Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Symmetry Group of Gaussian States in $L^2 (\mathbb{R}^n)$   This is a continuation of the expository article \cite{krp} with some new
remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert
space $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position
covariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ be
the group of all unitary operators in $L^2 (\mathbb{R}^n)$ conjugations by
which leave $S_n$ invariant. Here we prove the following results. $K_n$ is a
closed convex set for which a matrix $S$ is an extreme point if and only if
$S=\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n,
\mathbb{R}).$ Every element in $K_n$ is of the form $\frac{1}{2} (L^{T} L +
M^{T} M)$ for some $L,M$ in $Sp (2n, \mathbb{R}).$ Every Gaussian state in $L^2
(\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\mathbb{R}^{2n}).$
Any element $U$ in the group $\mathcal{G}_n$ is of the form $U = \lambda W
({\bm {\alpha}}) \Gamma (L)$ where $\lambda$ is a complex scalar of modulus
unity, ${\bm {\alpha}} \in \mathbb{C}^n,$ $L \in Sp (2n, \mathbb{R}),$ $W({\bm
{\alpha}})$ is the Weyl operator corresponding to ${\bm {\alpha}} $ and $\Gamma
(L)$ is a unitary operator which implements the Bogolioubov automorphism of the
Lie algebra generated by the canonical momentum and position observables
induced by the symplectic linear transformation $L.$

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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
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The categories of this paper are: econ
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
1
The categories of this paper are: cs
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The categories of this paper are: q-alg