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Acta Physica Sinica

ON THE HEISENBERC PICTURE IN QUANTUM ELECTRODYNAMICS

S. T.

Published Jan 1, 1945• 0 citations
Publisher

Abstract

The form of quantum theory of reaiation introdnced by Heiaonbarg is diarnaaed from the point of view of the transformation theory of quantum electrodynamics.A general investigation of the connection between Heisenberg's method and Dirac'smethod of variation of parameters is given.The exetension of Heisenberg's methodto eigenvalue problems.which was first carried out by Weissklpf for the self energyof the electron.is presented in such a way as to show more clearly its quantummechanical interpretation.Ageneral proof of the equivalence of We jsskopf's me-tbod and the method of the perturbation theory of stationary stntes in quantumicechanics is given.

DOI
10.7498/aps.6.36

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The form of quantum theory of reaiation introdnced by Heiaonbarg is diarnaaed from the point of view of the transformation theory of quantum electrodynamics.A general investigation of the connection between Heisenberg's method and Dirac'smethod of variation of parameters is given.The exetension of Heisenberg's methodto eigenvalue problems.which was first carried out by Weissklpf for the self energyof the electron.is presented in such a way as to show more clearly its quantummechanical interpretation.Ageneral proof of the equivalence of We jsskopf's me-tbod and the method of the perturbation theory of stationary stntes in quantumicechanics is given.
Some years have passed since the appearance of Dirac's striking interpretation of the Heisenberg-Schroedinger parallelism in the equations of motion, and yet it appears that there still remain obscure points which have led to misunderstanding. Although the Schroedinger method suffices to deal with many practical problems, the flexibility of the Dirac-Jordan representation theory, aside from its intrinsic unity and beauty, recommends it to general theoretical investigation. The present paper considers the invariant form of the equations of motion in relation to the Schroedinger-Heisenberg pictures of quantum mechanics insofar as the latter refer to the "pictured" intrinsic motion of states (vectors) and observables (Hermitian operators). The discussion is divided roughly into 5-sections: Operator and Vector Transforms, Contact Transformations, Heisenberg-Schroedinger Operators, Equations of Motion and Invariant Properties, and the Density Operator and Operator Spur. Where possible, quasi-geometrical diagrams are given to illustrate the relation and transition between the several "pictures of the motion."
In quantum electrodynamics one can obtain a Hamiltonian which gives reasonable field equations in the Heisenberg picture, but which does not allow of solutions of the wave equation to represent physical states in the Schr\"odinger picture. The inference is that the Heisenberg picture is a good picture, the Schr\"odinger picture is a bad picture, and the two pictures are not equivalent. The usual proof of the equivalence of the two pictures fails because the state vector of the Schr\"odinger picture does not remain in Hilbert space. One can set up quantum electrodynamics entirely in the Heisenberg picture and thereby avoid the worst difficulties encountered in the Schr\"odinger picture. The theory takes a logical form and is not merely an assembly of working rules. It can be applied to the calculation of the anomalous magnetic moment and the Lamb shift, and is then similar to the usual calculations of these effects, with a good deal of dead wood cut away. There is a problem concerned with the general interpretation of quantum mechanics when one cannot use the Schr\"odinger picture and some postulates are proposed for dealing with it.
It is pointed out that the notation of the theory of functionals and the theorem in it analogous to Taylor's theorem facilitates a compact discussion of perturbation and variation methods in quantum mechanics, both for bound state and scattering problems. With this aid it is shown (i) how stationary expressions required for variation methods may be sought and found, (ii) that the perturbation and variation methods are virtually identical, (iii) that second order perturbations and expressions stationary to the second order may be found similarly, though these are complicated. The form found for the second order perturbation of bound states differs from that commonly used in that it does not explicitly involve the eigenfunctions of states other than that perturbed.
Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.
This chapter presents the origins of the transformation theory and related concepts. It shows how, in 1925, a procedure initiated by Werner Heisenberg was developed by himself, Max Born, Pascual Jordan, and a little later by Paul Dirac, into a new system of quantum theory—the first complete system of quantum theory which physics has possessed. A little later Erwin Schrödinger developed the “wave mechanics” from an entirely different starting point. This accomplished the same ends, and soon proved to be equivalent to the Heisenberg, Born, Jordan, and Dirac system. On the basis of the Born statistical interpretation of the quantum theoretical description of nature, it was possible for Dirac and Jordan to join the two theories into one, the “transformation theory,” in which they make possible a grasp of physical problems which is especially simple mathematically.