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Physical Review

An Operator Calculus Having Applications in Quantum Electrodynamics

Richard P. Feynman

Published Oct 1, 1951• 1,185 citations
Publisher

Abstract

An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscript is introduced so that ${A}_{s}{B}_{{s}^{\ensuremath{'}}}$ means $\mathrm{AB}$ or $\mathrm{BA}$ depending on whether $s$ exceeds ${s}^{\ensuremath{'}}$ or vice versa. Then ${A}_{s}$ can be handled as though it were an ordinary numerical function of $s$. An increase in ease of manipulating some operator expressions results. Connection to the theory of functionals is discussed in an appendix. Illustrative applications to quantum mechanics are made. In quantum electrodynamics it permits a simple formal understanding of the interrelation of the various present day theoretical formulations.The operator expression of the Dirac equation is related to the author's previous description of positrons. An attempt is made to interpret the operator ordering parameter in this case as a fifth coordinate variable in an extended Dirac equation. Fock's parametrization, discussed in an appendix, seems to be easier to interpret.In the last section a summary of the numerical constants appearing in formulas for transition probabilities is given.

DOI
10.1103/physrev.84.108

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An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscript is introduced so that AsBs\ensuremath′{A}_{s}{B}_{{s}^{\ensuremath{'}}}As​Bs\ensuremath′​ means AB\mathrm{AB}AB or BA\mathrm{BA}BA depending on whether sss exceeds s\ensuremath′{s}^{\ensuremath{'}}s\ensuremath′ or vice versa. Then As{A}_{s}As​ can be handled as though it were an ordinary numerical function of sss. An increase in ease of manipulating some operator expressions results. Connection to the theory of functionals is discussed in an appendix. Illustrative applications to quantum mechanics are made. In quantum electrodynamics it permits a simple formal understanding of the interrelation of the various present day theoretical formulations.The operator expression of the Dirac equation is related to the author's previous description of positrons. An attempt is made to interpret the operator ordering parameter in this case as a fifth coordinate variable in an extended Dirac equation. Fock's parametrization, discussed in an appendix, seems to be easier to interpret.In the last section a summary of the numerical constants appearing in formulas for transition probabilities is given.
Although it was in fact the study of the Wigner distribution function that suggested to the author the general problem discussed in Part I, a consideration of the question of the formation of quantum mechanical operators corresponding to dynamical variables, being of fundamental importance to quantum theory, should properly precede a discussion of the Wigner distribution function. In both Parts I and II spin and relativistic effects will be ignored.\nA word about notation is in order. A symbol such as A will denote the quantum mechanical operator corresponding to the dynamical quantity A. For example, H is the Hamiltonian operator; q is the coordinate operator and p the conjugate momentum operator. Whenever there is a possibility of confusion, O(A) is used to represent the operator corresponding to the dynamical quantity A. For example, O(pq) is the operator corresponding to the product pq, and O(H2) is the operator corresponding to the square of the classical Hamiltonian.\nA subscript appearing on a differential operator, such as in 66PH , indicates the function on which the operator acts. Unless otherwise indicated all integrations are to be taken from -infinity to infinity. As usual, h denotes Planck's constant divided by 2pi.\nSince any physical investigation is directly dependent on the quality of the tools used in the investigation, mathematical rigor is much to be desired in physical theory. That rigor has not been achieved in this work is all too obvious, but it was nonetheless a constant goal.
A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto ccc-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and ccc-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a ccc-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto ccc-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.
Dirac introduced a very useful notation of state vectors Ψ i in terms of "bra" vectors. This notation allows to make the formal expressions of quantum physics more transparent and easier to manipulate. The bra-ket notation extends to action of operators on state vectors. The projection operator and the completeness relation are very important quantities in quantum physics. They are very useful in calculations and are often employed to represent an arbitrary operator and a wave function (state) in terms of orthonormal states. To illustrate the procedure of representations, this chapter considers an arbitrary operator  and shows how to represent the operator in terms of projection operators of known orthonormal states. The example in the chapter illustrates the concept of orthonormal states. It presents the application of the completeness relation. Using the completeness relation, that matrix elements of the product operator in the same orthonormal basis can be found from the multiplication of the matrix elements.
The present paper is based upon equations obtained in an earlier paper by the author devoted to a new formulation of quantum electrodynamics. The equations describe the structure of the electron as well as its motion in external fields, interaction with a measuring apparatus inclusive, in a deterministic manner without any jumps. Quantum mechanics is an approximate theory because its equations follow from the above equations upon neglecting the self-field of the electron itself. Just this leads to paradoxes, seeming contradictions and jumps in quantum mechanics. The quantum mechanical wavefunction has a dual interpretation. In some problems the square of its modulus represents a real distribution of the electronic density while in others the same square determines the probability distribution of coordinates. It is shown why, given the different interpretations of the wavefunction, it satisfies one and the same Dirac or Schrödinger equation. Description of many-electron systems is also considered in the starting approach as well as in quantum mechanics. Neutrinos are discussed in brief.
In a paper entitled ’’An Operator Calculus Having Applications in Quantum Electrodynamics’’ [Phys. Rev. 84, 108 (1951)], Feynman gives techniques and theorems for working with ordered operators. We believe that the entire development, because of its direct involvement in the transition from the Heisenberg–Schrödinger to Feynman’s path-integral formulation of quantum mechanics, is of general interest. We aim to make these less appreciated early techniques of Feynman more accessible to the general physicist.