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Communications in Theoretical Physics

The Representation Theory in Quantum Mechanics with Negative Metric

Ruan Tu-Nan, Fan Hong‐Yi

Published Mar 1, 1982• 0 citations
Publisher

Abstract

In this paper the coherent state for the negative metric boson is established, which, originating from its particle number representation provides a basis for constructing its coordinate and nomentum representations in the Euclidean space. However, it is demonstrated by us that these two representations do not exist in the Minkowski space. Thus this formulation provides a physical foundation for the path integral quantization theory of the negative metric boson in the Euclidean space.

DOI
10.1088/0253-6102/1/2/255

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In this paper the coherent state for the negative metric boson is established, which, originating from its particle number representation provides a basis for constructing its coordinate and nomentum representations in the Euclidean space. However, it is demonstrated by us that these two representations do not exist in the Minkowski space. Thus this formulation provides a physical foundation for the path integral quantization theory of the negative metric boson in the Euclidean space.
On the basis of our previous papers~[1,2], we discuss the representations in quantum mechanics with negative metric. With the aid of the properties of the coherent state of the abnormal boson (the boson with negative metric), in addition to the properties of the normal product, we derive the ordered product form of the metric operator, and them discuss the eigenstates of the coordinate and the momentum with the imaginary eigenvalues. With the coherent-state method we prove that these eigenstates cannot be normalized, but we can find metric-like operators which can normalize the eigenstates as the δ function. We also provide the completeness condition for these eigenstates with imaginary eigenvalues and their relation to the eigenstates with real eigenvalues, as well as the physical vacuum state's new representations.
In Quantum Physics, the term “Euclidean” means that one, compared to the conventional formulation, works in “imaginary time.” It is a general fact that objects associated with Euclidean as opposed to Minkowski space are easier to deal with; this is seen for instance when comparing the usual Laplacian, an elliptic operator, with its Minkowski space counterpart, the hyperbolic d’Alembertian. In Quantum Mechanics, the term “Euclidean” is usually associated with approaches based on the so-called Feynman-Kac formula. The latter originates in attempts to give Feynman’s ideas [10, 11] on path integrals a solid mathematical founding.
The notion of incompressible momentum observables is introduced. It is shown that when the metric in a manifold has a certain form, a set of canonically conjugate variables Xk and Pk in which Pk are incompressible, can be constructed. Based on this set of variables, the quantum mechanical description of the motion of a particle in a manifold, is identical to that associated with the familiar canonically conjugate variables xk and pk in an Euclidean space with Cartesian coordinates. The controversy related to non-uniqueness of momentum operators when the range of a coordinate is a finite interval is reduced to two possible extensions. This suggests relating these two types of extensions to the type of particle as being fermion or boson.
It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However, in order to be describable using the familiar framework of quantum mechanics, the spacetime structure of a quantum particle must be "quantised" by an introduction of the imaginary number iii. If a particle has a positive curvature then the quantisation is equivalent to turning the pseudo-Riemannian spacetime of the particle into a Riemannian spacetime. This means that it is assumed the particle is capable of measuring its temporal distance like its spatial distances. On the other hand, when a particle has a negative curvature and a negative energy density then quantising the spacetime structure of the particle is equivalent to viewing the particle as if it had a positive curvature and a positive energy density.
This chapter presents an analogous formalism which is developed for systems composed of many identical particles. Another extremely useful basis of the Fock space is the basis of coherent states which is analogous to the basis of position eigenstates in quantum mechanics. A crucial property of the coherent states is their overcompleteness in the Fock space, that is, the fact that any vector of the Fock space can be expanded in terms of coherent states. Boson coherent states are the physical states which emerge naturally when taking the classical limit of quantum mechanics or of a quantum field theory. In contrast, Fermion coherent states are not contained in the Fermion Fock space, they are not physically observable, and there are no classical fields of Fermions. In the subsequent development, this physical difference will give rise to some significant differences in the treatment of Bosons and Fermions.