Spacetime, space observables and time observables

Question

It appears to me that the concepts of space and time play a privileged role in Physical Theories.

If we look at classical non-relativistic theories such as point particle mechanics, rigid body mechanics and fluid mechanics, we immediately see that these theories rely on the assumption of the existence of a Spacetime Continuum on which Physical System exist. For example point particle mechanics relies on the idea that point particles move in $\mathbb{R}^{3}$ (the "space component" of Spacetime), rigid body mechanics uses $\mathbb{SE}(3)$ as the configuration space of a rigid body and this, in turn, is derived by the assumption that rigid bodies move in $\mathbb{R}^{3}$ mainteining the relative distances of the point particles of which are "composed" fixed, fluid mechanics uses the Lie group of all the volume-preserving diffemorphisms as the configuration space and this choice, again, arises because of the assumption that fluids move in $\mathbb{R}^{3}$.

Even non-relativistic quantum mechanics implicitely assume the existence of a Spacetime continuum otherwise the definition of the position observables would be meaningless.

Nevertheless the assumption of the existence of a Spacetime Continuum on which Physical Systems are placed seems to me rather underestimated in that very little (if none) attention is devoted to the structure of the underlying Spacetime and the consequences related to it.

I am particularly interested in the consequences related to the concepts of Space and Time arising after a Spacetime splitting is performed by means of a reference frame. I am led to think that, in a certain not yet precisely delined sense, there are observables directly connected with these concepts, and that such observables play a privileged role.

I think that such an attitude is not new, for example in non-relativistic quantum mechanics the Heisenberg group arises from the idea that space observables and their canonically conjugated momentum observables are the observables in terms of which all other observables are expressed.

Obviously nothing similar exists for a Time observable and this is precisely what I am interested in.

In any case I am trying to work out something of concrete from these rough thoughts and I would like to hear if someone knows of a similar, or related, attitude toward observables in the literature.

I apologyze anticipatedly if this question will not fit the rules of the forum.

Edit

I would like to know of any attempt to describe observables, both in classical and quantum theories, in terms of Spacetime. I know this is highly ambiguous but I am not able to put it differently. As a suggestion I found interesting the algebraic approach toward quantum theories of Araki, Haag and Kastler where the $C^{*}$-algebra of observables is thought of as being a net of subalgebras each of which is associated to a spacetime region.

Answer 1

Pretty much all of modern physics is based upon the assumption that spacetime can be described as a four dimensional manifold equipped with a metric. You say:

Nevertheless the assumption of the existence of a Spacetime Continuum on which Physical Systems are placed seems to me rather underestimated in that very little (if none) attention is devoted to the structure of the underlying Spacetime and the consequences related to it.

But assuming the metric tensor counts as a structure of the underlying Spacetime then this contradicts your statement. Special relativity and therefore quantum field theory rely on the signature of the metric, and of course the whole point of GR is to relate the metric to the distribution of matter.

If I've missed the point of what you are saying maybe you could be a bit more precise abut what you think modern physics is missing about spacetime.

Answer 2

[...] concepts of Space and Time arising [...]

An important guideline in this direction is surely Einstein's foundational principle that:

All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points.

The foundational "privileged observables" are consequently

• the identity of distinct recognizable participants in coincidences, and
• the distinctiveness of coincidence events due to different subsets of identifiable participants.

Arguably then any such coincidence event is in turn considered observable, and any of its observations (by any particular identified participant) is in turn considered a coincidence (of that participant with the corresponding signal front).

It remains, of course, the task to define further "space-time propositions" in terms of the foundational notions; such as

• the order of distinct coincidences in which any one identifiable participant took part,

• relations such as "mutual rigidity", or "mutual rest",

• metric propositions such as equality (or ratios) of durations, equality (or ratios) of distances, mutual speed, etc.

Answer 3

you should look for the theory of "Covariant Quantum Mechanics" originally introduced by M. Modugno and J. Jadcyk, in particular the "special algebra of quantizable functions". There is not so much literature since the theory is mathematically quite hard to enter, but afterwards it is worth. I have worked in that field for several years. The basic idea is to implement through a curved fibred spacetime over absolute time a principle of Galileian general relativity. This can be successfully done. The equivalence principle (acceleration -> gravity field -> curvature) is implemented by adapting, in an analogous way, the Einstein and Maxwell equations in their "Galileian general relativistic" limits as background fields. The theory is formulated independent of any choice of an observer and can be written down, for any coordinate systems adapted to any observer (even accelerated). In such a geometric framework, one gets a covariant classical and quantum mechanics (of interacting massive particles in gravitational and electromagnetic fields) that contains standard (flat) classical and quantum mechanics. In particular, through the investigation of the "algebra of special quadratic functions" and its subalgebras (all living on classical phase space), one finds out that thes algebras generate either the classical symmetries, the quantum symmetries, the quantum operators and the conserved quantities by means of isomorphisms of Liealgebras. I have to admit, that the literature on the is not so easy to collect. Several articles are written by Marco Modugno, Josef Janyska or Raffaele Vitolo. If you have further questions do not hesitate to ask. My diss-thesis might be taken as an introduction: https://ub-madoc.bib.uni-mannheim.de/33/1/33_1.pdf However, there are several smaller mistakes and some evolutions that can be done better. A monograph on the theory will appear in the near future. Best regards, Dirk