By Hitoshi Kitada
(http://xxx.lanl.gov/abs/gr-qc/9612043)
(Spectral and Scattering Theory, ed. by A. G. Ramm, Plenum Press, New York, pp.39-66 (1998). The talk was given at the first international ISAAC congress conference, 1997; 16:45 to 17:15 on June 3, 1997 at Room 202 in Smith Hall of University of Delaware.)
A part of the paper with math formulae removed follows ...
Abstract: In a framework of a stationary universe, 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 a finite number of particles. In this context, the total universe consisting of an infinite number of particles has no time associated, and quantum mechanics and general theory of relativity are united consistently. Relativistic Hamiltonians including gravitation are derived as a consequence of our treatment of observation. Related open problems in mathematical physics are presented.
Introduction.
Physics is a work to explain phenomena, i.e. a job to give a description of visible events. Insofar as we understand physics as such activities, it is neither surprise nor ridiculous thing if one takes other ways in explaining phenomena than the present physics: The problem of combining relativity and quantum theories might be able to be considered from a different viewpoint than the present trends where relativity theory is tried to be quantized or quantum mechanics is tried to be modified relativistically. It is enough if one can explain the relativistic phenomena or observations of them, in a systematic way. The purpose of the present paper is to give an attempt in this direction to explain relativistic quantum-mechanical phenomena. To make clear the contrast of our approach to the current physics, we briefly review the problems of physics in relation with relativity and quantum theories.
As is well-known, the solution of a Schroedinger equation for N particles is invariant with respect to Galilei transformation up to a factor of absolute value 1. This implies that the Schroedinger equation is not invariant under the Lorentz transformation. Therefore the quantum mechanics which is described by Schroedinger equation is understood, in the current physical context, as incompatible with special theory of relativity.
One of the features of the Schroedinger equation is that it yields the stability of matter, which is violated in the classical framework of Maxwell's equations and Rutherford model of atoms. However Schroedinger operator for the system mentioned above is bounded from below by some finite constant under a suitable assumption on the pair potentials. This means that the system does not collapse. In this respect, quantum mechanics remedies the difficulty of classical theory, while it is not Lorentz invariant.
In 1928, Dirac [Di] introduced a system of equations, which is invariant under Lorentz transformation, and could explain some of the relativistic quantum mechanical phenomena. However, Dirac operator is not bounded from below, and Dirac equation does not imply the stability of matter unlike the Schroedinger equation.
Dirac thus proposed an idea that the vacuum is filled with electrons with negative energy, and hence the electrons around the nucleus cannot fall into the negative energy anymore by Pauli exclusion principle, which explains the stability of matter. However, the idea of "filling the negative energy sea" becomes ambiguous in many particle case.
Quantum field theory is introduced to overcome this difficulty as well as to explain the annihilation-creation phenomena of particles. Quantum field theory is a theory of infinite degrees of freedom. This feature of treating infinite degrees of freedom yields the theory a difficulty, the difficulty of divergence. This sort of difficulty appears at almost every stage of the development of the theory. Mathematically, the difficulty of divergence has not been overcome yet at all. Physicists however found "renormalization" method which makes it possible to get finite quantities from infinite ones by extracting some infinite quantities from them systematically. They seemed to have succeeded in their explanation of Lamb shift going in this way and to have been able to give predictions outstandingly close to the experiment. However, the calculation done is up to the 6th or 8th order of a series giving Lamb shift or anomalous magnetic moment of electrons. Dyson noticed ([Dy]) that the series has symptom to diverge to infinity.
In the mathematical attempt, called "axiomatic quantum field theory," it is known that in some mathematical but important examples (see, e.g., [Fr]), renormalizability conditions and the axioms of quantum field theory yield that the theory is void as a physics, i.e. must not involve the interaction terms inside the theory.
These are the situation currently understood as an incompatibility problem between quantum theory and special theory of relativity. In the case of general theory of relativity and quantum mechanics, the situation seems similar or no better (see, e.g., [Ish]). The traditional attempts toward the unification of quantum theory and general relativity, as quantum gravity, superstring theory, and so on, are trying to find a way to unify them in a single layered theory where these two difficult theories should admit each other.
We present in the paper an attempt in a different direction, where general theory of relativity and quantum mechanics are considered as independent aspects of nature, but as playing complementary roles to each other. Our approach may be called two-layered theory, where these two theories have their own residences and they interfere only when observation is done. A procedure which describes the interference between them at observation will be our basis of explanation of relativistic quantum mechanical phenomena.
Our spirit behind the procedure we will introduce below for that interference is that what is intrinsic is the quantum-mechanical aspect of nature, while relativity plays a role of glasses to see nature. This attitude is contrary to the one adopted by current physics, which in origin comes from the spirit of Einstein [Ein]:
Thus, according to the general theory of relativity, gravitation occupies an exceptional position with regard to other forces, particularly the electromagnetic forces, since the ten functions representing the gravitational field at the same time define the metrical properties of the space measured.His position is that the metrical properties of space-time are intrinsic for nature, and space-time is a vessel of nature, into which other forces should be incorporated. In the framework of classical theory, electromagnetic forces can be treated in this direction in the sense that the equation for electromagnetic fields can be written as a tensor equation. In the framework of quantum theory, the characteristic of traditional approach is to treat gravity as a one which should be quantized, and the inclusion of other forces is a problem which is treated only after gravity is quantized successfully. In such attempts to quantize gravity or general theory of relativity, the canonical formalism of general theory of relativity is assumed usually, and this means that one has to introduce some global time coordinate which is common throughout the total universe. This itself produces a problem incompatible with the spirit of general theory of relativity that time is a local notion. If one would admit of introducing such a global time, it is difficult to reformulate general theory of relativity into canonical formalism even if gravitation is weak (see, e.g., [Ish]), and the quantization of gravity or space-time remains as a difficult problem even if one would defer to the global time.
To overcome these difficulties, we introduce a notion of local time tL which is proper to each local system L consisting of a finite number of quantum-mechanical particles. Our local time is a quantum-mechanical notion inasmuch as it is defined in each quantum-mechanical system as a parameter tL in the exponent of the propagator exp(-itL HL) describing the propagation of the local system L. It is a local notion defined for each local system L with a local Hamiltonian HL, and this will enable us to regard the time tL as a classical general relativistic local time, proper to the center of mass of the local system L, by identifying the classical particles with those centers of mass of local systems. We will show that these classical local times proper to the centers of mass of local systems constitute general relativistic notion of local times, compatible with the quantum mechanics inside each local system. The proof is, in part, a recall of the inclusion/exclusion assumption which has been adopted in physics, that the time tB of a bigger system LB which includes a smaller system LS dominates the time tS of LS, i.e. the assumption that tS must be equal to tB if the system LB includes LS. Apart from this traditional position on which physics has been founded, we retrieve the independence of each local system and its time coordinate among local systems, and liberate them from the bondage of inclusion/exclusion relation, which has been implicitly assumed for systems of physical particles. Geometrically expressed, our position may be formulated as a vector bundle with base space X representing the Riemannian manifold consisting of classical particles, identified with the centers of mass of local systems, and with the local system L which obeys the quantum mechanics on its own geometry being associated as a fibre to each point x in X, which is identified with the center of mass of the local system L.
There is a theory by Prugovecki [Pru] successful in a sense in quantizing general relativity, where he modifies quantum mechanics and general relativity so that the usual results are obtained as limits of his theory. His approach looks similar to ours in that he associates a Lorentzian quantum-mechanical world at each point of a Riemannian manifold as a fibre, regarding the total universe as a vector or fibre bundle. Our approach differs from his in the following points:
In our explanation of observation, we appeal to a procedure which transforms quantum-mechanical quantities into the classical quantities which obey the relativistic change of coordinates among the Euclidean quantum-mechanical worlds, which we call local systems. The quantum-mechanical world associated to each point of a Riemannian manifold has no relation with the Riemannian metric of the base space of our vector bundle, for we do not define any connections among the quantum-mechanical worlds. The procedure which transforms quantum-mechanical quantities into classical quantities is consistent with the two aspects of nature, i.e. with the quantum-mechanical aspect inside local systems and the general relativistic aspect outside local systems, because the results obtained by transformations are just concerned with observed facts. The relativity appears in our theory as "glasses," which deform quantum-mechanical quantities into classical relativistic quantities at each step of quantum-mechanical evolution, so that the resultant classical quantities accord with the observation. The intrinsic for our theory is quantum mechanics inside local systems, and relativity modifies quantum-mechanical calculations to accord with observations.
Summing up, our point is in the liberation of local systems from the inclusion/exclusion relation which has been an implicit assumption of physics. Instead of the inclusion/exclusion relation, we introduce a relation which transforms the quantum-mechanical values to classical relativistic values, as a procedure describing the interference between the two aspects of nature, the general relativistic aspect and the quantum-mechanical aspect.
In Part I of the paper, we give a presentation of our theory without using the notion of vector bundle. We first recall in section I.1 the basic notions related with the definition of local times from [Ki]. This notion of local times is a quantum-mechanical one defined in each local system consisting of a finite number of quantum-mechanical particles. In the sense that the local time is a local notion, it serves an ingredient which adheres the two layers: general theory of relativity and quantum mechanics. A result in many body quantum scattering is used to assign the usual meaning of time to our notion of local times. In section I.2, we review the proof of the consistency of the notion of local times with general theory of relativity. We give, in section I.3, a procedure of interpreting observation of quantum-mechanical process through the glasses of the relativity, yielding a relativistic quantum-mechanical Hamiltonian which explains gravitation and electric forces in quantum-mechanical way. In Part II, we treat two examples following the spirit of Part I. We present some open problems related with our formulation of physics in Part III.
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Part II. Examples
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Part III. Open Problems
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