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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

The geometric phase and the spin-statistics relation

P. Bandyopadhyay

Published May 5, 2010• 16 citations
Publisher

Abstract

The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.

DOI
10.1098/rspa.2010.0042

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The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.
The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.
This first show was that the spin system, the two nearest neighbour spin in the Entanglement of measuring compliance by the ground conditions are achieved Berry stage to the relation when it is a closed path developed is. It noted the need to share the stage in a kind of geometric phase as explained to go to which any deformations involved is not and it’s the first Stiefel-Whitney class which is to Zz–cohomology value takes. However for polarized fermions we can relate the exchange phase as the celebrated Berry phase as in this case the Zz-cohomology becomes irrelevant and is consistent with the first Chern class which involves curvature. This follows from the depiction of a fermion as a scalar particle attached with a magnetic flux line. As in this framework the measure of entanglement of two nearest neighbor spins in a spin system given by concurrence is found to be associated with the Berry phase acquired by a spin state when it evolves in a closed path we can consider that entanglement is a consequence of Fermi statistics. It has been noted that in terms of quantum field theory, the berry phase is related to the perpetual inconsistency caused by the breaking of the perpetual symmetry. As mentioned earlier the quantization procedure of a fermion in the framework of Nelson’s stochastic quantization procedure introduces an internal variable which appears as a direction vector and gives rise to spin degrees of freedom. It's pointing to is that when a spin- 1 state of two spin 1/2 state is a Entangle system as considered to be that, the most widely covered state longitudinal elements with the matching it.
We have studied the entanglement of identical fermions in two spatial regions in terms of the Berry phase acquired by their spins. The analysis is done from the viewpoint of the geometrical interpretation of entanglement, where a fermion is visualized as a scalar particle attached to a magnetic flux quantum. The quantification of spin entanglement in terms of their Berry phases is novel and generalizes the relationship between the entanglement of distinguishable spins and that of delocalized fermions.
We have studied the entanglement of identical fermions in two spatial regions in terms of the Berry phase acquired by their spins. The analysis is done from the viewpoint of the geometrical interpretation of entanglement, where a fermion is visualized as a scalar particle attached with a magnetic flux quantum. The quantification of spin entanglement in terms of their Berry phases is novel and generalises the relationship between the entanglement of distinguishable spins and that of delocalised fermions.
We study here the Berry phase from the point of view of stochastic quantization. The relativistic generalization of Nelson’s stochastic quantization procedure can be achieved when the Brownian motion process is taken into account in the internal space (apart from that in the external space). This effectively considers the particle as one that is stochastically extended, and nonrelativistic quantum mechanics is obtained in the sharp point limit. This can be formulated in terms of a gauge-field-theoretical extension of the particle. This inherent gauge field gives rise to the holonomy in a Hermitian line bundle, which appears as an extra phase factor in the adiabatic limit for a parameter-dependent Hamiltonian, and determines a unique connection. When the Hamiltonian is degenerate, the holonomy is defined in a complex vector bundle. In the case of a fermion, this Berry connection is found to be related to the Wess-Zumino term and can be considered to be of topological origin.
The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.