Abstract

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.

Highlights

  • 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 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

  • Quantum physics with the superposition principle is to be realized with states depicted by vectors on a Hilbert space, a complex vector space, usually endowed with a sesqulinear inner product with a positive definite signature, i.e. giving a positive definite norm

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Summary

Introduction

Quantum physics with the superposition principle is to be realized with states depicted by vectors on a Hilbert space, a complex vector space, usually endowed with a sesqulinear inner product with a positive definite signature, i.e. giving a positive definite norm. A proper symmetry transformation has to preserve the inner product, to be unitary. The latter is of central importance to the standard probability interpretation. Has an invariant inner product on which a Lorentz boost acts as a non-unitary transformation while a rotation acts as a unitary one. It is exactly the kind of pseudo-unitarity we suggest to be incorporated as a basic structure of a fully Lorentz covariant quantum mechanics

The Covariant Harmonic Oscillator
Theory from Symmetry Representation and the Geometric Picture
Final Remarks
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