Minimal quantities and measurability conception in quantum theory, gravity and thermodynamics

Abstract

At the present time the majority of researchers agree that a minimal length is involved at high (Planck’s) energies. But all the currently used low-energy theories (quantum mechanics and quantum field theory, gravity, etc.) are continuous, i. e. the minimal length in them is zero. This article presents an alternative approach when the hypothetical minimal length is nonzero at all the energy scales. By this approach the definition of measurability and of measurable quantities is given, within the scope of which there is no abstract infinitesimal increment of space-time coordinates. As a result, the initial low-energy continuous theory (quantum theory or general relativity) inevitably must be replaced by a discrete theory that gives very close results but operates with absolutely other mathematical apparatus. A real discreteness is exhibited only at high energies which are close to the Planck energies. A analogous concept (dual) of measurability is defined in thermodynamics on the basis of a hypothetical minimal inverse temperature. Based on this notions, some implications are obtained, in particular, for gravitational thermodynamics of black holes at all the energy scales, quantum corrections of the basic quantities in the general case. Besides, the measurable variant of General Relativity (GR) is constructed and it is shown that this variant represents its deformation. In the general form it is demonstrated that all the basic ingredients of GR have their measurable analogs.