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arXiv:0912.4409

Relativistic Model for Gravity-Induced Quantum State Reduction

Garrelt Quandt-Wiese

Published Dec 22, 2009• 0 citations
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Abstract

A Lorentz invariant model for gravity-induced quantum state reduction is presented, which is mainly developed from Penrose's argument that the time translation operator in a superposition of macroscopic states is ill-defined. The problem to define a Lorentz invariant stochastic dynamics for the wave-function is solved by assuming that the stochastic time flow is running orthogonal to the deterministic, unitary time evolution inside the four-dimensional space-time, which makes the direction of causality independent from the chosen reference frame. This new view allows to accept Bell's position on the implication of quantum non-locality on relativity, without having to give up the Lorentz invariance of the specified dynamics. It is shown that it is possible to formulate on the basis of this new view a meaningful physical model. The model is also checked for possible higher order effects, which provide new starting points for experimental research.

DOI
10.48550/arxiv.0912.4409

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A Lorentz invariant model for gravity-induced quantum state reduction is presented, which is mainly developed from Penrose's argument that the time translation operator in a superposition of macroscopic states is ill-defined. The problem to define a Lorentz invariant stochastic dynamics for the wave-function is solved by assuming that the stochastic time flow is running orthogonal to the deterministic, unitary time evolution inside the four-dimensional space-time, which makes the direction of causality independent from the chosen reference frame. This new view allows to accept Bell's position on the implication of quantum non-locality on relativity, without having to give up the Lorentz invariance of the specified dynamics. It is shown that it is possible to formulate on the basis of this new view a meaningful physical model. The model is also checked for possible higher order effects, which provide new starting points for experimental research.
The compatibility of special relativity and Quantum Mechanics has been questioned by several authors. The origin of this tension can be traced back mainly to the introduction of the measurement processes and the corresponding wave function reduction, which play a crucial role in Quantum Mechanics. We approach this problem with the help of a recent proposal for a model of Quantum Mechanics, where the measurement is explicitly described as a specific stochastic process. This implements ordinary Quantum Mechanics, where measurement and reduction are treated as phenomenological events of unknown origin without any physical justification. To state clearly the question in general, we first discuss and establish the effect of a Lorentz transformation on a generic wave function in space-time. Alongside the analysis we formulate the relativistic version of the model. We then consider few thought experiments in order to analyze to what extent Quantum Mechanics follows relativistic invariance and find the specific critical points where non invariance possibly occurs. The analysis can shade light to the interpretation of the existing experimental observations.
We analyze Penrose's ideas concerning the role of gravity in the quantum-mechanical state reduction from the perspective of analogue-gravity models. We first review a line of work which argues that it is not enough to find a satisfactory quantization scheme for gravity in the description of a matter-gravity system. Instead of that; the standard rules of quantum mechanics as they are actually applied to matter systems need to be modified when the gravitational interaction is relevant. Among these ideas we focus on Penrose's argument for the absence of macroscopic quantum superpositions. He argues that such phenomenon has its roots in the idiosyncratic aspects of gravity, which become relevant just when the system is macroscopic and hence able to generate a sufficiently strong gravitational field, and not in the environmental decoherence. We then present an analogue model consisting of a Bose-Einstein condensate in a double well potential, where it is possible to build states that would correspond to putative superpositions of spacetimes. Such configurations are unstable and the source of instability from a microscopic point of view can be related to the abscence of a well-defined causal structure in the effective geometric description. We also discuss some additional lessons that can be taken from these instabilities.
A gravity-induced approach to wavefunction collapse based on semiclassical gravity is enhanced by the hypothesis of a temporally expanding spacetime, which leads to a collapse model that can resolve the conflict between quantum nonlocality and relativity. It is postulated that the spacetime region on which the evolution of the state vector exists is bounded towards the future by a border that is dynamically moving towards the future, and at which the state vector must fulfil a boundary condition. Wavefunction collapse is represented in such a way that the evolution of the state vector changes abruptly at critical spacetime expansions to an evolution resembling a classical trajectory. This can explain the correlations in EPR experiments without coming into conflict with relativity, since the evolution of the state vector before and after the abrupt change is governed solely by local physical laws. This model leads to the same lifetimes of superpositions as the gravity-based approaches of Diosi and Penrose, and is characterised by the facts that energy is conserved at collapse and that the reduction point in time does not vary statistically. Some unique features of the model are that it naturally leads to stochastic behaviour and that it can predict reduction probabilities. It explains why all experiments performed so far behave in agreement with Born's rule, due to a property that they have in common. This gives rise to new experiments for checking Born's rule, which can be realised in the short term.
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This paper draws on a number of Roger Penrose's ideas—including the non-Hamiltonian phase-space flow of the Hawking Box, conformal cyclic cosmology, non-computability, and gravitationally induced quantum state reduction—in order to propose a radically unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: “quantum” process is associated with fine-scale fractal geometry, “gravitational” process with larger-scale heterogeneous geometry. With this, it becomes possible to explain the experimental violation of Bell inequalities without having to abandon key ingredients of general relativity: determinism and local causality. Ensembles in IST can be described by complex Hilbert states over a finite set ℂp of complex numbers, where p is a large finite integer. The quantum mechanics of finite-dimensional Hilbert spaces is emergent as a singular limit when p→∞. A small modification to the field equations of general relativity is proposed to make it consistent with IST.
After a review of the problems induced by the Lorentz signature of Minkowski space-time, like the need of a clock synchronization convention for the definition of 3-space and the complexity of the notion of relativistic center of mass, there is the introduction of a new formulation of relativistic quantum mechanics compatible with the theory of relativistic bound states. In it the zeroth postulate of non-relativistic quantum mechanics is not valid and the physics is described in the rest frame by a Hilbert space containing only relative variables. The non-locality of the Poincare' generators imply a kinematical non-locality and non-separability influencing the theory of relativistic entanglement and not connected with the standard quantum non-locality.