49.2 Other important mathematical physics topics

49.2 Other important mathematical physics topics

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie group · group action · U(1) · complexification

There are quite a few important mathematical topics which go beyond those discussed here, but which have significant connections to fundamental physical theories. These include:

• Higher rank simple Lie groups. The representation theory of groups like SU(3) has many applications in physics, and is also a standard topic in the graduate-level mathematics curriculum, part of the general theory of finite dimensional representations of semi-simple Lie groups and Lie algebras. This theory uses various techniques to reduce the problem to the cases of SU(2) and U(1) that we have studied. Historically, the recognition of the approximate $SU(3)$ symmetry of the strong interactions (because of the relatively light masses of the up, down and strange quarks) led to the first widespread use of more sophisticated representation theory techniques in the physics community.

• Euclidean methods. Quantum field theories, especially in the path integral formalism, are analytically best-behaved in Euclidean rather than Minkowski space-time signature. Analytic continuation methods can then be used to extract the Minkowski space behavior from the Euclidean space formulation of the theory. Such analytic continuation methods, using a complexification of the Lorentz group, can be used to understand some very general properties of relativistic quantum field theories, including the spin-statistics and CPT theorems.

• Conformal geometry and the conformal group. For theories of massless particles it is useful to study the group $SU(2,2)$ that acts on Minkowski space by conformal transformations, with the Poincar´e group as a subgroup. The complexification of this is the group $SL(4,\mathbf{C})$ . The complexification of (conformally compactified) Minkowski space turns out to be a well known mathematical object, the Grassmannian manifold of complex two dimensional subspaces of ${\mathbf{C}}^4$ . The theory of twistors exploits this sort of geometry of ${\mathbf{C}}^4$ , with spinor fields appearing in a “tautological” manner: a point of space-time is a ${\mathbf{C}}^2\subset {\mathbf{C}}^4$ , and the spinor field takes values in that ${\mathbf{C}}^2$

• Infinite dimensional groups. We have seen that infinite dimensional gauge groups play an important role in physics, but unfortunately the representation theory of such groups is poorly understood. Much is known if one takes space to be one dimensional. For periodic boundary conditions such one dimensional gauge groups are loop groups, groups of maps from the circle to a finite dimensional Lie group $G$. The Lie algebras of such groups are called afine Lie algebras and their representation theory can be studied by a combination of relatively conventional mathematical methods and quantum field theory methods, with the anomaly phenomenon playing a crucial role. The infinite dimensional group of difeomorphisms of the circle and its Lie algebra (the Virasoro algebra) also play a role in this context. From the two dimensional space-time point of view, many such theories have an infinite dimensional group action corresponding to conformal transformations of the space-time. The study of such conformal field theories is an important topic in mathematical physics, with representation theory methods a central part of that subject.

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Previous section: 49.1 Connecting quantum theories to experimental results

Appendix: conventions and notation

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原书 PDF · 印刷页 530、531、532

来源版本：2025-10-20

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