The relation between the symplectic group $Sp(4, \mathbb{R})$ and its Lie algebra: its application in polymer quantum mechanics

TL;DR

This paper explores the relationship between the symplectic group $Sp(4, \, \mathbb{R})$ and its Lie algebra, applying the findings to analyze squeezed states and polymer quantum mechanics, revealing dispersion similarities with traditional squeezed states.

Contribution

It establishes a connection between the symplectic group's Lie algebra and matrices, applying this to quantum mechanics and polymer states to understand their properties.

Findings

Polymer dispersions match those of traditional squeezed states.

Derived special cases of symplectic matrices relevant to quantum states.

Applied Lie algebra relations to polymer quantum mechanics.

Abstract

In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.