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Journal of Mathematical Physics

Statistical quantum probabilities in coherent bosonic and fermionic states

Abdulmuhsen H. Ali

Published Aug 1, 2006• 3 citations
Publisher

Abstract

A derivation of the statistical nature of quantum probabilities is presented by considering a system of bosonic particles in a coherent state. A comparison between the current approach and a former derivation is also discussed. The current approach, which is an evaluation of the field operator matrix in the coherent Glauber state, proved to be a concise method which can be applied to relativistic particles as well. An extension of the coherent state method to the fermionic type of particles is also presented and thus the generality of the coherent state method is established.

DOI
10.1063/1.2259583

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A derivation of the statistical nature of quantum probabilities is presented by considering a system of bosonic particles in a coherent state. A comparison between the current approach and a former derivation is also discussed. The current approach, which is an evaluation of the field operator matrix in the coherent Glauber state, proved to be a concise method which can be applied to relativistic particles as well. An extension of the coherent state method to the fermionic type of particles is also presented and thus the generality of the coherent state method is established.
Abstract The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.
The Glauber-Sudarshan P-representation is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. For simplicity only one type of boson or fermion will be considered.
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.
The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.
We show that fermionic particles emitted by a stochastic source develop a behaviour which is different from usual Fermi statistics. The particles are described by creation and anihilation operators which span a sl(2, R) algebra. The corresponding Glauber-coherent states,1 defined in an appropriate representation are identical to coherent states found earlier by Barut and Girardello2 in solving the eigenvalue equation of the step operators of the su(1, 1) algebra. This establishes a case in which the mathematically extended notion of coherent states as eigenstates of step operators of an arbitrary Lie algebra finds a physical interpretation.