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Progress of Theoretical Physics

Quantum Field Theory with Finite Time Interval: Application to QED

Kazumasa Nomoto, R. Fukuda

Published Jul 1, 1991• 15 citations
Publisher

Abstract

Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.

DOI
10.1143/ptp/86.1.269

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Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.
Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.
In order to study the validity of the functional integral formula of the Green's function, the large time limit of the time evolution kernel exp (-iHt) of any quantum system is discussed. With the Feynman prescription, giving an infinitesimally small negative imaginary part to quadratic term of the potential in the Hamiltonian, the correct formula for this limit is obtained. It differs from the naive formula and depends on the detailed form of the potential. The usual formula holds for the perturbative expansion in terms of the potential, the proof of which is given by the path integral method. The new expression for the ground state wave function is also obtained in the course of the proof.
Quantum field theory (QFT) is considered to be most correct for the description of quantum systems. For the reason, we study local physical quantities defined in QFT, such as local polarizability, local conductivity, local spin torque [1–4]. In these works, for quantum states, we used wave packets derived by first principles calculation based on quantum mechanics. To treat states and physical quantities consistently, quantum systems should be treated in QFT. Hence, we are developing a first principles calculation program package based on quantum electrodynamics (QED). We name this program package QEDynamics [5]. The purpose of this program package is to study time evolution of quantum systems consisting of molecules or condensed matter. This challenge has two points of difficulties. One is the non-perturbativity, and the other is the time evolution of quantum systems. The formulation of QFT has been developed in a perturbative method. Hence perturbative calculations are widely used. For non-perturbative systems i.e. bound states, many methods are explored, lattice quantum chromodynamics, NRQED [6], Bethe-Salpeter method, and so on. All methods have various advantages and disadvantages, and hence we should be careful with the treatment of non-perturbative systems. Moreover, in most perturbative calculations, the existence of asymptotic states is assumed and the time integration from −∞ to∞ is performed for calculations, such as cross section. Thus the time evolution i.e. the study of nonequilibrium systems is restricted in specific cases. In literature, nonequilibrium systems are studied by Boltzmann equation, closed time path formalism [7], thermo field dynamics [8] and so on. We solve directly Heisenberg equations of field operators in QED in the present version of QEDynamics for the study of time evolution. In computational simulations, operators are difficult to handle. In our program, the evolution of the operators is calculated as the transition between modes prepared by the method of first principles computations of quantum structures. This mode expansion is based on the method used in Furry picture. This choice of the expansion functions of field operator has an advantage representing bound states. In this article, we show the present status of our program. In particular, the computation method of the evolution of photon fields are explained. Hence most other detail information cannot be mentioned in this article. For the information, please see Ref. [5].
We first rewrite the perturbation expansion of the time evolution operator [An Min Wang, quant-ph/0611216] in a form as concise as possible. Then we derive out the perturbation expansion of the time-dependent complete Green operator and prove that it is just the Fourier transformation of the Dyson equation. Moreover, we obtain the perturbation expansion of the complete transition amplitude in the Feynman path integral formulism, and give an integral expression that relates the complete transition amplitude with the unperturbed transition amplitude. Further applications of these results can be expected and will be investigated in the near future.
This paper is devoted to establishing the relation between the holomorphic form of the functional integral and its Feynman form, as well as presenting a new way to evaluate the Feynman transformation matrix element by using the coherent state. The Feynman kernel for the harmonic oscillator with two time-dependent sources is thus derived. Besides these, by the use of the normal product form of the Wigner operator derived in Ref.[1] and the Weyl correspondence, the general holomorphic form of the functional integral and its evolution to the Feynman kernel are discussed.