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

• Published in: The journal of high energy physics 2020-11, Vol.2020 (11), p.1-26, Article 13
• Main Author: Padmanabhan, T.
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Evolution; Field theory; High energy physics; Models of Quantum Gravity; Nonperturbative Effects; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum mechanics; Quantum phenomena; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativistic effects; Relativity; Relativity Theory; Spacetime; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2020)013

A bstract 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. G NR ( x ) = 〈 x 2 | U NR ( t )| x 1 〉 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 G R ( 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 〈 x 2 | U R ( t )| x 1 〉 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 〈 x 2 | U QG ( t )| x 1 〉. I describe these results and explore several consequences. It turns out that the evolution operator U QG ( 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.