3.5 Symmetry reductions and quantization

Many problems in quantum mechanics reduce to the computation of transition probabilities. For instance, in the case of a free particle moving in three dimensions all the relevant information about the quantum evolution can be encoded in the propagator (x1,t1|x2,t2) giving the probability amplitude to find the particle at x2 in the time instant t2 if it was at x1 in the instant t1. A nice but somewhat heuristic way to obtain this amplitude is to use a path integral. The main contribution to it comes from the value that the action takes on the classical path connecting (x ,t ) 1 1 to (x ,t ) 2 2. However, we also have to consider the contributions given by other paths, especially those “close” to the classical trajectory. It is clear now that the amplitude will depend both on the class of paths used in the definition of the integral and the specific form of the action that has to be evaluated on these. Notice that it is possible to have different Lagrangians leading to the same equations of motion. Furthermore, these Lagrangians do not necessarily differ from each other in total derivative or divergence terms (an example of this phenomenon in the context of GR is provided by the self-dual action [127, 194] and the Holst action [114]). One expects that a modification – either in the class of allowed paths and/or in the action – will generically change physical amplitudes.

A natural way to think about a quantum symmetry reduction of a model (again in the heuristic setting provided by path integrals) would consist in first restricting ourselves to computing probability amplitudes between symmetric configurations and then considering only a restricted class of paths in the path integral: precisely those that are, themselves, symmetric. This would have two important effects. First, the value of the probability amplitude will generically differ from the one obtained by considering unrestricted trajectories connecting the two symmetric initial and final configurations. This is expected because we are ignoring paths that would be taken into account for the non-reduced system. Second, it will be generally impossible to recover the amplitudes corresponding to the full theory from the symmetry-reduced ones because information is unavoidably lost in the process of rejecting the non-symmetric trajectories (which can be thought of as a projection, see [85Jump To The Next Citation Point] and also [207] for a more general point of view). This is even more so because, in principle, completely different mechanical or field systems may have the same reduced sectors under a given symmetry.

Though it can be argued that we can learn very important lessons from a quantum symmetry-reduced model, and even get significant qualitative information about the full quantized theory, it will be generally impossible to recover exact results referring to the latter. This would be so even if we restricted ourselves to computing transition amplitudes between symmetric classical configurations. A nice discussion on this issue appears in [144]. There the authors compare in a quantitative way the physical predictions derived from two different symmetry reductions of GR such that one of higher symmetry (the Taub model) is embedded in the other (the mixmaster model). They do this by constructing appropriate inner products and comparing the probabilistic interpretations of wave functions in both models. Their conclusion is in a way expected: the respective behaviors are different. This result sends an important warning signal: one should not blindly extrapolate the results obtained from the study of symmetry reductions. On the other hand it does not exclude that in physically-relevant situations one can actually obtain interesting and meaningful predictions from the study of the quantization of symmetry reductions.

Finally it is also important to disentangle this last issue from the different one of understanding to what extent the processes of symmetry reduction and quantization commute. To see this, consider a certain classical field theory derived from an action principle and a symmetry reduction thereof obtained by restricting the action to symmetric configurations (this procedure will be consistent if the principle of symmetric criticality holds as we will discuss in the next section). One can consider at this point the quantization of the classical reduced model by using as the starting point the reduced action. Supposing that one has a consistent quantization of the full theory, one can try to see if it is possible to recover the results obtained by first reducing and then quantizing by a suitable restriction – requiring a correct and consistent implementation of the symmetry requirement – of the fully quantized model. This has been done in detail for the specific example of a rotationally symmetric Klein–Gordon field in [85]. The main result of this paper is that it is indeed possible to show that using a suitable “quantum symmetry reduction” both procedures give the same result, i.e., in a definite sense reduction and quantization commute.

Giving a general prescription guaranteeing the commutation of both procedures on general grounds would certainly be a remarkable result, especially if applicable to instances such as LQG. This is so because many details of the quantization of full GR in this framework are still missing. It would be very interesting indeed to know what LQG would say about concrete symmetry reductions of full quantum gravity that could conceivably be obtained by considering the comparatively simpler problem of loop-quantizing the corresponding reduced classical gravity model. This notwithstanding one should not forget what we said above. Even if this can be effectively done we would not learn the answer to the problem of computing the amplitudes predicted by LQG for transitions between symmetric configurations. This implies that the results derived in symmetry-reduced implementations of the full LQG program such as LQC, no matter how suggestive they are, cannot be extrapolated to completely trustable predictions of full quantum gravity.

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