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Reviews in Mathematical Physics

SYMMETRIES AND QUANTIZATION: STRUCTURE OF THE STATE SPACE

P. P. Divakaran

Published Apr 1, 1994• 9 citations
Publisher

Abstract

This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.

DOI
10.1142/s0129055x94000109

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This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
The thesis divides naturally into two parts. Part I raises, and in some cases answers, questions concerning symmetry in classical mechanics. The main result (Theorem 6.4) shows that the assumption of the existence of a realization puts an upper limit on the rank of the algebra. \n \n The heart of the thesis (covering three-quarters of the volume) is section II on the quantization of classical systems. §1 lists axioms desirable in any quantization rule for the 'functions of the q's'. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, into which conventional quantum mechanics fits nicely, we obtain in §7-§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions, which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule A assigning a unique self-adjoint operator to each such function f; a unitary operator WL corresponding to L and an equation \n \n𝛬(𝑓 ∘ 𝐿) = 𝑊[sub]𝐿⁻ 𝛬 𝑓 𝑊[sub]𝐿
It is proposed that an unconstrained quantum system is completely specified by its configuration space CCC and its group of symmetries GGG. It is then shown that the theory of projective unitary representations of GGG on a certain Hilbert space determined by CCC and GGG leads to a definitive resolution of supposed ambiguities relating to superselection rules, choice of wave functions, topological properties of GGG and CCC, anomalies, etc.
We study deterministic and quantum dynamics from a constructive “finite” point of view, since the introduction of the continuum or other actual infinities in physics poses severe conceptual and technical difficulties, and while all of these concepts are not really needed in physics, which is in fact an empirical science. Particular attention is paid to the symmetry properties of discrete systems. For a consistent description of the symmetries of dynamical systems at different time instants and the symmetries of various parts of such systems, we introduce discrete analogs of gauge connections. These gauge structures are particularly important to describe the quantum behavior. The symmetries govern the fundamental properties of the behavior of dynamical systems. In particular, we can show that the moving soliton-like structures are inevitable in a deterministic (classical) dynamical system, whose symmetry group breaks the set of states into a finite number of orbits of the group. We demonstrate that the quantum behavior is a natural consequence of symmetries of dynamical systems. This behavior is a result of the fundamental inability to trace the identity of indistinguish-able objects during their evolution. Information is only available on invariant statements and values related with such objects. Using general mathematical arguments, any quantum dynamics can be shown to reduce to a sequence of permutations. The quantum interferences occur in the invariant subspaces of permutation representations of the symmetry groups of dynamical systems. The observables can be expressed in terms of permutation invariants. We also show that in order to describe quantum phenomena it is sufficient to use cyclotomic fields—the minimal extensions of natural numbers suitable for quantum mechanics, instead of a non-constructive number system—the field of complex numbers. The finite groups of symmetries play the central role in this review. In physics there is an additional reason for such groups to be of interest. Numerous experiments and observations in particle physics point to the importance of finite groups of relatively low orders in a number of fundamental processes. The origin of these groups has no explanation within presently recognized theories, such as the Standard Model.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that arise from the non-simply connectedness of the classical configuration space. We define the quantum theory on the universal cover but restrict the algebra of observables ØØØ to the commutant of the algebra generated by deck-transformations. We apply standard superselection principles and construct the corresponding sectors. We emphasize the relevance of all sectors and not just the abelian ones.