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Physical Review

Quantum Mechanical Systems with Indefinite Metric. II

Howard J. Schnitzer, E. C. G. Sudarshan

Published Sep 15, 1961• 26 citations
Publisher

Abstract

Several simple models, similar to that of Lee, involving indefinite metric are studied in this paper. In this connection, a dispersion-theoretic treatment is applied to a simple "equal-mass" model. It is shown that, at least for these models, the scattering amplitude is analytic in the upper-half energy plane provided time-reversal invariance holds; the rules of the dispersion-theoretic formulation in the case of an indefinite metric theory are given. The solution is reinterpreted as the exact solution of a slightly different model, which can also be obtained by Hamiltonian techniques; further techniques are generalized to include recoil in a relativistic no-pair model. Certain basic questions of interpretation are discussed in some detail in the concluding section.

DOI
10.1103/physrev.123.2193

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The purpose of the following paper, which is published in three parts, is to show the possibility of making consistent use of an indefinite metric in the space of states and of the probability interpretation of quantum mechanics. In the part I bilinear forms, which are invariant under the transformations of symmetry groups, are constructed. The construction of the fundamental metric tensor for the various types of representations is exemplified for the case of inhomogenious LORENTZ group. The results are also applicable for other groups. The representations of the finite and compact group are however normal. In these cases the use of an indefinite metric does not bring about a new look for the theory of representations. In the parts II and III besides invariance of bilinear forms further conditions are stated, which are sufficient for the propability interpretation. The existence of superselection rules by which the space of states breaks up into coherent sectors leads to two procedures. In the part II the facts within one coherent sector are solely studied. These results are important for the representations of those symmetry groups, which leave the coherent sectors invariant (e. g. the inhomogenious LORENTZ group). Each sector is divided by a cut into a subspace, the elements of which represent physical systems, and a rest. The representations in the „subspace of physical systems“ of these symmetry groups, which leave the sectors invariant, are unitary. In the part III the space of states as a whole is investigated. The symmetry group of isotopic spin for a simple case is discussed there as an example of mapping from one coherent sector to another. At the end a possible generalization for the dual vector of a vector is discussed.

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Institute of Physics eBooks•Nov 26, 2004•2 citations
Several simple models, similar to that of Lee, involving indefinite metric are studied in this paper. In this connection, a dispersion-theoretic treatment is applied to a simple "equal-mass" model. It is shown that, at least for these models, the scattering amplitude is analytic in the upper-half energy plane provided time-reversal invariance holds; the rules of the dispersion-theoretic formulation in the case of an indefinite metric theory are given. The solution is reinterpreted as the exact solution of a slightly different model, which can also be obtained by Hamiltonian techniques; further techniques are generalized to include recoil in a relativistic no-pair model. Certain basic questions of interpretation are discussed in some detail in the concluding section.
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.
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.
Theories with indefinite metric have a long and very rich history. Such theories are being used time to time in searching the solutions for many outstanding problems in physics over the several decades. However much attentions were not paid for these as they were not mathematically consistent with usual quantum theories. Towards the end of last century this field of research received huge boost, when attempts had been made to accommodate certain class of non-Hermitian theories in rigorous mathematical formulation. Subsequently it has been shown that certain specific types of non-Hermitian theories can lead to fully consistent quantum theories with complete real spectrum, unitary time evolution and probabilistic interpretation in a modified Hilbert space. Due to this important realisation, theories with complex Hamiltonian have become the topic of frontier research over last two decades and enormous applicationswere found in various branches of physics. In this brief note we try to review certain aspects of such theories where we have contributed. In particular, we discuss PT phase transitionin non-relativistic, relativistic and quantum field theoretic systems and some new features of scattering in complex systems.
Theories with indefinite metric have a long and very rich history. Such theories are being used time to time in searching the solutions for many outstanding problems in physics over the several decades. However much attentions were not paid for these as they were not mathematically consistent with usual quantum theories. Towards the end of last century this field of research received huge boost, when attempts had been made to accommodate certain class of non-Hermitian theories in rigorous mathematical formulation. Subsequently it has been shown that certain specific types of non-Hermitian theories can lead to fully consistent quantum theories with complete real spectrum, unitary time evolution and probabilistic interpretation in a modified Hilbert space. Due to this important realisation, theories with complex Hamiltonian have become the topic of frontier research over last two decades and enormous applicationswere found in various branches of physics. In this brief note we try to review certain aspects of such theories where we have contributed. In particular, we discuss PT phase transition in non-relativistic, relativistic and quantum field theoretic systems and some new features of scattering in complex systems.
The purpose of the following paper, which is published in three parts, is to show the possibility of making consistent use of an indefinite metric in the space of states and of the probability interpretation of quantum mechanics. In the part I bilinear forms, which are invariant under the transformations of symmetry groups, are constructed. The construction of the fundamental metric tensor for the various types of representations is exemplified for the case of inhomogenious LORENTZ group. The results are also applicable for other groups. The representations of the finite and compact group are however normal. In these cases the use of an indefinite metric does not bring about a new look for the theory of representations. In the parts II and III besides invariance of bilinear forms further conditions are stated, which are sufficient for the propability interpretation. The existence of superselection rules by which the space of states breaks up into coherent sectors leads to two procedures. In the part II the facts within one coherent sector are solely studied. These results are important for the representations of those symmetry groups, which leave the coherent sectors invariant (e. g. the inhomogenious LORENTZ group). Each sector is divided by a cut into a subspace, the elements of which represent physical systems, and a rest. The representations in the „subspace of physical systems“ of these symmetry groups, which leave the sectors invariant, are unitary. In the part III the space of states as a whole is investigated. The symmetry group of isotopic spin for a simple case is discussed there as an example of mapping from one coherent sector to another. At the end a possible generalization for the dual vector of a vector is discussed.
The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation.Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate.In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial.A nontrivial result is obtained if the plane is not of this type.When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant.Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface.Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory.A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.