EVOLUTION EQUATIONS FOR FERMION SYSTEMS IN THE CONTINUAL REPRESENTATION (original) (raw)

A dynamic systems approach to fermions and their relation to spins

EPJ Quantum Technology, 2014

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental setup. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given setup can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.

Fermions and associated bosons of one-dimensional model

Communications in Mathematical Physics, 1967

The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fockrepresentation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly self adjoint operators. It provides the key to the solubility of LTJTTINGER'S model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.

On the Hamiltonian Form of Generalized Dirac Equation for Fermions with Two Mass States

arXiv: High Energy Physics - Phenomenology, 2006

Dynamical and non-dynamical components of the 20-component wave function are separated in the generalized Dirac equation of the flrst order, describing fermions with spin 1/2 and two mass states. After the exclusion of the non-dynamical components, we obtain the Hamiltonian Form of equations. Minimal and non-minimal electromagnetic interactions of particles are considered here. c

Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator

Doklady Mathematics, 2008

A Feynman formula is a representation of a solution to the Cauchy problem for an evolution differential or pseudodifferential equation in terms of a limit of integrals over the Cartesian degrees of some space E . A Feynman-Kac formula is a representation of a solution to the same problem in terms of a path integral. We assume that, on the path space, a countably additive measure or a pseudomeasure (of the type of the Feynman measure; see ) is defined, and the multiple integrals in the Feynman formulas coincide with integrals of finite multiplicity approximating integrals with respect to this measure or pseudomeasure.

Classical limit of fermions in phase space, Journal of Mathematical Physics 42 (2001) 4020–4030

Journal of Mathematical Physics

Using the mathematical structure of the Grassmann algebra, studied by Schönberg, we write down the Pauli equation and the Dirac equation in phase space. In addition, in order to investigate the physical nature of the spin degree of freedom inherent in these equations we set up a novel classical limiting process \!0. Thus we are able to derive relativistic and nonrelativistic classical statistical mechanics, for particle with spin 1/2, within a geometric algebra framework.

Fermionic extensions of KdV equation

2019

The integrable cases of fermionic extensions of the KdV equation are reviewed, using Hirota bilinear formalism extended to super space. The supersymmetric and the non-supersymmetric continuous and discrete KdV equations are presented with their super-bilinearisations and supersoliton solutions.

Gauged fermionic matrix quantum mechanics

Journal of High Energy Physics

We consider the gauged free fermionic matrix model, for a single fermionic matrix. In the large N limit this system describes a c = 1/2 chiral fermion in 1 + 1 dimensions. The Gauss’ law constraint implies that to obtain a physical state, indices of the fermionic matrices must be fully contracted, to form a singlet. There are two ways in which this can be achieved: one can consider a trace basis formed from products of traces of fermionic matrices or one can consider a Schur function basis, labeled by Young diagrams. The Schur polynomials for the fermions involve a twisted character, as a consequence of Fermi statistics. The main result of this paper is a proof that the trace and Schur bases coincide up to a simple normalization coefficient that we have computed.

A gauge field theory of fermionic continuous-spin particles

Physics Letters B

In this letter, we suggest a local covariant action for a gauge field theory of fermionic Continuous-Spin Particles (CSPs). The action is invariant under gauge transformations without any constraint on both the gauge field and the gauge transformation parameter. The Fang-Fronsdal equations for a tower of massless fields with all half-integer spins arise as a particular limit of the equation of motion of fermionic CSPs.

Nonlinear group representations and evolution equations

Czech J Phys, 1982

The theory of nonlinear evolution equations developed by M. Flato, J. Simon and a few others is reviewed. The method of construction of global solutions is described and the cohomological and analytic properties of linearizability of these equations are described.