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arXiv:hep-th/0210307

Fermionic functionals without Grassmann numbers

Hrvoje Nikolić

Published Oct 31, 2002• 0 citations
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Abstract

Since any fermionic operator ψcan be written as ψ=q+ip, where q and p are hermitian operators, we use the eigenvalues of q and p to construct a functional formalism for calculating matrix elements that involve fermionic fields. The formalism is similar to that for bosonic fields and does not involve Grassmann numbers. This makes possible to perform numerical fermionic lattice computations that are much faster than not only other algorithms for fermions, but also algorithms for bosons.

DOI
10.48550/arxiv.hep-th/0210307

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A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics from bosonic systems (systems for which T2=1{\mathcal{T}}^{2}=\mathbf{1}T2=1) to fermionic systems (systems for which ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT\mathcal{P}\mathcal{T}PT-symmetric matrix representations for operator algebras of the form ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, where $\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}$. It is easy to construct matrix representations for the Grassmann algebra ($\ensuremath{\alpha}=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\ensuremath{\alpha}\ensuremath{\ne}0$) if $\ensuremath{\alpha}=\ensuremath{-}1$; a matrix representation does not exist for the conventional value $\ensuremath{\alpha}=1$.
The phase-space description of bosonic quantum systems has numerous applications in such fields as quantum optics, trapped ultracold atoms, and transport phenomena. Extension of this description to the case of fermionic systems leads to formal Grassmann phase-space quasiprobability distributions and master equations. The latter are usually considered as not possessing probabilistic interpretation and as not directly computationally accessible. Here, we describe how to construct ccc-number interpretations of Grassmann phase-space representations and their master equations. As a specific example, the Grassmann BBB representation is considered. We discuss how to introduce ccc-number probability distributions on Grassmann algebra and how to integrate them. A measure of size and proximity is defined for Grassmann numbers, and the Grassmann derivatives are introduced which are based on infinitesimal variations of function arguments. An example of ccc-number interpretation of formal Grassmann equations is presented.
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The ground state and low excited states of liquid He3{\mathrm{He}}^{3}He3 (and other fermion systems) can be constructed from a set of basis functions $\ensuremath{\Psi}(|\mathrm{n})={{\ensuremath{\psi}}_{0}}^{B}\ensuremath{\Phi}(|\mathrm{n})$ in which ${{\ensuremath{\psi}}_{0}}^{B}$ is the ground-state boson-type solution of the Schr\"odinger equation and the model functions $\ensuremath{\Phi}(|\mathrm{n})$ are Slater determinants suitable for describing states of the noninteracting Fermion system. Diagonal and nondiagonal matrix elements of the identity and the Hamiltonian operator are evaluated by a cluster-expansion technique. An orthonormal basis system is constructed from $\ensuremath{\Psi}(|\mathrm{n})$ and used to express the Hamiltonian operator in quasiparticle form: a large diagonal component containing constant, linear, quadratic, and cubic terms in free-quasiparticle occupation-number operators and a nondiagonal component representing the residual interactions involved in collisions of two and three free quasiparticles.