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From S S SS-matrix theory to strings: scattering data and the commitment to non-arbitrariness
从 S S 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 ^(1){ }^{1}
罗伯特-范-利乌文 1 1 ^(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 S S 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 S S SS-matrix physics, aiming to clarify how modern string theory, as a theory detached from experimental data, grew out of an S S 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 S S 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 S S 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 S S 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 ^(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 ^(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 ^(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 ^(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 S S SS-matrix program for the strong interactions which was strongly dependent upon experimental results: the S S 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 ^(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 S S SS 强相互作用矩阵计划，该计划强烈依赖于实验结果： S S 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 ^(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 S S 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 S S 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 S S 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 S S 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 S S 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 S S SS-matrix program stuck to string theory”, Rickles hastens to make clear that string theory “soon became a very different structure”. 6 6 ^(6){ }^{6} In his view,
然而，在指出从 S S 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" S S 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 ^(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 S S 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 S S 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 S S 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 S S 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 S S 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 S S 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 ^(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 ^(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 ^(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 ^(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 ^(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 S S 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 ^(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 ^(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 ^(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 S S 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：）
  1. Analytic S S SS-matrix theory for hadrons (Chapter 2). Hadronic S S 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 S S SS-matrix program for quantum electrodynamics from the mid-1940s. The main aim of hadronic S S SS-matrix theory was to compute observable scattering amplitudes on the basis of a set of S S SS-matrix principles, while treating the dynamics of the scattering process as a black box. The virtue of non-arbitrariness was influential in hadronic S S SS-matrix theory through the notion of the “bootstrap” as advocated by Geoffrey Chew: the conjecture that imposing all S S 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 S S SS-matrix theory was developed while experimentally obtained values were used as input.
    强子的解析 S S SS 矩阵理论（第 2 章）。强子 S S 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 S S SS 矩阵计划。强子 S S SS 矩阵理论的主要目的是在一套 S S 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 S S SS 矩阵理论中产生了影响：将所有 S S 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， S S 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 S S SS-matrix theory. In S S 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 S S 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 S S SS 矩阵理论发展而来的一类模型。在 S S 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 S S 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 S S SS-matrix theory. The presupposition underlying the S S SS-matrix approach is that only mathematical relations between observable quantities (namely scattering amplitudes) are allowed. 11 11 ^(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 S S 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 S S 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 S S SS-matrix theorists were constructive empiricists in Van Fraassen’s sense. The reason I nevertheless emphasize the “empiricism” of S S 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 S S SS 矩阵理论展开。 S S 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 ^(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 S S 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 S S 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 S S 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 S S 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 S S 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 S S SS-matrix theory
2.解析 S S 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 ^(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 ^(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 S S SS-matrix theory. The aim of S S 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 S S SS-matrix approach to elementary particle theory originated with Werner Heisenberg, who proposed an S S 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 S S 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 S S SS 矩阵理论的研究计划的形成。 S S 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 S S 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 S S 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 S S SS 矩阵理论的研究，从而也间接地影响了弦理论。

2.1 Heisenberg's S S SS-matrix program
2.1 海森堡的 S S SS 矩阵程序

Heisenberg’s original S S 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 S S SS is determined, out of which the transition probabilities and cross sections can be calculated directly. Starting from the S S 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 ^(13){ }^{13}
海森堡最初提出 S S 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 S S SS 即可确定，由此可直接计算出转变概率和截面。从 S S 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 ^(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 ^(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 ^(14){ }^{14}
Almost twenty years after his successful formulation of matrix mechanics, Heisenberg returned to this empiricist approach for his S S 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 S S SS-matrix and used it to relate the total cross section σ T σ T sigma_(T)\sigma_{T} to the imaginary part of the forward scattering amplitude of elastic interactions. In addition, Heisenberg proposed to consider the S S 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 S S 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 S S 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 S S 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 S S SS-matrix. 15 15 ^(15){ }^{15}
在成功提出矩阵力学近二十年后，海森堡在其 S S 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 S S SS 矩阵的单位性，并用它将总截面 σ T σ T 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 S S 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 S S 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 S S 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 S S SS 矩阵系统：如果不引入哈密顿，就没有规则来指导 S S SS 矩阵的构造。 15 15 ^(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 S S SS-matrix program waned. 16 16 ^(16){ }^{16} Nevertheless, Heisenberg’s program strongly influenced postwar particle physics. Firstly, it firmly established the S S 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 S S 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 S S SS 矩阵计划的工作逐渐减弱。 16 16 ^(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 S S 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 S S SS 矩阵理论。
This S S 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 S S SS-matrix theory, identifies a repetition of events: after the success of renormalized QED, interest in Heisenberg’s original S S 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 S S SS-matrix program. With the success of gauge theory and the Standard Model in the 1970s, interest in S S SS-matrix theory as an independent program waned again. 17 17 ^(17){ }^{17} Yet, through dual resonance models, strong interaction S S SS-matrix theory would eventually lead to string theory.
这种强相互作用的 S S 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 S S 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 S S 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 S S SS 矩阵计划。随着70年代规理论和标准模型的成功，人们对 S S SS 矩阵理论作为一个独立项目的兴趣再次减弱。 17 17 ^(17){ }^{17} 然而，通过双共振模型，强相互作用 S S SS -矩阵理论最终导致了弦理论。

2.2 Analytic S S SS-matrix theory for the strong interactions
2.2 强相互作用的解析 S S SS 矩阵理论

Heisenberg’s original S S SS-matrix theory and the strong interaction S S 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 S S SS 矩阵理论和强相互作用 S S 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 ( ω ) f ( ω ) f(omega)f(\omega) is treated as a complex analytic function of the energy variable ω ω omega\omega, i.e., as a function that could be extended to the complex plane. 18 18 ^(18){ }^{18} When looking at the special case of θ = 0 θ = 0 theta=0\theta=0, called “forward scattering”, one can relate the real and imaginary part of f ( ω ) f ( ω ) 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 ( ω ) f ( ω ) f(omega)f(\omega) 被视为能量变量 ω ω omega\omega 的复解析函数，即可以扩展到复平面的函数。 18 18 ^(18){ }^{18} 在研究 θ = 0 θ = 0 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 ( ω ) f ( ω ) f(omega)f(\omega) 的实部和虚部联系起来：
Re [ f ( ω ) ] = 1 π P ∫ − ∞ + ∞ d ω ′ Im [ f ( ω ′ ) ] ω ′ − ω Re [ f ( ω ) ] = 1 π P ∫ − ∞ + ∞   d ω ′ Im f ω ′ ω ′ − ω 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 P P 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 ( ω ) f ( ω ) 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 / c l / c l//cl / c to reach a point at distance l l 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 S S SS-matrix, and if it could be extended to scattering processes where particles were created and annihilated. 19 19 ^(19){ }^{19}
P P 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 ( ω ) f ( ω ) 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 / c l / c l//cl / c 的时间才能到达距离 l l 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 S S 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 ^(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 S S SS-matrix must be unitary: S S † = 1 S S † = 1 SS^(†)=1S S^{\dagger}=1, where S † S † 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 S S SS 矩阵必须是单一的： S S † = 1 S S † = 1 SS^(†)=1S S^{\dagger}=1 ，其中 S † S † 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：
Im [ f ] ∝ σ T Im [ f ] ∝ σ T Im[f]propsigma_(T)\operatorname{Im}[f] \propto \sigma_{T}
The optical theorem relates the imaginary part of a forward scattering amplitude to the total cross section. 20 20 ^(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 ^(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){ }^{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 ^(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 S S SS-matrix program. They justified their approach on the basis of the statement of microcausality. This means that if two events x x xx and y y yy are spacelike separated, the corresponding local field operators ϕ ( x ) ϕ ( x ) phi(x)\phi(x) and ϕ ( y ) ϕ ( y ) 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 S S 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 x x xx 和 y y yy 在空间上相距甚远，则相应的局部场算子 ϕ ( x ) ϕ ( x ) phi(x)\phi(x) 和 ϕ ( y ) ϕ ( y ) 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 ^(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 ^(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 ] Im [ f ] Im[f]\operatorname{Im}[f] via the optical theorem. From there, the expression was inserted in the Kramers-Kronig integral for Re [ f ] Re [ f ] 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 ^(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 ^(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 ] Im [ f ] Im[f]\operatorname{Im}[f] 联系起来。在此基础上，将 Re [ f ] Re [ f ] 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 ^(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 ^(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 S S SS-matrix theory for the strong interactions was formed. With analyticity (via causality) and unitarity, two of the principles of S S SS-matrix theory have already been discussed; Lorentz invariance was another. The other key ideas underlying S S SS-matrix theory, as developed in the late 1950s, were 25 25 ^(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 S S SS 矩阵理论。前面已经讨论了 S S 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 S S SS 矩阵理论的其他关键思想是 25 25 ^(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 ∝ g 2 t − M π 2 A ∝ g 2 t − M π 2 A prop(g^(2))/(t-M_(pi)^(2))A \propto \frac{g^{2}}{t-M_{\pi}^{2}}
with t t tt the change in momentum from one of the incoming nucleons before and after emitting the pion, M π M π M_(pi)M_{\pi} the mass of the exchanged pion, and g 2 g 2 g^(2)g^{2} the strong interaction pionnucleon coupling constant. This term becomes infinite at the physical mass of the pion, when t = M π 2 t = M π 2 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 π 2 t = M π 2 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 = t=t= M π 2 M π 2 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， t t tt 是一个传入核子在发射先驱前后的动量变化， M π M π M_(pi)M_{\pi} 是交换先驱的质量， g 2 g 2 g^(2)g^{2} 是强相互作用先驱-核子耦合常数。当 t = M π 2 t = M π 2 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 π 2 t = M π 2 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 = t=t= M π 2 M π 2 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 S S SS-matrix program.
与特定交换的耦合常数。这一猜想在 S S 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 / 2 1 / 2 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 / 2 1 / 2 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 → c + d II. b + d ¯ → a ¯ + c III. b + c ¯ → a ¯ + d  I.       a + b → c + d  II.       b + d ¯ → a ¯ + c  III.       b + c ¯ → a ¯ + d {:[" 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 p_(a)p_{a} and p b p b p_(b)p_{b} and outgoing fourmomenta − p c − p c -p_(c)-p_{c} and − p d − p d -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 p_(a)p_{a} 和 p b p b p_(b)p_{b} 以及传出四动量 − p c − p c -p_(c)-p_{c} 和 − p d − p d -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：
s = ( p a + p b ) 2 = ( p c + p d ) 2 t = ( p a − p c ) 2 = ( p b − p d ) 2 u = ( p a − p d ) 2 = ( p b − p c ) 2 s = p a + p b 2 = p c + p d 2 t = p a − p c 2 = p b − p d 2 u = p a − p d 2 = p b − p c 2 {:[s=(p_(a)+p_(b))^(2)=(p_(c)+p_(d))^(2)],[t=(p_(a)-p_(c))^(2)=(p_(b)-p_(d))^(2)],[u=(p_(a)-p_(d))^(2)=(p_(b)-p_(c))^(2)]:}\begin{aligned} & s=\left(p_{a}+p_{b}\right)^{2}=\left(p_{c}+p_{d}\right)^{2} \\ & t=\left(p_{a}-p_{c}\right)^{2}=\left(p_{b}-p_{d}\right)^{2} \\ & u=\left(p_{a}-p_{d}\right)^{2}=\left(p_{b}-p_{c}\right)^{2} \end{aligned}
with units ℏ = c = 1 ℏ = c = 1 ℏ=c=1\hbar=c=1. In the center-of-mass frame, s s ss is also known as the total energy squared, and t t tt as the momentum transfer squared; the variables s , t s , t s,ts, t, and u u uu were quickly named “Mandelstam variables”. Using them one can write a full scattering amplitude A ( s , t , u ) A ( s , t , u ) 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 , t s , t s,ts, t, or u u 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 ℏ = c = 1 ℏ=c=1\hbar=c=1 。在质量中心框中， s s ss 也称为总能量平方， t t tt 称为动量传递平方；变量 s , t s , t s,ts, t 和 u u 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 ) 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 , t s , t s,ts, t 或 u u 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 J J JJ. A singularity that arises when treating J J 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 S S 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 J J JJ 。将 J J 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 S S 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 S S 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 S S SS-matrix program was still operating in Heisenberg’s empiricist spirit: observable S S SS-matrix elements were calculated without relying on a Hamiltonian or Lagrangian.
总之，在 20 世纪 60 年代初，解析 S S 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 S S SS 矩阵程序仍秉承海森堡的经验主义精神：可观测的 S S SS 矩阵元素的计算不依赖于哈密顿或拉格朗日。

2.3 The S S SS-matrix bootstrap in the 1960s
2.3 20 世纪 60 年代的 S S SS 矩阵自举法

In order to be able to properly understand how string theory grew out of S S SS-matrix physics, the conjecture denoted as the " S S 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 S S 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 ^(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 ^(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 S S 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 ^(28){ }^{28}
为了能够正确理解弦理论是如何从 S S SS 矩阵物理学发展而来的，被称为" S S 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 S S 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 ^(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 ^(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 S S 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 ^(28){ }^{28}
The bootstrap conjecture became an influential notion in hadronic S S 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 S S SS-matrix framework. This proposal was based on the hypothesis that the S S SS-matrix postulates could lead to a “complete and self-consistent theory of strong interactions”. 29 29 ^(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 S S 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 S S 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 S S SS 矩阵框架。这一提议基于这样一个假设： S S SS 矩阵假设可以导致 "完整和自洽的强相互作用理论"。 29 29 ^(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 S S 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 ρ ρ rho\rho meson, yielding an attractive force. When read from left to right, the diagram denotes the colliding of pions, forming an intermediate ρ ρ rho\rho meson bound state. Through a bootstrap calculation m ρ m ρ m_(rho)m_{\rho} and g ρ π π g ρ π π g_(rho pi pi)g_{\rho \pi \pi} of the bound state ρ ρ rho\rho meson was calculated self-consistently out of the exchange force m ρ m ρ m_(rho)m_{\rho} and g ρ π π g ρ π π 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 ρ ρ 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 ρ ρ rho\rho 介子束缚态。通过自举计算，结合态 ρ ρ rho\rho 介子的 m ρ m ρ m_(rho)m_{\rho} 和 g ρ π π g ρ π π g_(rho pi pi)g_{\rho \pi \pi} 从交换力 m ρ m ρ m_(rho)m_{\rho} 和 g ρ π π g ρ π π 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 ^(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 ^(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 ^(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 ^(31){ }^{31}
Both the bootstrap and the associated notion of nuclear democracy were central ideas around which the hadronic S S SS-matrix program of Chew and collaborators was developed in the 1960 s. 32 32 ^(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 S S 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 S S SS-matrix equations for all particle interactions (i.e., a complete bootstrap) was practically out of reach. For starters, the S S 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 [ S S 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 ^(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 S S 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 S S 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[ S S 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 ^(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 ρ ρ rho\rho meson; the exchanged meson yields an attractive force depending on the mass m ρ m ρ m_(rho)m_{\rho} and coupling constant γ ρ π π γ ρ π π 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 ρ ρ 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 ρ m ρ m_(rho)m_{\rho} 和耦合常数 γ ρ π π γ ρ π π 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 ρ ρ 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 ρ ρ rho\rho meson, obtaining “two relations between m ρ m ρ m_(rho)m_{\rho} and γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} from which both may be determined”. 34 34 ^(34){ }^{34} The ρ ρ 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 ^(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 ρ ρ 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 ρ ρ rho\rho 介子的质量和耦合常数，从而得到" m ρ m ρ m_(rho)m_{\rho} 和 γ ρ π π γ ρ π π 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 ρ m ρ m_(rho)m_{\rho} 和 γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} "。 34 34 ^(34){ }^{34} ρ ρ 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 ^(35){ }^{35}
This bootstrap calculation can thus (in principle) be designated as “self-consistent” because, when confronted with two equations relating m ρ m ρ m_(rho)m_{\rho} and γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi}, one can use the output of the first equation (e.g., m ρ m ρ m_(rho)\mathrm{m}_{\rho} and γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} of the bound state ρ ρ rho\rho meson) as input for the second equation to calculate the values for the exchanged ρ ρ 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 ρ m ρ m_(rho)m_{\rho} and γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} are uniquely determined by the two equations. However, in S S 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 ρ ρ rho\rho meson were used as input (and then shown to coincide with the output of the calculation taking the bound state ρ ρ 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 S S 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 ρ m ρ m_(rho)m_{\rho} 和 γ ρ π π γ ρ π π 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 ρ ρ rho\rho 介子的 m ρ m ρ m_(rho)\mathrm{m}_{\rho} 和 γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} ）作为第二个方程的输入，计算出交换态 ρ ρ rho\rho 介子的值、 ρ ρ rho\rho介子的 m ρ m ρ m_(rho)\mathrm{m}_{\rho}和 γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi}）作为第二个方程的输入，计算出交换的 ρ ρ 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 ρ m ρ m_(rho)m_{\rho} 和 γ ρ π π γ ρ π π gamma_(rho pi pi)\gamma_{\rho \pi \pi} 是由这两个方程唯一决定的。然而，在 S S 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 ρ ρ 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 ρ ρ 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 S S SS 矩阵计算的输入，而不是由它们决定的。
So, the conjecture driving the S S SS