The Path Integral Quantization corresponding to the Deformed Heisenberg Algebra

TL;DR

This paper explores a deformation of the Heisenberg algebra aligned with generalized uncertainty principles and doubly special relativity, deriving a path integral propagator that reveals significant differences from standard quantum mechanics.

Contribution

It introduces a novel approach to quantization with deformed algebra, providing explicit propagator expressions and highlighting differences from traditional quantum mechanics.

Findings

Derived the path integral propagator for the deformed Heisenberg algebra.

Explicitly evaluated the free particle propagator under the deformation.

Showed that the deformation leads to distinct results even for free particles.

Abstract

In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity shows very much different result.