39 Quantization of Infinite dimensional Phase Spaces
While finite dimensional Lie groups and their representations are rather wellunderstood mathematical objects, this is not at all true for infinite dimensional Lie groups, where only a fragmentary such understanding is available. In earlier chapters we have studied in detail what happens when quantizing a finite dimensional phase space, bosonic or fermionic. In these cases a finite dimensional symplectic or orthogonal group acts and quantization uses a representation of these groups. For the case of quantum field theories with their infinite dimensional phase spaces, the symplectic or orthogonal groups acting on these spaces will be infinite dimensional. In this chapter we’ll consider some of the new phenomena that arise when one looks for infinite dimensional analogs of the role these groups and their representations play in quantum theory in the finite dimensional case.
The most important diference in the infinite dimensional case is that the Stone-von Neumann theorem and its analog for Cliford algebras no longer hold. One no longer has a unique (up to unitary equivalence) representation of the canonical commutation (or anticommutation) relations. It turns out that only for a restricted sort of infinite dimensional symplectic or orthogonal group does one recover the Stone-von Neumann uniqueness of the finite dimensional case, and even then new phenomena appear. The arbitrary constants found in the definition of the moment map now cannot be ignored, but may appear in commutation relations, leading to something called an “anomaly”.
Physically, new phenomena due to an infinite number of degrees of freedom can have their origin in the degrees of freedom occurring at arbitrarily short distances (“ultraviolet divergences”), but also can be due to degrees of freedom corresponding to large distances. In the application of quantum field theories to the study of condensed matter systems it is the second of these that is relevant, since the atomic scale provides a cutof distance scale below which there are no degrees of freedom.
For general interacting quantum field theories, one must choose among inequivalent possibilities for representations of the canonical commutation relations, finding one on which the operators of the interacting field theory are well-defined. This makes interacting quantum field theory a much more complex subject than free field theory and is the source of well known dificulties with infinities that appear when standard calculational methods are applied. A proper definition of an interacting quantum field theory generally requires introducing cutofs that make the number of degrees of freedom finite so that standard properties used in the finite dimensional case still hold, then studying what happens as the cutofs are removed, trying to find a physically sensible limit (“renormalization”).
The reader is warned that this chapter is of a much sketchier nature than earlier ones, intended only to indicate some outlines of how certain foundational ideas about representation theory and quantization developed for the finite dimensional case apply to quantum field theory. This material will not play a significant role in later chapters.
Chapter contents
- 39.1 Inequivalent irreducible representations
- 39.2 The restricted symplectic group
- 39.3 The anomaly and the Schwinger term
- 39.4 Spontaneous symmetry breaking
- 39.5 Higher order operators and renormalization
- 39.6 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 417、418、419、420、421、422、423、424
来源版本：2025-10-20
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