The coherent state for fermion field and its classical correspondence

Charles M. Bowden

Fermion-like coherent states are obtained as projections from the Glauber coherent states for a boson field, and the two sets of states have an interesting correspondence. It is shown that because of the correspondence the equation of motion of the density operator for a fermion interacting with a radiation field maps into a classical-like transport equation in a hypothetical boson phase space. Besides being useful in the description of systems of fermions involved in cooperative phenomena, it is shown that the use of the continuous basis offers some unique advantages even for systems which are essentially Markovian. As an illustration, the method is applied to the description of the time evolution of a two-level atom interacting with a blackbody radiation field.

Chon‐Fai Kam, Wei-Min Zhang, D.H. Feng

Luca Salasnich

Fan Hong‐Yi, Ruan Tu-Nan

On the basis of the fermionic coherent state formulation we propose the Weyl-McCoy correspondence for fermion system, from which a functional integral form without the boundary term is derived. Moreover, the generating functional, the wigner theorem and the Moyal bracket for fermion system are also demonstrated. Thus the theory of quantum mechanics for Fermi operators and its pseudo-classical correspondence is established.

Monique Combescure, Didier Robert

Didier Robert, Monique Combescure

In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as \(\hbar \searrow 0\) of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit \(\hbar \rightarrow 0\), the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.