Go to previous page Go up Go to next page

4.7 Isotropy: Phenomenological higher curvature corrections

In addition to the behavior of effective densities, there is a further consequence of a discrete triad spectrum: Its conjugate c cannot exist as a self-adjoint operator because in that case it would generate arbitrary continuous translations in p. Instead, only exponentials of the form exp(iδc) can be quantized, where the parameter δ is related to the precise form of the discreteness of p. Corresponding operators in a triad representation of wave functions are finite differences rather than infinitesimal differentials, which also expresses the underlying discreteness. (This is analogous to quantum mechanics on a circle, although in loop quantum cosmology the discreteness is not based on a simple periodic identification but rather a more complicated compactification of the configuration space 5.2.)

Any classical expression when quantized has to be expressed in terms of functions of c, such as δ− 1sin (δc ), while common classical expressions, e.g., the Hamiltonian constraint (23View Equation), only depend on c directly or through powers. When δ −1sin(δc) occurs instead, the classical expression is reproduced at small values of c, i.e., small extrinsic curvature, but higher-order corrections are present [52Jump To The Next Citation Point36Jump To The Next Citation Point146Jump To The Next Citation Point299Jump To The Next Citation Point]. This provides a further discreteness effect in loop cosmology, which is part of the effective equations. The effect has several noteworthy properties:


  Go to previous page Go up Go to next page