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arXiv:quant-ph/0110169 · International Journal of Modern Physics A

SPIN-STATISTICS THEOREM AND GEOMETRIC QUANTIZATION

Charis Anastopoulos

Published Feb 20, 2004• 5 citations
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Abstract

We study the relation of the spin-statistics theorem to the geometric structures on phase space, which are introduced in quantization procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the nonrelativistic domain (in fact for any symmetry group including internal symmetries) by requiring that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates — the exchange then arises from suitable change of coordinate system. We then examine our construction in the geometric and the coherent-state-path-integral quantization schemes. In the appendix we apply our results to exotic systems exhibiting continuous "spin" and "fractional statistics." This gives novel and unusual forms of the spin-statistics relation.

DOI
10.1142/s0217751x04017860

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The Pauli exclusion principle---the reason atoms have shell structure and matter is stable against collapse---is conventionally explained through the spin-statistics theorem, whose every standard proof relies on the machinery of relativistic quantum field theory (Lorentz invariance and microcausality). Yet the principle governs entirely non-relativistic physics. This paper asks whether its non-relativistic content admits a correspondingly non-relativistic explanation, and argues that it does: the Fermi--Dirac statistics of a half-integer-spin particle can be obtained from the topology of three-dimensional space together with the M\"obius geometry of the particle, with no appeal to Lorentz invariance or microcausality. The relativistic input of the standard proof is replaced by a geometric one. Concretely, the proof rests on two algebraic-topological lemmas about three-dimensional space (the configuration space of two identical particles is homotopy equivalent to ^2; real line bundles over ^2 are classified by H^1(^2;_2) _2), together with an explicit equivariant bundle map identifying the relevant exchange line bundle with the normal bundle of a non-orientable surface in ^3. The construction realises the spin-statistics correspondence as a topological identity in the spirit of Kuckert--Mund~KuckertMund2004 and Anastopoulos~Anastopoulos2004, but using only standard algebraic topology without recourse to QFT axioms or geometric quantisation. If the argument stands, the Pauli exclusion principle emerges as a statement about the topology of three-dimensional space and the geometry of the particle, rather than about the causal structure of spacetime. The required geometric input on the particle is encapsulated as a single postulate (Section~sec:setup): identical fermionic particles are modelled as compact embedded surfaces in ^3 equipped with light-speed-constrained flow fields generating _2 normal holonomy () = -1. In this setting we prove: enumerate[label=(*)] The two-particle configuration space is homotopy equivalent to the real projective plane: C_2 ^2, so _1(C_2) _2. Real line bundles over ^2 are classified by H^1(^2;_2) _2: there are exactly two, the trivial bundle (holonomy +1) and the tautological Möbius bundle (holonomy -1). The exchange holonomy of the fermionic vortex-knot wavefunction equals -1 (a unit normal section traversing undergoes a sign reversal). An explicit equivariant bundle map : N|_L_{ex}|_, (s,) = (s,), is constructed and proved to be a well-defined bundle isomorphism if and only if () = -1 (Section~sec:equivariant, Proposition~prop:chi). The argument rests on one geometric axiom---the Normal Transport Axiom (NTA), that is connection-preserving---which selects =-1 from the two phases the configuration-space topology admits, playing the role microcausality plays in the standard relativistic proof (Section~sec:equivariant). The contribution is to show this selecting input can be geometric rather than relativistic. Therefore the two-particle wavefunction is antisymmetric under exchange: Fermi--Dirac statistics follows. enumerate Points (i)--(iii) are rigorous theorems. Point~(iv) is also rigorous (the bundle map is explicit and proved); the NTA is the one remaining axiom---a precisely stated geometric assumption that supplies the selection input required by the Leinaas--Myrheim classification, with the same logical status as microcausality in the standard proof. Point~(v) follows under the NTA. At the heart of the result is a striking geometric correspondence: the non-orientability of the Möbius particle (in ^3) and the non-orientability of the two-particle configuration space ^2 are the same _2 at two different scales---the scale of one particle and the scale of a pair---connected by the explicit bundle map . : spin-statistics theorem; holonomy; Möbius surface; real projective plane; configuration space; line bundle classification; exchange statistics; normal holonomy; Fermi--Dirac statistics; equivariant b

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These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a path-integral viewpoint. The group-theoretical underpinnings of relativistic fractional spin are discussed, then the path-integral quantization of spin and of massive fermions without using spinors is reviewed. The path integral for a system of n relativistic particles with fractional spin is constructed, the spin-statistics relation and the Lorentz and Poincare' representation content of physical states are discussed. Some problems in formulating a field theory with fractional statistics are presented, and their resolution in the operator cocycle approach is reviewed.

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We study the relation of the spin-statistics theorem to the geometric structures on phase space, which are introduced in quantization procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the nonrelativistic domain (in fact for any symmetry group including internal symmetries) by requiring that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates — the exchange then arises from suitable change of coordinate system. We then examine our construction in the geometric and the coherent-state-path-integral quantization schemes. In the appendix we apply our results to exotic systems exhibiting continuous "spin" and "fractional statistics." This gives novel and unusual forms of the spin-statistics relation.
A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
We examine historic formulations of the spin-statistics theorem from a point of view that distinguishes between the observable consequences and the ``symmetrization postulate''. In particular, we make a critical analysis of concepts of particle identity, state distinguishability and permutation, and particle ``labels''. We discuss how to construct unique state vectors and the nature of the full state descriptions required for this -- in particular the elimination of arbitrary 2π2π2π rotations on fermion spin quantization frames and argue that the failure to do this renders the conventional symmetrization postulate (and previous ``proofs'' of it) at best {\em incomplete}. We discuss particle permutation in a general way for any pairs of particles, whether identical or not, and make an essential distinction between exchange and pure permutation. We prove a revised symmetrization postulate that allows us to construct state vectors that are naturally symmetric under pure permutation, {\em for any spin}. The significance of particle labels (which, in the exchange operation, are not permuted along with other variables) is then that they stand in for any asymmetry (order dependence) that is present in the full state descriptions necessary for unique state vectors but not explicit in the regular state variables. {\em The exchange operation is then the physical transformation that reverses any asymmetry implicit in the labels}. We point out a previously unremarked geometrical asymmetry between all pairs of particles that is present whenever we choose a common frame of reference. We compute the exchange phase for various state vectors using different spin quantization frames, and prove the Pauli Exclusion Principle and its generalization to arbitrary spin.
A Geometric Origin for Fermi--Dirac Statistics: The Spin-Statistics Connection from Z2Z_2Z2​ Normal Holonomy Without Lorentz Invariance
The Pauli exclusion principle---the reason atoms have shell structure and matter is stable against collapse---is conventionally explained through the spin-statistics theorem, whose every standard proof relies on the machinery of relativistic quantum field theory (Lorentz invariance and microcausality). Yet the principle governs entirely non-relativistic physics. This paper asks whether its non-relativistic content admits a correspondingly non-relativistic explanation, and argues that it does: the Fermi--Dirac statistics of a half-integer-spin particle can be obtained from the topology of three-dimensional space together with the M\"obius geometry of the particle, with no appeal to Lorentz invariance or microcausality. The relativistic input of the standard proof is replaced by a geometric one. Concretely, the proof rests on two algebraic-topological lemmas about three-dimensional space (the configuration space of two identical particles is homotopy equivalent to ^2; real line bundles over ^2 are classified by H^1(^2;_2) _2), together with an explicit equivariant bundle map identifying the relevant exchange line bundle with the normal bundle of a non-orientable surface in ^3. The construction realises the spin-statistics correspondence as a topological identity in the spirit of Kuckert--Mund~KuckertMund2004 and Anastopoulos~Anastopoulos2004, but using only standard algebraic topology without recourse to QFT axioms or geometric quantisation. If the argument stands, the Pauli exclusion principle emerges as a statement about the topology of three-dimensional space and the geometry of the particle, rather than about the causal structure of spacetime. The required geometric input on the particle is encapsulated as a single postulate (Section~sec:setup): identical fermionic particles are modelled as compact embedded surfaces in ^3 equipped with light-speed-constrained flow fields generating _2 normal holonomy () = -1. In this setting we prove: enumerate[label=(*)] The two-particle configuration space is homotopy equivalent to the real projective plane: C_2 ^2, so _1(C_2) _2. Real line bundles over ^2 are classified by H^1(^2;_2) _2: there are exactly two, the trivial bundle (holonomy +1) and the tautological Möbius bundle (holonomy -1). The exchange holonomy of the fermionic vortex-knot wavefunction equals -1 (a unit normal section traversing undergoes a sign reversal). An explicit equivariant bundle map : N|_L_{ex}|_, (s,) = (s,), is constructed and proved to be a well-defined bundle isomorphism if and only if () = -1 (Section~sec:equivariant, Proposition~prop:chi). The argument rests on one geometric axiom---the Normal Transport Axiom (NTA), that is connection-preserving---which selects =-1 from the two phases the configuration-space topology admits, playing the role microcausality plays in the standard relativistic proof (Section~sec:equivariant). The contribution is to show this selecting input can be geometric rather than relativistic. Therefore the two-particle wavefunction is antisymmetric under exchange: Fermi--Dirac statistics follows. enumerate Points (i)--(iii) are rigorous theorems. Point~(iv) is also rigorous (the bundle map is explicit and proved); the NTA is the one remaining axiom---a precisely stated geometric assumption that supplies the selection input required by the Leinaas--Myrheim classification, with the same logical status as microcausality in the standard proof. Point~(v) follows under the NTA. At the heart of the result is a striking geometric correspondence: the non-orientability of the Möbius particle (in ^3) and the non-orientability of the two-particle configuration space ^2 are the same _2 at two different scales---the scale of one particle and the scale of a pair---connected by the explicit bundle map . : spin-statistics theorem; holonomy; Möbius surface; real projective plane; configuration space; line bundle classification; exchange statistics; normal holonomy; Fermi--Dirac statistics; equivariant b
These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a path-integral viewpoint. The group-theoretical underpinnings of relativistic fractional spin are discussed, then the path-integral quantization of spin and of massive fermions without using spinors is reviewed. The path integral for a system of n relativistic particles with fractional spin is constructed, the spin-statistics relation and the Lorentz and Poincare' representation content of physical states are discussed. Some problems in formulating a field theory with fractional statistics are presented, and their resolution in the operator cocycle approach is reviewed.