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International Journal of Modern Physics B

QUANTUM FIELD THEORETICAL FORMULATION OF EQUATION OF STATE FOR FERMION–BOSON MIXTURES

大輔 安間, Ken-ichi Takiuchi, Tadashi Toyoda

Published Dec 30, 2003• 0 citations
Publisher

Abstract

Using a quantum field theoretical canonical generator for the scale transformation of the second quantized Schrödinger fields describing a mixture of Fermion and Boson systems, the equation of state is derived. The derivation is based on the equal-time canonical commutation relations of the field operators and no approximation is employed. The result can be applied to liquid 3 He –4 He mixture.

DOI
10.1142/s0217979203023409

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Using a quantum field theoretical canonical generator for the scale transformation of the second quantized Schrödinger fields describing a mixture of Fermion and Boson systems, the equation of state is derived. The derivation is based on the equal-time canonical commutation relations of the field operators and no approximation is employed. The result can be applied to liquid 3 He –4 He mixture.
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In this chapter, we derived some elements of quantum mechanics, which are essential for the further development of our theory: the momentum of a system of Fermions in the second quantization, the coordinate and momentum of a harmonic oscillator as a unique operator at two different moments of time, Boson and Fermion operator algebra, coherent states, the electron-field interaction, the quantization of the electromagnetic field, Boson and Fermion distributions, and densities of states in a degenerate, or a non-degenerate system of Fermions. Our starting point is the wave nature of a quantum particle, the Hamiltonian equations were obtained as group velocities in the two conjugate spaces of the wave, of the coordinates and of the momentum. In this way, the Schr¨odinger equation and the electron-field potential of interaction are obtained from quantum equations generated by the particle wave function.
In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P)(Q,P)(Q,P), obtained by a canonical transformation upon the original phase space (q,p)(q,p)(q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^)(\hat{q},\hat{p})(q^​,p^​) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.