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8.1 Unique theories, unique solutions

It is often the case that, before quantitative concepts can be introduced into a field of science, they are preceded by comparative concepts that are much more effective tools for describing, predicting, and explaining than the cruder classificatory concepts.

Rudolf Carnap

An Introduction to the Philosophy of Science

The rise of loop quantum gravity presents an unprecedented situation in physics, in which full gravity is tackled in a background-independent and non-perturbative manner. Not surprisingly, the result is often viewed skeptically, since it is very different from other well-studied quantum field theories. Usually, intuition in quantum field theory comes either from models, which are so special that they are completely integrable, or from perturbative expansions around free field theories. Since no relevant ambiguities arise in this context, ambiguities in other frameworks are usually viewed with suspicion. A similar treatment is not possible for gravity because a complete formulation as a perturbation series around a free theory is unavailable and would anyway not be suitable in important situations of high curvature. In fact, reformulations as free theories exist only in special, non-dynamical backgrounds such as Minkowski space or planar waves, which, if used, immediately introduce a background.

If this is to be avoided in a background-independent formulation, it is necessary to deal with the full non-linear theory. This leads to complicated expressions with factor ordering and other ambiguities, which are usually avoided in quantum field theory but not unfamiliar from quantum theory in general. Sometimes it is said that such a theory loses its predictive power or it is even suggested that one stop working on applications of the theory until all ambiguities are eliminated. This view, of course, demonstrates a misunderstanding of the scientific process, in which general effects play important roles, even if they can be quantified only at later stages. What is important is to show that the qualitative effects are robust enough that their implications do not crucially depend on one choice among many.

So far, applications of loop quantum gravity and cosmology are in comparative stages, in which reliable effects can be derived from basic properties and remaining ambiguities preclude sharp quantitative predictions in general (notable exceptions are fundamental properties, such as the computation of γ through black-hole entropy [1112158225]). These ambiguities have to be constrained by further theoretical investigations of the overall consistency, or by possible observations.

Ambiguities certainly mean that a theory cannot be formulated uniquely, and uniqueness often plays a role in discussions of quantum gravity. In the many approaches different kinds of uniqueness have been advertised, most importantly the uniqueness of the whole theory, or the uniqueness of a solution appropriate for the one universe we can observe. Both expectations seem reasonable, though immodest. But they are conceptually very different and even, maybe surprisingly, inconsistent with each other as physical properties. For let us assume that we have a theory from which we know that it has one and only one solution. Provided that there is sufficient computational access to that theory, it is falsifiable by comparing properties of the solution with observations in the universe. Now, our observational access to the universe will always be limited and so, even if the one solution of our theory does agree with observations, we can always find ways to change the theory without being in observational conflict. The theory thus cannot be unique. Changing it in the described situation may only violate other, external conditions, which are not observable.

The converse, that a unique theory cannot have a unique solution, follows by logically reversing the above argument. However, one has to be careful about different notions of uniqueness of a theory. It is clear from the above argument that uniqueness of a theory can be realized only under external, such as mathematical, conditions, which always are a matter of taste and depend on existing knowledge. Nevertheless, the statement seems to be supported by current realizations of quantum gravity. String theory is one example, in which the supposed uniqueness of the theory is far outweighed by the non-uniqueness of its solutions. It should also be noted that the uniqueness of a theory is not falsifiable, and therefore not a scientific claim, unless its solutions are sufficiently restricted within the theory. Otherwise, one can always find new solutions if one comes in conflict with observations. A theory itself, however, is falsifiable if it implies characteristic effects for its solutions, even though it may otherwise be ambiguous.


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