Abstract
As it is well-known, Hermann Weyl pioneered two major conceptual trends in the mathematical and physical sciences. The first was the search for a unified theory of the forces of gravity and of electromagnetism. The second, closely related to the previous, was the search for a new geometrical framework appropriate for the elucidation of such a connection. According to Weyl, the first search is essentially dependent on the second, since a new theory of physical forces must rest upon the development of a new kind of geometry capable of explaining the structure of space-time at different scales. Two philosophical ideas underlies the Weyl’s program of geometrization of physics, namely that of emergence and that of the causal power of geometrical objects (see Wheeler JA: Am Sci 74:366–375, 1986; Penrose R: Hermann Weyl’s space-time and conformal geometry. In: Hermann Weyl 1885 – 1985 centenary lectures. Springer, Berlin/Heidelberg, 1985; Boi L: Le problème mathématique de l’espace, with a foreword of R. Thom. Springer, Berlin/Heidelberg, 1995, Boi L: Synthese 139:429–489, 2004a, Boi L: Int J Math Math Sci 2004(34):1777–1836, 2004b, 2019). The first amount to say that many kinds of physical phenomena in nature emerge out from changes that can occur in the structures and dynamics of space-time itself. The second stresses the fact that geometrical concepts are involved in, rather than applied to, natural phenomena. This new geometric theory, which was first introduced by Weyl in 1918 (Weyl H: Sitzungberichte der Königlichen Preussische Akademie der Wissenschaft, Berlin 26:465–480, 1918a) and thereafter in 1928 (Weyl H: Gruppentheorie und Quantenmechanik. Hirzel, Leipzig, 1928) within the context of quantum mechanics, was grounded on the idea of gauge invariance, or a non-integrable scale factor, which in some formulations of quantum mechanics, especially in those given by Aharonov and Bohm in 1959, can be translated in a phase factor. In 1954, the physicists Yang and Mills rediscovered the Weyl’s gauge principle and developed it within a different physical context and a new mathematical framework. They proposed that the strong nuclear interaction be described by a field theory like electromagnetism, which is exactly gauge invariant. They postulated that the local gauge group was the SU(2) isotopic-spin group. This idea was revolutionary because it changed the very concept of ‘identity’ of an elementary particle. The novel idea that the isotopic spin connection, and therefore the potential, acts like the SU(2) symmetry group is the most important result of the Yang-Mills theory. This concept shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields (see Atiyah 1979, 1997).
Nowhere do mathematics, natural sciences, and philosophy permeate one another so intimately as in the problem of space.
Hermann Weyl, PMNS, 1949.
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Notes
- 1.
When the Dirac spinors carry a reducible representation of the real subalgebra, elements of the irreducible subspaces are called Majorana spinors. This occurs when the real subalgebra is a real matrix algebra, or a sum of two such algebras, and this occurs when p – q = 0, 1, 2 mod 8. In these dimensions the space of Dirac spinors can be decomposed in eigenspaces of the charge conjugation operator. Thus a Majorana spinor is an eigenspinor of the charge conjugation operation ψ = ±ψ c. This can be written in terms of the Dirac and Majorana conjugates by using ψ c = \( C{\overline{\psi}}^{\Im} \) or ψ c = \( D{\overline{\psi}}^{\Im} \). In an even number of dimensions the irreducible representations of the complex Clifford algebra induce a reducible representation of the even subalgebra; the spinor representation splitting into two inequivalent semi-spinor representations of the even subalgebra. The central idempotents that project the even subalgebra into simple components are P ± = ½(1 ± ž) where either ž = z or ž = iz ensuring z 2 = 1, z denoting the volume n-form.
- 2.
If ψ is a Dirac spinor then it may be decomposed into subspaces that transform irreducibly under the even subalgebra, ψ = ψ + + ψ −, where ψ ± = P ± ψ. The semi-spinors ψ ± are called Weyl spinors, or chiral spinors. The Weyl spinors can carry a reducible representation of the real even subalgebra. This occurs when p – q = 0 mod 8. In this case the real even subalgebra is the direct sum of two real matrix algebras, having the real central idempotents P ± = ½(1 ± z). The “Majorana condition” can be consistently imposed together with the “Weyl condition” to decompose a Dirac spinor into subspaces transforming irreducibly under the real even subalgebra. The resulting spinors are called Majorana-Weyl spinors.
- 3.
See the landmark paper of T.T. Wu and C.N. Yang “Concept of nonintegrable phase factors and global formulation of gauge fields” (1975), in which they introduced the fundamental concept of nonintegrable (i.e., path-dependent) phase factor as the basis of a description of electromagnetism. Further this concept is made to correspond to the definition of a gauge field; to extend it to global problems, they analyzed, in relation with the original Dirac’s result, the field produced by a magnetic monopole. The monopole discussion leads to the recognition that in general the phase factor (and indeed the vector potential A μ) can only be properly defined in each of many overlapping regions of space-time. In the overlap of any two regions there exists a gauge transformation relating the phase factors defined for the two regions. The concept of monopole leads to the definition of global gauge and global gauge transformations. A surprising result is that the monopole types are quite different for SU(2) and SO(3) gauge fields and for electromagnetism. The mathematics underlying these results is fiber bundle theory. Furthermore gauge fields, including in particular the electromagnetic field, are fiber bundles, and all gauge fields are thus based on geometry. So maybe all the fundamental interactions of the physical world could be based on these geometric and topological structures.
- 4.
For a good overview of this subject, see Morandi (1992). As he pointed out (in the Introduction): “Dirac’s quantization condition (1935) is the first instance in Physics of ‘topological quantization’, i.e., of a quantization of consistency with quantum mechanics, and arising entirely from topology. At about the same time, H. Hopf discovered the fibration S 1 → S 3 → S 2 (1931). That the two structures were actually intimately related became clear only more than 30 years later, when it was realized that the fiber bundle corresponding to the Hopf fibration can be endowed with a natural connection whose curvature can be identified with the field of a magnetic monopole sitting at the center of the sphere S 2. Another instance in which nontrivial topological properties of space-time appear to play a relevant role is provided by the effect discussed by Y. Aharonov and D. Bohm in 1959. Although discussed originally as scattering event, the effect can also be described in different terms by saying that the wave function of a charged particle which is adiabatically dragged around an infinitely long solenoid enclosing a flux Φ acquires an extra phase of exp[2πiΦ/Φ0], where Φ0 = hc/q. In 1975, in an influential paper, T.T. Wu and C.N. Yang stressed that the proper language to describe quantum mechanics in the presence of electromagnetic couplings is that of U(1) principal fiber bundles, and that wave functions are to be properly seen as sections of such bundles. This paved the way to the development of non-Abelian gauge field theories. In the case of Aharonov-Bohm effect, the bundle is flat, but has nontrivial holonomy, and the phase acquired by the wave function is just a manifestation of the holonomy of the bundle”.
- 5.
According to T. Regge (1992), there is no difficulty in writing the modern (gauge) form of electromagnetism (with the compact group SO(1) or U(1) on a Riemannian manifold and it is possible to write à la Cartan general relativity as a SO(3, 1) gauge theory. Besides, it may be useful to recall that Cartan was largely responsible for the introduction of the concept of torsion in Physics. Torsion remains a very interesting idea. We need to use it, even by just declaring it to vanish, if we want to write general relativity as a gauge theory in which all fields and not only the spin connection appear as gauge potentials. The interesting feature of general relativity is that the associate curvature of the vierbein, i.e. torsion, vanishes as a consequence of the variational principle of Hilbert-Einstein-Cartan. And in fact the Lagrangian density is not invariant under all gauge transformations of the Poincaré group but only under those of the Lorentz subgroup. Although nature has prepared the gauge potential for the full group it end up by requiring invariance under a subgroup only. A world with torsion would appear inescapable if we have around enough density of high spin particles which acts as sources, but this density seems at the moment well below the limit of observability. Regards the kind of space in which torsion is supposed to appear, one can remark that it would not be any more a Riemannian manifold or, rather, none of the Riemannian structures existing on the manifold would be directly related to Physics and the theory would not be a geometrical theory in the sense envisaged by Einstein. One could yet consider general relativity as GL(4, R) theory with the Christoffel connection playing the role of a Yang-Mills potential. If the torsion vanishes it follows that the Christoffel symbol is symmetrical into the 2 lower indices whose role is however quite different. The first index is a GL(4, R) gauge index; the second labels instead the differentials on spacetime. We may relate them because of the accidental and marvellous fact that the Jacobian group of derivatives on a differentiable manifold is isomorphic to GL(4, R) and that we use the same indexing for differentials and vectors in GL(4, R). Once the symmetry is established the theory becomes almost by definition geometrical. If there is no symmetry but we can control torsion by introducing suitable norms and bounds then we may still speak of an almost geometrical theory whose exact mathematical definition is still lacking.
- 6.
The exact formulation of the concept of a nonintegrable phase factor depends on the definition of global gauge transformations, i.e., on the choice of the overlapping regions of R (where R is a region of space-time, precisely, all space-time minus the origin r = 0) and of the potential A μ in this region. Through a certain kind of operations, called distortions, one arrives at a large number of possibilities, each with a particular choice of overlapping regions and with a particular choice of gauge transformation from the original (A μ)a or (A μ)b to the new A μ in each region. Each of such possibilities will be called a gauge (or global gauge). This definition is a natural generalization of the usual concept, extended to deal with the intricacies of the field of a magnetic monopole. Notice that the gauge transformation factor in the overlap between R α and R β does not refer to any specific A μ. The gauge transformation in the overlap of the two regions is:
$$ (1)\kern1em S={S}_{\alpha \beta}=\exp \left(\hbox{--} i\alpha \right)=\exp \kern0.5em \left(2 ige/ hc\right)\upphi \Big). $$Thus two different gauges may share the same characterizations (a) and (b). In the case of the monopole field, one can attach to the gauge any (A μ)a and (A μ)b provided they are gauge-transformed into each other in the region of overlap. Thus a gauge is a concept not tied to any specific vector potential. Wu and Yang called the process of distortion leading from one gauge to another a global gauge transformation. It is also a concept not tied to any specific vector potential. The collection of gauges that can be globally gauge-transformed into each other will be said to belong to the same gauge type. The phase factor exp (ie/hc ∫ A μ dx μ) (which is nonintegrable, i.e., path-dependent) around a loop starts and ends at the same point in the same region. Thus it does not change under any global transformation, so that we have the, for Abelian gauge fields, the following
-
Theorem 1: The phase factor around any loop is invariant under a global gauge transformation.
-
It follows trivially from this, by taking an infinitesimal loop, that
-
Theorem 2: The field strength ƒ μν is invariant under a global gauge transformation.
-
And
-
Theorem 3: Between two gauge fields defined on the same gauge there exists a continuous interpolating gauge field defined on the same gauge.
-
Theorem 4: Consider gauge G D and define any gauge field on it. The total magnetic flux through a sphere around the origin r = 0 is independent of the gauge field and only depends on the gauge:
$$ (2)\kern1em \int \int {f}_{\upmu \upnu}{dx}^{\upmu}{dx}^{\upnu}=\left(\hbox{--} ihc/e\right)\int \partial /\partial {x}^{\upmu}\left(\ln {S}_{\upalpha \upbeta}\right){dx}^{\upmu}, $$where S is the gauge transformation defined by (1) for the gauge G D in question, and the integral is taken around any loop around the origin r = 0 in the overlap between R α and R β , such as the equation on a sphere r = 1.
As in the case of electromagnetism, in the non-Abelian gauge fields both the concept of a gauge and the concept of a global gauge transformation are not tied to any specific gauge potentials. The nonintegrable phase factor for a given path is now an element of the gauge group. Since these phase factor do not in general commute with each other, Theorems 1 and 2 for the Abelian case need to be modified as follows.
-
Theorem 5: Under a global gauge transformation, the phase factor around any loop remains in the same class. The class does not depend on which point is taken as the starting point around the loop.
-
Theorem 6: The field strength ƒ k μν is covariant under a global gauge transformation.
-
Theorem 5: defines the class of a loop. This concept is a generalization of the phase factor for electromagnetism around a loop with the magnetic flux as the exponent. It is a gauge-invariant concept.
-
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Boi, L. (2019). H. Weyl’s Deep Insights intotheMathematicalandPhysicalWorlds: His Important Contribution to the Philosophy of Space, Time and Matter. In: Bernard, J., Lobo, C. (eds) Weyl and the Problem of Space. Studies in History and Philosophy of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-11527-2_9
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