time_I.pdf

By Hitoshi Kitada

(http://xxx.lanl.gov/abs/astro-ph/9309051)

(published in Il Nuovo Cimento, Vol. 109 B, N. 3, March 1994, pp. 281-302)

A part of the paper follows ...

Abstract: 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.

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 with anticipating the naive refutation that the Euclidean geometry which quantum mechanics follows and the non-flat Riemannian geometry which relativity follows can never be united consistently.

Our trick of the consistent unification of these two theories is to adopt a ten-dimensional vector bundle X x R⁶ (the reason R⁶ is adopted instead of R⁴ will be touched below) as the total physics space, where the base space X and the fibre R⁶ are mutually orthogonal. Quantum mechanics is set on the Euclidean space R⁶ and relativity theory on the curved Riemannian space X. Each point (t,x) in X is correlated to the centre of mass of the local system consisting of finite number of (quantum-mechanical) particles, and these centres of mass are considered as the classical particles. These classical particles are regarded as moving following general relativity in the Riemannian manifold X on the one hand, and the particles inside the local systems are regarded as moving following quantum mechanics on the other hand.

In this sense each point (t,x) of the base Riemann space X of the vector bundle X x R⁶ corresponds to the local system consisting of finite number of particles which follow quantum mechanics in each fibre R⁶.

Because the fibre R⁶ where quantum mechanics holds and the base space X where relativity holds are mutually orthogonal in the total space X x R⁶, it can be shown that the principles of quantum mechanics and the principles of general relativity are united consistently in this formulation, with the aid of the notion of quantum-mechanical local time introduced after sect. 2 below.

The reader may ask: There are many sublocal systems in a local system (H, H), and the centres of mass of these sublocal systems follow classical relativity. But inside the local system (H,H), quantum mechanics holds. Then what mechanics do these sublocal systems follow? The answer is

These sublocal systems follow classical relativity as far as the observer observes them in accordance with the observer's own time, but if the time of the system (H,H) itself is adopted as the reference time, then the local system (H,H) follows quantum mechanics.

Our starting point is the recognition that the time t is given by the ratio |x|/|v| of the position x and the motion (velocity) v. This standpoint is fully explained in sect. 2. This formulation is justified by the result (theorem l) of the many-particle scattering theory. Because of this definition of time, the Euclidean space of quantum mechanics becomes R⁶ of (x,v) instead of R⁴ of (t,x), and the usual four-dimensional structure R⁴ is recovered as an approximation through the uncertainty principle stated in sect. 5 after definition 3.

Like this, quantum mechanics controls the mechanics inside each local system, and the relative motion of the centres of mass of these local systems follows general relativity. In actual observations or experiments, the observer's time must be transformed into the local time of the observed local system. The rules of this transformation are given by the covariance principle and the equivalence principle of general relativity. An actual example of this kind of explanation of the relativistic quantum phenomena is given in sect. 9.

As a consequence, the EPR paradox is explained without contradiction, and Hubble's red-shift is explained even though our model is constructed on the basis of a stationary universe. As another result, the dark matter is unnecessary in our theory to explain the stability of galaxies, clusters of galaxies, and so on.

2. What is time?

This question seems to have a history as long as that of mankind itself. In the literature of physics, in the seventeenth century time was defined by Newton as a kind of absolute notion. In the first decade of the twentieth century, a reflection on the notion of time was given by Einstein in his theory of special relativity. Later, Einstein gave a more profound reflection on space-time in his theory of general relativity. Even though these reflections required us to make a reconstruction of our notion of space-time on a very deep level, time together with space is still considered to give a reference frame according to which one measures the physical quantities like positions, motions, velocities, and so on: Matter determines the space-time structure of the universe according to the theory of general relativity. Nonetheless, the space-time remains as a frame according to which the position of each matter is determined.

What we propose in this paper is a different point of view on the notion of time based on the following simple but overlooked observation: time does not appear until it is measured by some equipments, clocks. We need tools to know the time. This situation is different from the perception of the positions and motions, which are recognized directly by our senses. Even when we use some tools like a rule to measure the length of a thing, what we actually do is to see which scales of the rule coincide with both sides of the thing. The fundamental recognition done here is the perception of this coincidence, which cannot be done without our senses. The positions and motions are recognized through our senses in this sense. On the contrary, if one reflects the measurement procedure of time by clocks, one notices that he or she compares several motions or movements of matters, and takes the quotient of position and velocity. In fact, the usual (analog) clock measures time by the motion of its hands. We look at the hands, and recognize that one second passes if the second hand "moves" one "scale." We do not measure time directly by our senses, but we know time by perceiving the positions and motions of the hands of clocks. In this sense time is neither a quantity nor a frame given a priori. What exists first are the positions and movements of the matters relative to our own position. The perception of the positions and motions indicates an introduction of the common parameter in each system of matters consisting of finite number of particles. This parameter is called time and it is a local notion by nature. This recognition is our starting point.

In the twentieth century several theories of the universe have been investigated. Most of these are related with the theory of (general) relativity. Many trials have been made to "quantize" the theory of relativity. But it seems that none of them can be said to have succeeded. (See Ashtekar's Introduction of Ashtekar and Stachel [1] and Streater's paper in Brown-Harre [2] for the history of many trials including the recent ones. See also Glimm-Jaffe [3] for the point of view that quantum field theory is successful to a certain extent.) One category of such theories is represented by the stationary theories of the universe. Another category consists of non-stationary ones. The typical theory of the latter category is the so-called "big-bang" theory. Tracing in this direction, Hawking introduced, for instance, imaginary time.

In this article we introduce one stationary model of universe. This enables us to define the notion of (local) times. According to our theory, there exists the total universe which has no time associated. It is a stationary bound state of the total Hamiltonian with infinite degrees of freedom. Our theory is quantized from the beginning in this context. Relative coordinates and momenta are postulated to exist according to our reflection of the measuring procedure of time stated above.

From these postulates we define time as a local (or "glocal") and quantum-mechanical quantity. Our theory may also be called a quantization of time in this sense. We further show that our theory is consistent with the theory of general relativity. According to the so-called EPR paradox (see Redhead [4], for instance), quantum mechanics and the theory of relativity cannot be consistently united, because the former denies the local causality which is a consequence of the latter, as far as we suppose that quantum mechanics is complete in the sense described below in the argument of the EPR paradox in sect. 8. In this sense the unification of these two theories is impossible on the same level or on the same "plane." In our unification of these theories, we "orthogonalize," so to speak, these two theories, or set these theories on mutually "orthogonal planes," with the introduction of local times. We do not try to quantize the theory of relativity itself, as it has been tried in many theories of quantum fields. We just leave the theory of general relativity as a classical theory. What we actually do is to show that our theory of local quantum-mechanical times is consistent with classical relativity.

Our theory is basically the non-relativistic quantum mechanics "orthogonalized" to general relativity. The actual way of this orthogonalization is shown in sect. 6. We take the standpoint that nature follows quantum mechanics intrinsically, and the relativistic effects only appear associated with observation. Namely, gravitation is nothing but an outlook; it is no intrinsic nature of the universe according to our theory. Note that this standpoint is different from the standpoint of quantum gravity being discussed recently, e.g., in Ashtekar and Stachel [1], where it is implicitly supposed that the quantum and gravitational aspects of the universe are possible to be discussed on the same "plane" or on the same level. Our theory differs from this traditional standpoint.

The point of our theory of orthogonalization of quantum mechanics and relativity theory is summarized as follows:

The observer's coordinate system (t⁰,x⁰) of R⁴ is independent of the coordinate system (t¹,x¹) of the observed system, according to our definition of local times t¹,t⁰ (sect. 5). Thus the observed system can follow quantum mechanics with respect to the coordinate system (t¹,x¹), as well as it can follow classical relativity with respect to the coordinate system (t⁰, x⁰). In this sense, any local system can follow classical relativity when observed, and at the same time, it can follow quantum mechanics inwardly or intrinsically.

Accordingly, any local system behaves in classical-mechanical way as well as in quantum-mechanical way, depending on the choice of coordinate systems. For the relation between these two looks of the behaviour of local systems, we make the following fundamental assumption:

The observer can see only the motions of centres of mass of local systems, and these motions are observed following classical relativity. The quantum effects inside the local systems are unobservable directly, but can be deduced from these classical observations of the sublocal systems.

The unification of quantum and relativity theories in the above form means that we abandon the relativistic quantum field theory. A positive reason which supports this abandonment is that there seems to be only a trivial model in the axiomatic quantum field theory in actual four-dimensional space-time (see Streater's paper in [2]). Owing to this abandonment of quantum field theory, we are free from such problems as renormalizations, divergence problems, and so on, which constitute the fundamental difficulties of quantum field theories. The classical fields are not quantized relativistically in our theory. They are left as classical notions. The quantum fields are auxiliary tools, which will be introduced to treat the creation and annihilation phenomena (see conjecture (2) below). The quantum fields are confined to the explanation of the non-relativistic quantum phenomena.

Summarizing, the most important result of the paper is that there exists at least one consistent theory which unites the non-relativistic quantum mechanics and general relativity. (As an example of other trials in this direction, we refer to Prugovecki [5], where a geometro-stochastic approach is adopted for the problem of the unification of quantum theory and general relativity.) Even in our approach, we can give an explanation of the so-called relativistic quantum phenomena. These phenomena will be explained as the consequences of the observation activities: the actual experimental data are different from their true quantum-mechanical values because of the relativistic effects of observations, and the observer has to calculate the true values from his experimental data through some relativistic considerations. This point will be discussed in sect. 9 as regards observation. The necessity of considering high-energy particle physics is not strong in our context, since the universe does not begin via something like the big bang. However, some of the high-energy phenomena related with relativistic quantum mechanics can be explained in our context. Hopefully, a full explanation of these phenomena would be given in the future.

3. Global axiom

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4. Local axioms

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5. Local times

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6. Relativity

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7. A paradox of cyclotron

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8. EPR paradox

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9. Observation

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10. Concluding discussions

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11. Some conjectures

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References

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[4] M. Redhead, Incompleteness, Nonlocality, and Realism, A Prolegomenon to the Philosophy of Quantum Mechanics, Clarendon Press, Oxford, 1990.

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