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Progress of Theoretical Physics

Hermitian Fermi Coordinates, Dimensions and Path Integrals

H. Nakajima

Published Mar 1, 1990• 0 citations
Publisher

Abstract

Quantized unconstrained fermi systems, Z2 graded Clifford algebras, of finite arbitrary dimension n are analyzed in terms of hermitian fermi operators, where 2[(n+1)/2] dimensional irreducible representations are discussed and odd dimensions are focussed upon. Path integral formula for the system is rigorously presented in terms of real Grassmann numbers via the reducible representation of the system.

DOI
10.1143/ptp.83.379

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Quantized unconstrained fermi systems, Z2 graded Clifford algebras, of finite arbitrary dimension n are analyzed in terms of hermitian fermi operators, where 2[(n+1)/2] dimensional irreducible representations are discussed and odd dimensions are focussed upon. Path integral formula for the system is rigorously presented in terms of real Grassmann numbers via the reducible representation of the system.
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Abstract In this chapter path integral formulae in various representations will be derived by starting from the Schrodinger equation. In order to obtain a c-number expression for the time evolution operator given in terms of the momentum and the coordinate, first we need to deal with the issue of operator ordering. Then a discussion on the Euclidean technique, with which path integration can prove its real worth, that is where path integration can be regarded as a powerful tool for acquiring quantum information, will be presented. Further, alternative path integral expressions other than those obtained by using the eigen states of the momentum and the coordinate will be presented by utilizing the coherent states. Path integral formulae for the Fermi system will be finally presented, together with guidance on Grass man numbers, the indispensable rudiments for doing this.
The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.
The theory of zero-range potentials is investigated in an arbitrary number of dimensions. Except for the trivial one-dimensional case the zero-range potentials are described by nonlocal operators called Fermi pseudopotentials. It is shown that in odd dimensions the Fermi pseudopotentials involve a very simple regularization operator. In even dimensions with the help of dimensional regularization, explicit formulas for the Fermi pseudopotentials are derived. The Green's functions, the propagators, and the exact solutions of the Lippmann-Schwinger equations are derived in explicit forms. In odd dimensions d=1 and 3 the Fermi pseudopotentials can be applied to describe multiphoton processes of atoms and molecules with very short-range interactions. In even dimension d=2 the Fermi pseudopotential can be applied to describe tunneling from laser-driven quantum wells. Physical applications involving higher d are also possible.
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.