Grassmann phase-space methods for fermions: uncovering classical probability structure

Grassmann representation in qubit informatics and superlogic

The Grassmann representation for the system of qubits, is considered. The treatment is based on natural description of the qubits system as fermions and uses coherent states of fermions. The quantum logic gates are represented in two forms - by symbols of operations and by partial differential operators on the symbols of the states. The considered representation of quantum logic is called as superlogic. The examples are given for classical logic operations of negation, conjunction, disjunction and for reversible three-bit Toffoli gate and its quantum generalization - three-qbit Deutsch gate. T…

二次量子化中的玻色子–费米子代数映射

Boson-fermion algebraic mapping in second quantization

本文提出玻色与费米产生、湮灭算符代数之间的代数映射，表明其结构对应变形 Grassmann 代数，并讨论规范不变性及谐振子 Hamilton 量应用。

费米子相空间分布的信息论与优超理论

Information and majorization theory for fermionic phase-space distributions

本文利用超数理论为费米子相空间分布提出信息论度量，证明其不确定性关系及费米子类比猜想，并指出相关度量可通过Berezin积分取实数值而具有物理意义。

Grassmann Variables and the Jaynes-Cummings Model

This paper shows that phase space methods using a positive P type distribution function involving both c-number variables (for the cavity mode) and Grassmann variables (for the two level atom) can be used to treat the Jaynes-Cummings model. Although it is a Grassmann function, the distribution function is equivalent to six c-number functions of the two bosonic variables. Experimental quantities are given as bosonic phase space integrals involving the six functions. A Fokker-Planck equation involving both left and right Grassmann differentiation can be obtained for the distribution function, an…

Non-Markovian Fermionic Stochastic Schrödinger Equation for Open System Dynamics

In this paper we present an exact Grassmann stochastic Schrödinger equation for the dynamics of an open fermionic quantum system coupled to a reservoir consisting of a finite or infinite number of fermions. We use this stochastic approach to derive the exact master equation for a fermionic system strongly coupled to electronic reservoirs. The generality and applicability of this Grassmann stochastic approach is justified and exemplified by several quantum open system problems concerning quantum decoherence and quantum transport for both vacuum and finite-temperature fermionic reservoirs. We sh…

费米子的张量重整化群

Tensor Renormalization Group for fermions

本文综述了张量重整化群方法在含费米子和Grassmann变量的格点场论中的应用，介绍了纠缠过滤、圈优化等新技术，并在二维Wilson--Majorana费米子、Gross--Neveu模型及Hubbard模型中验证了方法的有效性。

Coherent state of a weakly interacting ultracold Fermi gas

We examine the weakly interacting atoms in an ultracold Fermi gas leading to a state of macroscopic coherence, from a theoretical perspective. It has been shown that this state can be described as a fermionic coherent state. These coherent states are the eigenstates of fermionic annihilation operators, the eigenvalues being anti-commuting numbers or Grassmann numbers. By exploiting the simple rules of Grassmann algebra and a close kinship between relations evaluated for more familiar bosonic fields and those for fermionic fields, we derive the thermodynamic limit, the spontaneous symmetry brea…

Fermions as generalized Ising models

We establish a general map between Grassmann functionals for fermions and probability or weight distributions for Ising spins. The equivalence between the two formulations is based on identical transfer matrices and expectation values of products of observables. The map preserves locality properties and can be realized for arbitrary dimensions. We present a simple example where a quantum field theory for free massless Dirac fermions in two-dimensional Minkowski space is represented by an asymmetric Ising model on a euclidean square lattice.

Classical probabilities for Majorana and Weyl spinors

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function $q_τ(t)$ for the Ising states $τ$. The time dependent probability distribution of a generalized Ising model obtains as $p_τ(t)=q^2_τ(t)$. The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure…

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus enables first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness and positivity of the basis, and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring an…

Fermions from classical statistics

We describe fermions in terms of a classical statistical ensemble. The states $τ$ of this ensemble are characterized by a sequence of values one or zero or a corresponding set of two-level observables. Every classical probability distribution can be associated to a quantum state for fermions. If the time evolution of the classical probabilities $p_τ$ amounts to a rotation of the wave function $q_τ(t)=\pm \sqrt{p_τ(t)}$, we infer the unitary time evolution of a quantum system of fermions according to a Schrödinger equation. We establish how such classical statistical ensembles can be mapped to …