Snyder's quantized spacetime

The debate about the existence of the electromagnetic field has an interesting analogue: does empty space exist? In what sense is that a meaningful question?

There are numbers of theorists currently who take the view that there is something there, even when no one is looking. The programme of quantum gravity is predicated on the assumption that space and time do have some sort of dynamical structure, some dynamical degrees of freedom that could possibly be observed.

There are two problems with this. First, no one knows quite what the correct degrees of freedom are, and second, no one really knows how to test these ideas. The question arises, is there any sense in speculation?

Our answer is possibly. The problem with Wheeler’s participatory principle is that it does not actually explain anything. It gives us a guide for thinking about our relationship with the universe and how to go about thinking about physics, but it does not actual help us do that physics: it does not help us understand the information void.

By this we mean the following. Suppose we agree with Wheeler that the universe must be described empirically. So we set up our preparation devices, create quantum states, and then measure outcome probabilities in our detectors. That is all very fine, but on what basis can we predict those outcome probabilities?

Now it was argued in Chapter 2 that physics should be discussed only in terms of contextually complete generalized propositions. Such a proposition will have a relative internal context and a relative external context. The latter will usually be classical in its description, such as describing the observer as embedded in some four dimensional spacetime with curvature. On the other hand, the former, the internal context, will usually involve quantum theory, and we know that the world of the quantum is bizarre. That is also the world of the information void, the twilight world of ignorance in our theories between preparation devices and detectors.

Now in high-energy physics, the information void has traditionally been modelled by four-dimensional Minkowski spacetime over which quantum fields propagate. But perhaps that is not the best model. It is, after all, a metaphysical assumption to assert that empty spacetime is like Minkowski spacetime. We cannot in principle test whether space is empty without putting something in it to make that test.

Schwinger said that spacetime was a conceptual abstraction of our apparatus. Perhaps we could develop some alternative models for the information void. This is really what we believe Hartland Snyder did in 1947. He wrote two bold papers [Snyder, 1947a,b], setting out a vision of spacetime being associated with a set of

operators, satisfying a particular set of commutation relations. It is our view that this may make sense if we view his ideas as a model of the information void.

We shall use Standard International units in the following.

Now one of the principles of SR is that the laws of physics are invariant to standard Poincare transformations. These are coordinate transformations of the form

where [Л^] is a Lorentz transformation parameter matrix and the {ap} are translation parameters.

Now according to our discussion of canonical transformations in Chapter 14, infinitesimal transformations are represented by the action of generators of infinitesimal transformations, one generator for each parameter of the transformation. There are ten generators associated with the Poincare transformations: they can be represented by the differential operators [Hamermesh, 1962]

In relativistic quantum wave mechanics, these ten classical generators are replaced by the ten operators:

The satisfy the commutation rule

which has the interpretation that M4 is a flat spacetime. Assuming the spacetime coordinates can be represented by operators, viz., xp ^ Xp, then the phase-space coordinate operators {Xp, : p = 0,1, 2, 3 satisfy the algebra

where pp = npvpv. Snyder proposed replacing the algebra (27.5) with

where a is a fundamental length parameter [Snyder, 1947a,b]. In the limit a ^ 0, the algebra (27.6) reduces to (27.5).

Snyder introduced five new dimensionless coordinates вг: г = 0,1,2,3, and в4 for a new spacetime, which we denote S5. The conventional spacetime position operators хг are represented as vector fields over the tangent bundle TS5 as follows:

Then this representation satisfies the Snyder algebra (27.6).

Expanding out, we find for the coordinate operators

The line element in S5 is given by

Snyder stated that the X operators have spectra ma, where m is a positive, negative, or zero integer, whilst x0 has a continuous spectrum from - to +. To see this, consider the eigenvalue equation

for some real coordinates u, v. Changing coordinates from (u, v) to [r, в], where u = r cos в, v = r sin в, gives

Then the eigenvalue equation (27.10) becomes гадвф (r, в) = Хф (r, в), where ф (r, в) = ф (u, v). This equation is solved by the ansatz

Assuming v is a single valued function of u and v, then the periodicity of the coordinate в leads to the quantization condition k = ma, where m is some integer.

Comparing the form of the operators x, y, z in (27.8) to (27.10) leads to the conclusion that the spatial coordinate operators x, y, and z have discrete eigenvalues, which corresponds to spatial quantization.

On the other hand, consider the eigenvalue equation

This lead us to the coordinate transformation u = r cosh в, v = r sinh в, which gives

Now the eigenvalue equation (27.12) becomes гадвф (r, в) = Хф (r, в), which has solution ф (r, в) = R (r) в~гХв/а. Now, however, there is no condition on Х arising from periodicity and therefore no quantization. By inspection of (27.8), we deduce that x0 does not have a discrete spectrum.

The conclusion is that Snyder’s spacetime algebra does not amount to a ‘quantization’ of time, in that there is no chronon in this model. The reason why the spatial coordinate operators have a discrete spectrum whilst the time coordinate does not can be traced to the fact that the Minkowski spacetime metric has a Lorentzian signature.

Snyder’s model of spacetime remains of great interest to theorists: we speculate that it has not yet seen its best days.

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