Representation theory of Gaussian unitary transformations for bosonic and fermionic systems

TL;DR

This paper analyzes the representation theory of Gaussian unitary transformations in bosonic and fermionic systems, focusing on the sign ambiguities in their double cover groups and providing explicit formulas for related expectation values.

Contribution

It introduces a detailed analysis of the sign ambiguities in the metaplectic and spin group representations and offers closed-form formulas for expectation values involving quadratic Hamiltonians.

Findings

Closed formulas for expectation values of Gaussian states with quadratic Hamiltonians.

Efficient description of group multiplications in double cover groups.

Explicit parametrization of metaplectic and spin groups using symplectic and orthogonal elements.

Abstract

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$, where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.