The Immirzi ambiguity arises in loop quantum gravity when geometric operators are represented in terms of different connections that are related by means of an extended Wick transform. We analyze the action of this transform in gravity coupled with matter fields and discuss its analogy with the Wick rotation on which the Thiemann transform between Euclidean and Lorentzian gravity is based. In addition, we prove that the effect of this extended Wick transform is equivalent to a constant scale transformation as far as the symplectic structure and kinematical constraints are concerned. This equivalence is broken in the dynamical evolution. Our results are applied to the discussion of the black hole entropy in the limit of large horizon areas. We first argue that, since the entropy calculation is performed for horizons of fixed constant area, one might in principle choose an Immirzi parameter that depends on this quantity. This would spoil the linearity with the area in the entropy formula. We then show that the Immirzi parameter appears as a constant scaling in all the steps where dynamical information plays a relevant role in the entropy calculation. This fact, together with the kinematical equivalence of the Immirzi ambiguity with a change of scale, is used to preclude the potential non-linearity of the entropy on physical grounds.

very minor stylistic changes, version published in Phys. Rev. D