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Hermann Weyl, the Gauge Principle, and Symbolic Construction from the “Purely Infinitesimal”

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One Hundred Years of Gauge Theory

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 199))

Abstract

The modern gauge principle stipulates that every global symmetry of a quantum field theory be replaced by a local one. It has an unusual “context of discovery”: imposed in 1918 on philosophical grounds by Hermann Weyl, it led to a purely formal unification of Einstein’s gravitational theory and electromagnetism. This achievement prompted Weyl’s purely mathematical turn in 1925–6 to Lie theory and of course Lie groups and Lie algebras play prominent roles in the subsequent development of the gauge principle leading up to the Standard Model. I show that the gauge principle as well as Weyl’s predominant interest in Lie theory stem from two complementary philosophical interests: (i) phenomenological evidential requirements imposed on differential geometric construction, and (ii) the epistemological command of “Nahewirkungphysik”: to comprehend the world from its behavior in the infinitely small. (i) and (ii) productively meet in Weyl’s notion of “symbolic construction”. The gauge principle’s “radical locality” is the basis of Weyl’s constitution of objectivity as an intersubjectivity requiring local degrees of freedom. This leads to a necessary redundancy of physical description, a philosophical puzzle that might be elucidated by revisiting the philosophical underpinnings of the gauge principle.

Only in the infinitely small may we expect to encounter elementary uniform laws; hence the world must be comprehended through its behavior in the infinitely small.

(Weyl [32, 61, 44, 86])

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Notes

  1. 1.

    On the principle of continuity, see Leibniz’s letter to Varignon, 1702: “Assurément je pense que ce Principe [“mon Principe de Continuité”] est general, et qu’il tient bon, non seulement dans la Géometrie, mais encore dans la Physique. La Géometrie n’étant que la science des limites et de la grandeur du Continu, il n’est point étonnant, que cette loi s’y observe par-tout: car d’où viendroit une subite interruption dans un subject, qui n’en admet pas en vertu de sa nature? Aussi savons-nous bien, que tout est parfaitement lie dans cette science, et qu’on ne sauroit alléguer un seul exemple, qu’une propriété quelconque y cesse subitement, ou naisse de même, sans qu’on puisse assigner le passage intermédiaire de l’une à l’autre, les points d’inflexion et de rebroussement, qui rendent le changement explicable; de manière qu’une Equation Algébraique, qui représente exactement un état, en représente virtuellement tous les autres, qui peuvent convenir au même sujet. L’universalité de ce Principe dans la Géometrie m’a bientôt fait connoitre, qu’il ne sauroit manquer d’avoir lieu aussi dans le Physique: puisque je vois, que, pour qu’il y ait de la règle et d’ordre dans la Nature, it est necessaire, que la Physique harmonie constamment avec le Géométrique; …” “Brief von Leibniz an Varignon über das Kontinuiätsprinzip”, in E. Cassirer (ed.), G. W. Leibniz: Hauptschriften zur Grundlegung der Philosophie, Bd. II. Leipzig: Verlag der Dürr’schen Buchhandlung, 1906, 556.

  2. 2.

    Weyl [29, 9]: “Es beruht ja die Leistungsfähigkeit des in der Differentialrechnung, der Nahewirkungsphysik und der Riemannschen Geometrie zum Durchbruch kommenden Prinzips: die Welt nach Form und Inhalt aus ihrem Verhalten im Unendlichkleinen zu verstehen, eben darauf, dass alle Problem durch den Rückgang aufs Unendlichkleine linearisiert werden.” See also Weyl [24, 82, 32, 61, 44].

  3. 3.

    “Vorwort des Herausgebers”, in Weyl (ed.) [26].

  4. 4.

    A crucial feature of Einstein’s theory of gravitation is that it allows (pseudo-) Riemannian geometry (“pseudo” since time is treated differently than the three space dimensions) to be the appropriate mathematical framework for the concept of “local inertial frame” and so to uphold the “infinitesimal” validity of special relativity in that theory.

  5. 5.

    Weyl [29, 34]: “Die Ersetzung der endlichen Gruppe durch die infinitesimale-- das ist wieder der ‘Rückgang aufs Unendlichkleine’! - ist einer der Hauptgedanken der Lieschen Theorie.” Original emphasis. Hawkins [7] quotes from an 1879 paper of Lie, “In the course of investigations on first-order partial differential equations, I observed that the formulas that occur in this discipline become amenable to a remarkable conceptual interpretation by means of the concept of an infinitesimal transformation. In particular, the so- called Poisson-Jacobi theorem is closely connected with the composition of infinitesimal transformations. By following up on this observation I arrived at the surprising result that all transformation groups of a simply extended manifold can be reduced to the linear form by a suitable choice of variables, and also that the determination of all groups of an n-fold extended manifold can be achieved by the integration of ordinary differential equations. This discovery … became the starting point of my many years of research on transformation groups.”

  6. 6.

    Husserl, e.g. [12, 141], is careful to distinguish the usual (and “countersensical”) philosophical notion of evidence as the absolute criterion of truth from evidence as “that performance on the part of intentionality which consists in the giving of something-itself [die intentionale Leistung der Selbstgebung]. More precisely, it is the universal pre-eminent form of ‘intentionality’, of ‘consciousness of something’, in which there is consciousness of the intended-to objective affair in the mode itself-seized-upon, itself-seen—correlatively, in the mode: being with it itself in the manner peculiar to consciousness.

  7. 7.

    Husserl, ca. [10, 382]: “My transcendental method is transcendental-phenomenological. It is the ultimate fulfillment of old intentions, especially those of English empiricist philosophy, to investigate the transcendental-phenomenological ‘origins’. the origins of objectivity in transcendental subjectivity, the origin of the relative being of objects in the absolute being of consciousness.” Transcendental sense constitution of objective nature is founded on one of empathy; e.g., Husserl [13, 92]: “a transcendental theory of experience of the other (Fremderfahrung), the so-called empathy (Einfühlung)” has within its scope “the founding of a transcendental theory of the objective world … in particular, of objective Nature to whose existence sense (Seinsinn) belongs there-for-everyone (Für- jedermann-da).”

  8. 8.

    Weyl [38, 49]; compare this passage of Husserl [11, 121], “The existence of a Nature cannot be the condition for the existence of consciousness, since Nature itself turns out to be a correlate of consciousness: Nature is only as being constituted in regular concatenations of consciousness.”

  9. 9.

    Famously, Gödel in 1931 showed that any such consistency result could only be relative, since a consistency proof could only be carried out in a stronger theory.

  10. 10.

    Hilbert [9]: “The role that remains to the infinite is … merely that of an idea—if, in accordance with Kant’s words, we understand by an idea a concept of reason that transcends all experience and through which the concrete is completed so as to form a totality”. Weyl commented [36, 28]: “Hilbert himself says somewhat obscurely that infinity plays the role of an idea in the Kantian sense, by which the concrete is completed in the sense of totality. I understand this to mean something like the way in which I complete what is given to me as the actual content of my consciousness, into the totality of the objective world, which certain includes much that is not present to me. The scientific formulation of this objective concept of the world occurs in physics, which avails itself of mathematics as a means of construction. However, the situation we find before us in theoretical physics in no way corresponds to Brouwer’s idea of a science. That ideal postulates that every judgment has its own meaning achievable in intuition. The statements and laws of physics, nevertheless, taken one by one, have no content verifiable in experience; only the theoretical system as a whole allows itself to be confronted by experience. What is accomplished here is not the intuitive insight into singular or general contents and a description that truly renders what is given, but instead a theoretical, and ultimately purely symbolic, construction of the world.”

  11. 11.

    Weyl [33, 147–8]: “[The] individual assumptions and laws [of theoretical physics] have no separate fulfilling sense [that is] immediately realized in intuition (in der Anschauung unmittelbar zu erfüllender Sinn eigen); in principle, it is not the propositions of physics taken in isolation, but only the theoretical system as a whole that can be confronted with experience. What is achieved here is not intuitive insight [anschauende Einsicht] into particular or general states of affairs and a faithfully reproduced description of the given (das Gegebene), but rather theoretical, ultimately a purely symbolic, construction of the world.” Weyl goes on to state that if Hilbert’s view prevails over Brouwer’s, as indeed appears to be the case, then this represents “a decisive defeat of the philosophical attitude of pure phenomenology, as it proves insufficient to understand creative science in the one domain of knowledge that is most rudimentary and earliest open to evidence, mathematics.”

  12. 12.

    Weyl [31, 540], also [46, 645]. The latter is a lecture entitled “Erkenntnis und Besinnung (Ein Lebensrückblick)”. Besinnung, here translated ‘reflection’, is a technical term in Husserlian phenomenology, having the meaning of “sense-investigation”; e.g., Husserl [12, 8]: “Sense-investigation (Besinnung)… radically understood, is originary sense-explication (ursprüngliche Sinnesauslegung, orig. emphasis), transforming and above all striving to transform sense in the mode of unclear opinion into sense in the mode of full clarity or essential possibility (Wesensmöglichkeit).”

  13. 13.

    Weyl [40, 7]: “Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means.”

  14. 14.

    The influence of Husserl is also apparent in Weyl’s use of the term “natural attitude”.

  15. 15.

    Weyl [40, 82]. For example, the central underlying theoretical device of quantum mechanics, densities of a complex valued, infinite-dimensional wave function, can only be symbolically represented. Dirac, influenced by Weyl [34] offers the same philosophical message regarding the necessity of symbolic methods in the first sections of his well-known textbook on quantum mechanics.

  16. 16.

    Weyl [45, 627]: “… the words ‘in reality’ must be put between quotation marks; who could seriously pretend that the symbolic construct is the true real world?” The term ‘symbolic construct’ encompasses not merely the symbolic universe in which physical systems, states, transformations and evolutions are mathematically defined in terms of manifolds, functional spaces, algebras, etc., but also a symbolic specification of idealized procedures and experiments by which the basic physical quantities or observables of the theory are related to observation and measurement. It reflects an insistence, reinforced by quantum mechanics, that physical quantities (beginning with ‘inertial mass’) are not simply given, but “constructed” [37, 76, 41, 109ff].

  17. 17.

    In fact, general covariance is a purely formal requirement having nothing to do with a generalized principle of relativity.

  18. 18.

    Internal symmetries refer to the fact that particles occur in multiplets, members of which can be considered as “the same” under the symmetry of the interaction. Mathematically, the multiplets are realizations of an irreducible representation of some internal symmetry group. See further below.

  19. 19.

    Weyl [35, 246]. Cf. Weyl [37, 220]: “One can in fact take it as a general rule that an invariance property of the kind met in general relativity, involving an arbitrary function, gives rise to a differential conservation theory. In particular, gauge invariance is only to be understood from this standpoint.”

  20. 20.

    For a clear account of mathematical details in more modern terms, see Darrigol [3] and Scholz [21].

  21. 21.

    See [5, 7]. Cartan’s “structure theory” for infinitesimal Lie groups (today, Lie algebras) identifies isomorphic groups through their “structure constants”. Cartan worked exclusively at the level of abstract groups; Weyl would translate Cartan’s structure theory into the language of matrix groups, group representations by matrices, the language of most interest to physics.

  22. 22.

    Weyl [42, 4]. In modern terms, a Lie algebra to a Lie group G is usually denoted by the Gothic character g and is defined by three properties: (1) the elements X, Y, etc. of g form a linear vector space; (2) the elements of g close under a commutation relation [X,Y] = −[Y,X], ∀ X,Y ∈ g; (3) the Jacobi identity [X,[Y,Z]] +[Y,[Z,X]] + [Z,[X,Y]] = 0 is satisfied. Using Cartan’s “structure theory”, the structure of a Lie algebra is completely determined by its “structure constants” c kij that appear in the commutator of any two basis vectors [Xi, Xj] = c kij Xk.

  23. 23.

    Weyl [27, 329]: “auf diesem Felde mathematische Einfachheit und metaphysische Ursprünglichkeit in enger Verbindung miteinander stehen.”

  24. 24.

    Weyl [30]. A (non-abelian) Lie group whose Lie algebra cannot be factorized into two commuting subalgebras is called simple. A direct product of simple Lie groups is called semi-simple. To be sure, in these papers Weyl supplemented Cartan’s infinitesimal abstract group viewpoint with global and topological properties of Lie groups, thus in our view, “comprehending the world beginning from the infinitesimal”. Indeed, linearizing a Lie group G in the tangent space of the identity to form its Lie algebra g destroys G’s global properties, i.e., what happens far from the identity. Hence the need for integral and topological methods.

  25. 25.

    The second step is to determine the representations of fermions and scalars under the gauge symmetry; a third step is to postulate the pattern of spontaneous symmetry breaking.

  26. 26.

    A symmetry of a system is said to be “spontaneously broken” if its lowest energy state is not invariant under the operations of that symmetry. This is an extremely important concept in the weak interaction as the bosons introduced by gauge symmetries are massless, like the photon; their masses arise from the “spontaneous breaking” of the SU(2) x U(1) symmetry through couplings to the scalar Higgs field.

  27. 27.

    SSB plays two roles in the SM, giving mass to gauge bosons (other than the photon) and giving mass to fermions (leptons and quarks). In the electroweak theory (unifying the weak and electromagnetic interactions), SSB plays a crucial part, breaking the electroweak SU (2) X U (1) symmetry into individual electromagnetic and weak forces while enabling mass (the SU (2) multiplets) to emerge spontaneously in theories where initially, there is no mass. A standard story is that SSB invokes the Higgs mechanism to spontaneously break a local gauge symmetry; of course, this a symmetry connecting states that cannot be physically distinguished. In fact, in SSB it is not the local gauge transformations that are spontaneously broken. Rather it is a global symmetry (a unique vacuum state) that is spontaneously broken while the gauge symmetry is explicitly broken by gauge fixing in the Higgs mechanism in order to extract physical predictions. Elitzur [6] has showed the spontaneous breaking of a local symmetry is logically impossible.

  28. 28.

    Weyl [25, 72]: “The coordinate system is the unavoidable residue of the ego’s annihilation (das unvermeidliche Residuum der lch- Vernichtung) in that geometrico-physical world that reason sifts from the given under the norm of ‘objectivity’ - a final faint token in this objective sphere that existence (Dasein) is only given, and can only be given as the intentional content of the conscious experience of a pure, sense-giving ego.”

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Ryckman, T. (2020). Hermann Weyl, the Gauge Principle, and Symbolic Construction from the “Purely Infinitesimal”. In: De Bianchi, S., Kiefer, C. (eds) One Hundred Years of Gauge Theory. Fundamental Theories of Physics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-030-51197-5_7

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