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arXiv:hep-th/0301241 · Physics Letters B

Odd coset quantum mechanics

Evgeny Alexeevich Ivanov, Luca Mezincescu, Anatoly Pashnev, Paul K. Townsend

Published Jun 30, 2003• 11 citations
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Abstract

The standard quantum states of $n$ complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized $SU(n|1)$ symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space $SU(n|1)/U(n)$. We implement a quantization of this model for which the states with non-zero norm transform as a representation of $SU(n|1)$, the representation depending on the U(1) charge of the wave-function. For $n=2$ the wave-function can be interpreted as a BRST superfield.

DOI
10.1016/s0370-2693(03)00393-9

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The explicit forms of consistent eigenstate representations for finite dimensional fermionic quantum theories are considered in detail. In particular are the possible Grassmann characters of the eigenstates determined. A straightforward Schrödinger representation is shown to exist if they are even or odd. For an odd number of real eigenvalues, the eigenstates cannot be even or odd. Still a consistent Schrödinger picture is shown to exist provided the basic canonical operators are antilinearly represented. Since the wave functions within the Schrödinger picture are superfields, the class of superfields which also are first quantized wave functions is determined.
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SO(2N)U(N)\frac{{\rm SO}(2N)}{U(N)}U(N)SO(2N)​ Riccati–Hartree–Bogoliubov equation based on the SO (2N) Lie algebra of the fermion operators
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Properties and implications of the SO(2N+1) regular representation of a fermion system, which is a representation on the SO(2N+1) group proposed recently by us, are studied. By using the independent variables of the SO(2N+1) group, a representation of fermion annihilation-creation and pair operators is obtained which is an extension of the Dyson representation to cover both paired and unpaired modes. The structures of the SO(2N+1) spinor space and the SO(2N+1) wave function are determined. An integral and a differential Schrödinger equation on the SO(2N+1) spinor space are derived which are applicable to both even and odd fermion systems. The structures of the symmetry projected SO(2N+1) wave function and Schrödinger operator are determined. It is shown that odd fermion systems can be treated with an analogous way as even fermion systems by using a special SO(2N+1) canonical transformation which is very similar to the Bogoliubov transformation but corresponds to an improper rotation in the O(2N) group.