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Generally covariant quantum mechanics

Edwin Beggs, Shahn Majid

Abstract

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold $M$ as part of the construction of quantum geodesics on the algebra $D(M)$ of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein-Gordon operator, with stationary modes being solutions of the Klein-Gordon equation. As an application, we describe gravatom solutions of the Klein-Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.

Generally covariant quantum mechanics

Abstract

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold as part of the construction of quantum geodesics on the algebra of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein-Gordon operator, with stationary modes being solutions of the Klein-Gordon equation. As an application, we describe gravatom solutions of the Klein-Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.

Paper Structure

This paper contains 26 sections, 22 theorems, 230 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background and notation
  4. The Schrödinger representation and quantum geodesics flows
  5. The star operation
  6. Differential calculus on ${\mathcal{D}}(M)$
  7. Centrally extended one forms on $M$
  8. Commutator of differentials of functions and vector fields
  9. Commutator of functions and differentials of vector fields
  10. The form of commutator of vector fields and their differentials
  11. Schrödinger representation of the differential of a vector field
  12. Commutator of a vector field and the differential of one
  13. Check of Schrödinger representation of differential of a vector field
  14. Differential calculus to order $\lambda$
  15. Jacobiators
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let the operator $T$ be defined by $T(\psi)=\lambda^2\,M^{ij}\, \psi_{,i;j}$ where $M^{ij}$ is a matrix of real functions. Then

Figures (3)

  • Figure 1: (a) Evolution of $\psi(r)$ initially a Gaussian centred far from the horizon at 10 $r_s$ showing complex waves and diffusion of the Gaussian probability density with motion of the peak towards the horizon (b) the same model but in close up near the horizon showing appearance of horizon modes at time $s=0.65$. (c) Evolution of an initial Gaussian centred at $1.4 r_s$ close to the horizon. Units of $r_s=1$.
  • Figure 2: (a) Cross-sections of the model in Figure \ref{['figradial1']}(a)-(b) showing close-ups of the emergence of probability density waves when the Gaussian tail starts to interact with the horizon, at around $s=0.65$. Note the different scales in the plots. By $s=3$ these horizon modes are all that remain. (b) The same model in larger view showing the Gaussian bump absorbed at $s=1.4$ into the horizon modes. (c) The expected value ${\langle}r{\rangle}$ and (d) the probability density entropy both increase throughout the process.
  • Figure 3: Spherically symmetric $l=0$ evolution eigenfunctions for the pseudo gravatom with $p_t^2/2 m =0.5$. (a) shows oscilliatory mode (i) with eigenvalue $E_{KG}=-0.49$, exponentially divergent mode (ii) with $E_{KG}=-0.51$ and exponentially decaying 'atomic' mode (iii) with $E_{KG}=-0.51$. (b) and (c) shows the fractal nature of all three modes approaching the horizon, where successive close ups look the same.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 29 more