[Paper Review] Quantum Mechanics in General Relativity

E. A. Tagirov|ArXiv.org|Jul 14, 1998

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper constructs a general-covariant, quasinonrelativistic quantum mechanics for a spinless point particle in curved spacetime $V_{1,3}$ by deriving a Schrödinger-type equation with an asymptotically Hermitean Hamiltonian from the quantum field theory of a real scalar field. The key result is that coordinate and momentum operators do not commute in general, deviating from canonical quantization due to relativistic corrections, even in non-Minkowski spacetimes, except in the nonrelativistic limit or flat spacetime with Cartesian coordinates.

ABSTRACT

Having started with the general formulation of the quantum theory of the real scalar field (QFT) in the general Riemannian space--time $ V_{1,3} $, the general--covariant quasinonrelativistic quantum mechanics of a point-like spinless particle in $ V_{1,3} $ is constructed. To this end, for any normal geodesic 1+3--foliation of $ V_{1,3} $, a space $Φ^-$ of asymptotic in $c^{-1}$ solutions of the field equation is specified, which can be mapped to a space $Ψ$ of solutions of a Schrödinger equation with an (asymptotically) Hermitean hamiltonian and the Born probabilistic interpretation of the vectors of $Ψ$. The basic operators of the momentum and the spatial position of the particle acting in $Ψ$ generated by the corresponding observables of QFT include relativistic corrections, and therefore differ generally from those which follow for the geodesic motion in $ V_{1,3} $ from the canonical postulates of quantization. In particular, the operators of coordinates do not commute as well as the operators of the conjugate momenta, except the cases of Cartesian coordinates in the Minkowski space--time or of the exact nonrelativistic limit $(c^{-1} = 0)$. This approach provides QFT in the general $ V_{1,3} $ in the Fock representation with a particle interpretation based on the Born interpretation of wave functions.

Motivation & Objective

• To formulate a consistent quantum mechanics for a spinless point particle in general relativity-compatible curved spacetime.
• To derive a Schrödinger equation with an asymptotically Hermitean Hamiltonian from the framework of quantum field theory in curved spacetime.
• To establish a Born probabilistic interpretation of wave functions in a general spacetime $V_{1,3}$, extending Fock representation to curved backgrounds.
• To investigate how relativistic corrections modify standard quantum mechanical operators (position, momentum) in curved spacetime.
• To clarify the conditions under which standard non-relativistic quantum mechanics emerges from this general framework.

Proposed method

• Starts from the general formulation of quantum field theory (QFT) for a real scalar field in a general Riemannian spacetime $V_{1,3}$.
• Identifies a space $\Phi^-$ of asymptotic solutions in inverse powers of $c$ (speed of light) to the field equation, valid in the quasinonrelativistic limit.
• Maps $\Phi^-$ to a Hilbert space $\Psi$ of solutions of a Schrödinger equation with an asymptotically Hermitean Hamiltonian.
• Defines quantum operators for momentum and spatial position in $\Psi$ via the corresponding QFT observables, incorporating relativistic corrections.
• Uses a normal geodesic 1+3 foliation of spacetime to define time evolution and spatial structure in a general-covariant way.
• Applies the Born rule to interpret $\psi \in \Psi$ as probability amplitudes for particle localization and momentum.

Experimental results

Research questions

• RQ1 How can a consistent quantum mechanics for a point particle be derived from quantum field theory in a general curved spacetime?
• RQ2 What are the relativistic corrections to the standard quantum mechanical operators (position and momentum) in a general spacetime?
• RQ3 Under what conditions do the position and momentum operators commute, and when do they not?
• RQ4 How does the Born probabilistic interpretation of wave functions extend to curved spacetime in this framework?
• RQ5 In what limit does this construction reduce to standard nonrelativistic quantum mechanics?

Key findings

• The constructed quantum mechanics is general-covariant and based on the Fock representation of QFT in curved spacetime.
• The Hamiltonian in the Schrödinger equation is asymptotically Hermitean, ensuring unitary time evolution in the $c^{-1}$ expansion.
• The momentum and position operators derived from QFT observables include relativistic corrections and generally do not commute.
• Non-commutativity of coordinate and momentum operators arises in all spacetimes except when $c^{-1} = 0$ (nonrelativistic limit) or in Minkowski spacetime with Cartesian coordinates.
• The framework provides a particle interpretation of QFT in curved spacetime via the Born rule applied to wave functions in $\Psi$.
• The approach generalizes standard canonical quantization by incorporating spacetime curvature and relativistic effects directly into the operator algebra.