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arXiv:gr-qc/0110065

Local Time and the Unification of Physics Part I. Local Time

Hitoshi Kitada, Lancelot R. Fletcher

Published Oct 14, 2001• 8 citations
arXivHTMLPDF

Abstract

The notions of time in the theories of Newton and Einstein are reviewed so that the difficulty which impedes the unification of quantum mechanics (QM) and general relativity (GR) is clarified. It is seen that GR by itself contains an intrinsic difficulty relating to the definition of local clocks, as well as that GR still requires a kind of absolute that can serve as an objective reference standard. We present a new understanding of time, which gives a consistent definition of a local time associated with each local system in a quantum mechanical way, so that it serves the requirements of both GR as well as QM. As a consequence, QM and GR are reconciled while preserving the current mathematical formulations of both theories.

DOI
10.48550/arxiv.gr-qc/0110065

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The notions of time in the theories of Newton and Einstein are reviewed so that certain of their assumptions are clarified. These assumptions will be seen as the causes of the incompatibility between the two different ways of understanding time, and seen to be philosophical hypotheses, rather than purely scientific ones. The conflict between quantum mechanics and (general) relativity is shown to be a consequence of retaining the Newtonian conception of time in the context of quantum mechanics. As a remedy for this conflict, an alternative definition of time -- earlier presented in Kitada 1994a and 1994b -- is reviewed with less mathematics and more emphasis on its philosophical aspects. Based on this revised understanding of time it is shown that quantum mechanics and general relativity are reconciled while preserving the current mathematical formulations of both theories.

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I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio $|x|/|v|$ between the space coordinate $x$ and the velocity $v$. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function $ϕ$ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.

arXiv:quant-ph/0410061•Oct 8, 2004•0 citations
The notions of time in the theories of Newton and Einstein are reviewed so that the difficulty which impedes the unification of quantum mechanics (QM) and general relativity (GR) is clarified. It is seen that GR by itself contains an intrinsic difficulty relating to the definition of local clocks, as well as that GR still requires a kind of absolute that can serve as an objective reference standard. We present a new understanding of time, which gives a consistent definition of a local time associated with each local system in a quantum mechanical way, so that it serves the requirements of both GR as well as QM. As a consequence, QM and GR are reconciled while preserving the current mathematical formulations of both theories.
The notions of time in the theories of Newton and Einstein are reviewed so that certain of their assumptions are clarified. These assumptions will be seen as the causes of the incompatibility between the two different ways of understanding time, and seen to be philosophical hypotheses, rather than purely scientific ones. The conflict between quantum mechanics and (general) relativity is shown to be a consequence of retaining the Newtonian conception of time in the context of quantum mechanics. As a remedy for this conflict, an alternative definition of time -- earlier presented in Kitada 1994a and 1994b -- is reviewed with less mathematics and more emphasis on its philosophical aspects. Based on this revised understanding of time it is shown that quantum mechanics and general relativity are reconciled while preserving the current mathematical formulations of both theories.
Summary. — A model of a stationary universe is proposed. In this framework, time is defined as a local and quantum-mechanical notion in the sense that it is defined for each local and quantum-mechanical system consisting of finite number of particles. The total universe consisting of infinite number of particles has no time associated. It is a stationary bound state of the total Hamiltonian of infinite degrees of freedom. The quantum mechanics and the theory of general relativity are consistently united in this context if one uses this notion of local and quantum-mechanical time. As one of the consequences, the Einstein-Podolsky-Rosen paradox is resolved. The Hubble red-shift is explained as a consequence of general relativity which is consistent with quantum mechanics. This does not require us to argue on the beginning nor the end of the universe. The universe just exists without time. PACS O3.65.Bz- Foundations, theory of measurement, miscellaneous theories. 1. – Introduction. As stated in the abstract, the main theme of the paper is to present one possible consistent unification of quantum mechanics and general relativity. This is stated intentionally
As a continuation of Part I [8], a more precise formulation of local time and local system is given. The observation process is reflected in order to give a relation between the classical physics for centers of mass of local systems and the quantum mechanics inside a local system. The relation will give a unification of quantum mechanics and general relativity in some cases. The existence of local time and local motion is proved so that the stationary nature of the universe is shown to be consistent with the local motion.
In this review we present the problem of time in quantum physics, including a short history of the problem and the known objections about considering time as a quantum observable. The need to deal with time as an observable is elaborated through some unresolved problems. The lack of a consistent theory of time is currently hindering the formulation of a full-fledged theory of quantum gravity. It is argued that the proposal set forth by several authors of considering an intrinsic measurement of quantum time, besides having the conventional external time, is compelling. Recently several suggestions have been put forward to revive the proposal of Page and Wootters (1983), elaborating and resolving some of the main ambiguities of the original proposal and opening new scope for understanding its content. The approach followed in these new contributions exposes the need to go beyond the limitations enforced by the conventional approach of quantum physics. The attitude of covariant loop quantum gravity, in which it is called to completely ignore time, is also discussed. This review could be a step forward in an endeavour to reform our outlook of the unification of the theory of relativity and quantum physics by furnishing the conceptual ground needed for this goal. Intentionally, some technical details are avoided since we aim to present the approaches to resolve the problem in a simple way with the clearest possible outlook. These can be looked up in the original references provided.
Modern Physics rests on two pillars: General Relativity (GR) and Quantum Mechanics (QM). The former describes gravitation as the continuous geometry of spacetime, based on differential fields. The latter, in turn, operates in a discretized regime, where energy levels, frequencies, and experimental results arise in quantized packets. The tension between both arises because GR assumes absolute continuity of time and space, while QM presents intrinsically discrete phenomena. Historically, the search for a unified theory started from the assumption that one of the two formulations must be modified — either GR should be discretized, or QM should emerge from an underlying continuous theory. In this work, we argue that the conflict is not fundamental: it stems from the traditional interpretation of time in Special Relativity (SR). When analyzed correctly, the proper time of distinct observers does not evolve synchronously, and this produces beating and discretization effects that explain, without additional postulates, many behaviors labeled as "quantum."
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕϕϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.