Fleeting thoughts: From Time to Quantum Time, or Conservation of Information

There is a viewpoint, according to which the time itself is the one “side” of any quantum representation of the world, and the other one is the well-known coherent state of any quantum system before measurement. If that is the case, the concept of quantum time is redundant in a sense. It can be anyway introduced in that case also addressing the concept of variable time in general relativity. 

Furthermore, that understanding of quantum time implies the generalization of the conservation of energy-momentum as conservation of action, which is consistent with general relativity, on the one hand, and with the conservation of information as a physical quantity, quantum information.

These statements will be discussed in detail:
1. The concept of quantum invariance

Time is the only physical quantity featuring by its “arrow”. However what is the sense of that arrow after the concept of quantum time should be introduced? The arrow-like time means well-ordering: Indeed the transition of any coherent quantum state being fundamentally unorderable according to the Kochen – Specker theorem (1968) to the always well-ordered statistical ensemble after measurement requires necessarily the well-ordering theorem, which is equivalent to the axiom of choice. Then the concept of quantum invariance can be introduced in thus: Quantum “ontology” before measurement excludes the axiom of choice just because of the cited theorem. Nevertheless, quantum “epistemology” of measurement requires the axiom of choice as above. Thus ‘quantum invariance’ is necessary to reconcile the quantum ontology and epistemology in a single unified quantum reality.

Furthermore, one can demonstrate that Hilbert space as the fundamental mathematical structure of quantum mechanics satisfies that condition of quantum invariance: Any point in it (a wave function, or a state of a quantum system) can be interpreted both as an infinitelydimensional vector and as the characteristic function of the probabilistic distribution of the values of that vector. The former is well-ordered, and the latter is fundamentally unorderable; the structure of Hilbert space does not distinguish between them.

Consequently, time being arrow-like can be interpreted as the well-ordered hypostasis of the quantum world being opposed to the other one of coherent state, and the concept of quantum invariance serves to unify both.
2. Quantum time and relativity time

Once time is involved as the one side of all and any quantum system, quantum time can be already introduced as an ordinary quantum quantity being reversible and supplied with a corresponding self-adjoint operator as any other one featuring a given quantum system. The physical meaning can be that of the period of the de Broglie wave associated with the system and specific to it.

Then a given measured and random value of quantum time will correspond to exactly one value of time according to general relativity corresponding to the curvature of pseudo-Riemannian space in a point of it. Both quantum and relativity time correlate unambiguously with the energy (mass) of the system. The set of probabilities of all given values of quantum time for the quantum system is mapped into the set of all pseudo-Riemannian spaces sharing a measured space-time point of the quantum system.

3. Conservation in general relativity as conservation of action

The well-ordered side of the quantum world, which is embodied in the quantity of time (not that of quantum one) implies conservation in general (Noether, 1918) and conservation of energy in particular for the well-ordering of time imposes the equality of all moments of time as all units after counting are equal. However, once quantum time is introduced, quantum (or relativity) moments of time are not equal and conservation is questioned. It can be anyway restored as the generalized conservation of action corresponding to some dimensionless uniform counting. This means that the product of energy-momentum and space-time volume should be constant for a conservative system in pseudo-Riemannian space.

4. Conservation of information

In turn the concept of quantum invariance suggests some conservation with the following interpretation: Any quantity in a coherent state is the same as that in the corresponding statistical ensemble, or: The well-ordering of a “much” into any “many” cannot change any quantity. The ordering addresses information and thus conservation of information in quantum invariance. Indeed both Shannon’s information and the dimensionless thermodynamic quantity of entropy can be interpreted as the product of a quantity “by itself” and it encoding in a vector in a space with an orthonormal basis: The quantity is invariant to any non-degenerating change of the orthonormal basis. Even more, if information is introduced as mutual entropy, it will conserve under any non-degenerating change of any complete basis (neither orthogonal nor normal in general). Conservation of action as above can be also represented in the same way addressing the quantity of action as the mechanical equivalent of information defined in thus.

5. Conservation of quantum information

The vector space to be defined information as above can be that as over an infinite as a finite field. The classical definition of information is the latter case. Quantum information is a case of the former where the field is that of all complex numbers. Thus any wave function can be associated with a value of quantum information:

Any self-adjoint operator acting on any wave function cannot change the quantity of quantum information though the change of the corresponding physical quantity associated with that operator changes the energy of the system in general. Furthermore, even the operator to be an arbitrary one rather that self-adjoint, which is equivalent to some change of the well-ordering of the vector and thus to quantum time rather than to time, the quantity of quantum information is conserved. This is another representation of the above conservation of action.

Conclusions:

The concept of quantum invariance allows of quantum time to be introduced in a consistent way distinguishing it from the arrow-like time.

Quantum time implies conservation of quantum information in general.