Reparametrization form invariance

The original action integral (15.2) and the reparametrized version (15.10) look manifestly different. However, the reparametrized version has an extraordinary property: it is form invariant to further reparametrizations.

A function F of n independent variables q = (q1, q2,. . ., qn) is form invariant to the transformation q{ ^ qi (q), i =1,2,..., n, if F (q') = F (q).

Form invariance has been invoked as a fundamental principle in the development of relativistic equations. The basic idea is that all ‘reasonable’ frames of reference are ‘as good as each other’ in the description of physics, and therefore the laws of physics should take the same functional form in each of those frames.

We should take the following points into account when applying this idea:

  • 1. What constitutes a ‘reasonable’ frame of reference is contextual. For example, if we wish to ignore gravitation and/or curvature of spacetime, then we generally restrict our attention to transformations between inertial frames of reference. This is the SR scenario. If on the other hand we want to discuss the GR scenario involving curvature, we have to use generalized coordinates with suitable conditions on transformations between frames.
  • 2. Too much faith in the principle of general covariance might be misplaced. For example, the idea that all inertial frames of reference are equivalent in SR is at odds with the existence of the symmetry frame of the cosmic background radiation field at any given point. By this we mean that we can use the cosmic background radiation field to determine a preferred class of frames at any point in the universe. Since this is believed to be a product of the evolution of the universe, it is hard to accept the argument that such a frame is ‘accidental’, or not really special.
  • 3. We should be wary of any seductive arguments based on symmetry and/or intuitive logic. We have referred in previous chapters to the suggestion that the Ancient Greeks did not develop physics because they considered experiments, which are artificial, to be the wrong way to discover intrinsic truth. A more recent example of misplaced confidence in symmetry occurred in the 1950s, when experiments proved that the universe does not respect left- right symmetry, the so-called overthrow of parity conservation experiment [Lee & Yang, 1956; Wu et al., 1957].
  • 4. Form invariance as a principle of physics is based on a classical notion that SUOs have intrinsic properties independent of observers: on that basis, descriptions of their properties should be independent of any particular observers. However, there is a theorem in quantum mechanics due to Kochen and Specker [Kochen & Specker, 1967] that states that quantum states of SUOs cannot ‘have’ classical properties in the manner of classical states in CM. Observers must be taken into account. Therefore, it seems to us that form invariance is at best a property that we should build into classical descriptions of SUOs.
  • 5. There is no proof that the laws of physics have to be ‘beautiful’, a common theme of many mathematically inspired but as yet unverified approaches to physics, such as string theory. We should be prepared to find that the laws of physics are cumbersome and ugly, if that is what experiment eventually supports. In cosmology, for example, the cosmological principle states that there is no special place or direction in space, over large enough distance scales. Observations of galaxies at the furthest limit of observation may yet reveal this to be false. Form invariance to arbitrary coordinate transformations, otherwise known in GR as general covariance, is at best a guide for the development of potential theory.
  • 6. It is not the case that general covariance implies GR. SR can be rewritten in a generally covariant form but that theory does not discuss gravitation. True gravity comes about from spacetime curvature, which cannot be induced by mere coordinate transformations.
  • 7. Time is a phenomenon that cannot be separated from the observers of SUOs. It seems to us that too much adherence to the classical principle of general covariance, which does not take into account specific observers, might well be at the root of the failure to ‘quantize’ GR. We take the view that quantum mechanics is not a theory about bizarre particle/wave-like properties of SUOS, but a theory of observation. Form invariance is an emphasis on the properties of SUOs. Therefore, theorists who tend to focus too readily and too much on the construction of a generally covariant formulation of quantum gravity at the expense of looking at the observational physics of gravity stand of good chance of getting nowhere, as the history of the last seven decades has shown. We shall return to this point in later chapters.

To demonstrate the form invariance of (15.10) we consider a second repara- metrization, from X to X'. We take X = X(X') to be a differentiable function of X' such that dX/dX' > 0 and define

with X' running from 0 to 1. Then we find

which demonstrates explicitly the form invariance of the action integral under this particular class of reparametrizations of time. Note that on the right-hand side of (15.14), t = dt/dX whilst t' = dt'/dX', and so on.

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