为了避免量子电动力学场论模型中遇到的分歧。海森堡希望他的理论只基于有限的、可观测的量，避免提及哈密顿或运动方程。这并不是说他否认这些概念的物理意义，而是他认为从哈密顿或运动方程到实验可观测量的路径过于不明确，往往会导致无穷大。在量子场论中，一旦给出哈密顿方程，散射矩阵 SS 即可确定，由此可直接计算出转变概率和截面。从 SS 矩阵开始，海森堡希望从中提取他所预见的未来改进理论中所有与模型无关的一般特征。他认为未来的理论很有可能包含基本长度。 ^(13){ }^{13}
For Heisenberg, this was a return to the successful approach that had guided him in formulating matrix mechanics in 1925. Here, Heisenberg’s motivation had also been to restrict the theory to relations between observable quantities only (that is, what I call an “empiricist” approach). In the case of matrix mechanics, he criticized the old quantum theory because of the appearance of unobservable quantities, such as the position and orbital period of the electron, in the rules that were used to calculate observable quantities like the atom’s energy. Instead, Heisenberg proposed to reinterpret the Fourier expansion describing the periodic motion of a classical atom’s electron as an abstract set of numbers. These numbers were no longer thought to describe the electron’s orbit, but now represented the frequencies and amplitudes that defined transitions between atomic states, following up on work by himself and the Dutch physicist Hendrik Kramers on the dispersion of light. ^(14){ }^{14} 对海森堡来说，这是对他在 1925 年提出矩阵力学时所采用的成功方法的一种回归。在这里，海森堡的动机也是把理论局限于可观测量之间的关系（即我所说的 "经验主义 "方法）。就矩阵力学而言，他批评了旧量子理论，因为在用于计算原子能量等可观测量的规则中出现了不可观测量，如电子的位置和轨道周期。相反，海森堡提议将描述经典原子电子周期运动的傅立叶展开重新解释为一组抽象的数字。这些数字不再被认为是对电子轨道的描述，而是代表了定义原子态之间跃迁的频率和振幅，这也是海森堡本人和荷兰物理学家亨德里克-克莱默斯（Hendrik Kramers）在光的色散方面的研究成果。 ^(14){ }^{14}
Almost twenty years after his successful formulation of matrix mechanics, Heisenberg returned to this empiricist approach for his SS-matrix theory. In the years 1943-1946, Heisenberg published a series of papers developing this program. Among other things, Heisenberg proved the unitarity of the SS-matrix and used it to relate the total cross section sigma_(T)\sigma_{T} to the imaginary part of the forward scattering amplitude of elastic interactions. In addition, Heisenberg proposed to consider the SS-matrix as an analytic function of a complex energy variable, as suggested by Kramers (both unitarity and analyticity were also central concepts in strong interaction SS-matrix theory and will be more thoroughly discussed in Section 2.2). Using this method of analytic extension Heisenberg was able to construct a simple two-particle model in which the SS-matrix not only determined the scattering cross sections but also the bound-state energies of the system. However, it was difficult to formulate other SS-matrix systems apart from this two-particle model and some of its trivial extensions: without introducing a Hamiltonian there were no rules to guide the construction of the SS-matrix. ^(15){ }^{15} 在成功提出矩阵力学近二十年后，海森堡在其 SS 矩阵理论中重新采用了这种经验主义方法。1943-1946 年间，海森堡发表了一系列论文，发展了这一方案。其中，海森堡证明了 SS 矩阵的单位性，并用它将总截面 sigma_(T)\sigma_{T} 与弹性相互作用的前向散射振幅的虚部联系起来。此外，海森堡还提出将 SS 矩阵视为复能变量的解析函数，正如克拉默所建议的那样（单位性和解析性也是强相互作用 SS 矩阵理论的核心概念，将在第 2.2 节中进行更深入的讨论）。海森堡利用这种分析扩展方法构建了一个简单的双粒子模型，其中的 SS 矩阵不仅决定了散射截面，还决定了系统的边界态能量。然而，除了这个双粒子模型及其一些微不足道的扩展之外，很难提出其他的 SS 矩阵系统：如果不引入哈密顿，就没有规则来指导 SS 矩阵的构造。 ^(15){ }^{15}
Heisenberg’s proposal was further developed by a group of physicists, most notably Kramers, the Swiss physicists Ernst Stueckelberg and Res Jost, the Danish physicist Christian Møller, and Ralph Kronig, who was a professor in the Dutch town of Delft. With the success of renormalized QED in the late 1940s, work on the SS-matrix program waned. ^(16){ }^{16} Nevertheless, Heisenberg’s program strongly influenced postwar particle physics. Firstly, it firmly established the SS-matrix as a calculational tool in field theory. Most importantly, as historian of science Alexander Blum (2017) has argued, Heisenberg’s program, as embodied by his two-particle scattering model, constituted an important shift in perspective for quantum theory, because it was the first description of a scattering process that was not grounded in the concept of stationary states described by a time-independent wave function. Instead of the notion of stationary states, scattering became the primary concept, using asymptotic states to determine a system’s bound state energies. This approach of formulating relations between asymptotic states, while treating the dynamics of what happens in the scattering region as a black box, continued to define strong interaction SS-matrix theory in the 1950s and 1960s. 海森堡的提议得到了一批物理学家的进一步发展，其中最著名的有克拉默斯、瑞士物理学家恩斯特-斯图科尔伯格和雷斯-约斯特、丹麦物理学家克里斯蒂安-默勒，以及荷兰代尔夫特市的教授拉尔夫-克罗尼格。随着重规范化 QED 在 20 世纪 40 年代末取得成功，有关 SS 矩阵计划的工作逐渐减弱。 ^(16){ }^{16} 然而，海森堡的计划对战后粒子物理学产生了重大影响。首先，它牢固地确立了 SS 矩阵作为场论计算工具的地位。最重要的是，正如科学史家亚历山大-布卢姆（Alexander Blum，2017 年）所指出的，海森堡的计划，正如他的双粒子散射模型所体现的那样，构成了量子理论视角的重要转变，因为它是对散射过程的首次描述，而这种描述并不是基于与时间无关的波函数所描述的静止态概念。与静止态的概念不同，散射成为主要概念，它使用渐近态来确定系统的束缚态能量。这种在渐近态之间建立关系的方法，同时将散射区域中发生的动态视为黑箱，在 20 世纪 50 年代和 60 年代继续定义了强相互作用 SS 矩阵理论。
This SS-matrix theory for the strong interactions was, just like Heisenberg’s original approach, motivated by a desire to explore alternatives to field theory. Historian of science James Cushing, in his book on the history of strong interaction SS-matrix theory, identifies a repetition of events: after the success of renormalized QED, interest in Heisenberg’s original SS-matrix theory decreased; in the late 1950s and early 1960s, the problems encountered in attempts to construct a field theory for the strong interactions led to the new analytic SS-matrix program. With the success of gauge theory and the Standard Model in the 1970s, interest in SS-matrix theory as an independent program waned again. ^(17){ }^{17} Yet, through dual resonance models, strong interaction SS-matrix theory would eventually lead to string theory. 这种强相互作用的 SS 矩阵理论与海森堡最初的方法一样，都是出于探索场论替代方案的愿望。科学史学者詹姆斯-库欣（James Cushing）在他的《强相互作用 SS 矩阵理论的历史》一书中指出了事件的重复：在重正規化 QED 取得成功之后，人们对海森堡最初的 SS 矩阵理论的兴趣下降了；在 20 世纪 50 年代末和 60 年代初，人们在试图构建强相互作用场论时遇到的问题导致了新的分析 SS 矩阵计划。随着70年代规理论和标准模型的成功，人们对 SS 矩阵理论作为一个独立项目的兴趣再次减弱。 ^(17){ }^{17} 然而，通过双共振模型，强相互作用 SS -矩阵理论最终导致了弦理论。
2.2 Analytic SS-matrix theory for the strong interactions 2.2 强相互作用的解析 SS 矩阵理论
Heisenberg’s original SS-matrix theory and the strong interaction SS-matrix program are linked by work on dispersion relations in the 1950s. In the postwar years, there was a large amount of data on strong interaction scattering that was produced in new experiments, and dispersion relations seemed promising for describing it. The dispersion-theory approach to the strong interactions was grounded in two key aspects of the work by Kramers, Kronig, Heisenberg, and others on the 海森堡最初的 SS 矩阵理论和强相互作用 SS 矩阵计划是通过 20 世纪 50 年代的色散关系工作联系在一起的。战后几年，在新的实验中产生了大量关于强相互作用散射的数据，而色散关系似乎很有希望描述这些数据。强相互作用的色散理论方法是建立在克拉默斯、克罗尼格、海森堡等人关于强相互作用的两个关键方面的工作基础之上的
dispersion of light waves. The first was the analyticity of the expression for the scattering amplitude, the second was to impose unitarity on the amplitude. 光波的色散。首先是散射振幅表达式的解析性，其次是对振幅施加单位性。
Analyticity of the scattering amplitude means that the scattering amplitude f(omega)f(\omega) is treated as a complex analytic function of the energy variable omega\omega, i.e., as a function that could be extended to the complex plane. ^(18){ }^{18} When looking at the special case of theta=0\theta=0, called “forward scattering”, one can relate the real and imaginary part of f(omega)f(\omega) by the Kramers-Kronig relation: 散射振幅的解析性意味着散射振幅 f(omega)f(\omega) 被视为能量变量 omega\omega 的复解析函数，即可以扩展到复平面的函数。 ^(18){ }^{18} 在研究 theta=0\theta=0 的特殊情况（称为 "前向散射"）时，我们可以通过克拉默-克罗尼格关系将 f(omega)f(\omega) 的实部和虚部联系起来：
Re[f(omega)]=(1)/(pi)Pint_(-oo)^(+oo)domega^(')(Im[f(omega^('))])/(omega^(')-omega)\operatorname{Re}[f(\omega)]=\frac{1}{\pi} P \int_{-\infty}^{+\infty} d \omega^{\prime} \frac{\operatorname{Im}\left[f\left(\omega^{\prime}\right)\right]}{\omega^{\prime}-\omega}
with PP the Cauchy principal value. In the mid-1940s this calculation was well-established for describing the scattering of monochromatic light by atoms. In this case, the analyticity of f(omega)f(\omega) was justified on the basis of causality-that is, the requirement that a light wave propagates causally, taking at least a time l//cl / c to reach a point at distance ll. While the connection between causality and analyticity was well-known for the case of refracting light waves, in 1946 Kronig had raised the question if this relation could be used to determine Heisenberg’s SS-matrix, and if it could be extended to scattering processes where particles were created and annihilated. ^(19){ }^{19} PP 为柯西主值。20 世纪 40 年代中期，这种计算方法在描述原子对单色光的散射时得到了充分证实。在这种情况下， f(omega)f(\omega) 的解析性是以因果关系为基础的，也就是说，光波的传播是有因果关系的，至少需要 l//cl / c 的时间才能到达距离 ll 的点。虽然因果性和解析性之间的联系在光波折射的情况下是众所周知的，但克罗尼格在 1946 年提出了这样一个问题：这种关系是否可以用来确定海森堡的 SS 矩阵，以及是否可以扩展到粒子产生和湮灭的散射过程。 ^(19){ }^{19}
Unitarity is the condition in quantum theory that the time evolution of a quantum state is represented by a unitary operator, ensuring that the probabilities for a quantum process sum to 1 . In scattering processes, this implies that the SS-matrix must be unitary: SS^(†)=1S S^{\dagger}=1, where S^(†)S^{\dagger} denotes the conjugate transpose of the matrix. In dispersion theory, by imposing unitarity the so-called optical theorem can be derived: 单一性是量子理论中的一个条件，即量子态的时间演化由单一算子表示，确保量子过程的概率总和为 1。在散射过程中，这意味着 SS 矩阵必须是单一的： SS^(†)=1S S^{\dagger}=1 ，其中 S^(†)S^{\dagger} 表示矩阵的共轭转置。在色散理论中，通过施加单元性，可以推导出所谓的光学定理：
The optical theorem relates the imaginary part of a forward scattering amplitude to the total cross section. ^(20){ }^{20} In the 1950s the optical theorem was well-known as a general law for both classical and quantum wave scattering, among others in the Kramers-Kronig dispersion theory of light (although it became widely known under the name “optical theorem” only in the early 1960s). ^(21){ }^{21} 光学定理将前向散射振幅的虚部与总截面联系起来。 ^(20){ }^{20} 20世纪50年代，光学定理作为经典和量子波散射的一般定律而广为人知，其中包括克拉默-克罗尼光色散理论（尽管它在20世纪60年代初才以 "光学定理 "的名称广为人知）。 ^(21){ }^{21}
In the early 1950s, Marvin Goldberger and Murray Gell-Mann from the University of Chicago sought to understand dispersion relations of light on the basis of first principles, like causality-in this sense their approach was indirectly motivated by questions prompted by Heisenberg’s SS-matrix program. They justified their approach on the basis of the statement of microcausality. This means that if two events xx and yy are spacelike separated, the corresponding local field operators phi(x)\phi(x) and phi(y)\phi(y) must commute. Events that are timelike separated are quantum mechanically not independently observable and have a non-vanishing commutation relation. Using microcausality, in 20 世纪 50 年代初，芝加哥大学的马文-戈德伯格（Marvin Goldberger）和默里-盖尔曼（Murray Gell-Mann）试图根据第一性原理（如因果关系）来理解光的色散关系--从这个意义上说，他们的研究方法间接地受到了海森堡的 SS 矩阵计划所引发的问题的启发。他们根据微观因果关系的声明来证明自己的方法是正确的。这意味着，如果两个事件 xx 和 yy 在空间上相距甚远，则相应的局部场算子 phi(x)\phi(x) 和 phi(y)\phi(y) 必须相通。时间上分离的事件在量子力学上是不可独立观测的，并且具有不相等的换向关系。利用微观因果关系，在
1954 Goldberger and Gell-Mann, together with Walter Thirring from Vienna, were able to obtain the Kramers-Kronig relation for the forward scattering of light by a matter field, starting from a field theory of photons scattering off a fixed force center. ^(22){ }^{22} Although at this point only massless scattering had been satisfactorily handled, the dispersion relations of Gell-Mann, Goldberger and Thirring were then assumed to be valid for massive particle scattering and used to analyze experimental data. 1954 年，戈尔德伯格和盖尔曼与维也纳的沃尔特-瑟林（Walter Thirring）一起，从光子从固定力心散射的场论出发，得到了物质场对光的正向散射的克拉默-克罗尼格关系。 ^(22){ }^{22} 尽管此时只有无质量散射得到了令人满意的处理，但盖尔-曼、戈德伯格和蒂尔林的色散关系随后被假定为对大质量粒子散射有效，并被用来分析实验数据。
In the following years a growing group of physicists started to use dispersion relations in their investigations of strong interaction data. The steps in the procedure were analogous to the light wave case. First, analyticity of the amplitude function was justified by causality. Then physicists made assumptions on the basis of data about which states would make the largest contribution to the cross section, which could be related to Im[f]\operatorname{Im}[f] via the optical theorem. From there, the expression was inserted in the Kramers-Kronig integral for Re[f]\operatorname{Re}[f] to check whether the assumptions were correct. So, via this procedure a set of equations was generated that could be solved either perturbatively or in some non-perturbative manner. ^(23){ }^{23} In the late 1950s, a group of some forty physicists was working on the dispersion-theory program, about half of which were American while the other half consisted of European and Russian scholars. The overall attitude of dispersion theorists was pragmatic: this was not a case of “high theory” dictating experiment. Instead, dispersion relations were assumed to hold and then their use was justified by experimental success. ^(24){ }^{24} 随后几年，越来越多的物理学家开始在研究强相互作用数据时使用色散关系。这一过程的步骤与光波情况类似。首先，振幅函数的解析性通过因果关系得到证明。然后，物理学家根据数据假设哪些状态对截面的贡献最大，并通过光学定理将其与 Im[f]\operatorname{Im}[f] 联系起来。在此基础上，将 Re[f]\operatorname{Re}[f] 的表达式插入克拉默-克罗尼积分，以检验假设是否正确。因此，通过这个过程，就产生了一组方程，可以用微扰或非微扰方式求解。 ^(23){ }^{23} 20世纪50年代末，一个由大约40名物理学家组成的小组正在研究色散理论计划，其中大约一半是美国人，另一半由欧洲和俄罗斯学者组成。色散理论家们的总体态度是务实的：这不是 "高深理论 "支配实验的情况。相反，分散关系被假定为成立，然后通过实验的成功来证明其使用的合理性。 ^(24){ }^{24}
Around 1960, due to the contributions of physicists as Gell-Mann, Goldberger, Stanley Mandelstam, Francis Low, Tullio Regge, and Geoffrey Chew, out of the early work on dispersion relations a more or less well-defined SS-matrix theory for the strong interactions was formed. With analyticity (via causality) and unitarity, two of the principles of SS-matrix theory have already been discussed; Lorentz invariance was another. The other key ideas underlying SS-matrix theory, as developed in the late 1950s, were ^(25){ }^{25} : 1960 年前后，由于盖尔-曼、戈德伯格、斯坦利-曼德尔施塔姆、弗朗西斯-刘、图利奥-雷格和杰弗里-丘等物理学家的贡献，从早期关于色散关系的工作中形成了一个或多或少定义明确的强相互作用 SS 矩阵理论。前面已经讨论了 SS 矩阵理论的两个原则：解析性（通过因果关系）和单元性；另一个原则是洛伦兹不变性。20 世纪 50 年代后期发展起来的 SS 矩阵理论的其他关键思想是 ^(25){ }^{25} ：
The pole-particle conjecture. This was a new way, mainly due to Chew and Low, to interpret and analyze the Born term in a perturbation expansion of a scattering amplitude. For the scattering of nucleons through the exchange of a pion, this term in the amplitude was of the form 极点粒子猜想。这是一种解释和分析散射振幅扰动扩展中博恩项的新方法，主要归功于周和洛。对于通过交换先驱进行的核子散射，振幅中的这个项的形式是
A prop(g^(2))/(t-M_(pi)^(2))A \propto \frac{g^{2}}{t-M_{\pi}^{2}}
with tt the change in momentum from one of the incoming nucleons before and after emitting the pion, M_(pi)M_{\pi} the mass of the exchanged pion, and g^(2)g^{2} the strong interaction pionnucleon coupling constant. This term becomes infinite at the physical mass of the pion, when t=M_(pi)^(2)t=M_{\pi}^{2}. In perturbative field theory the corresponding singularity in the amplitude function is of no consequence, because scattering with momentum transfer t=M_(pi)^(2)t=M_{\pi}^{2} does not correspond to a physical situation (it corresponds to the exchange of a stable pion and has an unphysical scattering angle). So far, dispersion theorists had paid no particular attention to the singularities in their analytic scattering amplitudes. Chew, however, proposed to employ the analytic properties of the amplitude function by extrapolating to the unphysical point t=t=M_(pi)^(2)M_{\pi}^{2}. A singular point of a complex function is known as a pole; Chew proposed to associate the position of the pole on the real axis with the exchanged particles’ mass, and the coefficient that remained after contour integration around the pole (called the “residue”) 其中， tt 是一个传入核子在发射先驱前后的动量变化， M_(pi)M_{\pi} 是交换先驱的质量， g^(2)g^{2} 是强相互作用先驱-核子耦合常数。当 t=M_(pi)^(2)t=M_{\pi}^{2} 达到先驱的物理质量时，该项变为无穷大。在微扰场论中，振幅函数中的相应奇异性并不重要，因为带有动量转移 t=M_(pi)^(2)t=M_{\pi}^{2} 的散射并不对应于物理情形（它对应于稳定先驱的交换，并且有一个非物理的散射角）。迄今为止，色散理论家们并没有特别关注其解析散射振幅中的奇异性。然而，Chew 提议通过外推到非物理点 t=t=M_(pi)^(2)M_{\pi}^{2} 来利用振幅函数的解析性质。复变函数的奇异点被称为极点；Chew 提议将极点在实轴上的位置与交换粒子的质量联系起来，并将极点周围的等值线积分后剩余的系数（称为 "残差"）计算出来。
with the coupling constant of the particular exchange. This conjecture introduced a notion of force in the SS-matrix program. 与特定交换的耦合常数。这一猜想在 SS 矩阵程序中引入了力的概念。
Crossing. This refers to a symmetry property of scattering amplitudes that was extracted out of the structure of Feynman diagrams and first mentioned explicitly by Gell-Mann and Goldberger (1954) in a paper on the scattering of light off a spin 1//21 / 2 target. The idea of crossing symmetry is that the amplitude of a scattering process is invariant when swapping a pair of incoming and outgoing particles for antiparticles with opposite momentum. In the case of a two-body reaction with particles of zero spin, the same scattering amplitude then describes three “crossed” reactions: 交叉。这是指从费曼图结构中提取出来的散射振幅的对称特性，由盖尔-曼和戈尔德贝格尔（1954 年）在一篇关于自旋 1//21 / 2 目标的光散射的论文中首次明确提及。交叉对称的概念是，当把一对进出粒子换成动量相反的反粒子时，散射过程的振幅是不变的。在自旋为零的粒子的双体反应中，相同的散射振幅描述了三个 "交叉 "反应：
{:[" I. ",a+b rarr c+d],[" II. ",b+ bar(d)rarr bar(a)+c],[" III. ",b+ bar(c)rarr bar(a)+d]:}\begin{array}{ll}
\text { I. } & a+b \rightarrow c+d \\
\text { II. } & b+\bar{d} \rightarrow \bar{a}+c \\
\text { III. } & b+\bar{c} \rightarrow \bar{a}+d
\end{array}
where the bars denote antiparticles. Considering both uncrossed and crossed reactions and demanding the same scattering amplitude thus further constrained their calculation. 其中条形表示反粒子。考虑到非交叉反应和交叉反应，并要求相同的散射振幅，从而进一步限制了他们的计算。
Double dispersion relations. Mandelstam (1958) proposed to express dispersion relations not only in terms of the energy variable, but also in terms of the momentum transfer variable. For a two-body reaction with incoming four-momenta p_(a)p_{a} and p_(b)p_{b} and outgoing fourmomenta -p_(c)-p_{c} and -p_(d)-p_{d}, Mandelstam defined three Lorentz invariant variables: 双扩散关系。曼德尔施塔姆（1958 年）建议不仅用能量变量来表示频散关系，而且用动量传递变量来表示频散关系。对于具有传入四动量 p_(a)p_{a} 和 p_(b)p_{b} 以及传出四动量 -p_(c)-p_{c} 和 -p_(d)-p_{d} 的双体反应，曼德尔施塔姆定义了三个洛伦兹不变变量：
with units ℏ=c=1\hbar=c=1. In the center-of-mass frame, ss is also known as the total energy squared, and tt as the momentum transfer squared; the variables s,ts, t, and uu were quickly named “Mandelstam variables”. Using them one can write a full scattering amplitude A(s,t,u)A(s, t, u) with integrals representing different contributions corresponding to different possible intermediate states. Combined with crossing symmetry the Mandelstam representation was very useful when constraining the amplitude: by continuing energies from positive to negative values (corresponding to a particle-antiparticle swap), it became possible to switch between crossed reactions by allowing s,ts, t, or uu to play the role of energy or momentum transfer variable. It can be shown that only two of the Mandelstam variables are independent variables, so that any two of them suffices to construct scattering amplitudes. 单位为 ℏ=c=1\hbar=c=1 。在质量中心框中， ss 也称为总能量平方， tt 称为动量传递平方；变量 s,ts, t 和 uu 很快被命名为 "曼德尔施塔姆变量"。利用它们可以写出一个完整的散射振幅 A(s,t,u)A(s, t, u) ，其中的积分代表了与不同可能的中间状态相对应的不同贡献。结合交叉对称性，曼德尔施塔姆表示法在约束振幅时非常有用：通过将能量从正值延续到负值（对应于粒子-反粒子交换），允许 s,ts, t 或 uu 扮演能量或动量转移变量的角色，就有可能在交叉反应之间切换。可以证明，曼德尔斯塔姆变量中只有两个是独立变量，因此其中任何两个都足以构建散射振幅。
Regge poles. The use of complex variables of the scattering amplitude was also extended to the angular momentum JJ. A singularity that arises when treating JJ as a complex variable is called a “Regge pole”, after the Italian physicist Tullio Regge who first formulated them for nonrelativistic potential scattering. Regge poles were used to correlate the energy and spin values of resonances (that is, peaks in cross sections) that were associated with short-lived particles. The resonances were interpreted as “excitations” of hadrons such as the proton or the neutron. In particular, “families” of resonances of increasing spin were found and coined “Regge trajectories”. In SS-matrix calculations, Regge trajectories were crucial for calculating 雷格极点。散射振幅复变量的使用也扩展到了角动量 JJ 。将 JJ 作为复变量处理时出现的奇点被称为 "雷格极点"，这是以意大利物理学家图利奥-雷格（Tullio Regge）的名字命名的。雷格极点用于关联与短寿命粒子相关的共振（即截面中的峰值）的能量和自旋值。共振被解释为质子或中子等强子的 "激发"。特别是，人们发现了自旋递增的共振 "家族"，并将其命名为 "雷格轨迹"。在 SS 矩阵计算中，雷格轨迹对于计算
the high-energy amplitudes of scattering processes. The use of Regge poles will be more thoroughly discussed in Section 3.1. 散射过程的高能振幅。第 3.1 节将更深入地讨论雷格极点的使用。
Taken together, in the early 1960s the analytic SS-matrix program constituted an approach to strong interaction scattering that allowed for much fruitful contact with experimental data, employing a set of calculational tools (mostly related to the mathematics of complex variables) on the basis of a set of principles. Despite the introduction of a variety of new ideas and tools-some of which explicitly originated in field theory, such as crossing-the analytic SS-matrix program was still operating in Heisenberg’s empiricist spirit: observable SS-matrix elements were calculated without relying on a Hamiltonian or Lagrangian. 总之，在 20 世纪 60 年代初，解析 SS 矩阵方案构成了一种处理强相互作用散射的方法，它在一套原则的基础上，运用一套计算工具（大多与复变数学有关），与实验数据进行了富有成效的接触。尽管引入了各种新思想和新工具--其中一些明确源于场论，如交叉--但解析 SS 矩阵程序仍秉承海森堡的经验主义精神：可观测的 SS 矩阵元素的计算不依赖于哈密顿或拉格朗日。
2.3 The SS-matrix bootstrap in the 1960s 2.3 20 世纪 60 年代的 SS 矩阵自举法
In order to be able to properly understand how string theory grew out of SS-matrix physics, the conjecture denoted as the " SS-matrix bootstrap" or simply “bootstrap” is of central importance. Following Cushing (1985), I will define the bootstrap here as the conjecture that a “well-defined but infinite set of self-consistency conditions determines uniquely the entities or particles which can exist” (p. 31). In SS-matrix theory, these “self-consistency conditions” arise from the unitarity requirement. Recall that this requirement can be formulated as the statement that the probability of an initial state to evolve into any possible final state equals 1. Because of the possibility of particle creation at high energy, unitarity leads to an infinite set of nonlinear coupled equations. This set of equations allows the possibility that it has just one unique solution. According to the bootstrap conjecture, this one solution would determine all the masses, charges and other aspects of particles in nature. ^(26){ }^{26} Hence, such a theory would, if solved, contain no free parameters that need to be fitted to experiment. The virtue of a theory to lack free parameters is what I designate as “nonarbitrariness” - in the sense that all parameters are determined, so that none can be “arbitrarily” varied at will. ^(27){ }^{27} Furthermore, note that the term “self-consistency” here refers to the property that parameters appear both as input and output of a calculation; requiring that the input and output values match ensures that they are uniquely fixed. Such a “closed” set of equations can in principle be solved without external input. As we will see, in practice this was never the case in hadronic SS matrix calculations. In what follows I will reserve my use of the term “self-consistency” to its meaning in this “bootstrap” sense. ^(28){ }^{28} 为了能够正确理解弦理论是如何从 SS 矩阵物理学发展而来的，被称为" SS 矩阵自举 "或简称 "自举 "的猜想至关重要。按照 Cushing (1985)，我在这里将自举定义为 "定义明确但无限的自洽条件集合唯一地决定了可以存在的实体或粒子"（第 31 页）的猜想。在 SS -矩阵理论中，这些 "自洽条件 "源于单位性要求。回想一下，这一要求可以表述为：初始状态演变为任何可能的最终状态的概率等于 1。由于粒子有可能在高能量下产生，因此单一性导致了无限的非线性耦合方程组。这组方程有可能只有一个唯一的解。根据引导猜想，这一个解将决定自然界中粒子的所有质量、电荷和其他方面。 ^(26){ }^{26} 因此，这样的理论如果得到解决，将不包含任何需要与实验相匹配的自由参数。缺乏自由参数的理论的优点就是我所说的 "非任意性"--在这个意义上，所有参数都是确定的，因此没有一个参数可以 "任意 "改变。 ^(27){ }^{27} 此外，请注意，这里的 "自洽性 "指的是参数在计算中既作为输入又作为输出的特性；要求输入值和输出值相匹配，就能确保它们是唯一固定的。这种 "封闭 "方程组原则上可以在没有外部输入的情况下求解。正如我们将要看到的，在强子 SS 矩阵计算中，实际情况并非如此。 在下文中，我将保留使用 "自洽性 "一词在这种 "引导 "意义上的含义。 ^(28){ }^{28}
The bootstrap conjecture became an influential notion in hadronic SS-matrix physics mainly through physicist Geoffrey Chew. In the early 1960s, Chew, then at Berkeley, started to express the viewpoint (pioneered two years earlier by the Russian physicist Lev Landau) that for the strong interactions field theory should be abandoned altogether in favor of the SS-matrix framework. This proposal was based on the hypothesis that the SS-matrix postulates could lead to a “complete and self-consistent theory of strong interactions”. ^(29){ }^{29} Within Chew’s program, the bootstrap conjecture implied that all strongly interacting particles mutually generated all others through their interactions with one another. Instead of adopting as a starting point the field-theoretical notion to subscribe to a set of elementary particles and fields, the hypothesis was that in the SS-matrix framework all particles 自举猜想主要通过物理学家杰弗里-丘（Geoffrey Chew）成为强子 SS 矩阵物理学中一个有影响力的概念。20 世纪 60 年代初，当时在伯克利大学工作的 Chew 开始表达一种观点（两年前由俄罗斯物理学家 Lev Landau 首创），即对于强相互作用场论应该完全放弃，转而采用 SS 矩阵框架。这一提议基于这样一个假设： SS 矩阵假设可以导致 "完整和自洽的强相互作用理论"。 ^(29){ }^{29} 在Chew的方案中，引导猜想意味着所有强相互作用粒子通过彼此的相互作用相互产生其他粒子。他没有把场论的概念作为出发点来认同一组基本粒子和场，而是假设在 SS 矩阵框架中，所有粒子
Figure 1: diagram for pion-pion scattering. When read upwards, the diagram denotes scattering of pions through exchange of a rho\rho meson, yielding an attractive force. When read from left to right, the diagram denotes the colliding of pions, forming an intermediate rho\rho meson bound state. Through a bootstrap calculation m_(rho)m_{\rho} and g_(rho pi pi)g_{\rho \pi \pi} of the bound state rho\rho meson was calculated self-consistently out of the exchange force m_(rho)m_{\rho} and g_(rho pi pi)g_{\rho \pi \pi}. Figure from Zachariasen (1961, p. 112). 图 1：先锋-负离子散射示意图。从上往下读时，图中表示的是通过交换一个 rho\rho 介子而产生的引力的质子散射。从左往右读时，该图表示的是质子碰撞，形成中间的 rho\rho 介子束缚态。通过自举计算，结合态 rho\rho 介子的 m_(rho)m_{\rho} 和 g_(rho pi pi)g_{\rho \pi \pi} 从交换力 m_(rho)m_{\rho} 和 g_(rho pi pi)g_{\rho \pi \pi} 中自洽地计算出来。图自 Zachariasen (1961, p. 112)。
could be treated on an equal footing, and idea that was coined “nuclear democracy”. ^(30){ }^{30} As Chew, Gell-Mann, and Rosenfeld put it in a 1964 Scientific American article: 这种想法被称为 "核民主"。 ^(30){ }^{30} 正如Chew、Gell-Mann和Rosenfeld在1964年《科学美国人》的一篇文章中所说的那样：
[The bootstrap hypothesis] may make it possible to explain mathematically the existence and properties of the strongly interacting particles. According to this hypothesis all these particles are dynamical structures in the sense that they represent a delicate balance of forces; indeed, they owe their existence to the same forces through which they mutually interact. ^(31){ }^{31} [引导假说]可以用数学方法解释强相互作用粒子的存在和性质。根据这一假说，所有这些粒子都是动力学结构，因为它们代表了力的微妙平衡；事实上，它们的存在要归功于它们相互作用的同一种力。 ^(31){ }^{31}
Both the bootstrap and the associated notion of nuclear democracy were central ideas around which the hadronic SS-matrix program of Chew and collaborators was developed in the 1960 s. ^(32){ }^{32} 自举和与之相关的核民主概念都是 20 世纪 60 年代 Chew 和合作者提出的强子 SS 矩阵计划的核心思想。
From the outset it was however made clear by practitioners that finding a unique solution of the SS-matrix equations for all particle interactions (i.e., a complete bootstrap) was practically out of reach. For starters, the SS-matrix bootstrap that was advocated by Chew and others was solely concerned with strong interactions: Chew readily acknowledged that he had no sharp convictions about electromagnetic and weak interactions, and could “not see how leptons and photons can emerge from the [ SS-matrix] principles”. But even when restricting the bootstrap to hadronic particles, Chew stressed that “[w]e shall, in fact, never have a complete solution; it would be far too complicated, since all [strongly interacting] particles would have to be considered simultaneously”. ^(33){ }^{33} 然而，从一开始，实践者们就清楚地认识到，要为所有粒子相互作用找到 SS 矩阵方程的唯一解（即完全自举）实际上是不可能的。首先，Chew 等人提倡的 SS 矩阵自举法只涉及强相互作用：Chew 坦然承认，他对电磁相互作用和弱相互作用没有明确的信念，而且 "看不出轻子和光子是如何从[ SS -矩阵]原理中产生的"。但是，即使把自举法限制在强子粒子上，Chew 也强调说："事实上，我们永远不会有一个完整的解决方案；这将太复杂，因为必须同时考虑所有[强相互作用]粒子"。 ^(33){ }^{33} .
Instead, the bootstrap ideal was applied in calculations of specific scattering processes. These calculations relied on simplifications; in particular, multiparticle intermediate states were usually neglected. A well-studied example of such a bootstrap calculation, among others researched by Caltech physicist Fredrik Zachariasen, considered pion scattering (see Figure 1). When reading the figure upwards, the pions interact through the exchange of a single rho\rho meson; the exchanged meson yields an attractive force depending on the mass m_(rho)m_{\rho} and coupling constant gamma_(rho pi pi)\gamma_{\rho \pi \pi}. When reading 相反，引导理想被应用于特定散射过程的计算。这些计算依赖于简化；特别是，多粒子中间态通常被忽略。在加州理工学院物理学家弗雷德里克-扎卡里亚森（Fredrik Zachariasen）的研究中，有一个关于这种自举计算的例子，即先驱散射（见图 1）。当向上读图时，离子通过交换单个 rho\rho 介子而相互作用；交换的介子产生的吸引力取决于质量 m_(rho)m_{\rho} 和耦合常数 gamma_(rho pi pi)\gamma_{\rho \pi \pi} 。当读取
Figure 1 from left to right (the crossed reaction) the pions collide to form an intermediate bound state rho\rho meson. In a bootstrap calculation one uses the mass and coupling constant from the attractive force between the two pions (which were in this case known from experiment) to calculate the mass and coupling constant of the bound state rho\rho meson, obtaining “two relations between m_(rho)m_{\rho} and gamma_(rho pi pi)\gamma_{\rho \pi \pi} from which both may be determined”. ^(34){ }^{34} The rho\rho meson calculation thus reflected the idea that a system of particles produces itself. As described by Zachariasen together with his Berkeley collaborator Charles Zemach in a paper on the same scattering process, “various particles give rise to forces among themselves making bound states which are the particles”. ^(35){ }^{35} 图 1 从左到右（交叉反应），两个负离子碰撞形成中间束缚态 rho\rho 介子。在自举计算中，我们使用来自两个小离子之间吸引力的质量和耦合常数（在这种情况下，它们是实验中已知的）来计算束缚态 rho\rho 介子的质量和耦合常数，从而得到" m_(rho)m_{\rho} 和 gamma_(rho pi pi)\gamma_{\rho \pi \pi} 之间的两种关系，从这两种关系中可以确定 m_(rho)m_{\rho} 和 gamma_(rho pi pi)\gamma_{\rho \pi \pi} "。 ^(34){ }^{34}rho\rho 介子计算由此反映了粒子系统产生自身的思想。正如扎卡里阿森和他在伯克利的合作者查尔斯-泽马赫（Charles Zemach）在一篇关于同一散射过程的论文中所描述的那样，"各种粒子之间产生作用力，形成束缚态，而束缚态就是粒子"。 ^(35){ }^{35}
This bootstrap calculation can thus (in principle) be designated as “self-consistent” because, when confronted with two equations relating m_(rho)m_{\rho} and gamma_(rho pi pi)\gamma_{\rho \pi \pi}, one can use the output of the first equation (e.g., m_(rho)\mathrm{m}_{\rho} and gamma_(rho pi pi)\gamma_{\rho \pi \pi} of the bound state rho\rho meson) as input for the second equation to calculate the values for the exchanged rho\rho meson, and vice versa, and check whether the outcomes are in agreement. Together with the nonlinearity of the unitarity equation this ensures that m_(rho)m_{\rho} and gamma_(rho pi pi)\gamma_{\rho \pi \pi} are uniquely determined by the two equations. However, in SS-matrix calculations it was in practice always the case that some values obtained from experiment were used as input. In the case of pion scattering, as said the experimental value for the mass and coupling of the exchanged rho\rho meson were used as input (and then shown to coincide with the output of the calculation taking the bound state rho\rho meson as input). More generally, the slope and the point of interception with the vertical axis (usually called the “intercept”) of the Regge trajectories that correlated energy and spin values of hadronic resonances were parameters obtained from experiment that were used as input in SS-matrix calculations, instead of being determined by them. 因此，这种自举计算（原则上）可以被称为 "自洽"，因为当面对两个有关 m_(rho)m_{\rho} 和 gamma_(rho pi pi)\gamma_{\rho \pi \pi} 的方程时，我们可以使用第一个方程的输出（例如，束缚态 rho\rho 介子的 m_(rho)\mathrm{m}_{\rho} 和 gamma_(rho pi pi)\gamma_{\rho \pi \pi} ）作为第二个方程的输入，计算出交换态 rho\rho 介子的值、rho\rho介子的m_(rho)\mathrm{m}_{\rho}和gamma_(rho pi pi)\gamma_{\rho \pi \pi}）作为第二个方程的输入，计算出交换的rho\rho介子的值，反之亦然，并检查结果是否一致。加上单位性方程的非线性，这就确保了 m_(rho)m_{\rho} 和 gamma_(rho pi pi)\gamma_{\rho \pi \pi} 是由这两个方程唯一决定的。然而，在 SS 矩阵计算中，从实验中获得的一些数值实际上总是被用作输入。在先驱散射的例子中，如前所述，交换的 rho\rho 介子的质量和耦合的实验值被用作输入（然后证明与以束缚态 rho\rho 介子为输入的计算输出相吻合）。更一般地说，与强子共振的能量和自旋值相关的雷格轨迹的斜率和与纵轴的截距点（通常称为 "截距"）都是从实验中获得的参数，这些参数被用作 SS 矩阵计算的输入，而不是由它们决定的。
So, the conjecture driving the SS-matrix program in the 1960 s was that it would be possible to generate, on the basis of a number of postulates (with a central role for unitarity), a set of equations that in principle could uniquely determine all parameters of hadronic interactions, but in practice, the SS-matrix program was elaborated through simplified calculations that used experimentally obtained values as input and were compared to the results of scattering experiments. The corresponding attitude among contributors was therefore that SS-matrix theory was “work-inprogress”. As Chew made clear while presenting his program in the early 1960s, one of the most attractive features of SS-matrix theory was precisely the possibility of many checks with experiment at different levels, and he urged both experimenters and theorists (in particular addressing “[a]ll the physicists who never learned field theory”) to join in the effort of further investigating these contacts between theory and experiment. Chew and collaborators actively campaigned for this effort throughout the 1960s, giving lectures and publishing textbooks, in which the material often was presented such that no acquaintance with field theory was needed. During the decade, more and more models and particle interactions were gradually included in the SS-matrix program. The pragmatic style associated with this way of working was well illustrated by particle physicist John D. Jackson, who commented in a review talk that “[t]he true believer in Regge poles leaves no application untried, no challenge unaccepted. If there is structure in a cross section (…), he will fit it” ^(36){ }^{36} 因此，在 20 世纪 60 年代，推动 SS 矩阵计划的猜想是，有可能在一些假设的基础上（以单位性为核心）生成一组方程，原则上可以唯一地确定强子相互作用的所有参数，但实际上， SS 矩阵计划是通过简化的计算来制定的，这些计算使用实验获得的值作为输入，并与散射实验的结果进行比较。因此， SS -矩阵理论是 "正在进行中的工作"。正如 Chew 在 20 世纪 60 年代初介绍他的计划时明确指出的那样， SS 矩阵理论最吸引人的特点之一，恰恰是可以在不同层次上与实验进行多次检查，他敦促实验人员和理论人员（特别是 "所有从未学过场论的物理学家"）一起努力，进一步研究理论与实验之间的这些联系。在整个 20 世纪 60 年代，周与合作者积极开展这项工作，举办讲座和出版教科书，其中的材料往往无需了解场论。在这十年间，越来越多的模型和粒子相互作用逐渐被纳入 SS 矩阵计划。粒子物理学家约翰-杰克逊（John D. Jackson）很好地诠释了与这种工作方式相关的务实风格，他在一次评论演讲中评论说："雷格极点的真正信徒不会放过任何未经尝试的应用，不会放过任何不被接受的挑战。如果横截面上有结构（......），他就会去适应它" ^(36){ }^{36} 。
2.4 "Empiricism" in 1960s particle theory 2.4 20世纪60年代粒子理论中的 "经验主义
As in Heisenberg’s original 1940s SS-matrix proposal, hadronic SS-matrix theory in the 1960s can be designated as “empiricist”: the theory only provided rules for calculating observable scattering 正如海森堡在 20 世纪 40 年代最初提出的 SS 矩阵理论一样，20 世纪 60 年代的强子 SS 矩阵理论可以被称为 "经验主义 "理论：该理论只提供了计算可观测散射的规则。
amplitudes, instead of striving for a description of the evolution of unobservable states as in field theory. For SS-matrix theorists, “particles” were the asymptotic states observed as the incoming and outgoing states of a scattering process. When considering the actual scattering, SS-matrix theorists did sometimes also speak of “particles” or “families of particles”, but this only reflected cross section resonances (experimentally) or singularities of the SS-matrix equations (theoretically). Further physical interpretation was of no concern. Instead, the central use of these “particles” lay in the calculation of scattering amplitudes (especially at high energies), without caring which objects were precisely exchanged in the scattering. ^(37){ }^{37} This stands in stark contrast to field theory, which assumes a spacetime continuum with field operators defined at each spacetime point, and dynamical equations to govern their time evolution. ^(38){ }^{38} 振幅，而不是像场论那样努力描述不可观测状态的演化。对于 SS 矩阵理论家来说，"粒子 "是散射过程中作为入射和出射状态观察到的渐近状态。在考虑实际散射时， SS -矩阵理论家有时也谈到 "粒子 "或 "粒子族"，但这只反映了截面共振（实验上）或 SS -矩阵方程的奇异性（理论上）。进一步的物理解释并不重要。相反，这些 "粒子 "的核心用途在于计算散射振幅（尤其是在高能量下），而不关心散射中哪些物体发生了精确交换。 ^(37){ }^{37} 这与场论形成了鲜明的对比，场论假定时空是连续的，每个时空点都定义了场算子，并用动力学方程来控制它们的时间演化。 ^(38){ }^{38}
Yet, not all aspects of strong interaction physics could be described by the analytic SS-matrix. In particular, the conserved quantities of the “symmetry approach” to hadrons, which provided a successful classification scheme of low-energy features of the strong interactions, could not be derived from SS-matrix arguments. This classification scheme was constructed out of exploiting the relation between symmetries and conservation laws, resulting in a list of conserved quantities (baryon number AA, angular momentum JJ, parity PP, isotopic spin II, and strangeness SS ) with corresponding quantum numbers. ^(39){ }^{39} The symmetry calculations were grounded in field theory: hadron currents were extracted out of an effective Hamiltonian and their matrix elements were used to describe and predict physical scattering processes. A description in terms of quarks was suggested on the basis of SU(3)S U(3) symmetry. At the same time the symmetry approach was to a large extent independent from field theory: the currents were primarily regarded as symmetry representations and were not derived from field-theoretical dynamical models, since those could not be solved at the time. ^(40){ }^{40} The conserved quantum numbers of the symmetry approach were added to the SS-matrix theory postulates, something that was “less satisfying from an aesthetic point of view” but seemed “unavoidable”, according to Chew. ^(41){ }^{41} 然而，并非强相互作用物理学的所有方面都可以用解析的 SS 矩阵来描述。特别是强子 "对称方法 "的守恒量，它为强相互作用的低能特征提供了一个成功的分类方案，但却不能从 SS 矩阵论证中推导出来。这种分类方案是利用对称性和守恒定律之间的关系而构建的，由此产生了一个具有相应量子数的守恒量列表（重子数 AA 、角动量 JJ 、奇偶性 PP 、同位素自旋 II 和奇异性 SS ）。 ^(39){ }^{39} 对称性计算以场论为基础：从有效哈密顿中提取出强子电流，并用它们的矩阵元素来描述和预测物理散射过程。在 SU(3)S U(3) 对称性的基础上，建议用夸克来描述。同时，对称方法在很大程度上独立于场论：电流主要被视为对称表示，而不是从场论动力学模型中推导出来的，因为这些模型在当时无法求解。 ^(40){ }^{40} 对称方法的守恒量子数被添加到了 SS 矩阵理论公设中，Chew 认为这 "从美学角度来看不太令人满意"，但似乎 "不可避免"。 ^(41){ }^{41}
In spite of Chew’s continuous efforts from the early 1960s onward to advocate his conviction that strongly interacting particles should not be thought of as “elementary” hadrons in field theory but instead should be viewed as observable structures calculable through the SS-matrix equations, a large part of the physicists working on SS-matrix theory was of the opinion that its results should in the end be derivable from field theory. As German physicist George Wentzel for example put it at the 1961 Solvay Conference, to abandon field theory in favor of an SS-matrix scheme “seems to me similar in spirit to abandoning statistical mechanics in favor of phenomenological thermodynamics”. Statistical mechanics was the “comprehensive theory”; only when the calculation of a partition function in statistical mechanics was too difficult should one resort to apply thermodynamics, “at the cost of feeding in more experimental data”. The same was true for the SS-matrix case, according to Wentzel: field theory should be the “superior discipline”, whereas SS-matrix theory could merely offer a description of scattering results when field-theoretical calculations were too hard. ^(42){ }^{42} At the same conference, Mandelstam made clear that he too doubted the superiority of analytic scattering amplitudes over field theoretical concepts: 尽管 Chew 从 20 世纪 60 年代初开始就不断努力宣传他的信念：强相互作用粒子不应被视为场论中的 "基本 "强子，而应被视为可通过 SS 矩阵方程计算的可观测结构，但大部分从事 SS 矩阵理论研究的物理学家都认为，其结果最终应该可以从场论中推导出来。例如，德国物理学家乔治-温策尔（George Wentzel）在 1961 年索尔维会议上指出，放弃场论而采用 SS 矩阵方案，"在我看来，其精神与放弃统计力学而采用现象热力学相似"。统计力学是 "综合理论"；只有当统计力学中的分区函数计算过于困难时，人们才会求助于热力学，"代价是输入更多的实验数据"。温策尔认为， SS -矩阵的情况也是如此：场论应该是 "高级学科"，而 SS -矩阵理论只能在场论计算过于困难时描述散射结果。 ^(42){ }^{42} 在同一次会议上，曼德尔施塔姆明确表示，他也怀疑解析散射振幅优于场论概念：
[T]he possibility of analytically continuing a function into a certain region is a very mathematical notion, and to adopt it as a fundamental postulate rather than a derived theorem appears to be rather artificial. The concept of local field operators, though it may well have to be modified or abandoned in the future, seems more physical. ^(43){ }^{43} [把一个函数分析地延续到某个区域的可能性是一个非常数学化的概念，把它作为一个基本公设而不是一个导出定理似乎是相当人为的。局部场算子的概念，虽然将来很可能要修改或放弃，但似乎更符合物理原理。 ^(43){ }^{43}
Murray Gell-Mann, one of the physicists who made substantial contributions to both SS-matrix theory and the “symmetry physics” of the 1960s, also was of the opinion that SS-matrix theory and field theory in the end should be complementary. ^(44){ }^{44} 默里-盖尔-曼（Murray Gell-Mann）是对 SS -矩阵理论和 20 世纪 60 年代的 "对称物理学 "都做出过重大贡献的物理学家之一，他也认为 SS -矩阵理论和场论最终应该是互补的。 ^(44){ }^{44}
It should be stressed, however, that despite this appreciation of field theory all of particle physics in the 1960s came with a degree of ontological vagueness. It was only during the mid-1970s that the modern dynamical understanding of particles as “symmetry carriers” in gauge field theory became established; in 1960s particle physics, there was no well-defined theoretical notion of a particle, even if one did expect these to populate the elementary world. ^(45){ }^{45} Scattering data showed excitations of protons and neutrons with ever increasing spins, suggesting that protons and neutrons could not be elementary. Regge poles in the SS-matrix formalism offered a description of these excitations, but the idea that the analytic SS-matrix should be the fundamental description of hadronic interactions was controversial. On top of that it became increasingly clear during the 1960s that extending the SS-matrix bootstrap calculations to ever more scattering processes led to an unfeasibly complex calculational scheme relying on many assumptions and approximations that were far from self-evident. ^(46){ }^{46} In addition, symmetry properties were used to classify the hadron spectrum at low energies, but this approach lacked a precise field-theoretical formulation, and could not account for the high-energy behavior of scattering processes. Overall, experiment outstripped theory, and neither “symmetry physics” and field theory nor the SS-matrix program was successful in providing a solid description of the experimental data. 然而，应该强调的是，尽管对场论有了这种认识，但 20 世纪 60 年代的所有粒子物理学都带有一定程度的本体论模糊性。直到 20 世纪 70 年代中期，人们才确立了对粒子的现代动力学理解，即粒子是规量场理论中的 "对称载体"；在 20 世纪 60 年代的粒子物理学中，没有明确定义的粒子理论概念，即使人们确实期望粒子遍布基本世界。 ^(45){ }^{45} 散射数据显示，质子和中子的激发自旋不断增加，这表明质子和中子不可能是基本粒子。 SS 矩阵形式主义中的雷格极点提供了对这些激发的描述，但关于解析的 SS 矩阵应该是强子相互作用的基本描述的观点却引起了争议。此外，在 20 世纪 60 年代，人们越来越清楚地认识到，将 SS 矩阵自举计算扩展到更多的散射过程，会导致计算方案过于复杂，而这种复杂性依赖于许多远非不言自明的假设和近似。 ^(46){ }^{46} 此外，人们还利用对称特性对低能强子谱进行分类，但这种方法缺乏精确的场论表述，无法解释散射过程的高能行为。总之，实验超过了理论，无论是 "对称物理学 "和场论，还是 SS 矩阵程序，都未能成功地对实验数据进行可靠的描述。
One can thus conclude that hadronic SS-matrix theory had two sides to it. On the one hand, the program was strongly principle-driven, requiring solutions that satisfied all the SS-matrix principles; on the other hand, it was developed in continuous contact with experimental results. SS-matrix theory’s principal nature was most strongly embodied in the bootstrap conjecture that from the SS-matrix postulates an infinite set of equations could be generated that allowed for a unique solution, thereby pinning down all parameters of hadronic interactions. 由此我们可以得出结论，强子 SS 矩阵理论具有两面性。一方面，该计划具有强烈的原理驱动性，要求解决方案满足所有 SS 矩阵原理；另一方面，它是在与实验结果不断接触的过程中发展起来的。 SS 矩阵理论的主要性质在自举猜想中得到了最有力的体现，即从 SS 矩阵公设中可以产生无限的方程组，这些方程组允许唯一的解，从而确定了强子相互作用的所有参数。
The bootstrap ideal is of particular interest for our purposes, since the theoretical virtue of finding a “non-arbitrary” description of nature is not limited to SS-matrix theory, but is reminiscent of various unified theory attempts, including those of Einstein, Eddington, and string theorists. What links them is a shared ideal of deriving experimental results from a fundamental theory that determines the parameters, instead of obtaining them from experiment. As Van Dongen (2010) has discussed, Einstein’s quest for a unified theory for electromagnetism and gravity, which occupied most of his later life, was among others grounded in the epistemological belief that the probabilistic 自举理想对我们的目的特别重要，因为找到对自然的 "非任意 "描述这一理论优点并不局限于 SS -矩阵理论，而是让人想起各种统一理论的尝试，包括爱因斯坦、爱丁顿和弦理论家的尝试。将它们联系在一起的是一个共同的理想，即从决定参数的基础理论推导出实验结果，而不是从实验中获取参数。正如 Van Dongen（2010 年）所论述的，爱因斯坦对电磁学和引力统一理论的探索占据了他晚年生活的大部分时间。
quantum nature of matter should not be accepted as a mere empirical fact, but was instead to be deduced from a mathematical theory of field equations. One of the desirable properties of such a theory was that it was free from arbitrary dimensionless constants. For example, when arbitrary constants appeared in a Kaluza-Klein theory proposal, Einstein was quite uncomfortable with this. ^(47){ }^{47} Another example of an attempt to construct a unified theory without arbitrary parameters can be found in the work of Arthur Eddington, who spent the latter part of his life, from the 1920s onwards, searching for an overarching framework that would determine all values of physical constants without relying on quantitative data from experiment. ^(48){ }^{48} 物质的量子性质不应被视为一个单纯的经验事实，而应该从场方程的数学理论中推导出来。这种理论的理想特性之一，就是不存在任意的无量纲常数。例如，当任意常数出现在卡卢扎-克莱因理论的提议中时，爱因斯坦对此感到非常不舒服。 ^(47){ }^{47} 试图构建一个没有任意参数的统一理论的另一个例子可以在阿瑟-爱丁顿的工作中找到，爱丁顿从 20 世纪 20 年代开始，用他的后半生寻找一个总体框架，这个框架可以确定所有物理常数的值，而不依赖于实验的定量数据。 ^(48){ }^{48}
Most importantly for our purpose of linking SS-matrix theory to string theory, a bootstrap-like “non-arbitrariness” argument has also been prominent in assessments of modern string theory. Throughout the 1980s, the alleged uniqueness of the string models that were known was often cited as a reason for its viability, an argument that is intimately related to string theory’s mathematical structure that contains no free parameters to tweak. ^(49){ }^{49} In the more recent philosophical defense of string theory by Richard Dawid (2013), he called string theory “the first physical theory that does not contain or allow any fundamental free parameters”. ^(50){ }^{50} This property he labeled “structural uniqueness”, in contrast to the freedom of the gauge field theory of the standard model, which “allow[s] a nearly unlimited number of models with different interaction structures and particle contents”. 对于我们将 SS -矩阵理论与弦理论联系起来的目的来说，最重要的是，类似于自举的 "非任意性 "论点在对现代弦理论的评估中也很突出。在整个20世纪80年代，人们经常把已知弦模型的所谓唯一性作为弦理论可行的理由，这种论点与弦理论的数学结构密切相关，因为弦理论不包含可随意调整的参数。 ^(49){ }^{49} 理查德-达维德（Richard Dawid，2013年）在最近为弦理论进行的哲学辩护中，称弦理论是 "第一个不包含或不允许任何基本自由参数的物理理论"。 ^(50){ }^{50} 他把这一特性称为 "结构唯一性"，这与标准模型的规量场理论的自由性形成了鲜明对比，后者 "允许具有不同相互作用结构和粒子内容的近乎无限数量的模型"。
At this point it is however far from evident how string theory’s lack of free parameters is precisely related to the hadronic SS-matrix bootstrap: as discussed, in practice hadronic SS-matrix theory came nowhere near the full bootstrap ideal. Instead, SS-matrix physicists were working to include an increasing number of scattering processes in the SS-matrix framework. While doing so they were unable to avoid the use of values obtained from experiment (such as the slopes and intercepts of Regge trajectories) as input in their calculations. In order to understand how one gets from here to string theory and its alleged uniqueness, the next Chapter is concerned with the construction of dual resonance models, which constitutes the link between SS-matrix theory and string theory. As will become clear, adding the new principle of “duality” to the list of SS-matrix postulates seemingly made it plausible that on the basis of the SS-matrix principles a full theory of hadrons could be constructed with almost all parameters determined on a theoretical basis. With this, the practice of theory construction became rapidly disconnected from experiment. 然而，弦理论缺乏自由参数与强子 SS -矩阵自举的关系还远未显现出来：如上所述，在实践中，强子 SS -矩阵理论远未达到完全自举的理想状态。相反， SS -矩阵物理学家正在努力将越来越多的散射过程纳入 SS -矩阵框架。在这样做的同时，他们无法避免使用从实验中获得的数值（例如雷格轨迹的斜率和截距）作为计算的输入。为了理解如何从这里进入弦理论及其所谓的唯一性，下一章将讨论双共振模型的构建，它构成了 SS 矩阵理论与弦理论之间的联系。显而易见，在 SS -矩阵公设列表中加入新的 "二重性 "原理，似乎就可以在 SS -矩阵原理的基础上，构造出一个完整的强子理论，而且几乎所有参数都是在理论基础上确定的。由此，理论构建的实践迅速与实验脱节。
3. Dual resonance models for hadrons 3.强子的双共振模型
Dual resonance models were a class of models that grew out of SS-matrix theory. They were grounded in a theoretical formulation of an approximation known as “duality” that suggested an equivalence between the low- and high-energy descriptions of hadronic scattering. Dual resonance models were essential for the formation of string theory, since string theory’s mathematical structure originated with them. In the mid-1970s, physicists Joël Scherk and John Schwarz (as well as Tamiaki Yoneya) suggested to reinterpret dual resonance models for hadrons as models for a unified theory of all fundamental interactions, which would eventually lead to modern string theory. The main purpose of this Chapter is to demonstrate how particle physicists with the construction of hadronic dual resonance models drifted away from experiment, already before the models were reinterpreted as a potential unified quantum gravity theory. The notion of non-arbitrariness was crucial in this development. 双共振模型是由 SS 矩阵理论发展而来的一类模型。它们以一种被称为 "二元性 "的近似理论表述为基础，这种近似认为强子散射的低能和高能描述之间是等价的。双共振模型对于弦理论的形成至关重要，因为弦理论的数学结构就起源于双共振模型。20 世纪 70 年代中期，物理学家乔尔-舍克（Joël Scherk）和约翰-施瓦茨（John Schwarz）以及米谷玉明（Tamiaki Yoneya）建议把强子的双共振模型重新解释为所有基本相互作用统一理论的模型，这最终导致了现代弦理论的产生。本章的主要目的是论证粒子物理学家在构建强子双共振模型时是如何偏离实验的，在这些模型被重新解释为潜在的统一量子引力理论之前就已经偏离了实验。非任意性概念在这一发展过程中至关重要。
First, I will discuss how duality was elevated from an approximation used to analyze scattering data in SS-matrix theory to a theoretical principle underlying a new type of model building. Next, I highlight some key developments in the construction of dual resonance models that illustrate how the set of principles underlying dual models made it possible to determine almost all parameters on the basis of theoretical reasoning, in line with the virtue of non-arbitrariness. Finally, I discuss how the string picture of dual resonance models arose, how dual model theorists dealt with the physical interpretation of strings, and how string theory was then reinterpreted as a potential unified theory. 首先，我将讨论二重性如何从 SS -矩阵理论中用于分析散射数据的近似值提升为一种新型模型构建所依据的理论原则。接下来，我将重点介绍构建对偶共振模型的一些关键进展，这些进展说明了对偶模型的基本原理是如何使我们有可能根据理论推理来确定几乎所有参数，从而符合非任意性的美德。最后，我讨论了双共振模型的弦图景是如何产生的，双模型理论家是如何处理弦的物理解释的，以及弦理论随后是如何被重新解释为一种潜在的统一理论的。
3.1 Duality and the Veneziano amplitude 3.1 对偶性和威尼斯振幅
The “duality” in dual resonance models stemmed from an extrapolation of the smooth asymptotic high-energy behavior of scattering cross sections to low energies. At low energies the cross section generally is not smooth, but exhibits resonances: narrow peaks associated with the formation of short-lived particles. As was worked out in the late 1960s, the curve that results when extrapolating the high-energy behavior to low energies represented an average of the low-energy resonances (see Figure 2). This extrapolation was called a duality, because it suggested a correspondence between the different descriptions used to calculate the low- and high-energy parts of the amplitude: the lowenergy behavior was calculated from direct channel resonances, the high-energy behavior from exchanged Regge trajectories. It is important to first discuss these two descriptions in more detail. 双共振模型中的 "二元性 "源于将散射截面的平滑渐近高能行为外推至低能。在低能情况下，散射截面通常并不平滑，而是表现出共振：与短寿命粒子的形成有关的狭窄峰值。正如 20 世纪 60 年代末所研究的那样，将高能行为外推到低能时产生的曲线代表了低能共振的平均值（见图 2）。这种外推法被称为二元性，因为它表明了用于计算振幅的低能和高能部分的不同描述之间的对应关系：低能行为是通过直接沟道共振计算出来的，而高能行为则是通过交换的雷格轨迹计算出来的。首先有必要对这两种描述进行更详细的讨论。
At low energies, a bump in the cross section was described by a pole in the amplitude A(s,t)A(s, t) at the associated energy, corresponding to a resonance being produced with mass m_(R)m_{R}. For scattering of spinless particles in the ss-channel (that is, with the Mandelstam variable ss denoting the center-ofmass energy squared), the amplitude in the vicinity of the resonance pole was written as 在低能量时，截面的凹凸由相关能量下的振幅 A(s,t)A(s, t) 的极点来描述，这与质量 m_(R)m_{R} 产生的共振相对应。对于无自旋粒子在 ss - 信道中的散射（即曼德尔施塔姆变量 ss 表示质量中心能量的平方），共振极附近的振幅写为
which has a pole at s=m_(R)^(2)-im_(R)Gamma_(R)s=m_{R}^{2}-i m_{R} \Gamma_{R}. Here A_(if)A_{i f} denotes the residue at the pole (i.e., the coefficient remaining after contour integration). On the basis of unitarity, the residue (associated with the coupling) should be positive; states with negative residue are referred to as “ghosts”. Gamma_(R)\Gamma_{R} represents the width of the cross section bump and is inversely proportional to the resonance lifetime: tau=\tau=1//Gamma_(R)1 / \Gamma_{R}. The resonance amplitude A(s,t)A(s, t) is called a Breit-Wigner amplitude and had been in use as a 这里的 A_(if)A_{i f} 表示极点处的残差（即轮廓积分后剩余的系数）。根据单位性，（与耦合相关的）残差应为正；残差为负的状态被称为 "幽灵"。 Gamma_(R)\Gamma_{R} 表示截面凹凸的宽度，与共振寿命成反比： tau=\tau=1//Gamma_(R)1 / \Gamma_{R} 。共振振幅 A(s,t)A(s, t) 称为布赖特-维格纳振幅，曾被用作
Figure 2: schematic depiction of phenomenological duality. The red line represents the high-energy behavior (calculated in terms of exchanged Regge poles) that is extrapolated to low energies. This on average smooths out the resonance bumps in the low-energy region (calculated in terms of direct channel resonances). Figure from Schwarz (1975, p. 64). 图 2：现象学二重性示意图。红线代表高能行为（以交换的雷格极点计算），并外推至低能。这平均抹平了低能区的共振颠簸（以直接通道共振计算）。图自 Schwarz（1975 年，第 64 页）。
formula for resonance scattering since the 1930s. ^(51){ }^{51} For low-energy hadronic scattering, cross sections were described well by a sum of these Breit-Wigner amplitudes, with every term representing a single intermediate hadron state. However, this ceased to work at higher energies, because then more and more hadrons (including multiparticle states) are created. ^(52){ }^{52} The low-energy resonances were called “direct channel resonances” because they were associated with intermediate resonance states formed out of a collision between incoming particles and subsequently decayed, as is schematically depicted in Figure 3 (right). 自 20 世纪 30 年代以来，布赖特-维格纳一直是共振散射的计算公式。 ^(51){ }^{51} 对于低能强子散射，用这些布雷特-维格纳振幅的总和可以很好地描述截面，每个项都代表一个中间强子态。然而，在更高的能量下，这种方法就不再适用了，因为这时会产生越来越多的强子（包括多粒子态）。 ^(52){ }^{52} 低能共振被称为 "直接通道共振"，因为它们与进入的粒子之间碰撞形成的中间共振态有关，并在随后发生衰变，如图 3（右）所示。
At high energies scattering amplitudes were not computed out of direct-channel BreitWigner contributions, but analyzed in terms of exchanged Regge poles (see the left diagram of Figure 3). As already mentioned in Section 2.2, a Regge pole is the name for a singularity in the scattering amplitude that may arise when treating the angular momentum JJ as a complex variable, the location of the pole in the complex plane being related to the energy. In the 1960s, hadronic mass spectra had been grouped in “Regge trajectories”: groups of resonances of increasing spin and energy values, but with otherwise the same internal quantum numbers. Regge trajectories were found to approximately obey the linear relation 在高能量下，散射振幅不是从直接信道布雷特-维格纳贡献中计算出来的，而是根据交换的雷格极点来分析的（见图 3 左图）。如第 2.2 节所述，雷格极点是将角动量 JJ 视为复变时可能出现的散射振幅奇点的名称，极点在复平面上的位置与能量有关。20 世纪 60 年代，人们曾把强子质谱按 "雷格轨迹 "分组：即自旋和能量值递增但内部量子数相同的共振组。人们发现雷格轨迹近似服从线性关系
where again s=m^(2)s=m^{2} in the center-of-mass frame. Depending on the specific particle, the intercept alpha_(0)\alpha_{0} differed and was determined on the basis of experimental results, but the slope alpha^(')\alpha^{\prime} appeared to be universal and roughly equal to 1GeV^(-2)1 \mathrm{GeV}^{-2} for all mesons and baryons. An example of an experimentally well-studied case was the rho\rho meson trajectory, with resonances at spin values J=1,2,3,dotsJ=1,2,3, \ldots and 其中 s=m^(2)s=m^{2} 也是在质量中心框架内。根据具体粒子的不同，截距 alpha_(0)\alpha_{0} 也不同，是根据实验结果确定的，但斜率 alpha^(')\alpha^{\prime} 似乎是通用的，大致等于所有介子和重子的 1GeV^(-2)1 \mathrm{GeV}^{-2} 。实验研究得比较清楚的一个例子是介子轨迹 rho\rho ，它在自旋值 J=1,2,3,dotsJ=1,2,3, \ldots 和rho\rho 处有共振。
Figure 3 (to be read upwards): schematic depiction of “exchanged” and “direct-channel” resonances. The figure on the left represents two particles interacting through an exchange of resonances, which was calculated from high-energy exchanged Regge poles. The figure on the right depicts two particles interacting due to the formation of an intermediate resonance state that then decays, which was described by a low-energy Breit-Wigner amplitude. Figure from Schwarz (1975, p. 64). 图 3（向上读）："交换 "和 "直接通道 "共振的示意图。左图表示两个粒子通过交换共振相互作用，这是由高能交换雷格极计算得出的。右图描述的是两个粒子由于形成中间共振态而相互作用，然后衰减，这是用低能布雷特-维格纳振幅描述的。图自 Schwarz（1975 年，第 64 页）。
Figure 4: schematic diagram of the rho\rho meson trajectory. The dots represent resonances, with on the vertical axis their spin value, J=alpha(s)=1,2,3J=\alpha(s)=1,2,3, and on the horizontal axis the corresponding mass-squared values s=m^(2)=1,2,3GeV^(2)s=m^{2}=1,2,3 \mathrm{GeV}^{2} (with convention ℏ=c=1\hbar=c=1 ). So, the first dot (denoted with an " ee ") represents the spin-1 rho\rho meson with m_(e)^(2)=1GeV^(2)m_{e}^{2}=1 \mathrm{GeV}^{2}, the " ff " dot a spin- 2 resonance with m_(f)^(2)=2GeV^(2)m_{f}^{2}=2 \mathrm{GeV}^{2}, etc. In the rho\rho meson case Regge trajectories were found experimentally to continue up to at least J=6J=6. Figure from Jacob\operatorname{Jacob} (1969, p. 128). 图 4： rho\rho 介子轨迹示意图。点代表共振，在纵轴上代表其自旋值 J=alpha(s)=1,2,3J=\alpha(s)=1,2,3 ，在横轴上代表相应的质量平方值 s=m^(2)=1,2,3GeV^(2)s=m^{2}=1,2,3 \mathrm{GeV}^{2} （按惯例为 ℏ=c=1\hbar=c=1 ）。因此，第一个点（用" ee "表示）代表自旋-1 的 rho\rho 介子与 m_(e)^(2)=1GeV^(2)m_{e}^{2}=1 \mathrm{GeV}^{2} ，" ff "点代表自旋-2 的共振与 m_(f)^(2)=2GeV^(2)m_{f}^{2}=2 \mathrm{GeV}^{2} 等。实验发现，在 rho\rho 介子的情况下，Regge 轨迹至少会持续到 J=6J=6 。图自 Jacob\operatorname{Jacob} (1969, p. 128)。
approximately at masses s=m^(2)=1,2,3,dotsGeV^(2)s=m^{2}=1,2,3, \ldots \mathrm{GeV}^{2} (see Figure 4). ^(53){ }^{53} Apart from their use in classifying resonance trajectories from hadronic scattering data, Regge poles were crucial in SS-matrix theory for calculating the asymptotic behavior of scattering. A regime that was explored extensively in experiments of the 1960s was the case of high energy ( s rarr oos \rightarrow \infty ) and fixed and negative |t||t|, corresponding to forward scattering (i.e., zero scattering angle). 近似于质量 s=m^(2)=1,2,3,dotsGeV^(2)s=m^{2}=1,2,3, \ldots \mathrm{GeV}^{2} （见图 4）。 ^(53){ }^{53} 除了用于从强子散射数据中对共振轨迹进行分类之外，雷格极点在 SS 矩阵理论中对计算散射的渐近行为也至关重要。在 20 世纪 60 年代的实验中，对高能量（ s rarr oos \rightarrow \infty ）和固定负 |t||t| 的情况进行了广泛的探索，这与前向散射（即散射角为零）相对应。
The Regge trajectories were interpreted as families of exchanged force particles, but instead of a one-by-one description of the exchanged particles (as in field theory) all the resonances lying on the trajectory were viewed as being exchanged together, which was a very mathematical notion. These exchanged trajectories were sometimes called “Reggeons” to distinguish them from elementary particles. ^(54){ }^{54} In scattering processes where only one type of exchanged particle was involved, the high-energy behavior of the amplitude could be calculated in terms of a single Regge trajectory, whereas in other cases combinations of Regge trajectories were involved. The trajectories were in general grounded in measurement of resonances: linear plots relating spin to mass squared of resonances (like Figure 4) were verified experimentally, at least for the first number of resonances. ^(55){ }^{55} 雷格轨迹被解释为交换力粒子族，但不是对交换粒子进行逐一描述（如在场论中），而是将轨迹上的所有共振视为一起被交换，这是一个非常数学化的概念。这些交换的轨迹有时被称为 "Reggeons"，以区别于基本粒子。 ^(54){ }^{54} 在只涉及一种交换粒子的散射过程中，振幅的高能行为可以用单一的雷格轨迹来计算，而在其他情况下，则涉及雷格轨迹的组合。这些轨迹一般以共振的测量为基础：共振的自旋与质量平方之间的线性图（如图 4）已在实验中得到验证，至少对第一批共振是如此。 ^(55){ }^{55}
The exception to this was a specific trajectory called the “Pomeron trajectory” that was assumed solely to fit the data on total cross sections, and was only motivated by indirect and equivocal evidence. The Pomeron trajectory differed from all the other known trajectories, whose slopes were roughly alpha^(')∼1GeV^(-2)\alpha^{\prime} \sim 1 \mathrm{GeV}^{-2}, because it had a slope of approximately alpha^(')∼0.5GeV^(-2)\alpha^{\prime} \sim 0.5 \mathrm{GeV}^{-2}. No resonances were detected that provided evidence that the Pomeron trajectory could be associated in any way with a physical particle, but it nevertheless was essential for fitting the data. It was associated with elastic scattering, where no intermediate states of higher energies were formed (also called “vacuum” scattering). ^(56){ }^{56} 但有一种被称为 "波美拉尼亚轨迹 "的特殊轨迹是个例外，这种轨迹完全是为了拟合总截面数据而假定的，其动机只是间接和模棱两可的证据。波美拉尼亚轨迹与所有其他已知轨迹不同，它们的斜率大致为 alpha^(')∼1GeV^(-2)\alpha^{\prime} \sim 1 \mathrm{GeV}^{-2} ，因为它的斜率大约为 alpha^(')∼0.5GeV^(-2)\alpha^{\prime} \sim 0.5 \mathrm{GeV}^{-2} 。没有发现任何共振能证明波美拉尼亚轨迹与物理粒子有任何关联，但它对拟合数据至关重要。它与弹性散射有关，在弹性散射中没有形成更高能量的中间态（也称为 "真空 "散射）。 ^(56){ }^{56} .
So, at low energies the amplitude was computed out of sums of direct resonances in the ss channel; at high energies in terms of exchanged Regge trajectories in the tt-channel (see Figure 3). The complete amplitude is then obtained by summing both contributions. The extrapolation of the high-energy Regge curve to the low-energy region (i.e., Figure 2) now suggested an equivalence between the two. As Maurice Jacob from CERN did put it: “The exchanged Regge trajectories can thus be considered as built up from direct channel resonances. Conversely, Regge exchange already includes the resonances in an average sense.” ^(57){ }^{57} It is this equivalence that became known as “duality”. 因此，在低能量时，振幅由 ss 通道中的直接共振之和计算得出；在高能量时，振幅由 tt 通道中交换的雷格轨迹计算得出（见图 3）。然后，将这两个贡献相加就得到了完整的振幅。将高能雷格曲线外推到低能区域（即图 2），现在表明两者之间是等价的。正如欧洲核子研究中心的莫里斯-雅各布（Maurice Jacob）所言："因此，交换的雷格轨迹可以被认为是由直接通道共振建立起来的。相反，雷格交换已经包含了平均意义上的共振"。 ^(57){ }^{57} 正是这种等价性被称为 "对偶性"。
Initially, the duality was employed as a tool in the analysis of scattering data: since it related the low-energy properties to the high-energy Regge exchange, it gave “fruitful constraints on the possible parameters used to describe high energy processes”…^(58)^{58} There was no consensus on a precise definition of duality. Nevertheless, again in the words of Jacob (1969, p. 127), “the scheme has (…) predictive value, can be tested and meets success”. It went by the name of “DHS duality” (after the 最初，二重性被用作分析散射数据的工具：因为它把低能特性与高能的雷格交换联系起来，所以它 "对用于描述高能过程的可能参数提供了富有成效的约束"...... ^(58)^{58} 关于二重性的精确定义，人们还没有达成共识。尽管如此，用雅各布（1969 年，第 127 页）的话说，"该方案具有（......）预测价值，可以进行测试并取得成功"。该方案被命名为 "DHS 对偶性"（以 "DHS "命名）。
Figure 5: the mathematical structure of a Veneziano amplitude A(s,t)A(s, t). The dots denote resonances that are exchanged in particle reactions; alpha(s)=J\alpha(s)=J is the spin value and s=m^(2)s=m^{2} in the center-of-mass frame (with convention ℏ=c=1\hbar=c=1 ). Below the leading (“parent”) trajectory there are “daughter” trajectories. A resonance at a pole alpha(s)=J\alpha(s)=J was associated with the exchange of particles of spin J,J-1,dots,0J, J-1, \ldots, 0. In this sense the amplitude described “towers of hadrons”. Figure from Jacob (1969, p. 129). 图 5：威尼斯振幅 A(s,t)A(s, t) 的数学结构。点表示粒子反应中交换的共振； alpha(s)=J\alpha(s)=J 是自旋值， s=m^(2)s=m^{2} 在质量中心框中（按惯例为 ℏ=c=1\hbar=c=1 ）。在前导（"父"）轨迹的下方有 "子 "轨迹。极点 alpha(s)=J\alpha(s)=J 的共振与自旋 J,J-1,dots,0J, J-1, \ldots, 0 的粒子交换有关。从这个意义上说，振幅描述了 "强子塔"。图自雅各布（1969 年，第 129 页）。
physicists Dolen, Horn and Schmid who introduced it) or “global duality”. 59 或 "全局对偶性"。59
In this guise duality was simply a new tool in the toolbox of SS-matrix theorists grappling with scattering data. However, mainly following up on a result from Gabriele Veneziano (1968), a theoretical definition of “duality” emerged that was added to the SS-matrix postulates and underpinned a new class of models. These “dual resonance models”, as they became known, were soon researched by a large number of physicists, and ushered in a new practice of theory construction that quickly became, in contrast to the SS-matrix tradition from which it sprang, disconnected from experiment. 在这种情况下，对偶性只是 SS -矩阵理论家处理散射数据的工具箱中的一个新工具。然而，主要是为了跟进加布里埃尔-威尼斯诺（Gabriele Veneziano，1968 年）的一项成果，"对偶性 "的理论定义出现了，它被添加到 SS 矩阵公设中，并成为一类新模型的基础。这些后来被称为 "二元共振模型 "的模型很快被大量物理学家研究，并开创了一种新的理论构建实践，与 SS 矩阵传统形成鲜明对比的是，这种实践很快就与实验脱节了。
Veneziano’s result consisted of an amplitude function for the particular meson scattering process of pi pi rarr pi omega\pi \pi \rightarrow \pi \omega that exhibited the duality between Regge poles and resonances: the function contained poles in families of linear trajectories, and had the right asymptotic behavior. Veneziano and collaborators had been systematically looking for a function with these properties for some months. ^(60){ }^{60} Following Veneziano’s lead, physicists quickly started to formulate similar amplitudes for other scattering processes. In the simple general case of identical spinless bosons, such an amplitude reads 委内瑞拉诺的成果包括一个用于 pi pi rarr pi omega\pi \pi \rightarrow \pi \omega 特定介子散射过程的振幅函数，它展现了雷格极点和共振之间的二重性：该函数包含线性轨迹族中的极点，并且具有正确的渐近行为。委内瑞拉诺和合作者几个月来一直在系统地寻找具有这些特性的函数。 ^(60){ }^{60} 在威尼斯诺的引领下，物理学家们很快开始为其他散射过程制定类似的振幅。在相同的无自旋玻色子的简单一般情况下，这样的振幅为
A gamma function Gamma(n)\Gamma(n) exhibits simple poles when nn is zero or a negative integer, so this amplitude has poles when either alpha(s)\alpha(s) or alpha(t)\alpha(t) equals n=0,1,2,dotsn=0,1,2, \ldots. Most importantly, the residue of a pole in ss is a polynomial in tt. For alpha(s)rarr n=0,1,2,dots\alpha(s) \rightarrow n=0,1,2, \ldots, one can write: 当 nn 为零或负整数时，伽马函数 Gamma(n)\Gamma(n) 显示出简单极点，因此当 alpha(s)\alpha(s) 或 alpha(t)\alpha(t) 等于 n=0,1,2,dotsn=0,1,2, \ldots 时，该振幅具有极点。最重要的是， ss 中极点的残差是 tt 中的多项式。对于 alpha(s)rarr n=0,1,2,dots\alpha(s) \rightarrow n=0,1,2, \ldots ，我们可以写出：
using the limit Gamma(x)rarr((-1)^(n))/(n!)(1)/(x+n)\Gamma(x) \rightarrow \frac{(-1)^{n}}{n!} \frac{1}{x+n} for x rarr-nx \rightarrow-n. Then, using the property Gamma(x)=(x-1)Gamma(x-1)\Gamma(x)=(x-1) \Gamma(x-1) iteratively, it can be shown that the last term in the above equation is a polynomial in tt : 利用 x rarr-nx \rightarrow-n 的极限 Gamma(x)rarr((-1)^(n))/(n!)(1)/(x+n)\Gamma(x) \rightarrow \frac{(-1)^{n}}{n!} \frac{1}{x+n} 。然后，利用 Gamma(x)=(x-1)Gamma(x-1)\Gamma(x)=(x-1) \Gamma(x-1) 的迭代性质，可以证明上式的最后一项是 tt 的多项式：
Defining ((-1)^(n))/(n!)(-alpha(t)-1)(-alpha(t)-2)dots(-alpha(t)-n)-=c_(n)(t)\frac{(-1)^{n}}{n!}(-\alpha(t)-1)(-\alpha(t)-2) \ldots(-\alpha(t)-n) \equiv c_{n}(t), it then follows that the full amplitude for alpha(s)rarr n=0,1,2,dots\alpha(s) \rightarrow n=0,1,2, \ldots is given by the sum: 根据 ((-1)^(n))/(n!)(-alpha(t)-1)(-alpha(t)-2)dots(-alpha(t)-n)-=c_(n)(t)\frac{(-1)^{n}}{n!}(-\alpha(t)-1)(-\alpha(t)-2) \ldots(-\alpha(t)-n) \equiv c_{n}(t) 的定义，可以得出 alpha(s)rarr n=0,1,2,dots\alpha(s) \rightarrow n=0,1,2, \ldots 的全振幅由总和给出：
that is, an exchange of resonances in tt. However, the same result is obtained if one starts with poles a(t)rarr n=0,1,2,dotsa(t) \rightarrow n=0,1,2, \ldots, leading to an exchange of resonances in ss. The amplitude is thus built up from either a set of ss-channel resonances or a set of tt-channel resonances; it is no longer needed to sum the low-energy ss-channel contributions and the high-energy tt-channel Regge pole exchange. It is in this sense that the amplitude A(s,t)A(s, t) exhibits duality between the ss - and tt-channel. ^(61){ }^{61} Mathematically this duality was expressed as 然而，如果从极点 a(t)rarr n=0,1,2,dotsa(t) \rightarrow n=0,1,2, \ldots 开始，也会得到相同的结果，从而导致 ss 中的共振交换。因此，振幅是由一组 ss 沟道共振或一组 tt 沟道共振建立起来的；不再需要对低能量 ss 沟道贡献和高能量 tt 沟道雷格极点交换进行求和。正是在这个意义上，振幅 A(s,t)A(s, t) 显示了 ss - 和 tt - 信道之间的二重性。 ^(61){ }^{61} 这种二重性的数学表达式为
where, as said, the c_(n)c_{n} coefficients denote the residue polynomials at each pole. ^(62){ }^{62} Note that an infinite sum of resonances was needed to yield the right asymptotic behavior, which meant the Regge trajectories were conjectured to keep on rising infinitely. Two other things must be noted about A(s,t)A(s, t) : 其中， c_(n)c_{n} 系数表示每个极点的残差多项式。 ^(62){ }^{62} 请注意，要产生正确的渐近行为，需要无限的共振总和，这意味着雷格轨迹被推测为无限上升。关于 A(s,t)A(s, t) 还必须注意两点：
There are no double poles in A(s,t)A(s, t) : when both alpha(s)\alpha(s) and alpha(t)\alpha(t) are equal to zero or a positive integer, i.e., if there is a double pole in the numerator, then the denominator has a pole as well, so one is again left with a simple pole. ^(63){ }^{63} A(s,t)A(s, t) 中没有双极点：当 alpha(s)\alpha(s) 和 alpha(t)\alpha(t) 都等于零或正整数时，也就是说，如果分子中有一个双极点，那么分母也有一个极点，因此又只剩下一个简单极点。 ^(63){ }^{63}
The pole structure of A(s,t)A(s, t) represents an infinite set of parallel trajectories. As said, for a pole at alpha(s)=J=0,1,2,dots\alpha(s)=J=0,1,2, \ldots, the residue of the pole is a polynomial of degree JJ in tt. It can be decomposed as a series of Legendre polynomials of degrees J,J-1dots0J, J-1 \ldots 0. These polynomials were associated with the exchange of particles of spin J,J-1,dots0\operatorname{spin} J, J-1, \ldots 0. For example, in Figure 5 the dot labeled with gg in the upper right denotes a pole of the rho\rho meson with spin J=3J=3, represented by a polynomial of degree 3 . By decomposing it, Legendre polynomials of degrees J=3,2,1,0J=3,2,1,0 contribute to the amplitude, associated with the A(s,t)A(s, t) 的极点结构代表了一个无限平行轨迹集。如前所述，对于 alpha(s)=J=0,1,2,dots\alpha(s)=J=0,1,2, \ldots 处的极点，极点的残差是 tt 中 JJ 度的多项式。它可以分解为一系列度数为 J,J-1dots0J, J-1 \ldots 0 的 Legendre 多项式。这些多项式与 spin J,J-1,dots0\operatorname{spin} J, J-1, \ldots 0 的粒子交换有关。例如，在图 5 中，右上方标有 gg 的圆点表示自旋为 J=3J=3 的 rho\rho 介子的一个极点，用 3 度多项式表示。将其分解后， J=3,2,1,0J=3,2,1,0 度的 Legendre 多项式对振幅有贡献，它与
exchange of particles of spin 3,2,1,0.^(64)3,2,1,0 .{ }^{64} In this sense the Veneziano-type amplitude was interpreted as describing “towers of hadrons”. 3,2,1,0.^(64)3,2,1,0 .{ }^{64} 从这个意义上说，威尼斯式振幅被解释为描述了 "强子塔"。
In short, the Veneziano amplitude provided a description of “DHS duality” (which was essentially an approximation) in a single formula. 简而言之，威尼斯振幅用一个公式描述了 "DHS 对偶性"（本质上是一种近似）。
Veneziano’s original amplitude described scattering processes with two incoming and two outgoing particles, but it was soon generalized to the NN-point case. The main drawback was that the amplitude violated unitarity. This was due to the fact that the amplitude described poles lying on exactly linear Regge trajectories. However, Regge trajectories can only be exactly linear in an approximation where the resonances have zero width, called a “narrow-resonance approximation”. Since the width is inversely related to the resonance lifetime, this approximation implies that intermediate states do not decay. ^(65){ }^{65} This also meant that the Veneziano amplitude described resonances that were exchanged one at a time, with no interactions between them (as this would lead to unstable intermediate states). It was in tackling the unitarity problem that the idea took hold that the Veneziano amplitude could perhaps lead to a full theory of hadronic interactions. ^(66){ }^{66} The main idea behind this was that Veneziano-type amplitudes violated unitarity in a similar manner as the Born approximation in field theory. This inspired an approach in which the Veneziano amplitude was considered as the Born term (or “tree diagram”) in a perturbation expansion, analogous to the role of the Born term in QED. ^(67){ }^{67} As Alessandrini, Amati, Le Bellac, and Olive explained in a 1971 Physics Report: Veneziano 最初的振幅描述的是两个入射粒子和两个出射粒子的散射过程，但很快就被推广到 NN 点的情况。其主要缺点是振幅违反了单位性。这是由于振幅描述了位于完全线性的雷格轨迹上的极点。然而，雷格轨迹只有在共振宽度为零的近似情况下才能完全线性，这种近似被称为 "窄共振近似"。由于宽度与共振寿命成反比，这种近似意味着中间状态不会衰变。 ^(65){ }^{65} 这也意味着威尼斯诺振幅描述的共振是一个一个交换的，它们之间没有相互作用（因为这会导致不稳定的中间态）。正是在解决单位性问题的过程中，人们产生了这样的想法，即威尼斯振幅或许能带来强子相互作用的完整理论。 ^(66){ }^{66} 这背后的主要想法是，威尼斯式振幅违反了单整性，其方式与场论中的玻恩近似类似。这启发了一种方法，即在扰动展开中，把威尼斯振幅看作是Born项（或 "树图"），类似于QED中Born项的作用。 ^(67){ }^{67} 正如亚历山德里尼、阿玛蒂、勒贝拉克和奥利弗在 1971 年的《物理学报告》中解释的那样：
What we are exploring with this dual construction is a new approximation scheme to hadron physics. Instead of considering the construction of one state after another (as was done up to now under the general appellation of nearby singularities) we consider a coherent infinite set of states even in the first approximation. The next step will perturb coherently the infinite set and so on. This is the novel idea underlying the dual perturbative approach and we must still learn if we are able to construct a consistent disease-free theory with this approximation scheme. ^(68){ }^{68} 我们通过这种双重构造探索的是强子物理学的一种新近似方案。我们不考虑一个接一个状态的构建（就像迄今为止在 "附近奇点 "的一般称谓下所做的那样），而是考虑一个连贯的无限状态集，甚至在第一近似中也是如此。下一步将对这个无限集进行相干扰动，依此类推。这就是二重扰动方法的新思想，我们还必须了解我们是否能够用这种近似方案构建出一致的无病理论。 ^(68){ }^{68}
This aimed-for theory became known as “dual resonance theory” or shortly “dual theory”. As we will see in the next Section, with the attempts to construct a full theory of hadrons starting from the Veneziano amplitude the practice of theory construction soon became primarily theory-driven and detached from experimental input. 这一目标理论后来被称为 "双共振理论"，简称 "双理论"。我们将在下一节中看到，由于试图从 Veneziano 振幅出发构建一个完整的强子理论，理论构建的实践很快变得主要由理论驱动，而脱离了实验输入。
3.2 Dual resonance models and the "model world" 3.2 双共振模型与 "模型世界"
To understand the shift to a theory-driven practice that accompanied the construction of dual resonance models, we need to once again turn to the SS-matrix bootstrap. Recall that the bootstrap hypothesis behind hadronic SS-matrix theory in the 1960 s was that a unique solution of the equations generated by the SS-matrix principles would determine all parameters of hadronic interactions. A full hadronic bootstrap was clearly out of reach, as an exact fulfillment of all SS-matrix postulates was practically impossible. Instead, the bootstrap was implemented in simplified calculations that contained free parameters, such as the intercepts and slopes of Regge trajectories. This, as Veneziano (1974) put it in a Physics Report on dual models, had led to a theory that was "too loose, 要理解伴随着双共振模型的构建而出现的向理论驱动实践的转变，我们需要再次转向 SS -矩阵自举。回想一下，20 世纪 60 年代强子 SS 矩阵理论背后的自举假设是，由 SS 矩阵原理产生的方程的唯一解将决定强子相互作用的所有参数。完全的强子自举显然是不可能的，因为精确地实现所有 SS 矩阵假设实际上是不可能的。取而代之的是在包含自由参数（如雷格轨迹的截距和斜率）的简化计算中实施自举法。正如 Veneziano（1974 年）在一份关于二元模型的物理学报告中所说，这导致了一种 "过于松散 "的理论、
since there are many free parameters which can be chosen in order to fit the data in various kinematical regions" (p. 18). What was missing, according to Veneziano, was 因为有许多自由参数可供选择，以拟合不同运动学区域的数据"（第 18 页）。Veneziano 认为，缺少的是
an idea of how to really arrive at a fairly unique and simple first-order SS-matrix. Either there is too much freedom of choice (if we do not demand crossing and unitarity, for instance), or there is no simple solution (if those requirements are enforced). (p. 18) 如何才能真正得到一个相当独特和简单的一阶 SS 矩阵？要么是选择的自由度太大（例如，如果我们不要求交叉性和单位性），要么是没有简单的解决方案（如果强制执行这些要求）。(p. 18)
Adding duality as an extra principle to the already listed SS-matrix postulates was a way out of this stalemate, promising the possibility “that theory can find its way through even before a number of detailed experiments will be done to provide theoretical hints”. In contrast to the single-particle intermediate states that were usually considered in hadronic SS-matrix theory, in dual resonance models an infinite set of resonances (as appearing in Veneziano-type amplitudes) was invoked. 在已经列出的 SS -矩阵公设之外，增加二重性作为额外的原理，是摆脱这种僵局的一种方法，有望 "在进行大量详细实验以提供理论提示之前，理论就能找到突破口"。与强子 SS -矩阵理论通常考虑的单粒子中间态不同，在双共振模型中，引用了一组无限的共振（如在委内瑞拉型振幅中出现的那样）。
In the late 1960s there was an explosion of work on dual resonance models-in the words of Rickles, a “near-industrial scale refinement”. ^(69){ }^{69} The intense interest went accompanied by the hope that dual models could actually lead to a full theory of hadronic interactions, that is, that the “approximation scheme” could indeed be turned into a “consistent disease-free theory”. The choice to pursue dual models was also made because of the promise of professional advancement: many contributors were early-career scholars and motivated by the fact that the dual models were different. The center of activity for dual model research was the CERN Theory Division, where Daniele Amati was active in gathering enthusiasts. David Olive (2012, pp. 349-350) recalled the mood in the group: “We were driven by our shared common belief that we were working on the theory of the future and we were trying to work out what that was specifically. It was clearly something new and unlike conventional quantum field theory and that was attractive to us.” There were many interactions between the CERN group and the US physicists working on dual models. In the US John Schwarz was a central figure, collaborating for instance with the French physicists André Neveu and Joël Scherk on numerous occasions in the early 1970s at Princeton, Caltech, and CERN. Around the same time, Sergio Fubini and Gabriele Veneziano were working on dual models at MIT. 20 世纪 60 年代末，关于双共振模型的研究出现了爆炸式增长--用里克斯的话说，这是 "近乎工业规模的改进"。 ^(69){ }^{69} 这种浓厚的兴趣伴随着一种希望，即双重模型能够真正导致一个完整的强子相互作用理论，也就是说，"近似方案 "确实可以变成一个 "一致的无病理论"。选择研究二元模型的另一个原因是职业晋升的前景：许多贡献者都是初出茅庐的学者，他们的动机是二元模型与众不同。欧洲核子研究中心理论部是双重模型研究的活动中心，达尼埃莱-阿马蒂在那里积极召集爱好者。大卫-奥利弗（David Olive）（2012，第 349-350 页）回忆了当时的情景："我们被共同的信念所驱使，认为我们正在研究未来的理论，并试图找出它的具体内容。这显然是新的东西，与传统的量子场论不同，这对我们很有吸引力"。欧洲核子研究中心小组与美国研究二元模型的物理学家之间有许多互动。在美国，约翰-施瓦茨是一个核心人物，例如，20 世纪 70 年代初，他在普林斯顿大学、加州理工学院和欧洲核子研究中心与法国物理学家安德烈-内维（André Neveu）和若埃尔-舍克（Joël Scherk）多次合作。大约在同一时期，塞尔吉奥-富比尼（Sergio Fubini）和加布里埃尔-威尼斯诺（Gabriele Veneziano）在麻省理工学院研究二元模型。
The work carried out on dual resonance models in roughly the years 1968-1975 is nowadays mostly known because the models allowed for an interpretation in terms of a quantum-relativistic string. Dual models were constructed making use of both an harmonic oscillator operator formalism and of an oscillating string picture. By the mid-1970s it was clear that the two approaches amounted to the same mathematical structure (Scherk, 1975), and it was this “hadronic string theory” that was recast and further developed as a quantum gravity string theory. In the string picture, the Veneziano model described bosonic open strings; the “Ramond-Neveu-Schwarz model” described fermionic open strings, and the Pomeron “Virasoro-Shapiro model” described closed strings. Thus, dual models are the link between analytic SS-matrix theory and modern string theory. This is true for the theoretical framework involved, but, more importantly for our purposes, it also reshaped the attitude of practitioners towards the role of experimental data in the theory’s development. 大约在 1968-1975 年间进行的关于双重共振模型的工作如今已广为人知，因为这些模型可以用量子相对论弦来解释。双共振模型是利用谐振子算子形式主义和振荡弦图构建的。到 20 世纪 70 年代中期，这两种方法显然具有相同的数学结构（Scherk，1975 年），正是这种 "强子弦理论 "被重新构建并进一步发展为量子引力弦理论。在弦图中，威尼斯模型描述的是玻色开弦；"拉蒙德-奈维-施瓦茨模型 "描述的是费米开弦，而波美拉尼亚 "维拉索罗-沙皮罗模型 "描述的是闭弦。因此，对偶模型是解析 SS 矩阵理论与现代弦理论之间的纽带。这对相关理论框架来说是正确的，但对我们的目的来说更重要的是，它还重塑了实践者对实验数据在理论发展中的作用的态度。
As said, in the new scheme all postulates (now including duality) were imposed on the SS matrix, unitarity being enforced through loop corrections. ^(70){ }^{70} Starting from there, dual model physicists became concerned with the theory-driven construction of a class of mathematical models: specific values for quantities and parameters that were previously determined on an experimental basis could now be derived from theoretical reasoning on the basis of the dual model postulates. This was in line with the (bootstrap-inspired) theoretical virtue of non-arbitrariness, reducing the number 如前所述，在新方案中，所有公设（现在包括二重性）都被强加在 SS 矩阵上，通过环路修正实现了单一性。 ^(70){ }^{70} 从那时起，二元模型物理学家开始关注理论驱动的一类数学模型的构建：以前在实验基础上确定的量和参数的具体值，现在可以在二元模型公设的基础上通过理论推理得出。这符合（自举法启发的）非任意性的理论美德，减少了
of experimentally determined “arbitrary” values in the theory. The theoretical outcomes were, however, inconsistent with known empirical results. It is instructive to briefly discuss two examples here: the value of the intercept of the leading Regge trajectory of the Veneziano amplitude, and the value of the number of spacetime dimensions. ^(71){ }^{71} 实验确定的 "任意 "理论值。然而，理论结果与已知的经验结果并不一致。在此简要讨论两个例子很有启发：威尼斯振幅的前导雷格轨迹截距值和时空维数的值。 ^(71){ }^{71}
Let us first discuss the case of the intercept of the leading Regge trajectory of the Veneziano amplitude. Recall that the intercept is the point where the Regge trajectory intersects the vertical axis; as such, it determines the mass squared of the lowest-lying state. For the Veneziano amplitude, the intercept could be varied; experimental results suggested a value of around 0.5 . However, in order to establish the unitarity of the Veneziano model, it had to be demonstrated that the amplitude was “factorizable”. In general, factorizability means that the coupling, represented by the residue at a pole in the amplitude, can be written as a product of two terms. One term corresponds to the amplitude for creating an excited particle (or resonance) out of the incoming particles; the other term represents the decay amplitude of the intermediate state into the final, outgoing particles. For an arbitrary pole in the Veneziano amplitude, it was demonstrated that the residue could be written as a sum of a finite number of factorized terms. Each of these terms was then matched with a different excited state (of the “towers of hadrons”, see Figure 5) to construct the spectrum. ^(72){ }^{72} To study the factorization properties and corresponding spectrum an infinite set of harmonic oscillators a_(mu)^(n)a_{\mu}^{n} and a_(mu)^(n†)a_{\mu}^{n \dagger} was introduced obeying the commutation relations [a_(mu)^(n),a_(v)^(n†)]=\left[a_{\mu}^{n}, a_{v}^{n \dagger}\right]=delta_(mn)eta_(mu v)^(73)A\delta_{m n} \eta_{\mu v}{ }^{73} \mathrm{~A} major problem was that states with negative norm (so-called “ghosts”) appeared in the spectrum, thereby violating unitarity. The appearance of ghost was well-known from QED, in which case it is resolved by imposing a condition for physical states (the “Fermi condition”) that follows from QED’s gauge invariance. In the dual model case a similar approach was tried. Here the conditions to keep only the physical states were based upon an infinite-dimensional gauge algebra first suggested (in the dual model context) by Miguel Virasoro, then at the University of Wisconsin. ^(74){ }^{74} However, and that is the point here, the “Virasoro conditions” were only satisfied if the intercept alpha_(0)\alpha_{0} of the leading Regge trajectory was equal to 1 . This was inconsistent with the experimentally suggested value of 0.5 ; an intercept of 1 implied that the spectrum contained a massless boson of spin one, which was contradictory to the short-range reach of the strong force. In addition, it implied that the lowest-lying particle (that is, the ground state) had a negative mass squared: m^(2)=-1//alpha^(')m^{2}=-1 / \alpha^{\prime}. Such an imaginary-mass particle is called a “tachyon” and it violates causality. The Virasoro conditions thus were, as Joël Scherk (1975, p. 124) some years later stated in a review article, “clearly a step away from reality”, where “an intercept of 0.5 would be much preferred”. However, Scherk continued, “it also led to a more satisfactory situation from the theoretical point of view”. Apparently, saving unitarity (by making the spectrum of the Veneziano-model ghost-free) had priority in theory construction, even if it implied a massless boson and a causality-violating tachyon. Consequently, with the Virasoro conditions, the intercept of the leading Regge trajectory became fixed on a theoretical basis. 让我们首先讨论委内瑞拉振幅的雷格轨迹截距的情况。回顾一下，截距是雷格轨迹与纵轴相交的点；因此，它决定了最底层状态的质量平方。对于威尼斯振幅来说，截距是可以改变的；实验结果表明截距值在 0.5 左右。然而，为了建立威尼斯模型的单位性，必须证明振幅是 "可因式分解 "的。一般来说，可因式分解意味着耦合（由振幅中某一极的残差表示）可以写成两个项的乘积。其中一个项对应于从输入粒子中产生一个激发粒子（或共振）的振幅；另一个项则代表中间态衰变为最终输出粒子的振幅。对于 Veneziano 振幅中的任意极点，实验证明残差可以写成有限个因数项的总和。然后，每个项都与不同的激发态（"强子塔 "的激发态，见图 5）相匹配，从而构建出频谱。 ^(72){ }^{72} 为了研究因式分解的特性和相应的频谱，我们引入了一组无限的谐振子 a_(mu)^(n)a_{\mu}^{n} 和 a_(mu)^(n†)a_{\mu}^{n \dagger} ，它们服从换向关系 [a_(mu)^(n),a_(v)^(n†)]=\left[a_{\mu}^{n}, a_{v}^{n \dagger}\right]=delta_(mn)eta_(mu v)^(73)A\delta_{m n} \eta_{\mu v}{ }^{73} \mathrm{~A} ，主要问题是频谱中出现了负规范的状态（即所谓的 "幽灵"），从而违反了统一性。幽灵的出现在 QED 中是众所周知的，在这种情况下，可以通过对物理状态施加一个条件（"费米条件"）来解决，这个条件来自 QED 的量规不变性。在对偶模型中，我们也尝试了类似的方法。 在这里，只保留物理状态的条件是基于当时在威斯康星大学工作的米格尔-维拉索罗（Miguel Virasoro）首次提出的无穷维规代数（在对偶模型背景下）。 ^(74){ }^{74} 然而，这也是这里的重点，"维拉索罗条件 "只有在前导雷格轨迹的截距 alpha_(0)\alpha_{0} 等于1时才能满足。这与实验建议的 0.5 值不一致；截距为 1 意味着谱包含一个自旋为 1 的无质量玻色子，这与强力的短程作用范围相矛盾。此外，这还意味着最低层粒子（即基态）的质量平方为负： m^(2)=-1//alpha^(')m^{2}=-1 / \alpha^{\prime} 。这种虚质量粒子被称为 "超速粒子"，它违反了因果关系。因此，正如乔尔-舍克（Joël Scherk，1975 年，第 124 页）几年后在一篇评论文章中所说，维拉索罗条件 "显然与现实相去甚远"，"截距最好是 0.5"。不过，Scherk 继续说，"从理论角度看，这也带来了更令人满意的结果"。显然，在理论建构中，保存统一性（通过使威尼斯模型的频谱无幽灵）具有优先权，即使这意味着一个无质量玻色子和一个违反因果关系的超速子。因此，有了维拉索罗条件，雷格轨迹的截距就在理论基础上固定下来了。
The second example concerns how starting from the dual model constraints the value of the number of spacetime dimensions was determined. Again, it was the requirement of unitarity driving the result. As said, attempts to construct a full theory out of the Veneziano model were grounded in 第二个例子涉及如何从对偶模型约束条件出发确定时空维数的值。同样，这也是单位性要求驱动的结果。如前所述，试图从维尼希亚诺模型中构建完整理论的基础是
considering the Veneziano amplitude as the Born term in a perturbation expansion. Then, on the basis of the spectrum of states and the corresponding set of rules for connecting initial states with final states (analogous to the Feynman rules in QED) that were found at tree level, higher order “loop” terms were calculated. Some of the one-loop amplitudes contained new singularities that were inconsistent with unitarity. The expectation was that the singularities were related to the Pomeron trajectory. ^(75){ }^{75} Claude Lovelace from Rutgers University was interested in the properties of the Pomeron because of its application in the analysis of scattering data. He further investigated the problematic one-loop singularity and calculated that it could be turned into a series of factorizable poles (associated with the exchange of the Pomeron trajectory), but only for the spacetime dimension d=26d=26. In the words of Veneziano (1974, p. 46): the “singularity is quite sick except for the critical value of dd [i.e., d=26d=26 ] when it becomes a pole with intercept alpha_(P)(0)=2\alpha_{P}(0)=2 !” So, despite being clearly inconsistent with empirical results—or, in the words of Lovelace himself (1971, p. 502), “obviously unworldly” -in order to retain unitarity at the one-loop level the value for the number of spacetime dimensions of the Veneziano model had to be 26.^(76)26 .{ }^{76} 将 Veneziano 振幅视为扰动扩展中的 Born 项。然后，根据在树水平上发现的状态谱以及连接初始状态和最终状态的相应规则（类似于 QED 中的费曼规则），计算出高阶 "一环 "项。一些一环振幅包含了新的奇点，与单位性不符。人们期望这些奇点与波美子轨迹有关。 ^(75){ }^{75} 罗格斯大学的克劳德-洛夫莱斯（Claude Lovelace）对波美子的特性很感兴趣，因为它可用于分析散射数据。他进一步研究了存在问题的一环奇异性，并计算出它可以转化为一系列可因子化的极点（与波美拉尼亚轨迹的交换有关），但仅限于时空维度 d=26d=26 。用 Veneziano（1974 年，第 46 页）的话说："奇点是相当病态的，除了 dd 的临界值[即 d=26d=26 ]，这时它变成了截距为 alpha_(P)(0)=2\alpha_{P}(0)=2 的极点！"因此，尽管与经验结果明显不符--或者用拉夫罗斯自己的话说（1971 年，第 502 页），"显然不符合世界规律"--为了在一环水平上保持单一性，威尼斯模型的时空维数值必须是 26.^(76)26 .{ }^{76} 。
In subsequent years, the requirement for the number of spacetime dimensions of d=26d=26 in the Veneziano model was rederived in a number of independent ways. Apart from Lovelace’s calculation, it appeared as the critical dimension required for the physical states (i.e., without ghosts) of the Veneziano model to span the complete Hilbert space; this latter result was also obtained using the string formalism that was in development. ^(77){ }^{77} Philosopher Elena Castellani (2019) has argued that the independent ways of arriving at d=26d=26 can point at a “convergence argument” in theory construction: independent ways of arriving at the same result motivate the acceptance of a theory-in-development, because it aligns with theoretical virtues such as consistency and cohesiveness. What Castellani does not address, however, is that the argument in this case only holds when virtues such as consistency or cohesiveness are themselves central to the practitioners. In other words: dual model theorists’ temporary acceptance of d=26d=26 (and of the problematic particles in the spectrum) points out that striving for empirical adequacy, for the moment, had been pushed to the back. 在随后的几年里，人们通过多种独立的方法重新得出了威尼斯模型中 d=26d=26 的时空维数要求。除了洛夫莱斯的计算之外，它还被认为是威尼斯模型的物理状态（即没有幽灵）跨越完整的希尔伯特空间所需的临界维数；后一个结果也是利用当时正在发展的弦形式主义得到的。 ^(77){ }^{77} 哲学家埃莱娜-卡斯特拉尼（Elena Castellani，2019）认为，得出 d=26d=26 的独立方法可以指向理论构建中的 "趋同论证"：得出相同结果的独立方法促使人们接受正在发展中的理论，因为它符合一致性和内聚性等理论优点。然而，卡斯特拉尼没有提到的是，只有当一致性或内聚性等美德本身就是实践者的核心时，这种情况下的论证才会成立。换句话说：二元模型理论家暂时接受了 d=26d=26 （以及光谱中的问题粒子），这说明对经验充分性的追求暂时被推到了后面。
As these examples illustrate, theoretical reasoning on the basis of the set of dual model principles determined almost all parameters of the models. This can be seen as an echo of the bootstrap conjecture, according to which a unique solution consistent with the SS-matrix principles would fix all “arbitrary” parameters of the theory. In the examples discussed above it is hard to judge whether we can speak about unique solutions of the equations following from the dual model postulates (that is, in the precise sense of the self-consistency conditions related to the bootstrap, see Section 2.3). However, it was at least the case that a chain of theoretical reasoning starting from enforcing the principles determined the values for the intercept and the number of spacetime dimensions. With that, only the value of the slope of the Regge trajectories could still be varied. 正如这些例子所说明的，基于二元模型原则集的理论推理几乎决定了模型的所有参数。根据这一猜想，符合 SS 矩阵原理的唯一解将固定理论的所有 "任意 "参数。在上文讨论的例子中，我们很难判断我们是否可以谈论根据对偶模型公设（即与自引导相关的自洽条件的确切含义，见第 2.3 节）的方程的唯一解。然而，至少可以这样说，从执行原则开始的一连串理论推理决定了截距和时空维数的值。这样，只有雷格轨迹的斜率值还可以变化。
The restrictiveness following from the postulates underlying dual resonances models was a recurring theme in the early 1970s—Fubini even called dual models “a beautiful but sophisticated way of obtaining a solution for almost all the constraints that any reasonable theory should satisfy”. ^(78){ }^{78} Clavelli and Shapiro (1973, p. 491) wrote in a similar vein that “dual models are extremely ‘tight’ in the sense that modifications are very difficult to make without destroying the theory”. Despite this rigidity, they continued, “there are now several theoretically satisfactory theories”. Theoretically indeed, since “none are good in detail when compared to data”. The comparison of theory to experimental data pointed at here is markedly different from the situation in the “old” SS-matrix theory. There, the theory could be satisfactorily applied to a variety of scattering processes, but in the end the experimental data outstripped the available theoretical descriptions. In the dual model case, it were the empirical predictions of a potential full hadronic theory that were inconsistent with data. That satisfying the constraints underlying this potential theory came at the expense of empirical adequacy was perhaps best articulated by theorists David Olive and Joël Scherk, who wrote in the introduction to a paper (on constructing a ghost-free spectrum of Pomeron states) that 在 20 世纪 70 年代初，双共振模型的基本假设所产生的限制性是一个反复出现的主题--傅比尼甚至称双模型为 "一种美丽而复杂的方法，可以为任何合理的理论所应满足的几乎所有约束条件求得一个解决方案"。 ^(78){ }^{78} Clavelli 和 Shapiro（1973 年，第 491 页）同样写道："双重模型极其'严密'，因为很难在不破坏理论的情况下进行修改"。尽管如此，他们继续说，"现在有几种理论上令人满意的理论"。理论上确实如此，因为 "在与数据比较时，没有一个理论在细节上是令人满意的"。这里所说的理论与实验数据的比较与 "旧的" SS 矩阵理论的情况明显不同。在那里，理论可以令人满意地应用于各种散射过程，但最终实验数据还是超过了可用的理论描述。在二元模型中，潜在的全强子理论的经验预测与数据不一致。理论学家戴维-奥利弗（David Olive）和乔尔-舍克（Joël Scherk）在一篇论文（关于构建波美拉尼亚态的无幽灵谱）的引言中写道："满足这种潜在理论的约束条件是以牺牲经验的充分性为代价的。
[ t ]he existing dual models seem to be more a self-consistent alternative to polynomial local field theories than a phenomenological approach to describe the real world of hadrons. The main advantage over field theories is that at each order of the perturbation expansion an infinite number of particles of any spin is included, while maintaining the basic properties of duality, Regge behaviour, positive definiteness of the spectrum of resonances (absence of ghosts), and perturbative unitarity. The drawback is that this set of conditions is so constraining that it can be realized only for unphysical values of the number of dimensions of spacetime and at the expense of having, in general, tachyons. However, this critical dimension of spacetime, DD, varies from model to model ( D=26D=26 in the conventional [Veneziano, bosonic] model), 10 in the Neveu-Schwarz [fermionic] model) so that in the spirit of the bootstrap one may think that a realistic and unique dual model would yield critical dimension 4.^(79)4 .{ }^{79} [现有的对偶模型似乎更像是多项式局部场论的自洽替代方案，而不是描述强子真实世界的现象学方法。与场论相比，二元模型的主要优点是在扰动扩展的每个阶都包含了无限数量的任意自旋粒子，同时保持了二元性、雷格行为、共振谱的正确定性（不存在幽灵）和扰动单位性等基本特性。缺点是这组条件的约束性太强，只能在时空维数的非物理值下才能实现，而且一般情况下要以拥有超速子为代价。然而，时空的临界维度 DD 在不同的模型中是不同的（在传统的[维尼希亚诺，玻色]模型中为 D=26D=26 ，在内维尔-施瓦茨[费米子]模型中为10），因此，根据自举法的精神，我们可以认为一个现实的、独特的对偶模型会产生临界维度 4.^(79)4 .{ }^{79} 。
So, although seeing a way forward towards more realistic models, it was made clear by Olive and Scherk that so far the dual model conditions had led them away from “the real world of hadrons”. 因此，尽管奥利弗和舍克看到了通向更现实模型的道路，但他们明确表示，迄今为止，双重模型条件使他们远离了 "真实的强子世界"。
Other physicists were also very explicit about the shift away from experiment that accompanied the introduction of dual models, in particular when compared to the “traditional” SS matrix approach. For example, in the only textbook on dual models for hadrons, Paul Frampton’s “Dual Resonance Models” (first published in 1974), the introductory chapter is devoted to “background material” from SS-matrix theory. ^(80){ }^{80} This included kinematical definitions, resonances, Regge poles, and the concept of duality as developed in interplay with experiment-essentially reviewing the situation as it stood by mid-1968. The subsequent chapters then discuss technical and theoretical aspects of dual resonance models. As Frampton made clear in his introduction, the step from SS-matrix theory to dual models corresponded to a step away from empirical data. He wrote that "[b]roadly speaking we shall be concerned here [in the introductory chapter] with the real world while later on we shall be concerned almost entirely with a model world (the Veneziano model 其他物理学家也非常明确地指出，伴随着双重模型的引入，特别是与 "传统的" SS 矩阵方法相比，实验发生了转变。例如，在唯一一本关于强子对偶模型的教科书--保罗-弗兰普顿（Paul Frampton）的《双共振模型》（Dual Resonance Models）（1974年首次出版）中，引言一章专门介绍了 SS 矩阵理论的 "背景材料"。 ^(80){ }^{80} 其中包括运动学定义、共振、雷格极点，以及在与实验的相互作用中发展起来的二元性概念--基本上回顾了 1968 年中期的情况。随后的章节讨论了双共振模型的技术和理论方面。正如弗兰普顿在引言中明确指出的，从 SS 矩阵理论到二元模型，相当于远离经验数据的一步。他写道："从理论上讲，我们在这里（导言部分）关注的是现实世界，而在后面的章节中，我们将几乎完全关注一个模型世界（威尼斯模型）。
world)." ^(81){ }^{81} The “real world” and the “model world” are related, Frampton continued, but only because concepts from the real world provide a “vocabulary” for features of the model world. Concepts such as duality or linearly rising Regge trajectories were part of this vocabulary: instead of being used as tools to analyze scattering data they were now considered as building blocks of the model world. 世界）"。 ^(81){ }^{81} 弗兰普顿继续说，"现实世界 "和 "模型世界 "是相关的，但这只是因为现实世界的概念为模型世界的特征提供了 "词汇"。二元性或线性上升的雷格轨迹等概念就是这些词汇的一部分：它们不再被用作分析散射数据的工具，而是被视为模型世界的构件。
As a result of this turn towards the construction of a “model world”, the close relation between theorists and experimenters, which had been at the heart of progress in SS-matrix theory in the 1960s, faded. The theoretical progress that dual model theorists were making was of no direct use for experimentalists. An example of how this played out in practice is the management of the National Accelerator Laboratory (now Fermilab) abruptly dismissing three group members (Pierre Ramond, David Gordon, and Lou Clavelli) who were working on dual models. The three were hired in the fall of 1969 to foster dialogue between theory and experiment, but the formal advances they made in dual theory were not what the laboratory management, in the person of its Director Bob Wilson, desired. As Clavelli phrased it in hindsight, the management “seemed to think we should have been working on more short term solutions”. In the fall of 1970, their contracts were abruptly terminated, “with the explanation that interactions between us and the experimental physicists had not developed as fully as had been expected”. ^(82){ }^{82} 由于转向构建 "模型世界"，理论家与实验者之间的密切关系逐渐消失，而这种关系曾是 20 世纪 60 年代矩阵理论进展的核心。二元模型理论家们在理论上取得的进展对实验家们没有直接用处。国家加速器实验室（现费米实验室）的管理层突然解雇了研究二元模型的三位成员（皮埃尔-拉蒙德、戴维-戈登和卢-克拉维利），这就是实际情况的一个例子。他们三人是在 1969 年秋天受聘来促进理论与实验之间的对话的，但他们在对偶理论方面取得的正式进展并不是实验室主任鲍勃-威尔逊（Bob Wilson）所希望的。正如克拉克维利事后所说，管理层 "似乎认为我们应该研究更多的短期解决方案"。1970 年秋天，他们的合同被突然终止，"理由是我们与实验物理学家之间的互动没有像预期的那样充分发展"。 ^(82){ }^{82}
Yet, despite this expression of discontent from experimentally-minded physicists, dual theory was not to return to experiment. It would have been possible for dual model physicists to steer back to experimental applications, but then they had to let go of one or more of the “constraints that any reasonable theory should satisfy” (dixit Fubini) - and the theorists were simply not inclined to do so in their further development of dual models. This was for instance pointed out by Stanley Mandelstam. The mass spectra of dual models, he noted, were qualitatively in good agreement with observation, and with all mass ratios being determined by a single coupling strength (related to the Regge slope alpha^(')\alpha^{\prime} ), the models were very restrictive. It was however the same restrictiveness that led to dual models “embarrassing” features, such as the prediction of the massless particles (in conflict with the strong force’s short range). Empirically more adequate models (without unwanted symmetries) could be obtained “[b]y dropping some of the consistency requirements”, but although such models could be useful, the drawback was that “they no longer enable us to make parameter-free predictions” ^(83){ }^{83} As it turned out, dual model physicists were not willing to give up on this. Paul Frampton, in the final chapter of his 1974 textbook that was devoted to experimental applications, articulated this conviction most explicitly. Only for “mutilations” of the Veneziano model, fits to experimental data could be obtained, he noted. With this Frampton referred to modifications and approximations, such as neglecting unitarity violation and accepting the presence of ghost states. In other words, in order to be of use for experimentalists one had to work with the Veneziano amplitude, which was seen as the Born term in the full theory that was being developed. Although acknowledging that such fits had proved significant “in their own right” by providing “a stimulus to the experimentalists”, it was of no use in theory construction. As Frampton put it: 然而，尽管具有实验精神的物理学家表达了这种不满，但二元理论并没有回归实验。二元模型物理学家本来有可能回到实验应用上来，但这样他们就不得不放弃一个或多个 "任何合理理论都应满足的约束条件"（富比尼语）--而理论家们在进一步发展二元模型时根本不愿意这样做。斯坦利-曼德尔施塔姆（Stanley Mandelstam）就指出了这一点。他指出，二元模型的质谱在质量上与观测结果十分吻合，而且由于所有质量比都是由单一的耦合强度（与雷格斜率 alpha^(')\alpha^{\prime} 有关）决定的，因此模型的限制性非常强。然而，正是这种限制性导致了双模型的 "尴尬 "特征，如对无质量粒子的预测（与强力的短程相冲突）。如果 "放弃一些一致性要求"，就可以得到经验上更充分的模型（没有不需要的对称性），但尽管这些模型可能有用，其缺点是 "它们不再使我们能够做出无参数的预测" ^(83){ }^{83} 事实证明，二元模型物理学家并不愿意就此放弃。保罗-弗兰普顿（Paul Frampton）在其1974年教科书中专门论述实验应用的最后一章，最明确地阐述了这一信念。他指出，只有对威尼斯模型进行 "修改"，才能获得与实验数据的拟合。为此，弗兰普顿提到了修改和近似，如忽略违反单元性和接受幽灵态的存在。 换句话说，要想对实验者有用，就必须使用威尼斯振幅，而威尼斯振幅被视为正在发展的完整理论中的伯恩项。虽然这种拟合 "本身 "已经证明了其重要性，为 "实验者 "提供了 "刺激"，但在理论构建中却毫无用处。正如弗兰普顿所说：
The mathematically-oriented theorist may easily regard as misguided any attempt to compare such an incomplete [i.e., modified, approximative] theory to experiment; he may 以数学为导向的理论家很容易认为，将这种不完整的[即修正的、近似的]理论与实验进行比较的任何尝试都是误导；他可能会
ask: what do we learn from such work that is relevant to building a better theory? He anticipates the answer which is: very little. ^(84){ }^{84} 他问：我们从这些工作中学到了什么，而这些工作又与建立更好的理论有关？他预想的答案是：很少。 ^(84){ }^{84}
The point is clear, then: by not letting down any of the dual model principles, dual model physicists no longer had anything to gain from experiment in their search for a full theory of hadrons. 那么，问题就很清楚了：通过不放过任何一个二元模型原理，二元模型物理学家在寻找完整的强子理论时，不再能从实验中获得任何好处。
3.3 Strings and hadrons 3.3 弦和强子
So far we have focused on how adding the constraint of duality to the SS-matrix postulates led to the construction of models that, in a step towards the bootstrap ideal, contained almost no free parameters. The construction of these models was no longer taking place in direct contact with experimental results. Earlier I have designated SS-matrix theory as “empiricist”, restricting the theory to mathematical relations between observable scattering amplitudes. With the shift from SS-matrix theory to dual models, we see that physicists started to attach some physical interpretation to the mathematical framework, in the form of an infinite set of harmonic oscillator operators. These oscillators, in turn, could be understood as being generated by an underlying string system. Yet, as also argued by Rickles (2014, pp. 16, 71-72, 92), dual model physicists used the string picture mostly as a heuristic tool for understanding the mathematical structure of dual models, while talking only in rather non-committal terms on how the string could reflect the realistic properties of hadrons. 到目前为止，我们主要讨论了如何在 SS 矩阵公设中加入二重性约束，从而构建出几乎不含任何自由参数的模型。这些模型的建立不再与实验结果直接联系。早些时候，我曾把 SS 矩阵理论称为 "经验主义"，把理论局限于可观测散射振幅之间的数学关系。随着 SS 矩阵理论向二元模型的转变，我们看到物理学家开始为数学框架附加一些物理解释，其形式是一组无限的谐振子算子。反过来，这些振荡器又可以被理解为由底层弦系统产生的。然而，正如里克尔斯（2014，第16、71-72、92页）所论证的那样，二元模型物理学家大多将弦图景用作理解二元模型数学结构的启发式工具，而对于弦如何能够反映强子的现实特性，他们只是用相当不明确的措辞来谈论。
The idea that the infinite towers of resonances underlying the Veneziano amplitude could be generated by string-like oscillatory motions was independently suggested by Holger Nielsen from Copenhagen, Leonard Susskind, then at Yeshiva University, and Yoichiro Nambu from the University of Chicago. ^(85){ }^{85} In the cases of Nambu (“the internal energy of a meson is analogous to that of a quantized string of finite length”, p. 275) and Susskind (the Veneziano spectrum “agrees exactly with the form of spectrum postulated on the basis of a harmonic continuum model with cyclic boundary conditions, or in other words, a rubber band”, p. 547) the string picture was clearly suggested on the basis of an analogy with an oscillating system. Nielsen’s proposal was different, as he arrived at a string picture for hadrons through an approximation using higher-order Feynman diagrams representing point-like particles interacting with their nearest neighbors, thereby forming “threads or chain molecules” (p. 2). The first main steps taken in 1970 to work out the idea of a string system for the Veneziano model were the introduction of the concept of a worldsheet (the two-dimensional surface that is swept out by strings in spacetime) by Susskind, and the introduction of an action functional (due to Nambu and, independently, the Japanese physicist Tetsuo Gotō) for a onedimensional string, analogous to the action for a point particle. ^(86){ }^{86} 哥本哈根的霍尔格-尼尔森（Holger Nielsen）、当时在耶什华大学的伦纳德-苏斯金德（Leonard Susskind）和芝加哥大学的难武洋一郎（Yoichiro Nambu）分别提出了这样的想法：维尼希亚诺振幅所蕴含的无限共振塔可能是由类似于弦的振荡运动产生的。 ^(85){ }^{85} 在南布（"介子的内能类似于长度有限的量子化弦"，第 275 页）和苏斯金德（威尼斯谱 "与根据具有循环边界条件的谐波连续模型推测的谱形式完全一致，换句话说，就是橡皮筋"，第 547 页）的例子中，弦的图景显然是根据与振荡系统的类比而提出的。尼尔森的提议则不同，他通过使用高阶费曼图来近似地表示点状粒子与其近邻粒子相互作用，从而形成 "线状或链状分子"（第 2 页），从而得出了强子的弦图。1970年，在为维尼西亚诺模型设计弦系统方面所采取的第一个主要步骤是，苏斯金德引入了世界表（弦在时空中扫出的二维表面）的概念，并为一维弦引入了与点粒子作用类似的作用函数（由南武和日本物理学家后藤哲夫分别独立提出）。 ^(86){ }^{86}
During the first half of the 1970s the string picture for dual models was further developed. It is important to note that this development took place in constant interplay with the operator formalism (i.e., factorizing amplitudes using creation and annihilation operators). Results from the operator approach were leading-derivations on the basis of the string picture were checked against those results. Around 1974 some physicists, for example Joël Scherk, Claudio Rebbi from CERN, or Paul Frampton in his textbook on dual resonance models, started to express the idea that out of strings all output from dual models could be computed. The main motivation for this was a result by Goddard, Goldstone, Rebbi, and Thorne (1973), who proposed a procedure for removing the ghost states from the spectrum on the basis of the string. In short, by demanding the string action to be invariant under reparametrizations of the worldsheet, the authors were able to eliminate all the 20 世纪 70 年代上半叶，对偶模型的弦图谱得到了进一步发展。值得注意的是，这一发展是在与算子形式主义（即使用创生和湮灭算子对振幅进行因式分解）不断相互作用的过程中进行的。在弦图景的基础上对算子方法的结果进行前导衍生，并与这些结果进行核对。大约在1974年，一些物理学家，例如欧洲核子研究中心的约埃尔-舍克（Joël Scherk）、克劳迪奥-雷比（Claudio Rebbi），或者保罗-弗兰普顿（Paul Frampton）在其关于对偶共振模型的教科书中，开始表达这样的观点：从弦中可以计算出对偶模型的所有输出。其主要动机是戈达德、戈德斯通、雷比和索恩（1973 年）的一项成果，他们提出了一种在弦的基础上从频谱中去除幽灵态的程序。简而言之，通过要求弦作用在世界表的重拟态下是不变的，作者们能够消除所有的
ghost states; afterwards, only the remaining physical states were quantized. ^(87){ }^{87} It was on the basis of this result that, in the words of Frampton, “one may hope (…) [to] simplify the formulation of the [dual resonance] theory from the postulation of a complicated set of on-mass-shell tree amplitudes to the postulation of a limited number of axioms in the language of a string.” ^(88){ }^{88} 之后，只有剩余的物理状态被量子化了。 ^(87){ }^{87} 正是基于这一结果，用弗兰普顿的话说，"人们才有希望（......）[简化][双共振]理论的表述，从假设一组复杂的质量壳上树状振幅，简化为假设数量有限的弦语言公理"。 ^(88){ }^{88}
Frampton’s designation of the possible use of strings in dual models is rather formal: it points at a mathematical procedure to rederive in a simpler manner the results from dual theory. As said, it was in this way that the string formalism was developed-it helped clarify the mathematical structure of dual models. ^(89){ }^{89} Due to its success in this, the string increasingly took center stage in work on dual models-notwithstanding that all the main results from dual models so far had been obtained independent of the string picture. ^(90){ }^{90} As Sergio Fubini (1974, p. 5) remarked: 弗兰普顿关于弦在对偶模型中的可能用途的描述是相当形式化的：它指出了一种以更简单的方式重新解读对偶理论结果的数学过程。如前所述，弦形式主义正是以这种方式发展起来的--它有助于澄清对偶模型的数学结构。 ^(89){ }^{89} 由于弦形式主义在这方面的成功，它在对偶模型的研究中日益占据中心位置--尽管迄今为止对偶模型的所有主要结果都是在独立于弦图景的情况下获得的。 ^(90){ }^{90} 正如塞尔吉奥-富比尼（Sergio Fubini）（1974 年，第 5 页）所说：
The string model, which has been recently the object of much attention and interest provides an extremely fruitful and inspiring visualization of dual models. The idea of a string model for elementary particles has been first suggested by the analogy of the mass spectrum of dual models with the energy spectrum of a vibrating string. The present success of the string model is due to the fact that it leads to a general interpretation of all the detailed features of the Veneziano model and provides important indications about the most promising avenues of progress. 弦模型是近来备受关注和兴趣的对象，它为对偶模型提供了一个极富成果和启发性的可视化模型。关于基本粒子弦模型的想法，最早是通过把对偶模型的质量谱与振动弦的能量谱进行类比而提出来的。弦模型目前之所以取得成功，是因为它导致了对威尼斯模型所有细节特征的一般解释，并为最有希望取得进展的途径提供了重要指示。
A similar opinion was expressed by Veneziano (1974, p. 47), who stated that “in search for new physical ideas, we may make appeal to [string] models visualizing the existing dual theories”. So, strings were no longer a just a tool to rederive already known results obtained with the operator approach, but a starting point for further developing dual theory. 威尼斯诺（Veneziano，1974 年，第 47 页）也表达了类似的观点，他说："在寻找新的物理思想时，我们可以利用[弦]模型将现有的对偶理论形象化"。因此，弦不再仅仅是重现用算子方法获得的已知结果的工具，而是进一步发展对偶理论的起点。
With their depiction of strings as a “visualization” of dual models, both Fubini and Veneziano were rather vague about the physical status of strings. Others more explicitly viewed the string as the physical object underlying dual models. Rebbi (1974, p. 224), for example, in a review article on the string picture, stated that “it should be manifest that the physics of the dual models is the physics of one-dimensional relativistic extended objects”, noting that “it has been customary to call such a system a ‘string’”. So dual models, unlike the old SS-matrix theory which were essentially cast in the form of a set of mathematical equations relating scattering amplitudes, could be explained on the basis of an underlying physical object-the hadronic string. At the same time, “empiricist” convictions kept playing a role. Most importantly, despite the use of an action functional (the Nambu-Goto action) and the corresponding Lagrangian formalism, and the formulation of an interacting picture for the bosonic string by Mandelstam (1973), a complete dynamic picture of dual model strings was not at hand, nor was it demanded as a criterium for a successful theory. This was explicitly motivated by John Schwarz (1974), who remarked that “[e]ven though a complete spacetime picture formulation is not required, it should at least be possible to construct weak, electromagnetic, and gravitational currents, since these are experimentally observable” (p.156). 富比尼和维尼恰诺都把弦描述为二元模型的 "可视化"，但他们对弦的物理地位却含糊其辞。其他人则更明确地把弦视为二元模型的物理对象。例如，雷比（1974 年，第 224 页）在一篇关于弦图景的评论文章中指出："应该明确的是，对偶模型的物理学是一维相对论扩展对象的物理学"，并指出 "人们习惯于把这样的系统称为'弦'"。因此，二元模型与旧的矩阵理论不同，后者基本上是以一组与散射振幅有关的数学方程的形式出现的，而二元模型则可以在一个基本物理对象--强子弦--的基础上得到解释。与此同时，"经验主义 "信念也在不断发挥作用。最重要的是，尽管曼德尔施塔姆（Mandelstam，1973 年）使用了作用函数（南布-后藤作用）和相应的拉格朗日形式主义，并提出了玻色弦的相互作用图景，但手头并没有双模型弦的完整动态图景，也没有要求把它作为成功理论的标准。约翰-施瓦茨（John Schwarz，1974 年）明确提出了这一点，他说："尽管不需要完整的时空图景表述，但至少应该可以构建弱流、电磁流和引力流，因为这些都是实验可以观测到的"（第 156 页）。
As said, the relation between dual model strings and the properties of real hadrons could not be made precise. This, of course, had everything to do with the incorrect predictions made by dual models. As such, even though the string was a physical interpretation of the models, these models at best provided a highly flawed account of the empirical properties of hadrons. The noticeable exception was the case of quark confinement: here, the string picture yielded a (qualitatively) 如前所述，二元模型弦与真实强子特性之间的关系无法做到精确。当然，这与二元模型的错误预测不无关系。因此，尽管弦是对模型的物理解释，但这些模型充其量只是对强子的经验特性提供了一个漏洞百出的解释。值得注意的例外是夸克禁闭的情况：在这里，弦图景产生了一种（定性的）............
satisfactory description of experimental results in hadron scattering. The idea was to incorporate quarks in the string picture by attaching them to the endpoints of strings, with the quarks carrying the hadronic quantum numbers (somewhat similar to how quantum numbers were added to SS matrix theory, see Section 2.4). Simply put, confinement follows from the strings-with-quarks picture because an increase in string length (increasing the distance between quarks) requires a corresponding increase in string energy. ^(91){ }^{91} The string picture was also very fruitful for the early understanding of quark confinement in QCD, in particular through work by 't Hooft on twodimensional gauge theories in the large NN limit and Kenneth Wilson’s construction of lattice gauge theories. ^(92){ }^{92} However, in this case the dual model string was combined with the QCD framework to advance understanding of a particular feature of hadronic interactions, instead of using the string as the constituent for a complete theory of the experimental properties of hadrons. 对强子散射实验结果的描述令人满意。我们的想法是通过将夸克附着在弦的端点来将其纳入弦图景中，夸克携带强子量子数（有点类似于将量子数添加到 SS 矩阵理论中的方法，见第2.4节）。简单地说，禁闭是由带夸克的弦图景产生的，因为增加弦的长度（增加夸克之间的距离）需要相应增加弦的能量。 ^(91){ }^{91} 弦图景对于早期理解 QCD 中的夸克禁闭也很有帮助，特别是通过 't Hooft 在大 NN 限度下对二维规规理论的研究，以及 Kenneth Wilson 对晶格规规理论的构建。 ^(92){ }^{92} 然而，在这种情况下，二元模型弦与 QCD 框架相结合，是为了推进对强子相互作用某一特征的理解，而不是把弦作为强子实验特性完整理论的组成部分。
It is safe to say that by 1975 the string was established as the fundamental mathematical entity of dual resonance models. This turned them into a class of models with an ontology that was clearly distinct from field theory. In the words of Schwarz, “elementary particles described by a fieldtheory model are pointlike, whereas those in a dual-resonance model are spatially extended” (1975, p. 64). However, apart from their use in lattice QCD and quark confinement, there was no precise connection between the string-like particles from dual models and elementary hadrons. In other words, practitioners were not (yet) ontologically committed to fundamental strings; instead, strings were foremost seen as a feature of the mathematical models. This was made particularly clear by Scherk (1975) in his closing remarks of a review paper on dual theory: 可以说，到 1975 年，弦已被确立为双共振模型的基本数学实体。这使它们成为一类本体论明显有别于场论的模型。用施瓦茨的话说，"场论模型描述的基本粒子是点状的，而双共振模型中的粒子是空间扩展的"（1975 年，第 64 页）。然而，除了在格子 QCD 和夸克束缚中的应用之外，双共振模型中的弦状粒子与基本强子之间并没有确切的联系。换句话说，实践者们（尚未）从本体论上对基本弦做出承诺；相反，弦首先被视为数学模型的一个特征。舍克（Scherk，1975 年）在一篇关于对偶理论的评论文章的结束语中特别明确地指出了这一点：
We hope we have achieved our aim to convince the reader that the covariant treatment of dual models in the operator formalism, and the transverse string picture are two complementary faces of a single mathematical structure, having a high and maybe perfect degree of self-consistency. Whether or not these mathematical structures have anything to do with the real world is still unclear, and one has to wait to see whether more realistic models can be built or not. At the worst, it seems that the existing mathematical structures can be to hadron physics [i.e., through lattice QCD] what the two-dimensional Ising model is to the theory of ferromagnetism. (p. 163) 我们希望我们已经达到了我们的目的，让读者相信算子形式主义中对偶模型的协变处理和横弦图景是单一数学结构的两个互补面，具有高度甚至完美的自洽性。这些数学结构是否与现实世界有关，目前还不清楚，我们只能拭目以待，看看是否能建立起更现实的模型。在最坏的情况下，现有的数学结构对于强子物理学（即通过晶格 QCD）的作用似乎就像二维伊辛模型对于铁磁理论的作用一样。(p. 163)
Thus, the investigation of dual resonance models in the first half of the 1970s had resulted in the construction of a mathematical structure that described string-like particles, but-as a whole-did not relate to the “real world”. 因此，20 世纪 70 年代前半期对双重共振模型的研究，最终构建了一种描述弦状粒子的数学结构，但整体上与 "真实世界 "无关。
3.4 Dual models as a unified theory 3.4 作为统一理论的双重模型
From the mid-1970s onward, work on hadronic dual models declined. During those years, the theory of quantum chromodynamics was formed, in particular following the results by 't Hooft and Veltman on the renormalizability of Yang-Mills theories and on asymptotic freedom by Gross, Wilczek, and Politzer. ^(93){ }^{93} Subsequently, the Standard Model became established as the leading framework for strong and electroweak interactions, which it remains to this day. An important argument in favor of the QCD approach and against dual models followed from the results of various experiments at SLAC and Brookhaven that were carried out in the late 1960s and early 1970s. The results suggested hard, point-like constituents of hadrons (i.e., quarks) and were in conflict with dual model’s soft behavior. With the establishment of the Standard Model, the intense activity in research on dual models for 从 20 世纪 70 年代中期开始，关于强子对偶模型的研究逐渐减少。在这些年里，量子色动力学理论形成了，特别是在't Hooft和Veltman关于杨-米尔斯理论重规范化的结果以及Gross、Wilczek和Politzer关于渐近自由度的结果之后。 ^(93){ }^{93} 随后，标准模型被确立为强相互作用和弱电相互作用的主要框架，并一直沿用至今。支持QCD方法、反对二元模型的一个重要论据来自20世纪60年代末和70年代初在SLAC和布鲁克海文进行的各种实验的结果。这些结果表明，强子（即夸克）是硬的、点状的成分，与双重模型的软行为相冲突。随着标准模型的建立，对二元模型的研究活动日趋激烈。
hadrons from the first half of the 1970s waned, although, as said, elements of the dual string picture were integrated in QCD and turned out fruitful in the understanding of quark confinement. ^(94){ }^{94} 尽管如前所述，双弦图景的元素被整合到了 QCD 中，并在理解夸克禁闭方面取得了丰硕成果，但自 20 世纪 70 年代前半期以来，强子的研究就日渐式微了。 ^(94){ }^{94}
In 1974, however, Joël Scherk and John Schwarz proposed a reinterpretation of dual models as a unified theory, instead of a description of hadrons. The essential component of the proposal was that the Pomeron trajectory-or, equivalently, the closed string-in dual models could be interpreted as the graviton, yielding a low-energy behavior equivalent to Einsteinian gravity, a feature that was also pointed out by the Japanese physicist Tamiaki Yoneya (1973). In order to identify the Pomeron as the graviton, the Regge slope parameter alpha^(')\alpha^{\prime} (the fundamental parameter of the string models) had to be rescaled to alpha^(')∼10^(-34)GeV^(-2)\alpha^{\prime} \sim 10^{-34} \mathrm{GeV}^{-2}, corresponding to an elementary string length of 10^(-30)cm.^(95)10^{-30} \mathrm{~cm} .{ }^{95} 然而，1974 年，乔尔-舍克（Joël Scherk）和约翰-施瓦茨（John Schwarz）提出将对偶模型重新解释为统一理论，而不是对强子的描述。该建议的主要内容是，二元模型中的波美拉尼亚子轨迹--或者，等同于封闭弦--可以被解释为引力子，产生与爱因斯坦引力等效的低能行为，日本物理学家米谷玉明（Tamiaki Yoneya，1973 年）也指出了这一特征。为了把波美拉尼亚子确定为引力子，必须把雷格斜率参数 alpha^(')\alpha^{\prime} （弦模型的基本参数）重新标度为 alpha^(')∼10^(-34)GeV^(-2)\alpha^{\prime} \sim 10^{-34} \mathrm{GeV}^{-2} ，对应于基本弦长度 10^(-30)cm.^(95)10^{-30} \mathrm{~cm} .{ }^{95} 。
With the reinterpretation of string theory as a candidate unified quantum gravity theory, the original experimental motivations for the Veneziano amplitude were no longer meaningful. Scherk and Schwarz (1974, p. 119) explicitly stressed in their proposal for unified dual models that “[o]bviously, there is no empirical evidence for duality or Regge behavior in nonhadronic interactions.” However, they continued, “the idea of having string-like rather than point-like particles is so general that it might extend to nonhadrons as well”, as long as the string’s fundamental length was so small as to not be in conflict with the limits imposed by successful theories such as QED. With direct experimental evidence supporting nor refuting nonhadronic strings, what motivated the proposal were first of all the previously problematic massless particles predicted by dual models, which were now suddenly deemed desirable: 随着弦理论被重新解释为一种候选统一量子引力理论，威尼斯振幅的原始实验动机就不再有意义了。舍克和施瓦茨（1974 年，第 119 页）在他们的统一对偶模型提案中明确强调："显然，在非重子相互作用中没有对偶性或雷格行为的经验证据。不过，他们继续说，"具有弦状而非点状粒子的想法是如此普遍，以至于它也可能扩展到非中子"，只要弦的基本长度非常小，不会与成功的理论（如 QED）所施加的限制相冲突。由于直接的实验证据既不支持也不反驳非重子弦，促使人们提出这一建议的首先是二元模型所预言的先前存在问题的无质量粒子，而现在人们突然认为这种粒子是可取的：
Viewing the existing dual models as candidates for nonhadronic schemes, one notices that at least one defect, namely the presence of massless particles, becomes a virtue. Both the 26dimensional Veneziano model (VM) and the 10-dimensional meson-fermion [Ramond-NeveuSchwarz] model (MFM) have a massless ‘photon’, the nonplanar Virasoro-Shapiro model (VSM) has a massless ‘graviton’, and the MFM has a massless ‘lepton’. (p. 119) 将现有的对偶模型视为非重子方案的候选者，我们会发现至少有一个缺陷，即无质量粒子的存在，成为了一个优点。26 维 Veneziano 模型（VM）和 10 维介子-费米子[Ramond-Neveu-Schwarz]模型（MFM）都有一个无质量的 "光子"，非平面的 Virasoro-Shapiro 模型（VSM）有一个无质量的 "引力子"，而 MFM 有一个无质量的 "轻子"。(p. 119)
Apart from the presence of the massless particles, the constraining nature of dual models (in line with the notion of non-arbitrariness) was also emphasized as a virtue of the new proposal. As Scherk and Schwarz for example wrote in a 1975 paper, even if particles (in this specific case, quarks and gluons) are “practically pointlike [at] normal energies”, an advantage of nevertheless describing them in terms of a unified dual string model was that “the dual theory itself is so tightly constrained that it is much more specific in many of its predictions”. ^(96){ }^{96} 除了无质量粒子的存在，二元模型的约束性质（与非任意性概念一致）也被强调为新建议的优点。例如，舍克和施瓦茨在 1975 年的一篇论文中写道，即使粒子（具体到夸克和胶子）"[在]正常能量下实际上是点状的"，但用统一的对偶弦模型来描述它们的一个好处是，"对偶理论本身受到如此严格的约束，以至于它的许多预言要具体得多"。 ^(96){ }^{96}
Throughout the late 1970s and early 1980s, a small group of physicists (all of whom had also worked on hadronic dual models) kept working on this proposal-important figures were Schwarz, Scherk until his untimely death in 1980, the British physicist Michael Green, and the Swedish theorist Lars Brink. Some of the problems that were tackled had already been present in the hadronic case: in particular, the problem of tachyons in the spectrum was a major point of concern. A solution for this was proposed by Gliozzi, Scherk, and Olive (1977), who imposed a specific projection restricting the number of states. This resulted in a tachyon-free theory, but also required the full ten-dimensional interacting theory to possess local supersymmetry. The construction of such a theory was achieved 在整个 20 世纪 70 年代末和 80 年代初，一小群物理学家（他们都曾研究过强子对偶模型）一直在研究这一提议--重要人物包括施瓦茨、舍克（直到 1980 年英年早逝）、英国物理学家迈克尔-格林和瑞典理论家拉尔斯-布林克。所要解决的一些问题在强子情况下已经存在：特别是频谱中的超光子问题是一个主要关注点。格里奥齐、谢克和奥利弗（1977 年）提出了一个解决方案，他们施加了一个特定的投影来限制状态的数量。这就产生了一种无速子理论，但同时也要求完整的十维相互作用理论具有局部超对称性。这种理论的构建实现了
by Green and Schwarz in the early 1980s. ^(97){ }^{97} Another major issue was the presence of chiral anomalies in superstring theories. In short, a chiral anomaly is a breakdown of gauge invariance when quantizing the theory, for theories where left- and right-handed fermions enter asymmetrically (that is, a chiral theory). In a landmark 1984 paper, Green and Schwarz demonstrated that string theory’s anomalies cancelled out for the specific gauge groups SOS O (32) and E_(8)xxE_(8)E_{8} \times E_{8} (Green & Schwarz, 1984). It was this result that initiated the significant increase in interest in superstring theory as a unified quantum gravity theory (what is often called the “first superstring revolution”), because it allowed for the possibility to construct renormalizable perturbative string models containing Standard Model symmetry groups. 格林和施瓦茨在20世纪80年代初提出的。 ^(97){ }^{97} 另一个主要问题是超弦理论中存在手性反常现象。简而言之，手性反常是指左手费米子和右手费米子不对称进入的理论（即手性理论），在量子化时破坏了理论的规整不变性。格林和施瓦茨在 1984 年发表的一篇里程碑式的论文中证明，弦理论的反常现象在特定的量规群 SOS O (32) 和 E_(8)xxE_(8)E_{8} \times E_{8} 中被抵消了（格林和施瓦茨，1984 年）。正是这一结果使人们对作为统一量子引力理论的超弦理论的兴趣大增（这就是人们常说的 "第一次超弦革命"），因为它使人们有可能构建包含标准模型对称群的可重规范化微扰弦模型。
4. Conclusions: string theory and non-arbitrariness 4.结论：弦理论与非任意性
We have seen how with the work on dual models in the first half of the 1970s not only the mathematical structure of string theory was formulated, but also a practice of theory construction was shaped that would condition important elements of the relation between theory and experimental data in modern string theory. Most importantly, hadronic dual model physics became theory-driven, with the dual model principles determining almost all parameters of the model. This resulted in a number of predictions that were incompatible with experimental results on hadronic scattering (or, in the case of 26 spacetime dimensions, with everyday experience). Significantly, rather than modifying the theory to accommodate experimental results, dual model theorists maintained all dual model constraints, particularly duality, hoping that the theory would eventually align with a realistic model for hadrons. After its reinterpretation as a quantum gravity theory, this theory-driven attitude and the emphasis on the virtue of non-arbitrariness remained in place. 我们已经看到，20 世纪 70 年代前半期的对偶模型工作不仅制定了弦理论的数学结构，而且形成了理论构建的实践，这将成为现代弦理论中理论与实验数据之间关系的重要因素。最重要的是，强子对偶模型物理学变成了理论驱动，对偶模型原理决定了模型的几乎所有参数。这导致许多预言与强子散射的实验结果不符（或者，在 26 维时空的情况下，与日常经验不符）。重要的是，二元模型理论家并没有修改理论以适应实验结果，而是保留了所有二元模型约束，尤其是二元性，希望理论最终能与现实的强子模型相一致。在其被重新解释为量子引力理论之后，这种理论驱动的态度和对非任意性美德的强调依然存在。
Before moving on it is worthwhile to briefly pause here to point out that the historical analysis presented in this paper is especially of importance because the early development and establishment of unified quantum gravity string theory was characterized by a lack of explicitly voiced motivations guiding the theory’s construction. This becomes particularly clear when comparing string theory with other historical examples of unified theory attempts; we confine ourselves here to the cases of Heisenberg and Einstein. In the late 1950s, Heisenberg attempted to reduce all of physics to the dynamics of a fundamental spinor field. Heisenberg’s work was grounded in a philosophical principle of “reductive monism”. As Blum (2019, pp. 9-11) describes, Heisenberg pictured the history of science as a series of “epistemological concessions”: already the Greek atomists had stripped atoms of most qualities (e.g., color, smell), and this program had been almost completed by quantum mechanics, stripping the atoms of their mechanical qualities. The last epistemological concession to be made, according to Heisenberg, was to reduce the elementary protons, neutrons, and electrons to a single substance, embodied by his proposed spinor field. ^(98){ }^{98} Einstein’s attempts in his later life to construct a unified theory of gravity and electrodynamics were justified and guided by his theoretical maxims of mathematical naturalness and logical simplicity. There is no final definition of these terms to be found in Einstein’s writings, but important elements of mathematical naturalness, while also involving an aesthetic judgment, included unification and the use of classical geometric fields to describe particles. For Einstein, logical simplicity usually referred to a theory’s degree of unity, on the basis of as few independent assumptions or axioms as possible. ^(99){ }^{99} The main point is that both Einstein and Heisenberg voiced what I will refer to here as “epistemic motivations”: motivations for their choices in theory construction that explicated the reasons for them and that were related to the epistemological beliefs in which their theorizing was grounded. On the one hand this made their theoretical attempts vulnerable: both Einstein and Heisenberg have been accused of being blinded by, if you want, grand methodological principles. On the other hand, Einstein’s and Heisenberg’s explicit articulation at least allowed for an assessment of the epistemic motivations for their respective proposals. 在继续讨论之前，我们值得在此稍作停顿，指出本文的历史分析尤为重要，因为统一量子引力弦理论的早期发展和建立的特点是缺乏明确的动机来指导理论的构建。这一点在将弦理论与其他历史上的统一理论尝试进行比较时尤为明显；我们在此只讨论海森堡和爱因斯坦的案例。20 世纪 50 年代末，海森堡试图将所有物理学还原为一个基本旋量场的动力学。海森堡的工作以 "还原一元论 "的哲学原理为基础。正如布卢姆（2019，第9-11页）所描述的，海森堡将科学史描绘成一系列 "认识论让步"：希腊原子论者已经剥离了原子的大部分特质（如颜色、气味），量子力学几乎完成了这一计划，剥离了原子的机械特质。海森堡认为，认识论上的最后让步是把基本质子、中子和电子还原为单一物质，这就是他提出的自旋场。 ^(98){ }^{98} 爱因斯坦晚年试图构建万有引力和电动力学的统一理论，他的理论格言是数学自然性和逻辑简洁性，并以此为指导。在爱因斯坦的著作中找不到这些术语的最终定义，但数学自然性的重要元素，同时也涉及美学判断，包括统一和使用经典几何场来描述粒子。 对爱因斯坦来说，逻辑简单性通常是指理论的统一程度，它以尽可能少的独立假设或公理为基础。 ^(99){ }^{99} 主要的一点是，爱因斯坦和海森堡都表达了我在这里所说的 "认识论动机"：他们在理论建构中做出选择的动机，这些动机解释了做出这些选择的原因，并且与他们的理论建构所基于的认识论信念有关。一方面，这使得他们的理论尝试容易受到攻击：爱因斯坦和海森堡都被指责被宏大的方法论原则所蒙蔽。另一方面，爱因斯坦和海森堡的明确表述至少允许我们对其各自提议的认识论动机进行评估。
In contrast, the early development of unified string theory is distinctive because of the lack of such articulated epistemic motivations, apart from a general appeal to the idea of a unified description of the fundamental interactions. The main motivation was that the mathematical structure was already at hand and that, bluntly put, it just worked. This pragmatic attitude, focusing 相比之下，统一弦理论的早期发展之所以与众不同，是因为除了对统一描述基本相互作用这一理念的普遍呼吁之外，缺乏这种明确的认识论动机。主要动机是数学结构已经唾手可得，直截了当地说，它就是行之有效的。这种务实的态度侧重于
on calculations instead of on foundational reflection, is mirrored by the manner in which Scherk, Schwarz, and Green framed their papers on unified string theory in the 1970s and early 1980s. Their main message was that it was an “attractive possibility” (Green & Schwarz, 1981, p. 504) that could turn out successful, if various theoretical problems were to be overcome step-by-step. After having taken another step towards a consistent supersymmetric string theory, Green and Schwarz (1982a, pp. 267-268) for example calmly write: “Much work remains to be done, but we remain enthusiastic about the possibility of establishing that this theory is a well-defined relativistic quantum theory free from any pathology.” String theory’s rise to prominence and corresponding high hopes only emerged after the 1984 anomaly cancellation result-with the result itself providing the main motivation to further pursue the theory. I do explicitly not want to suggest that the anomaly cancellation result was not a good reason to pursue the theory further, nor do I mean this as a judgment on string theory’s viability. What the foregoing does imply, however, is that string theorists like Green and Schwarz were in the first place concerned with the technical arguments and problems and then justified their work on the basis of success. This way of working was in line with the practices of SS-matrix and dual model physics out of which the unified string theory work grew (and, in the case of dual models at least, in which Green and Schwarz had a background themselves). Here theoretical progress and the presentation of new results were also mostly concerned with the often highly technical mathematical applications—but now the ties to experimental practices were severed. ^(100){ }^{100} This pragmatist style must arguably be understood in the larger context of American particle physics as it was shaped during and after World War II, fostering an approach to theory that prioritized calculation over philosophical considerations (Schweber, 1986); an attitude that was furthermore reinforced, as historian of physics David Kaiser (2002) has argued, by the significant growth in physics Ph. D. students in 1950s and 1960s Cold War America. In line with this, philosophical reflection was largely absent in string theory’s early development. 舍克、施瓦茨和格林在 20 世纪 70 年代和 80 年代初发表的关于统一弦理论的论文，也反映了他们的观点：统一弦理论是一种 "有吸引力的可能性"（格林和施瓦茨，1981 年，第 504 页），只要逐步克服各种理论问题，就有可能取得成功。他们的主要观点是，如果能逐步克服各种理论问题，统一弦理论是一种 "有吸引力的可能性"（格林和施瓦茨，1981 年，第 504 页），有可能取得成功。格林和施瓦茨（1982a, 第 267-268 页）在向一致的超对称弦理论又迈进了一步之后，冷静地写道："还有许多工作要做："还有许多工作要做，但我们仍然热衷于确定这一理论是一种定义明确的相对论性量子理论，不存在任何病态"。弦理论的崛起和相应的殷切希望只是在 1984 年的反常取消结果之后才出现的--该结果本身为进一步研究该理论提供了主要动力。我明确不想暗示，异常取消结果并不是进一步研究该理论的充分理由，我也无意以此来评判弦理论的可行性。不过，上述内容确实意味着，格林和施瓦茨这样的弦理论家首先关注的是技术论证和问题，然后在成功的基础上为自己的工作辩护。这种工作方式与 SS 矩阵物理学和二元模型物理学的做法是一致的，而统一弦理论的工作正是从这些工作中发展起来的（至少就二元模型而言，格林和施瓦茨本身就有这方面的背景）。在这里，理论进展和新成果的展示也主要涉及通常技术性很强的数学应用--但现在与实验实践的联系被切断了。 ^(100){ }^{100} 可以说，这种实用主义风格必须在美国粒子物理学的大背景下加以理解，因为美国粒子物理学是在二战期间和二战之后形成的，它培养了一种将计算置于哲学考虑之上的理论方法（Schweber, 1986）；正如物理学史学家大卫-凯泽（David Kaiser, 2002）所认为的那样，这种态度因20世纪50年代和60年代冷战时期美国物理学博士生的显著增长而得到进一步强化。因此，在弦理论的早期发展过程中，哲学思考在很大程度上是缺失的。
My historical analysis of how particle physicists developed string theory out of SS-matrix physics, however, does bring forward some important motivations that were guiding early string theorist’s practice of theory construction, and shaped string theory’s relation to experimental data. As said, the most important was the commitment of the involved physicists to non-arbitrariness, that is, to theoretically determining the theory’s parameters. We have seen that the ideal of a unique solution consistent with all SS-matrix principles originated with the bootstrap conjecture in Chew’s SS matrix program and remained prominent in dual model theory construction. For dual models, adding duality to the list of SS-matrix principles resulted in a framework with only one free parameter, luring theorists away from experimental input towards the theory-driven construction of mathematical models. In contrast, the competing field theory approach that gave rise to QCD was much more flexible in adjusting parameters and accommodating data. In terms of describing experimental results for strong interactions, QCD proved to be more empirically adequate. 然而，我对粒子物理学家如何从 SS 矩阵物理学中发展出弦理论的历史分析，确实提出了一些指导早期弦理论家理论建构实践的重要动机，并塑造了弦理论与实验数据的关系。如前所述，最重要的是相关物理学家对非任意性的承诺，即从理论上确定理论参数。我们已经看到，符合所有 SS 矩阵原理的唯一解的理想，起源于周氏 SS 矩阵计划中的自举猜想，并在对偶模型理论的构建中一直占据重要地位。对于对偶模型来说，在 SS 矩阵原理列表中加入对偶性，就形成了一个只有一个自由参数的框架，从而诱使理论家从实验输入转向理论驱动的数学模型构建。相比之下，产生 QCD 的竞争场论方法在调整参数和适应数据方面要灵活得多。在描述强相互作用的实验结果方面，QCD 被证明更符合经验。
However, after the successful formulation of the Standard Model, circumstances became much more favorable for a lack of free parameters to be appreciated as a virtue guiding theory construction, essentially for two reasons. Firstly, what physicists wanted after the Standard Model’s establishment was an explanation for it. This was a widely shared wish among particle physicists. As voiced by Murray Gell-Mann in his opening talk at the second Shelter Island Conference of 1983: “As usual, solving the problems of one era has shown up the critical questions of the next era.” He then listed “the very first [critical questions] that come to mind, looking at the standard theory of today.” ^(101){ }^{101} All these questions can be united under the banner “why the Standard Model?”: e.g., why 然而，在标准模型成功提出之后，缺乏自由参数的情况变得更加有利，人们开始将其视为指导理论构建的一种美德，这主要有两个原因。首先，标准模型建立后，物理学家们想要的是对它的解释。这是粒子物理学家的普遍愿望。正如默里-盖尔-曼（Murray Gell-Mann）在 1983 年第二次避难岛会议的开幕演讲中所说："像往常一样，解决了一个时代的问题，就会发现下一个时代的关键问题"。随后，他列举了 "从当今的标准理论出发，首先想到的[关键问题]"。 ^(101){ }^{101} 所有这些问题都可以统一在 "为什么是标准模型？"的旗帜下：例如，为什么是标准模型？
the particular structure for families, why three families, why SU(3)xx SU(2)xx U(1)?^(102)S U(3) \times S U(2) \times U(1) ?^{102} In this search for an explanation, the free parameters in the Standard Model, essential in providing an empirically adequate description, came to weigh heavily against it. String theory’s promise of a possibly unique description with no arbitrary parameters then promised a powerful alternative. SU(3)xx SU(2)xx U(1)?^(102)S U(3) \times S U(2) \times U(1) ?^{102} 在寻找解释的过程中，标准模型中的自由参数对提供充分的经验描述至关重要，但却对标准模型产生了严重的负面影响。弦理论承诺提供一种可能是唯一的、没有任意参数的描述，这为我们提供了一个强有力的替代方案。
The second reason why circumstances were more favorable to non-arbitrariness as theoretical virtue, is the practical difficulty of empirical input in quantum gravity theory construction. With the energy scales of quantum gravity out of reach for any conceivable particle acceleratorwhat Rickles (2014(2014, pp. 16, 171) has called the “tyranny of experimental distance” -internal consistency became a guiding principle in theory construction. As Green, Schwarz, and Witten (1987, p. 14) put it in their textbook on superstring theory, quantum gravity “has always been a theorist’s puzzle par excellence”. The hope for testing it, they contended, lay in learning “how to make a consistent theory” of quantum mechanics and gravity and then see what its implications are. We are now in a position to better appreciate how this theory-driven attitude was not only conditioned by the search for a quantum gravity theory, as often presupposed, but was at the same time already inherent to the practice of string theorists before quantum gravity. Surely, the move from hadron physics to unified quantum gravity rigorously expanded the scope of the theory, and opened up new interpretations and possibilities (for example, the need for the compactification of higher spacetime dimensions was now seen as less problematic or even desirable). However, as we have aimed to demonstrate, the theory construction from the physicists developing hadronic dual models was already driven by theoretical progress constrained by the dual model conditions; that is to say, hadronic dual model physics also was a “theorist’s puzzle par excellence”. This important result firmly establishes dual models as the missing link in understanding how string theory, as a theory detached from empirical data, grew out of an SS-matrix theory that was strongly dependent upon observable quantities-a transition that is sometimes found puzzling. ^(103){ }^{103} Moreover, it is of crucial importance when assessing string theory’s relation to experimental results: it was the commitment of practitioners to the dual model principles, making it possible to determine all but one of the theory’s parameters on the basis of theoretical reasoning, which led to string theory’s initial detachment from experimental input-and not the reinterpretation of dual models as a candidate unified quantum gravity theory. 环境更有利于将非任意性作为理论美德的第二个原因，是量子引力理论构建中的经验输入的实际困难。量子引力的能量尺度对于任何可以想象的粒子加速器来说都是遥不可及的，这就是里克斯（Rickles (2014(2014 , 第 16 和 171 页）所说的 "实验距离的暴政"--内部一致性成为理论构建的指导原则。正如格林、施瓦茨和威滕（1987 年，第 14 页）在他们的超弦理论教科书中所说，量子引力 "一直是理论家的卓越难题"。他们认为，检验量子引力的希望在于学会 "如何建立一个一致的 "量子力学和引力理论，然后看看它的影响是什么。我们现在能够更好地理解，这种理论驱动的态度不仅受制于量子引力理论的探索（这一点经常被预设），同时也是量子引力之前弦理论学家的固有做法。当然，从强子物理学到统一量子引力严格地扩大了理论的范围，并开辟了新的解释和可能性（例如，现在人们认为对更高时空维度的压缩需求问题不大，甚至是可取的）。然而，正如我们旨在证明的那样，从物理学家发展强子对偶模型开始的理论建构已经受到了对偶模型条件制约的理论进步的推动；也就是说，强子对偶模型物理学也是一个 "卓越的理论家难题"。 这一重要结果牢固地确立了二元模型是理解弦理论这一脱离经验数据的理论如何从强烈依赖于可观测量的 SS 矩阵理论发展而来的缺失环节--这种转变有时令人费解。 ^(103){ }^{103} 此外，在评估弦理论与实验结果的关系时，这一点也至关重要：正是实践者对二元模型原理的承诺，使我们有可能在理论推理的基础上确定除一个参数之外的所有理论参数，才导致弦理论最初脱离了实验输入--而不是将二元模型重新解释为候选的统一量子引力理论。
Acknowledgements: I am very grateful to Karel Gaemers and Jeroen van Dongen for discussions and valuable feedback on drafts; to Erik Verlinde and Manus Visser for stimulating conversations; and to three anonymous reviewers for extremely helpful comments. Furthermore, I want to thank the organizers of the 2022 Wuppertal Spring School on the History, Philosophy and Sociology of Large Physics Experiments and the 2022 Strings, Cosmology and Gravity Student Conference for giving me the opportunity to present and discuss this work as it was being shaped. 致谢：我非常感谢卡雷尔-盖默斯（Karel Gaemers）和耶罗恩-范东恩（Jeroen van Dongen）的讨论和对草稿的宝贵反馈；感谢埃里克-韦林德（Erik Verlinde）和马努斯-维瑟（Manus Visser）激励人心的谈话；感谢三位匿名审稿人极有帮助的评论。此外，我要感谢 2022 年伍珀塔尔大型物理实验的历史、哲学和社会学春季学校以及 2022 年弦、宇宙学和引力学生会议的组织者，感谢他们给我机会在这项工作形成的过程中进行展示和讨论。
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Veneziano, G. (2012). Rise and fall of the hadronic string. In A. Cappelli, E. Castellani, F. Colomo, & P. D. Vecchia (Eds.), The birth of string theory (pp. 17-36). Cambridge University Press. Veneziano, G. (2012).强子弦的兴衰。In A. Cappelli, E. Castellani, F. Colomo, & P. D. Vecchia (Eds.), The birth of string theory (pp. 17-36).剑桥大学出版社。
Virasoro, M. Á. (1970). Subsidiary conditions and ghosts in dual-resonance models. Physical Review D, 1(10), 2933-2936. https://doi.org/10.1103/PhysRevD.1.2933 Virasoro, M. Á.(1970).双共振模型中的附属条件和幽灵.Physical Review D, 1(10), 2933-2936. https://doi.org/10.1103/PhysRevD.1.2933
Wilson, K. G. (1974). Confinement of quarks. Physical Review D, 10(8), 2445-2459. Wilson, K. G. (1974).夸克的禁闭。物理评论 D》，10（8），2445-2459。 https://doi.org/10.1103/PhysRevD.10.2445
Yoneya, T. (1973). Quantum gravity and the zero-slope limit of the generalized Virasoro model. Lettere al Nuovo Cimento (1971-1985), 8(16), 951-955. https://doi.org/10.1007/BF02727806 Yoneya, T. (1973).Quantum gravity and the zero-slope limit of the generalized Virasoro model.Lettere al Nuovo Cimento (1971-1985), 8(16), 951-955. https://doi.org/10.1007/BF02727806
Yoneya, T. (1974). Connection of dual models to electrodynamics and gravidynamics. Progress of Theoretical Physics, 51(6), 1907-1920. https://doi.org/10.1143/PTP.51.1907 Yoneya, T. (1974).二元模型与电动力学和重力流体力学的联系.Progress of Theoretical Physics, 51(6), 1907-1920. https://doi.org/10.1143/PTP.51.1907
Yoneya, T. (1975). Dual string models and quantum gravity. In H. Araki (Ed.), International Symposium on Mathematical Problems in Theoretical Physics (Vol. 39, pp. 180-183). Springer-Verlag. Yoneya, T. (1975).Dual string models and quantum gravity.In H. Araki (Ed.), International Symposium on Mathematical Problems in Theoretical Physics (Vol. 39, pp. 180-183).Springer-Verlag. https://doi.org/10.1007/BFb0013318
Zachariasen, F. (1961). Self-consistent calculation of the mass and width of the =1,t=1,pi pi=1, t=1, \pi \pi resonance. Physical Review Letters, 7(3), 112-113. https://doi.org/10.1103/PhysRevLett.7.112 Zachariasen, F. (1961). =1,t=1,pi pi=1, t=1, \pi \pi 共振的质量和宽度的自洽计算.物理评论快报》，7（3），112-113。https://doi.org/10.1103/PhysRevLett.7.112。
^(1){ }^{1} E-mail address: r.a.vanleeuwen@uva.nl ^(1){ }^{1} 电子邮件地址：r.a.vanleeuwen@uva.nl ^(2){ }^{2} Full postal address: P.O. Box 94485, 1090 GL Amsterdam, The Netherlands ^(2){ }^{2} 详细邮政地址：P.O. Box 94485, 1090 GL Amsterdam, The Netherlands
^(3){ }^{3} See Scherk and Schwarz (1974, 1975), Yoneya (1973, 1974, 1975). ^(3){ }^{3} 见 Scherk 和 Schwarz (1974, 1975), Yoneya (1973, 1974, 1975)。 ^(4){ }^{4} More precisely, Green and Schwarz demonstrated that for superstring theories with two specific gauge groups ( SO (32) and E_(8)xxE_(8)\mathrm{E}_{8} \times \mathrm{E}_{8} ) so-called chiral anomalies (a breakdown of gauge invariance when quantizing theories in which left- and right-handed fermions enter asymmetrically) cancelled. ^(4){ }^{4} 更确切地说，格林和施瓦茨证明，对于具有两个特定轨距群（SO (32) 和 E_(8)xxE_(8)\mathrm{E}_{8} \times \mathrm{E}_{8} ）的超弦理论，所谓的手性反常现象（在量子化理论中，左手和右手费米子不对称地进入时，轨距不变性的破坏）会消失。 ^(5){ }^{5} Cushing (1990). ^(5){ }^{5} 库欣（1990 年）。 ^(6){ }^{6} Rickles (2014, p. 16). ^(6){ }^{6} 里克斯（2014 年，第 16 页）。
^(7){ }^{7} Recently, however, Lauren Greenspan (2022) has urged the history and philosophy of science and STS communities to consider a new epistemic shift from the early 2000s, when string theorists started to use holography not only in quantum gravity research but also as “a tool for making real-world predictions” (p. 74), that is, by starting with assuming string theory as a framework and then apply its results in specific contexts. ^(7){ }^{7} 然而，最近，劳伦-格林斯潘（Lauren Greenspan，2022 年）敦促科学史与科学哲学界和 STS 界考虑从 2000 年代初开始的一种新的认识论转变，即弦理论学家开始不仅在量子引力研究中使用全息技术，而且还将其作为 "一种进行现实世界预测的工具"（第 74 页），也就是说，从假设弦理论为框架开始，然后将其结果应用于特定环境。
^(8){ }^{8} A well-known example of an early criticism of string theory is the Physics Today article “Desperately Seeking Superstrings?” by theoretical physicists Paul Ginsparg and Sheldon Glashow (1986), in which they heavily criticized string theory on the basis that the gap between Planck scale strings and observable particles is unbridgeable and that superstring theory’s search for unification is almost certainly fruitless in the absence of experimental data. In contrast, string theorist John Schwarz (in Davies & Brown, 1988, pp. 70-89) for example invoked the theory’s mathematical consistency, including its success in providing provisional versions of a unified description of the four fundamental forces and the absence of free parameters, to favorably judge the theory’s prospects. See also Galison (1995). ^(8){ }^{8} 早期批评弦理论的一个著名例子是理论物理学家保罗-金斯帕格（Paul Ginsparg）和谢尔顿-格拉肖（Sheldon Glashow）（1986年）发表的《今日物理学》（Physics Today）文章《急切地寻找超弦？相反，弦理论家约翰-施瓦茨（John Schwarz）（见 Davies & Brown, 1988, pp.另见 Galison (1995)。 ^(9){ }^{9} This is also pointed out by Camilleri and Ritson (2015). ^(9){ }^{9} Camilleri 和 Ritson（2015 年）也指出了这一点。 ^(10){ }^{10} Related to this, Gilbert and Loveridge (2021) have sought to identify different physical, epistemic, and professional “tastes” in the communities of string theory and loop quantum gravity (on the basis of a statistical analysis of a collection of semi-structured interviews with physicists from both camps), shining light on how the two communities have “developed distinct epistemic standards reflecting different commitments to and realizations of objectivity” (p. 75). ^(10){ }^{10} 与此相关，Gilbert 和 Loveridge（2021 年）试图在弦理论和环量子引力社区中找出不同的物理、认识论和专业 "品味"（基于对来自两个阵营的物理学家的半结构式访谈的统计分析），揭示这两个社区如何 "发展出不同的认识论标准，反映出对客观性的不同承诺和实现"（第 75 页）。
^(11){ }^{11} Here I adopt some standard use of jargon in physics: a “physical theory” ascribes numerically measurable properties to an object (such as a particle). These properties (e.g., energy, or position) are called quantities, and the amounts that are ascribed to quantities (mostly real numbers) are called values. The state of an object is then the list of values for the various quantities that apply to an object. Over time, the state changes; commonly, in physics a theory gives a description of these changes. In field theory, the idea is that the equations of motion (usually derived from a Hamiltonian or Lagrangian) give an exact description of how the state changes over time-what is often called the dynamics. ^(11){ }^{11} 在此，我采用物理学中的一些标准术语："物理理论 "为物体（如粒子）赋予了可以用数字测量的属性。这些属性（如能量或位置）被称为量，而赋予量的金额（大多为实数）被称为值。物体的状态就是适用于物体的各种量的值的列表。随着时间的推移，状态会发生变化；在物理学中，理论通常会对这些变化做出描述。在场论中，运动方程（通常由哈密顿方程或拉格朗日方程导出）给出了状态随时间变化的精确描述--这就是通常所说的动力学。
^(12){ }^{12} Rickles (2014, p. 22). ^(12){ }^{12} 里克斯（2014 年，第 22 页）。 ^(13){ }^{13} See Heisenberg (1943a, 1943b, 1944, 1946); also Cushing (1990, pp. 29-34). ^(13){ }^{13} 参见 Heisenberg (1943a, 1943b, 1944, 1946)；另见 Cushing (1990, pp. 29-34)。 ^(14){ }^{14} See Duncan and Janssen (2007a, 2007b) and Blum et al. (2017) for more on the developments that led to Heisenberg’s reinterpretation of quantum theory. For the original papers, see Van der Waerden (1967). ^(14){ }^{14} 关于海森堡重新诠释量子理论的进展，请参阅 Duncan 和 Janssen (2007a, 2007b) 以及 Blum 等人 (2017)。原始论文见 Van der Waerden (1967)。
^(15){ }^{15} Cushing (1990, pp. 30-39); Blum (2017, p. 56). ^(15){ }^{15} 库欣（1990 年，第 30-39 页）；布卢姆（2017 年，第 56 页）。 ^(16){ }^{16} Cushing (1990, pp. 48-49); Schweber (1994, pp. 154-155). ^(16){ }^{16} Cushing（1990 年，第 48-49 页）；Schweber（1994 年，第 154-155 页）。 ^(17){ }^{17} See Cushing (1990, pp. 26, 48-49); also Cao (1991). ^(17){ }^{17} 见 Cushing（1990 年，第 26、48-49 页）；另见 Cao（1991 年）。
^(18){ }^{18} As Blum (2017, p. 54) notes, analyticity at the time was used in a somewhat loose manner, implying an extension to a more or less well-behaved function of complex variables; some singularities were not at all problematic. ^(18){ }^{18} 正如 Blum (2017, 第 54 页)所指出的，当时的解析性是以一种略微宽松的方式使用的，它意味着扩展到复变函数或多或少的良好状态；一些奇异性根本不成问题。 ^(19){ }^{19} Cushing (1990, pp. 57-58). ^(19){ }^{19} 库欣（1990 年，第 57-58 页）。 ^(20){ }^{20} When directing a beam of incoming particles at a particle target, the differential cross section d sigma//d Omegad \sigma / d \Omega is a measure for the proportion of particles that is scattered into a solid angle d Omegad \Omega, as detected in the laboratory system, and has the dimensions of an area. It is related to the scattering amplitude ff as (d sigma)/(d Omega)(theta,phi)=|f(theta,phi)|^(2)\frac{d \sigma}{d \Omega}(\theta, \phi)=|f(\theta, \phi)|^{2}. Here, theta\theta denotes the scattering angle and phi\phi the azimuthal angle in the laboratory frame. Using particle collision kinematics, ff can be expressed in terms of energy and momentum variables in the center-of-mass frame. ^(20){ }^{20} 当将射入粒子束对准粒子目标时，微分截面 d sigma//d Omegad \sigma / d \Omega 是对散射到实角 d Omegad \Omega 的粒子比例的测量，如在实验室系统中检测到的那样，具有面积的尺寸。它与散射振幅 ff 的关系为 (d sigma)/(d Omega)(theta,phi)=|f(theta,phi)|^(2)\frac{d \sigma}{d \Omega}(\theta, \phi)=|f(\theta, \phi)|^{2} 。 这里， theta\theta 表示散射角， phi\phi 表示实验室框架中的方位角。利用粒子碰撞运动学， ff 可以用质量中心框中的能量和动量变量来表示。
When integrated over all scattering angles, one obtains the total cross section sigma_(T)\sigma_{T}. Total cross sections are thus equal to the sum of all cross sections for a particular collision process. See Eden (1971, pp. 1003-1005). 对所有散射角进行积分，就得到了总截面 sigma_(T)\sigma_{T} 。因此，总截面等于特定碰撞过程的所有截面之和。参见 Eden (1971，第 1003-1005 页)。 ^(21){ }^{21} Newton (1976). ^(21){ }^{21} 牛顿（1976 年）。
^(22){ }^{22} Gell-Mann et al. (1954). ^(22){ }^{22} 盖尔-曼等人（1954 年）。 ^(23){ }^{23} See Cao (1997, p. 222); Kaiser (2005, pp. 284-285). ^(23){ }^{23} 见 Cao (1997, p. 222)；Kaiser (2005, pp. 284-285)。 ^(24){ }^{24} Cushing (1990, pp. 80-88). ^(24){ }^{24} 库欣（1990 年，第 80-88 页）。 ^(25){ }^{25} See Cao (1997, pp. 222-225); Cushing (1990, Chapters 3-5); Kaiser (2005, pp. 298-299). ^(25){ }^{25} 见 Cao (1997, pp. 222-225)；Cushing (1990, Chapters 3-5)；Kaiser (2005, pp. 298-299)。
^(26){ }^{26} Cushing (1985, p. 39). ^(26){ }^{26} 库欣（1985 年，第 39 页）。 ^(27){ }^{27} Note that the appreciation of non-arbitrariness-i.e., the virtue of theoretically determining “arbitrary” parameters-is not at all limited to SS-matrix physics or string theory. We will for example encounter an emphasis on non-arbitrariness in the work of Einstein and Eddington at the end of this Chapter. However, in this paper my main concern is how the notion of non-arbitrariness historically connects SS-matrix physics and modern string theory, and how this affected string theory’s relation to experimental data. ^(27){ }^{27} 请注意，对非任意性的理解--即从理论上确定 "任意 "参数的优点--并不局限于 SS 矩阵物理学或弦理论。例如，在本章结尾，我们将看到爱因斯坦和爱丁顿的研究对非任意性的强调。然而，在本文中，我主要关注的是非任意性概念如何在历史上将 SS 矩阵物理学与现代弦理论联系起来，以及这如何影响弦理论与实验数据的关系。 ^(28){ }^{28} See Cushing (1985, p. 40). ^(28){ }^{28} 见 Cushing (1985, 第 40 页)。 ^(29){ }^{29} Chew (1962, p. 395). ^(29){ }^{29} Chew（1962 年，第 395 页）。
^(30){ }^{30} Cushing (1990, p. 135). ^(30){ }^{30} 库欣（1990 年，第 135 页）。 ^(31){ }^{31} Chew et al. (1964, p. 79). ^(31){ }^{31} Chew 等人（1964 年，第 79 页）。 ^(32){ }^{32} The technical details of Chew’s program, its development, and its wider influence in the physics community are extensively discussed elsewhere; see, e.g., Cushing (1990, Chapters 6, 7); Kaiser (2005, Chapters 8, 9). See Cushing (1985) for a discussion of the epistemological and ontological underpinnings of the bootstrap conjecture. ^(32){ }^{32} 关于Chew计划的技术细节、发展及其在物理学界的广泛影响，我们已在其他地方进行了广泛讨论；例如，参见Cushing (1990, Chapters 6, 7); Kaiser (2005, Chapters 8, 9)。有关引导猜想的认识论和本体论基础的讨论，请参阅 Cushing (1985)。 ^(33){ }^{33} Chew (1962, p. 400). ^(33){ }^{33} Chew（1962 年，第 400 页）。
^(34){ }^{34} Zachariasen (1961, p. 113). ^(34){ }^{34} Zachariasen（1961 年，第 113 页）。 ^(35){ }^{35} Zachariasen and Zemach (1962, p. 849). ^(35){ }^{35} Zachariasen 和 Zemach（1962 年，第 849 页）。 ^(36){ }^{36} Jackson (1969, p. 73). Note that SS-matrix theory was sometimes also referred to as “Regge pole theory”. ^(36){ }^{36} Jackson (1969, p. 73)。请注意， SS 矩阵理论有时也被称为 "雷格极点理论"。
^(37){ }^{37} See, e.g., Frautschi (1963, p. 101). ^(37){ }^{37} 参见 Frautschi (1963, p. 101)。 ^(38){ }^{38} See Cushing (1985). ^(38){ }^{38} 见 Cushing (1985)。 ^(39){ }^{39} See Chew et al. (1964). ^(39){ }^{39} 见 Chew 等人（1964 年）。 ^(40){ }^{40} Cao (1997, pp. 229-230); see also Lipkin (1969, p. 53). ^(41){ }^{41} Chew (1962, p. 395). ^(41){ }^{41} Chew（1962 年，第 395 页）。 ^(42){ }^{42} See the proceedings of the Twelfth Solvay Conference (Stoops, 1962, pp. 204-205). ^(42){ }^{42} 见第十二届索尔维会议记录（Stoops，1962 年，第 204-205 页）。
^(43){ }^{43} Stoops (1962, p. 215) ^(44){ }^{44} Cushing (1990, pp. 144-145). ^(44){ }^{44} 库欣（1990 年，第 144-145 页）。 ^(45){ }^{45} I thank Arianna Borelli for pointing out this vague notion of the concept of a particle in pre-Standard Model high-energy physics and the (fruitful) implications of this vagueness for the practices of particle physicists, as shared in a presentation of yet unpublished work. ^(45){ }^{45} 我感谢阿里安娜-博雷利（Arianna Borelli）指出前标准模型高能物理中粒子概念的模糊性，以及这种模糊性对粒子物理学家的实践所产生的（富有成效的）影响。 ^(46){ }^{46} See Cushing (1990, p. 164). Chew’s program was particularly successful in calculations for scattering processes with just two incoming and outgoing particles, and had more difficulties coping with multiparticle production, which became increasingly problematic with new accelerator experiments operating at ever higher energies. ^(46){ }^{46} 见 Cushing (1990, p. 164)。Chew 的程序在计算只有两个进出粒子的散射过程时特别成功，但在应对多粒子产生时却遇到了更多困难，而随着新的加速器实验在更高能量下运行，多粒子产生变得越来越成问题。
^(47){ }^{47} See Van Dongen (2010), especially Chapter 6. ^(47){ }^{47} 见 Van Dongen (2010)，尤其是第 6 章。 ^(48){ }^{48} See Kilmister (1994); Cushing (1985, pp. 33-34). ^(48){ }^{48} 见 Kilmister (1994); Cushing (1985, pp. 33-34)。 ^(49){ }^{49} See, e.g., string theorist John Schwarz in Davies and Brown (1988, p. 86). ^(49){ }^{49} 参见弦理论家约翰-施瓦茨（John Schwarz）在《戴维斯与布朗》（1988 年，第 86 页）中的论述。 ^(50){ }^{50} Dawid (2013, pp. 141-142). ^(50){ }^{50} 达维德（2013 年，第 141-142 页）。
^(51){ }^{51} The Breit-Wigner amplitude also arises in field-theory perturbation, with the width being included by loop corrections. ^(51){ }^{51} 布赖特-维格纳振幅也出现在场论扰动中，其宽度包含在环路修正中。 ^(52){ }^{52} See Frampton (1986, pp. 13-14); also Capelli et al. (2012, p. 92); Cushing (1990, p. 30). ^(52){ }^{52} 见 Frampton (1986, 第 13-14 页)；另见 Capelli 等人 (2012, 第 92 页)；Cushing (1990, 第 30 页)。
^(53){ }^{53} See 't Hooft (2004, p. 4); Jacob (1969); Rickles (2014, p. 33). ^(53){ }^{53} 见 't Hooft (2004, 第 4 页)；Jacob (1969); Rickles (2014, 第 33 页)。 ^(54){ }^{54} See, e.g., Eden (1971, p. 1026). ^(54){ }^{54} 参见伊登（1971 年，第 1026 页）。 ^(55){ }^{55} Note that for some hadrons, resonances were only found at odd or even values of ss, which was incorporated in the mathematical formalism ^(55){ }^{55} 请注意，对于某些强子，只有在 ss 的奇数或偶数值时才会出现共振，这一点已被纳入数学形式主义中。 ^(56){ }^{56} See Eden (1971, pp. 1023-1032), also Chew (1967). ^(56){ }^{56} 见 Eden (1971, pp. 1023-1032)，另见 Chew (1967)。 ^(57){ }^{57} Jacob (1969, p. 127). ^(57){ }^{57} 雅各布（1969 年，第 127 页）。 ^(58){ }^{58} Jackson (1969, p. 85). ^(58){ }^{58} 杰克逊（1969 年，第 85 页）。
^(59){ }^{59} See Dolen et al. (1967). Schwarz (1975, p. 62) in retrospect designated DHS duality as “phenomenological duality”, in contrast to the “theoretical duality” that was used as a principle underlying dual resonance model building. ^(59){ }^{59} 见 Dolen 等人 (1967)。施瓦茨（1975 年，第 62 页）回过头来把 DHS 的对偶性称为 "现象学对偶性"，与 "理论对偶性 "形成鲜明对比。 ^(60){ }^{60} Ademollo et al. (1967, 1968); see also Veneziano (2012, pp. 22-24). ^(60){ }^{60} Ademollo 等人（1967 年，1968 年）；另见 Veneziano（2012 年，第 22-24 页）。
^(61){ }^{61} See Jacob (1969). ^(61){ }^{61} 见雅各布（1969 年）。 ^(62){ }^{62} See Fubini and Veneziano (1969, p. 813); Fubini (1974, p. 3); also 't Hooft (2004, p. 6); Green et al. (1987, p. 7). ^(62){ }^{62} 见 Fubini 和 Veneziano (1969, 第 813 页)；Fubini (1974, 第 3 页)；还有 't Hooft (2004, 第 6 页)；Green 等人 (1987, 第 7 页)。 ^(63){ }^{63} 't Hooft (2004, p. 6); Green et al. (1987, p. 7), ^(63){ }^{63} 't Hooft (2004, 第 6 页)；Green 等人 (1987, 第 7 页)、
^(64){ }^{64} Jacob (1969). ^(64){ }^{64} 雅各布（1969 年）。 ^(65){ }^{65} Mandelstam (1968, 1974). See also Ademollo (2012). ^(65){ }^{65} 曼德尔施塔姆（1968 年，1974 年）。另见 Ademollo (2012)。 ^(66){ }^{66} See, e.g., Chan (1970, p. 23). ^(66){ }^{66} 参见 Chan（1970 年，第 23 页）。 ^(67){ }^{67} Kikkawa et al. (1969, p. 1701), see also Rickles (2014, pp. 59-62). ^(67){ }^{67} Kikkawa 等人（1969 年，第 1701 页），另见 Rickles（2014 年，第 59-62 页）。 ^(68){ }^{68} Alessandrini et al. (1971, p. 272). ^(68){ }^{68} Alessandrini 等人（1971 年，第 272 页）。
^(69){ }^{69} Rickles (2014, p. 56). ^(69){ }^{69} 里克斯（2014 年，第 56 页）。 ^(70){ }^{70} From now on, I will refer to “dual model postulates” (or “principles”, “constraints”, “conditions”) when referring to the “old” SS-matrix postulates plus duality. ^(70){ }^{70} 从现在起，在提到 "旧的" SS --矩阵公设加二重性时，我将提到 "二重模型公设"（或 "原则"、"约束"、"条件"）。
^(71){ }^{71} Both examples are also discussed by Castellani (2019). For a treatment of the procedures to get rid of ghosts in dual theory, see Di Vecchia (2012); for an historical overview see Rickles (2014, pp. 63-67; 84-86); for the one-loop singularity problem, see Cappelli et al. (2012, pp. 143-145) and Rickles (2014, pp. 88-92). ^(71){ }^{71} 卡斯特拉尼（2019）也讨论了这两个例子。关于在对偶理论中摆脱幽灵的程序，见 Di Vecchia (2012)；关于历史概述，见 Rickles (2014, pp. 63-67; 84-86)；关于一环奇点问题，见 Cappelli 等人 (2012, pp. 143-145) 和 Rickles (2014, pp. 88-92)。 ^(72){ }^{72} Bardakçi and Mandelstam (1969); Fubini and Veneziano (1969). For a treatment of factorizability in dual models, see Di Vecchia (2012, pp. 160-163). ^(72){ }^{72} Bardakçi and Mandelstam (1969); Fubini and Veneziano (1969).关于二元模型中可因性的处理，见 Di Vecchia (2012, pp. 160-163)。 ^(73){ }^{73} Fubini et al. (1969). ^(73){ }^{73} Fubini 等人（1969 年）。 ^(74){ }^{74} Virasoro (1970). ^(74){ }^{74} 维拉索罗（1970 年）。
^(75){ }^{75} See Gross et al. (1970); Frye and Susskind (1970). Recall that the Pomeron trajectory (see Section 3.1) was posited in SS-matrix theory to account for scattering data (in particular for “vacuum” scattering), but there was no direct evidence for it to correspond to a physical particle or resonance. ^(75){ }^{75} 见 Gross et al. (1970); Frye and Susskind (1970)。回想一下， SS 矩阵理论提出了波美拉尼亚轨迹（见第 3.1 节）来解释散射数据（尤其是 "真空 "散射），但没有直接证据表明它对应于一个物理粒子或共振。 ^(76){ }^{76} A spacetime dimension of d=26d=26 is clearly in conflict with empirical data, but this was in fact also the case for the intercept value alpha_(P)(0)=2\alpha_{P}(0)=2 that as well followed from Lovelace’s calculation. In the “old” SS-matrix theory, a value for the intercept of the Pomeron trajectory alpha_(P)(0)=1\alpha_{P}(0)=1 was expected. As incorporated in dual models (i.e., with intercept 2), the ground-state of the Pomeron trajectory was (again) a tachyon, followed by a massless spin-2 meson as first excited state. In unified string theory, the Pomeron became the closed string/graviton, and its appearance at one-loop level was understood as the formation of a closed-string intermediate state in the scattering of open strings. ^(76){ }^{76}d=26d=26 的时空维度显然与经验数据相冲突，但事实上，洛夫莱斯计算得出的截距值 alpha_(P)(0)=2\alpha_{P}(0)=2 也是如此。在 "旧的" SS 矩阵理论中，波美拉尼亚轨迹的截距值 alpha_(P)(0)=1\alpha_{P}(0)=1 是预期的。在二元模型中（即截距为 2），波美子轨迹的基态（同样）是一个超速子，其次是一个作为第一激发态的无质量自旋-2 介子。在统一弦理论中，波美子成为闭弦/引力子，它在一环水平的出现被理解为开弦散射中形成的闭弦中间态。 ^(77){ }^{77} Goddard et al. (1973). Nowadays the requirement for d=26d=26 is understood as being the critical dimension to retain conformal symmetry when quantizing classical string theory, see Rickles (2014, p. 91). ^(77){ }^{77} 戈达德等人（1973 年）。如今，对 d=26d=26 的要求被理解为在量子化经典弦理论时保留保角对称性的临界维度，见 Rickles (2014, p. 91)。
^(78){ }^{78} Fubini (1974, p. 5). ^(78){ }^{78} Fubini（1974 年，第 5 页）。 ^(79){ }^{79} Olive and Scherk (1973, p. 296). ^(79){ }^{79} Olive 和 Scherk（1973 年，第 296 页）。 ^(80){ }^{80} The textbook was first published in 1974 under the title Dual Resonance Models and went out of print in 1979. It was then reprinted in 1986 with a supplement on superstrings under the name Dual Resonance Models and Superstrings-that is, after the rise of interest in string theory in the mid-1980s (see Frampton, 1986, pp. v -vii). ^(80){ }^{80} 这本教科书以《双共振模型》为题于1974年首次出版，1979年绝版。1986年，在弦理论的兴趣在20世纪80年代中期兴起之后，这本教科书以《双共振模型与超弦》为名再版，并附有关于超弦的补编（见Frampton, 1986, pp.v-vii）。
^(81){ }^{81} Frampton (1986, pp. 3-4 underlining in original). ^(81){ }^{81} Frampton (1986, pp. 3-4 underlining in original). ^(82){ }^{82} Clavelli (2012, pp. 194-195). See also Rickles (2014, pp. 97-98), Van Dongen (2021, p. 173). ^(82){ }^{82} Clavelli (2012, pp. 194-195).另见 Rickles（2014 年，第 97-98 页）、Van Dongen（2021 年，第 173 页）。 ^(83){ }^{83} Mandelstam (1974, p. 298). ^(83){ }^{83} 曼德尔施塔姆（1974 年，第 298 页）。
^(84){ }^{84} Frampton (1986, p. 363). ^(84){ }^{84} Frampton（1986 年，第 363 页）。 ^(85){ }^{85} See Nambu (1970); Nielsen (1970); Susskind (1969). ^(85){ }^{85} 见 Nambu (1970)；Nielsen (1970)；Susskind (1969)。 ^(86){ }^{86} See Rickles (2014, pp. 80-83), also Cappelli et al. (2012, pp. 228-231). ^(86){ }^{86} 参见 Rickles (2014, pp. 80-83)，以及 Cappelli 等人 (2012, pp. 228-231)。
^(87){ }^{87} See Cappelli et al. (2012, pp. 233-234). ^(87){ }^{87} 见 Cappelli 等人（2012 年，第 233-234 页）。 ^(88){ }^{88} Frampton (1986, p. 235). ^(88){ }^{88} Frampton（1986 年，第 235 页）。 ^(89){ }^{89} See, e.g., Rebbi (1974). ^(89){ }^{89} 参见 Rebbi (1974)。 ^(90){ }^{90} See also Rickles (2014, p. 127). ^(90){ }^{90} 另见 Rickles（2014 年，第 127 页）。
^(91){ }^{91} See Nambu (1976, p. 58). ^(91){ }^{91} 见 Nambu (1976, 第 58 页)。 ^(92){ }^{92} See 't Hooft (1974); Veneziano (1976); Wilson (1974); Rickles (2014, pp. 124-126). ^(92){ }^{92} 见 't Hooft (1974); Veneziano (1976); Wilson (1974); Rickles (2014, pp. 124-126)。 ^(93){ }^{93} See 't Hooft and Veltman (1972), Gross and Wilczek (1973, 1974), Politzer (1973). ^(93){ }^{93} 见 't Hooft 和 Veltman (1972)，Gross 和 Wilczek (1973, 1974)，Politzer (1973)。
^(94){ }^{94} See Rickles (2014, pp. 122-126). It must also be mentioned that work applying dual model tools to QCD continued throughout the following years. Especially, the work by Polyakov (1981a, 1981b) on QCD strings and Liouville theory led to important advancements in the understanding of string theory’s mathematical properties; see Rickles (2014, pp. 151-153). ^(94){ }^{94} 参见 Rickles (2014, pp. 122-126)。还必须一提的是，把对偶模型工具应用于 QCD 的工作在随后几年一直在继续。尤其是波利亚科夫（Polyakov，1981a，1981b）关于 QCD 弦和柳维尔理论的工作，使人们对弦论数学性质的理解有了重要的进步；见 Rickles (2014, pp. 151-153)。 ^(95){ }^{95} Scherk and Schwarz (1974, p. 120). ^(95){ }^{95} Scherk 和 Schwarz（1974 年，第 120 页）。 ^(96){ }^{96} Scherk and Schwarz (1975, p. 463). ^(96){ }^{96} Scherk 和 Schwarz（1975 年，第 463 页）。
^(97){ }^{97} See Green and Schwarz (1981, 1982a, 1982b). ^(97){ }^{97} 见 Green 和 Schwarz (1981, 1982a, 1982b)。
^(98){ }^{98} For a full discussion of Heisenberg’s philosophical views and the subsequent formulation of his program, see Blum (2019). ^(98){ }^{98} 有关海森堡哲学观点的全面论述及其后来纲领的制定，请参阅 Blum (2019)。 ^(99){ }^{99} For a detailed account of the motivation and implementation of the maxims of logical simplicity and mathematical naturalness in Einstein’s work, see Van Dongen (2010, pp. 58-63). ^(99){ }^{99} 关于爱因斯坦著作中逻辑简洁性和数学自然性格言的动机和实施情况，详见 Van Dongen (2010, pp. 58-63)。
^(100){ }^{100} See also Cushing (1990, pp. 215-216). ^(100){ }^{100} 另见 Cushing (1990, pp. 215-216)。 ^(101){ }^{101} Gell-Mann (1985, p. 4). ^(101){ }^{101} 盖尔-曼（1985 年，第 4 页）。
^(102){ }^{102} Others who made remarks on a sense of unease about the Standard Model’s arbitrariness include Steven Weinberg, at the 1986 International Conference on High Energy Physics in Berkeley; see Galison (1995, pp. 369-370). ^(102){ }^{102} 其他对标准模型的任意性感到不安的人包括史蒂文-温伯格（Steven Weinberg），他于 1986 年在伯克利举行的国际高能物理会议上发表了讲话；见 Galison (1995, pp. 369-370)。 ^(103){ }^{103} See Rickles (2014, p. 17). ^(103){ }^{103} 见 Rickles（2014 年，第 17 页）。
