The Problem of Time

The present physics meets the problem: "Time cannot be defined."

To see what the problem is, let us compare the notions of time in classical and quantum mechanics.

Time in classical mechanics
-- There are two kinds of time --

In Newtonian mechanics, there is a unique "absolute time" which dominates the whole universe.

In special and general relativity, we cannot choose a special time and space coordinates. All space-time coordinates are equivalent by the principle of special and general relativity: the postulate for covariance of physical laws under the change of coordinates.

Time in quantum mechanics

Quantum mechanics is constructed on Newtonian, i.e., Euclidean geometry; hence it describes phenomena according to one special time coordinate: the absolute time.

That is, the state f(t) of a system follows the Schroedinger equation: df/dt(t) + iH f(t) = 0, f(0) = g.
Here H is the Hamiltonian of the system, g is the initial state of the system, and t is the time of the system.

Why can't we leave classical time and quantum time as they are?

Is it not possible for us to leave classical and quantum mechanics as the physics for macroscopic and microscopic worlds, respectively?

There are two reasons why we should not do so . . .

The reason 1 why physics must be described by quantum mechanics

As far as we know, microscopic phenomena follow quantum mechanics precisely.

Thus if we can extend quantum mechanical description to the phenomena of a large scale, we will be able to have a "unified view" to the world.

The reason 2 why physics must be described by quantum mechanics

In the present physics, the universe is assumed to have begun from an infinitesimally small point. Namely Big Bang theory is assumed.

According to this theory, the initial state of the universe is a resident of microscopic world; hence, if we want to explain the generation of the universe by this theory, we have to apply "quantum mechanics" to the initial state of the universe.

Thus we need relativistic quantum mechanics

The large-scale structure of the universe follows the general theory of relativity in classical approximation.

Therefore, the quantum mechanics that explains Big Bang, etc. must be relativistic. That is, physicists think, "we have to build relativistic quantum mechanics."

Then is there any possibility for the relativistic quantum mechanics?

To see this, let us compare again the features of time in quantum mechanics and classical mechanics.

Time in quantum mechanics and classical mechanics

As we stated, quantum mechanics is constructed on Euclidean geometry; therefore, there must be defined a special time coordinate without any ambiguity. And physical phenomena must be described according to this time coordinate.

There is a large arbitrariness of choosing space-time coordinates in relativity. And any of them are equivalent in describing phenomena. Namely the physical laws are covariant under the transformations between those coordinates.

At a glance, these two notions are not compatible . . . Let us see more . . .

Incompatible aspect 1

If we quantize relativity by taking a special time coordinate, the resulting relativistic quantum mechanics must be unitarily equivalent with each other, independently of the time coordinate chosen at the quantization, according to the postulate of covariance.

However, there is no evidence for that by now.

Incompatible aspect 2

Conversely, if we take the standpoint that we do not start with a special time coordinate at the quantization as in the string theory, it seems that we cannot choose a special time coordinate that is effective in describing quantum mechanics. That is, by the postulate of covariance, any time coordinate becomes equivalent to each other, and we cannot choose a special one after quantization.

Two attempts to resolve this problem
-- corresponding to the difficulties --

Canonical quantization

In this attempt, to preserve the framework of quantum mechanics, one quantizes the relativity after choosing a special time coordinate.
Thus in general, covariance is lost.
Then the subject in this direction is to recover the covariance.

To start with no special time coordinate

In this attempt, one generalizes or modifies quantum mechanics in order to make it covariant with respect to the change of coordinates.
Afterwards, one tries to define a special time coordinate that can recover the quantum mechanical description.

All these are the attempts to try to assure the existence of some special time coordinate to make quantum mechanical description effective.

In this sense, the present physics meets "the problem of defining time," as we stated at the title.

The current status of these attempts

Canonical quantization

The introduction of new canonical variables by Ashtekar (Ashtekar variables).
non-superstring approach, might open novel non-perturbative techniques

To start with no special time coordinate

Feynman's sum-over-histories method.
One can feel a tendency toward a localization of the notion of time in the current attempts in this direction.
However, the notion of "locality" seems not clear yet.

The problem is the conflict between Euclidean geometry and Riemannian geometry

As we have seen, these attempts can be regarded as the attempts to try to reconcile the two incompatible geometries: Euclidean and Riemannian ones.

Therefore any usual thought might not work.

My attempt is to introduce "Local systems:"

Local system 1

Local system is an inner system equipped with Euclidean geometry and quantum mechanics, associated to each point on the Riemannian manifold.

We thus assure the Euclidean geometry for the quantum mechanics as an inner space.

We adopt Riemannian geometry for the outside; thus, the general relativity holds for the outer space.

By the construction these two geometries are compatible. (Remark: We do not assume any connections among the Euclidean structures of local systems so that we avoid the incompatibility of the inner Euclidean structures with the outer Riemannian geometry. This is the main point of the theory of local times.)

Local system 2

We define the proper time of a point on the Riemannian manifold as the local quantum mechanical time of the local system corresponding to the point.

We assume that each local system can observe the outside by using this local time.

And we try to explain relativistic quantum mechanics from this process of "observation."

Local system 3

As a result, time is proper to each local system, and we need not think a global time that dominates the total universe.

The difficulty of the previous attempts is in trying to construct a global time which is effective both in quantum mechanics and in relativity.

Localization of time resolves this difficulty.

Local system 4

Local times belong to the observers, and do not demand a global time effective throughout the universe.

Time is with observers, and cannot be separated from the observation.

An example of recent research of the problem of time
A paper by S. Kauffman and L. Smolin

"A Possible Solution for the Problem of Time in Quantum Cosmology"

The argument of Kauffman-Smolin

They start with no special time, and argue as follows:

"It is an undecidable, hence unreachable problem, for the finite beings to know if there is no global time." (They do not present any proof, but they just conjecture.)

"Thus it is inconsequential for the finite beings if there is a logical conflict between the nonexistence of global time and the existence of local time."

"In this sense the problem can be avoided."

Their argument may be summarized like this.

The implicit assumption hidden behind their argument

Let us reconsider how it is inconvenient if there is no global time.

It is because, even if there would be no global time, there surely exists time locally.

It is therefore a problem, because they assume: "if there is no global time, then there is no local time."

The problem of time

As they start with no special time in quantizing relativity, they have to make a certain time coordinate, which can describe quantum mechanics. However, if, as they assume, "it follows the nonexistence of local time from the nonexistence of global time," then they cannot build any local time even after quantization, and hence this is a problem for them.

However, is there no local time if there is no global time?

Kauffman and Smolin think, "the problem arises because nonexistence of global time implies the nonexistence of local time."

Are they correct?
A mathematical result that assures the existence of local timeeven if there is no global time

A mathematically known result tells us that "we can define local time even if there is no global time."

This is a result in functional analysis.

Let us see it more precisely.

First, what does it mean that there is no global time?

What do physicists mean when they say there is no global time?

Let f be a quantum mechanical wave function of the universe.

When physicists say that there is no time in the universe, they mean that this wave function f is an eigenfunction of the total Hamiltonian H of the universe:

H f = 0 (or H f = a f, a is a constant).

What then does it mean that there is no local time?

Let L be a local system consisting of the N particles 1,2,...,N inside the universe.

Let H_L be the Hamiltonian corresponding to the local system L .

Then that there is no time for this local system means that any state g of this local system is an eigenfunction of H_L . Namely the following holds:

H_L g = 0 (or H_L g = b g, b is a constant).

Then physicists think as follows:

"H_L, g are parts of H, f , therefore it follows from H f = 0 naturally that H_L g = 0. "
However. . . . Mathematics knows a curious fact:The following has been proved:

"Even if H f = 0 holds, it does not necessarily holds that H_L g = 0." That is, "Even if the total Universe has no time, subsystems can have their own times." ( Comments on the Problem of Time --- A response to "A Possible Solution to the Problem of Time in Quantum Cosmology" by Stuart Kauffman and Lee Smolin, (with L. Fletcher) (gr-qc/9708055) (1997). A possible solution for the non-existence of time, (gr-qc/9910081) (1999). )

Remark: This is easily seen if one formulates quantum mechanics in a mathematically rigorous way. Just nobody has ever tried to apply this fact in the context of the problem of time.

Thus we can define time locally.

If a local state is not an eigenfunction, it varies, and thus using the change of that wave function, one can define a local clock of the local system.

In this sense, the problem of time is not a true problem, but is an incorrectly formulated problem.

There just remains now to construct local times and to explain local phenomena.

An attempt in this direction is the afore-mentioned theory of local times.

A consequence of this mathematical observation

From the context of quantizing general relativity, we found that time can be defined only locally. (Note that the "local" here has a slightly different meaning from what is usually meant by "near" in topological sense.)

This casts a question to the field equation in the following sense.

The field equation in general relativity is an equation by solving which one knows the total universe from the past to the whole future (in this sense it has been problematic even in classical regime).

Therefore, if time (and also space) is a local notion and any global space-time cannot be defined, there cannot be any quantum version of the field equation: Namely, in quantum mechanics one cannot presuppose any global space-time structure as a manifold, and consequently there is no way to grasp the universe as a manifold. Thus it follows that there is no means to construct a field equation in quantum mechanical way.

In this sense, the usual proposition that the quantization of general relativity is the quantization of the field equation is an incorrect formulation.

There has been an opinion that to clarify the meaning of the field equation will make clear what the problem of time means.

The mathematical result that we have just seen is telling us to abandon the field equation. It might be a correct way to abandon the things that have no clear meanings . . . as Einstein has ever discarded the notion of 'ether.'

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