Fleeting thoughts: The symmetry implied by general relativity as the mapping of local into global symmetries

Background: All global symmetries meant in the first theorem of Emmy Noether (1918) have the direct empirical status (DES). They are connected to space-time, which is empirically observable in principle. If the model of Minkowski space represents space-time exhaustedly, other symmetries cannot exist. The Lorentz symmetries meant in special relativity have a certain peculiar empirical status. The system of two or more observers who watch each other after the relative velocity comparable to that of light in vacuum cannot be realized technically yet. Anyway, slight mutual effects after lower relative speeds are visible by a few observers by meditation of devices, which only amplify the invisible effects. The direct empirical status can be assigned to them as well after the above reservations anyway. Then one can introduce the following postulate in the framework of the above restrictions:

Postulate 1: All global symmetries have DES.

On the contrary, the local symmetries meant in the second theorem of Emmy Noether (1918) do not possess DES. They need a macroscopic apparatus obeys the global symmetries. Unlike the Lorentz symmetries, the necessary measuring device do not amplify solely a slight input signal. The symmetries refer only to the probability distribution of a set of measurements of the same quantity, which is not constant under equal conditions. However, all measurements being states of macroscopic apparatuses obey the global symmetries. The mapping of the measured quantity onto the device is not one-to-one or unambiguous. Anyway, the reverse mapping is a function though not one-to-one. Consequently, one can define the local symmetries as symmetries of subsets of global symmetries and as symmetries of sets having DES.

Postulate 2: All local symmetries are symmetries of sets of elements having DES.

General relativity conserves only energy-momentum and thus the physical quantity of action. If its corresponding symmetry is global for it is a theory about macroscopic phenomena, it should refer to a dimensionless physical quantity. The relative quantities of probability distribution, entropy, and information met that condition. Another reason one to choose any of them as the symmetric counterpart of the conserving action is the following: probability distribution can be the same as that associated with a certain value of a local symmetry.

One can reformulate quantum mechanics thoroughly in terms of quantum information and thus a natural mapping of any wave function to a certain trajectory in the space-time of general relativity appears. That mapping is grounded into: (1) the isomorphism of a qubit into the usual 3D ball with two chosen points of it, and (2) the homomorphism of that isomorphism; that isomorphism is interpretable as the deformation for both entanglement and gravity.

The local symmetries known in physics describe what one should mean as “wave-particle duality” in terms well-defined mathematically: U(1) determines the equivalence of a sphere (for the “wave”) and a point on it (for the “particle”). SU(2) determines the equivalence of a sphere (for the “particle) truly within the unit ball (such as that of a qubit) and its surface (for the “wave”), and by means of SU(2) also of two points on each of them. SU(3) decomposed to the tensor product of three symmetries SU(2) would generalize SU(2) to any ball after the decomposition of any (“deformed”) unit ball of the space-time of general relativity to two independent unit balls of the space-time of special relativity after the “Hamilton representation”. The latter two unit balls would correspond as follows: the energy-momentum unit ball to the “wave”, and the space-time unit ball to the “particle”.

Postulate 3: There is a one-to-one mapping of each group of local symmetries (i.e. U(1), SU(2), SU(3)) and a certain group trajectories of the space-time of general relativity (modeled by pseudo-Riemannian space).

The thesis: Those three postulate are consistent to both each other and (Standard model & general relativity).

A few comments of the thesis:

The local and global symmetries are distinguished from each other in relation to DES: The latter refers to the symmetries of single experiments in a single point of the universe, and the former to those of huge sets of experiments meaning the relation of a whole space-time trajectory in the universe and a single point within it. Thus, one considers the universe as a whole after the local symmetries. Any point within the universe turns out to be determinable only dually: (1) within the universe to the other points after the global symmetries identifiable with DES; (2) to the universe as a whole after the local symmetries. That duality originating from the fundamental wave-particle duality in quantum mechanics therefore explains the way for the universe being all in definition to contain its externality within itself right by means of the local symmetries.

The Standard model from the same viewpoint determines a unique “complex constant” for the universally valid local symmetries directly grounded on the wave-particle duality. The sense of the same “constant” and thus of the Standard model is to privilege a group space-time trajectories (even possibly a single one), to which the local symmetries are restricted for the validity of the Standard model. That group of trajectories might be interpreted as those originating from the Big Bang, but far not only in thus. The Big Bang interpretation corresponds rather to the Christian and European tradition and to the popularization of those theories than to both mathematical formalism and experimental data.

The “quantum origin” of gravity should be searched in entanglement for it deforms homeomorphically any qubit, whose deformation is equivalent to the deformation of a unit ball of the pseudo-Riemannian space of general relativity to the correspondent ball in Minkowski space. Furthermore, that deformation right corresponds unambiguously to gravity according to general relativity. Thus, gravity turns out to be in a meta-position to the other known physical interactions. It is what transfers the universe into each point within it as the local symmetries “curved” relevantly by it to the local symmetries in any other point within the universe.