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It is question of specifying mathematical definitions which are cummunes in several theories. In classical physics, in special relativity, in quantum theories (wave functions and state vectors) and in general relativity, we can assert :

(a) The universe U is a topological space whose elements are called events and as each event has a neighborhood homeomorphic to R^4.

(b) A local coordinate system is a homeomorphism between an open subset of U and a bounded subset of R.

(c) A world line segment is a continuous function which is defined on an open subset of R and takes values in U.

(d) A generalized physical space (a set of spatial positions) is a particular family of world lines of material bodies. For example, in general relativity, a generalized physical space of Rindler consists of world lines of a family of Rindler observers. http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers

(e) To define a time variable in a generalized physical space we just have to choose a particular parametrization along each of his world lines or (in a corpuscular model) we just have to choose a particular parametrization along the world line of the body whose movement is studied.For example, a Poincaré-Einstein dating carried out by an experimenter P is a temporal variable (t) and a Poincaré-Einstein dating carried out by an experimenter P' is another temporal variable (t'). A Poincaré-Einstein dating carried out by an experimenter P is a temporal variable obtained by this method : the date associated with an event A is the arithmetic mean of the dates of issuance and receipt by P of a light signal which is reflected in A.

(f) A physical space is a particular family of world lines of material bodies that is associated to a unique observer. Each of these world lines seems to him continuously immobile. We use a physical space to define a wave function in quantum mechanics, we use it to define the motion of a body in a corpuscular model, we use it to define the Doppler effect (which results from the motion of the source in the physical space of the receiver).

The (f) is used in all other theories but is not yet specified in general relativity.See observational frames of reference : http://en.wikipedia.org/wiki/Frame_of_reference

A theory which defines all the physical spaces of nature (some with regard to the others) is compatible with all the quantum mechanics.

(1) In classical kinematics we postulate the existence of a universal chronology and we suppose that all the experimenters notice the same spatial distances between pairs of events that are simultaneous with respect to this chronology. We establish then strictly the possible states of movement between two physical spaces R and R'. We obtain that R'can be uniform or not niform translation with respect to R, this movement being coupled to a uniform or not uniform rotation.

(2) In a relativist theory where there are several possible chronologies (for example, the datings of Poincaré-Einstein realized by diverse observers) and where none of them is favored, it is still necessary to specify the possible states of the movement between two physical spaces R and R'.

Best regards.

Rommel Nana Dutchou

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# Rigorous definitions in quantum mechanics

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