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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.

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, hence to be unitary. The latter is of central importance to the standard probability interpretation. However, there has been important theoretical development on understanding quantum mechanics from a symmetry/spacetime and symplecto-geometric perspective that can get around the probability interpretation [

The kind of quantum theory we have in mind can easily be appreciated in the covariant harmonic oscillator problem, which has been among the first studies of a Lorentz covariant quantum mechanics. It is important to note that the problem actually goes beyond the setting of Poincaré symmetry. The proper symmetry behind the problem is that of H R ( 1,3 ) given as

[ J μ ν , J ρ σ ] = i ℏ ( η ν σ J μ ρ + η μ ρ J ν σ − η μ σ J ν ρ − η ν ρ J μ σ ) , [ J μ ν , X ρ ] = i ℏ ( η μ ρ X ν − η ν ρ X μ ) , [ J μ ν , P ρ ] = i ℏ ( η μ ρ P ν − η ν ρ P μ ) , [ X μ , P ν ] = i ℏ η μ v I , (1)

where we have adopted η μ ν = diag { − 1,1,1,1 } . Naively, one wants to think about the operator representation with X ^ μ given by x μ , P ^ μ by − i ℏ ∂ ∂ x μ , J ^ μ ν = X ^ μ P ^ ν − X ^ ν P ^ μ , while I represented by the identity operator. The representation is unitary and does not work so well as the case of the familiar H R ( 3 ) setting at all [

There is a parallel problem for any H R ( l , m ) . The Hermitian operator N ^ representing 1 2 ℏ ( X μ X μ + P μ P μ ) , plus a constant, commutes with all J ^ μ ν . Each fixed n-level, for n being the eigenvalue of N ^ , corresponds to a representation of S O ( l , m ) . We want the n = 0 level to be the trivial representation and the n = 1 level to be the defining vector representation. The latter is to say, the real span of n = 1 Fock states is essentially a ( l + m ) -dimensional pseudo-Euclidean space of signature ( l , m ) . The higher n levels then naturally correspond to symmetric Cartesian/pseudo-Euclidean tensors each of which splits into irreducible representations of S O ( l , m ) corresponding to the rank of the tensors. Of course, all such representations at any finite n are finite dimensional and non-unitary. The S O ( l , m ) transformations are to be represented by “rotations” on the pseudo-Euclidean space preserving the pseudo-Euclidean inner product. The Hilbert space as the space spanned by all Fock states can be seen as the natural complex extension of it. We will soon report on a detailed analysis with explicit Fock state wavefunctions and the pseudo-unitary inner product along the line.

Basic quantum mechanics is really a representation theory of group H R ( 3 ) [

One lesson from above is that there is no need at all to think about a negative effective ℏ value. We have one theory of quantum mechanics the one particle phase space of which is a Hilbert space for one value of ζ , for which we know [ X ^ i , P ^ j ] = i ℏ δ i j I ^ . Moreover, the free particle phase space can be seen as the proper quantum model of the physical space on which quantum mechanics is the associated symplectic mechanics. Under the proper formulation, the physical space model and the dynamical theory reduce back exactly to the Newtonian ones at the classical limit [

The situation is however different in the case of H R ( 1,3 ) . H ( 1,3 ) and H ( 4 ) are isomorphic, i.e. really the same so long as we do not have a priori identification of the generators with physical observables. H R ( 1,3 ) and H R ( 4 ) are definitely different as (real) Lie groups/algebras though. The relative sign in η μ ν says that the X 0 - P 0 pair maintaining the mathematical nature as the components of the X μ - P μ four-vectors has a commutator of a different sign from the X i - P i pairs, which has to be preserved in an representation of H R ( 1,3 ) with the position and momentum operators being Minkowski four-vectors. Hence, we cannot avoid having the commutator [ X 0 , P 0 ] , or actually in terms of the corresponding operators in the physical representation [ X ^ 0 , P ^ 0 ] , being equal to − i ℏ I ^ , analogous to an effective ℏ value being negative.

Quantum mechanics can completely be described by the symplectic or Kähler geometry of its phase space, the infinite dimensional projective Hilbert space. The observable algebra corresponds to an algebra of the so-called Kählerian functions and Schrödinger dynamics is given by their Hamiltonian flows [

It is important to note that the covariant harmonic oscillator problem and the formulation of the quantum mechanics itself are much the same. For the usual quantum mechanics, as the unitary representation of H R ( 3 ) with Hermitian position and momentum operators, for example, the true Hilbert space is not that of the square-integrable functions even for the ϕ ( x i ) wavefunction formulation. It is a dense subspace of rapidly decreasing functions, the most ready explicit picture of which is the span of the harmonic oscillator Fock states [

We have mentioned above that the projective Hilbert space should be seen as the proper model of the physical space behind quantum mechanics, from which one can retrieve the correct classical limit. It is also true that the submanifold of the coherent states is exactly like a copy of the classical phase space sitting inside the quantum one. The classical phase space is naively a simple product of the space/configuration part and the momentum part with the same Euclidean geometry. In fact, their metrics are simply given by restrictions of the metric for the projective Hilbert space [

Special thanks go to Suzana Bedi’c for discussions and assistance in editing the manuscript, as well as collaboration on related studies. The author is partially supported by research grant number 107-2119-M-008-011 of the MOST of Taiwan.

The authors declare no conflicts of interest regarding the publication of this paper.

Kong, O.C.W. (2020) The Case for a Quantum Theory on a Hilbert Space with an Inner Product of Indefinite Signature. Journal of High Energy Physics, Gravitation and Cosmology, 6, 43-48. https://doi.org/10.4236/jhepgc.2020.61005