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Journal of Physics A Mathematical and General

Some aspects of the boson-fermion (in)equivalence: a remark on the paper by Hudson and Parthasarathy

Piotr Garbaczewski

Published Apr 1, 1987• 8 citations
Publisher

Abstract

For original paper see Commun. Math. Phys., vol.104, p.457-70, 1986. The author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson-fermion equivalence (duality, reciprocity, etc.) for quantum fields. Even if via the Fock construction the common Fock space for bosons and fermions can be introduced, it still does not allow for the unrestricted boson-fermion equivalence for field theory models. All local fermion field theory models thus have boson equivalents (violating the weak local commutativity condition for space dimension three). The reverse statement is not valid: not all boson models admit a pure fermion reconstruction.

DOI
10.1088/0305-4470/20/5/036

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For original paper see Commun. Math. Phys., vol.104, p.457-70, 1986. The author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson-fermion equivalence (duality, reciprocity, etc.) for quantum fields. Even if via the Fock construction the common Fock space for bosons and fermions can be introduced, it still does not allow for the unrestricted boson-fermion equivalence for field theory models. All local fermion field theory models thus have boson equivalents (violating the weak local commutativity condition for space dimension three). The reverse statement is not valid: not all boson models admit a pure fermion reconstruction.
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The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincaré group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bundle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are synthesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus.