Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

Abstract

The propagator which evolves the wave-function in non-relativistic quantum mechanics, can be expressed as a matrix element of a time evolution operator: i.e. GNR(x) = 〈x2|UNR(t)|x1〉 in terms of the orthonormal eigenkets |x〉 of the position operator. In quantum field theory, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator GR(x) as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in quantum field theory, as a matrix element 〈x2|UR(t)|x1〉 for a suitably defined time evolution operator and (non-orthonormal) kets |x〉 labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this quantum-gravity-corrected propagator can be expressed as a matrix element 〈x2|UQG(t)|x1〉. I describe these results and explore several consequences. It turns out that the evolution operator UQG(t) becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalized to any ultrastatic curved spacetime.

Using the Evolution Operator to Classify Evolution Algebras

“…Among the most recent references referring to the applications of the evolution operator (it cannot be said that there are many in the literature), the two following ones can be cited: that by Absalamov and Rozikov [22], who investigate in the field of biology the dynamical system generated by a gonosomal evolution operator of sex linked inheritance depending on parameters and the one by Padmanabhan, related to physics, who studies the Planck scale by the modification of the time evolution operator due to the quantum structure of spacetime [23]. We have tried to find some applications of this operator related to both scientific and non-scientific fields, as they might be in the latter case economics and commerce, for example, although we have not quite achieved what we wanted.…”

Discrete heat kernel, UV modified Green’s function, and higher derivative theories

“…In this case, the Green's function containing higher order terms in the momentum space is obtained. It coincides with Pauli-Villars' Green function in its simplest form [14][15][16] 4 . We identify the higher derivative theory as the effective field theory that yields the UV modified Green's function.…”

“…Taking the effects of quantum gravity into account, we can expect the emergence of fundamental lengths in effective field theories. In recent years, various types of UV deformed Green's functions incorporating a fundamental length have been proposed based on the expression of the Green's function obtained by the heat kernel method [1][2][3][4][5][6][7][8][9][10][11][12] 3 . Putting a lower limit of integral on the Schwinger's proper-time parameter in the standard expression is known as the easiest way to regularize the Green's function in the heat kernel method.…”

“…In our previous paper [13], we calculated the vacuum expectation values of the stress tensor in nontrivial background spaces using the UV deformed Green's function [1][2][3][4][5][6][7][8][9][10][11][12]. The dependence on the cutoff constant (which may be identified with the Planck length) was clarified by using numerical calculations.…”

“…In each case, attention is paid to the cutoff dependence of the physical quantity. The dimensionful cutoff constant can be taken as a parameter in a mere regularization, but similarly to the introduction of the aforementioned UV deformed Green's function [1][2][3][4][5][6][7][8][9][10][11][12], it may be resulted from yet-unknown quantum gravity theory (possibly directly related to the Planck length). Thus, the cutoff constant can be considered as a fundamental constant in nature.…”