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Canadian Journal of Physics

QUANTUM MECHANICS WITH AN INDEFINITE METRIC

Suraj N. Gupta

Published Aug 1, 1957• 20 citations
Publisher

Abstract

The formalism of quantum mechanics in a vector space with an indefinite metric is developed in a simplified way, and the properties of the indefinite vector space are discussed in detail with particular attention to vectors of negative and vanishing norms. It is shown how the present formalism simplifies the use of the indefinite metric in physical applications.

DOI
10.1139/p57-105

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The formalism of quantum mechanics in a vector space with an indefinite metric is developed in a simplified way, and the properties of the indefinite vector space are discussed in detail with particular attention to vectors of negative and vanishing norms. It is shown how the present formalism simplifies the use of the indefinite metric in physical applications.
We present the theoretical considerations for the case of looking into a generalization of quantum theory corresponding to having an inner product with an indefinite signature on the Hilbert space. The latter is essentially a direct analog of having the Minkowski spacetime with an indefinite signature generalizing the metric geometry of the Newtonian space. In fact, the explicit physics setting we have in mind is exactly a Lorentz covariant formulation of quantum mechanics, which has been discussed in the literature for over half a century yet without a nice full picture. From the point of view of the Lorentz symmetry, indefiniteness of the norm for a Minkowski vector may be the exact correspondence of the indefiniteness of the norm for a quantum state vector on the relevant Hilbert space. That, of course, poses a challenge to the usual requirement of unitarity. The related issues will be addressed.
Indefinite metric vectors are absolutely required as the physical states in Minkowski space because that is indefinite metric space and the physical space-time. For example, Maxwell equations are wave equations in Minkowski space. However, traditional Quantum theory ordinarily has been studied only in definite metric space, i.e., Hilbert space. There are no clear expression for indefinite metric vectors. Here we show a wave function example using Dirac's delta function for indefinite metric vectors in Minkowski space. In addition, we show the vectors can interfere with itself. This example also suggests indefinite metric will be absolutely required.
Abstract There are many problems of relativistic quantum mechanics which cannot be treated by working with a positive definite metric in the vector space of the fully quantized system. In this paper it is shown that, independently of the signs of energy and charge in the c number theory, any relativistic wave equation can be quantized and interpreted in terms of positive energies by use of an indefinite metric of suitable signature. A powerful method of obtaining positive definite transition probabilities from an indefinite metric is introduced and the concept of ‘negative probability’ is entirely avoided.The transition probabilities are defined as the squared matrix elements of a unitary matrix T, which is obtained from the solution of the Schrodinger equation by means of a special transformation.These techniques make it possible to treat relativistic wave equations which describe particles with several mass and spin states and to give a satisfactory account of the transition between two such mass states, a problem which is otherwise quite intractable. The relationship of spin and statistics is discussed and it is concluded that, apart from the special cases of spin 0, ½, 1, there is no necessary connexion between spin and statistics.
The structure of quantum theories with indefinite metric is studied with the aid of several simple models. It is shown that the pseudo-unitary scattering matrices entering such a theory are not inconsistent with physical interpretation provided a suitable invariant projection of physical state is carried out from among all the states. A relativistic quantum theory of interacting fields is outlined and is suggested as a basis for a dynamical theory of elementary particles. It is argued that the formal introduction of an indefinite metric together with supplementary interpretive postulates may help to reinstate the principle of simplicity in a consistent theory of elementary particles.