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

Guillermo Chacón-Acosta<EMAIL_ADDRESS>Departamento de Matemáticas
Aplicadas y Sistemas, Universidad Autónoma Metropolitana Cuajimalpa, Vasco de
Quiroga 4871, Ciudad de México 05348, MEXICO  Angel Garcia-Chung
<EMAIL_ADDRESS>Departamento de Física, Universidad Autónoma
Metropolitana - Iztapalapa,
San Rafael Atlixco 186, Ciudad de México 09340, México  Universidad
Panamericana,
Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, C.P. 01050 Ciudad de
México, México

###### 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.

###### Contents

  1. I Introduction
  2. II $Sp(4,\mathbb{R})$ group analysis and $sp(4,\mathbb{R})$ Lie algebra
  3. III Quantum relations and examples
    1. III.1 Relation between $sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$
    2. III.2 Examples
      1. III.2.1 Case ${\bf a},{\bf c}\neq{\bf 0}$ and ${\bf b}={\bf 0}$.
      2. III.2.2  Case ${\bf c}={\bf a}=\mbox{diag}(a_{11},a_{22})$ and ${\bf b}=$diag$(b_{11},b_{22})$.
      3. III.2.3 Case ${\bf a}={\bf c}={\bf 0}$ and ${\bf b}\neq{\bf 0}$.
  4. IV Quantum representation and its applications
    1. IV.1 Schrödinger representation of $Sp(2n,\mathbb{R})$
    2. IV.2 Schrödinger representation of the squeeze operator for a bi-partite system.
    3. IV.3 Covariance matrix for squeezed states
  5. V Squeezed states in polymer quantum mechanics
  6. VI Conclusions
  7. VII Acknowledgments
  8. VIII Appendix: calculation of ${\bf m}^{2n}$
  9. IX Appendix: Series analysis
  10. X Appendix: Covariance matrix coefficients

##  I Introduction

Squeezed states are broadly used in many areas of physics walls1983squeezed ;
schnabel2017squeezed ; walls2007quantum ; adesso2014continuous ;
braunstein2005quantum . Of particular interest is the use of these squeezed
states in cosmology grishchuk1990squeezed ; polarski1996semiclassicality ;
lesgourgues1997quantum ; kiefer1998quantum ; guth1985quantum ;
martin2016quantum ; albrecht1994inflation ; gasperini1993quantum ;
grain2019squeezing , specifically when arguing for the emergence of semi-
classical behavior in the early universe. Loop Quantum Cosmology (LQC)
ashtekar2003mathematical ; bojowald2010canonical ; bojowald2011quantum ;
ashtekar2006quantum ; ashtekar2006quantum2  is another scenario in which
squeezed states are relevant. There, squeezed states for a single-mode show
some of the features of the quantum bounce and closely approximate solutions
to the classical Einstein equations taveras2008corrections ;
mielczarek2012gaussian ; gazeau2013quantum ; diener2014numerical ;
diener2014numerical2 ; corichi2011coherent . The squeezed states used in LQC
are constructed by hand, imposing the Gaussian form of the states to obtain
the squeezing nature of the dispersion relations. Moreover, the states
describe systems with only one degree of freedom, i.e., single-mode squeezed
states milburn1984multimode ; gerry2005introductory .

In quantum optics, squeezed states can be used to improve the sensitivity of
measurement devices beyond the usual quantum noise limits simon1988gaussian ;
braunstein2005quantum ; walls2007quantum ; adesso2014continuous ;
ma1990multimode ; schnabel2017squeezed ; pirandola2009correlation ;
weedbrook2012gaussian . They are defined by the squeeze operators’ action on
coherent states, or the vacuum state walls2007quantum ; adesso2014continuous ;
braunstein2005quantum . These operators are defined within the Fock
representation using the annihilation and creation operators, or in the Wigner
representation, using the Wigner functional. A particular squeezed state used
in quantum optics is the two-mode squeezed state which plays a prominent role
in the study of entanglement for bipartite systems. Particularly in the limit
when the amount of squeezing is infinitely large, the states become EPR-like
states EPRpaper .

Based on the relevance that squeezed states play in cosmology, LQC, and
quantum optics, one might ask whether there is a relation between them and
whether it is possible to obtain squeezed states in LQC the same way squeezed
states are defined in quantum optics. Recall that the construction used in LQC
for the squeezed states is somewhat artificial and does not correspond to any
mechanism in the cosmological events. Hence, exploring whether LQC formulation
admits an operator similar to the squeeze operator and whose action on some
state yields a squeezed state might pave the way to construct such a mechanism
in LQC.

To do so, one must consider that in the LQC, the representation of the
operators is not weakly continuous, hence the Fock representation is not
suitable for the physical description. Instead, the Schrödinger
representation, which is the scheme inherited from the quantization procedure,
seems to be the natural scheme to be considered ashtekar2003quantum ;
corichi2007polymer ; flores2013propagators ; velhinho2007quantum ;
pawlowski2014separable ; Garcia-Chung:2020cag . Despite the Schrödinger
representation of the squeeze operator might be obtained using the
representation of the infinitesimal squeeze operator via the exponential map
hall2018theory , in LQC this cannot be done. In addition to the mathematical
challenge that this operation requires in standard quantum mechanics, this is
not possible in LQC because there is no infinitesimal representation of the
squeeze operator. Therefore, in the present work, we will use the
representation of the symplectic group $Sp(2n,\mathbb{R})$ in polymer quantum
mechanics Garcia-Chung:2020cag . It is worth mentioning that Polymer Quantum
Mechanics (PQM) can be considered as a “toy model” of LQC because they share
Hilbert spaces with the same mathematical structures. As a result, the
representation of the symplectic group in PQM is mathematically the same as in
LQC.

To analyze the squeeze operator corresponding to the bipartite squeezed states
in LQC, we will provide the relation between the Lie algebra of the symplectic
group $sp(4,\mathbb{R})$ and the Lie group $Sp(4,\mathbb{R})$. As far as the
authors’ knowledge, this relation has not been reported before. With this
result, we show some specific cases and then move to the analysis within
polymer quantum mechanics. Also, this relation allows us to describe the
single-mode squeeze operator (specifically, the product of two single-mode
operators) as a particular case of a symplectic matrix in $Sp(4,\mathbb{R})$.

We will show that the squeezed states derived in this way for LQC share the
same features as those used in quantum optics. In particular, the
correlations’ structure is the same for both the single-mode and the two-mode
squeezed states. However, there is no need for a Gaussian-like structure for
the initial states upon which the polymer squeeze operators act and such
structure is absent in the polymer squeezed states.

This paper is organized as follows: in section (II) we calculate the relation
between $sp(4,\mathbb{R})$ and $Sp(4,\mathbb{R})$. In section (III) we discuss
the isomorphism between $sp(4,\mathbb{R})$ and the second-order polynomial
operators ${\cal P}(2,\mathbb{R})$ and provide some examples. In Section (IV),
we show some of the applications of the results given in section III; in
particular, we determine the covariance matrix for the squeezed states in
standard quantum mechanics. In Section (V), we analyze the squeeze operators’
representation in polymer quantum mechanics and construct the polymer squeezed
state. We also calculate the dispersion relations and show that they are equal
to those obtained for the standard squeezed states. We give our conclusion in
Section (VI).

##  II $Sp(4,\mathbb{R})$ group analysis and $sp(4,\mathbb{R})$ Lie algebra

In this section we will detail the relation between an arbitrary element of
$sp(4,\mathbb{R})$ and its corresponding element in the group
$Sp(4,\mathbb{R})$. This relation is the main result of this section and has
not been reported as far as we know. First, let us introduce some preliminary
concepts and notation, which we will use throughout the paper, to make the
presentation self-contained.

Let us begin by considering the Poisson manifold $(\mathbb{R}^{2n},\\{,\\})$
with Poisson bracket for the coordinates $q_{j}$ and momenta $p_{j}$
($j=1,2,3,\dots,n$) given by

$\left\\{q_{j},q_{k}\right\\}=0,\qquad\left\\{p_{j},p_{k}\right\\}=0,\qquad\left\\{q_{j},p_{k}\right\\}=\delta_{jk}.$
(1)

These coordinates are collected using the array
$\vec{Y}^{T}=(q_{1},\,p_{1},\,q_{2},\,p_{2},\,\ldots,\,q_{n},\,p_{n})$ for
which the Poisson bracket (1) takes the form

$\left\\{\vec{Y},\vec{Y}^{T}\right\\}=\left(\begin{array}[]{cccc}{\bf J}&{\bf
0}&\cdots&{\bf 0}\\\ {\bf 0}&{\bf J}&\cdots&{\bf 0}\\\
\vdots&\vdots&\vdots&\vdots\\\ {\bf 0}&{\bf 0}&\cdots&{\bf
J}\end{array}\right)={\bf 1}_{n\times n}\otimes{\bf J},$ (2)

where the ${\bf 0}$ is the $2\times 2$ null matrix, ${\bf 1}_{n\times n}$ is
the identity matrix and the matrix ${\bf J}$ is given by

${\bf J}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right).$ (3)

The group action over the manifold $\mathbb{R}^{2n}$ is

$Sp(2n,\mathbb{R})\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n};\left({\bf
M},\vec{Y}\right)\mapsto\vec{Y}^{\prime T}={\bf M}\;\vec{Y}^{T},$ (4)

provided that the matrix ${\bf M}$ satisfies the condition

$\left({\bf 1}_{n\times n}\otimes{\bf J}\right)={\bf M}\left({\bf 1}_{n\times
n}\otimes{\bf J}\right){\bf M}^{T},$ (5)

where ${\bf M}^{T}$ is the transpose matrix. That is, the symplectic group
$Sp(2n,\mathbb{R})$ can be defined as the set of $2n\times 2n$ real matrices
satisfying (5) and, additionally, its group action on the Poisson manifold
$(\mathbb{R}^{2n},\\{,\\})$ is given by (4). Note that a “coordinatization” of
$({\mathbb{R}^{2n}},\\{,\\})$ different from $\vec{Y}$ yields a condition for
the symplectic group matrices different to that in (5). To show this, consider
now the array $\vec{X}^{T}=(\vec{q}^{\;T}\;,\;\vec{p}^{\;T})$ where
$\vec{q}^{\;T}=(q_{1},q_{2},\dots,q_{n})$ and
$\vec{p}^{\;T}=(p_{1},p_{2},\dots,p_{n})$ are the coordinates on the space
$\mathbb{R}^{2n}$. The Poisson bracket for this array is given by

$\left\\{\vec{X},\vec{X}^{T}\right\\}=\left(\begin{array}[]{cc}{\bf 0}&{\bf
1}_{n\times n}\\\ -{\bf 1}_{n\times n}&{\bf 0}\end{array}\right)={\bf
J}\otimes{\bf 1}_{n\times n}.$ (6)

The group action is now given by
$Sp(2n,\mathbb{R})\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n};\left(\widetilde{\bf
M},\vec{X}\right)\mapsto\vec{X}^{\prime}$ where $\vec{X}^{\prime}$ is

$\vec{X}^{\prime T}=\widetilde{\bf M}\,\vec{X}^{T},$ (7)

and the matrix $\widetilde{\bf M}$ satisfies

$\left({\bf J}\otimes{\bf 1}_{n\times n}\right)=\widetilde{\bf M}\left({\bf
J}\otimes{\bf 1}_{n\times n}\right)\widetilde{\bf M}^{T}.$ (8)

Hence, both conditions (5) and (8), can be considered as definitions for the
symplectic group in different “coordinatizations” of the phase space
$\mathbb{R}^{2n}$. Naturally, both group actions $\widetilde{\bf M}$ and ${\bf
M}$ are related via the similarity transformation ${\bf\Gamma}(n)$
adesso2014continuous  as

$\widetilde{\bf M}={\bf\Gamma}(n)\;{\bf M}\;{\bf\Gamma}^{-1}(n),$ (9)

where ${\bf\Gamma}(n)$ is given by

$\vec{X}^{T}={\bf\Gamma}(n)\,\vec{Y}^{T},$ (10)

and is such that ${\bf\Gamma}^{T}(n)={\bf\Gamma}^{-1}(n)$. Since the present
work concerns the case where $n=2$, it is worth showing the explicit form of
${\bf\Gamma}(2)$ which is

${\bf\Gamma}(2)=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{array}\right).$ (11)

Having provided the two group actions over the manifold $\mathbb{R}^{2n}$
using different “coordinatizations” and their relation for arbitrary $n$, let
us now focus on the symplectic group $Sp(4,\mathbb{R})$. According to (5) this
group is given by $4\times 4$ real matrices ${\bf M}$ for which the following
condition holds

$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right)={\bf
M}\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right){\bf
M}^{T},$ (12)

The matrix ${\bf M}$ can be written in block form as

${\bf M}:=\left(\begin{array}[]{cc}{\bf A}&{\bf B}\\\ {\bf C}&{\bf
D}\end{array}\right),$ (13)

where the $2\times 2$ block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf
D}$ satisfy the conditions

${\bf J}={\bf A}{\bf J}{\bf A}^{T}+{\bf B}{\bf J}{\bf B}^{T}={\bf C}{\bf
J}{\bf C}^{T}+{\bf D}{\bf J}{\bf D}^{T},\qquad{\bf 0}={\bf A}{\bf J}{\bf
C}^{T}+{\bf B}{\bf J}{\bf D}^{T},$ (14)

which result from (12).

The Lie algebra of $Sp(4,\mathbb{R})$, denoted as $sp(4,\mathbb{R})$, is given
by $4\times 4$ matrices ${\bf m}$ such that the exponential map hall2018theory
of the Lie algebra element ${\bf m}$ yields symplectic matrices ${\bf M}$
close to the identity, i.e.,

${\bf M}=e^{\bf m}:={\bf 1}+{\bf m}+\frac{1}{2}{\bf
m}^{2}+\dots+\frac{1}{n!}{\bf m}^{n}+\dots$ (15)

It can be shown that the matrices in $sp(4,\mathbb{R})$ can be written as the
product

${\bf m}=\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right){\bf
L},$ (16)

where ${\bf L}$ is a real symmetric matrix written in block form as

${\bf L}=\left(\begin{array}[]{cc}{\bf a}&{\bf b}\\\ {\bf b}^{T}&{\bf
c}\end{array}\right),$ (17)

and where ${\bf b}$ is a $2\times 2$ real matrix, whereas ${\bf a}$ and ${\bf
c}$ are also real but $2\times 2$ symmetric matrices.

If a matrix ${\bf M}$ can be written as in (15), then its inverse ${\bf
M}^{-1}$, its transpose ${\bf M}^{T}$ and the $n$-power matrix $({\bf
M})^{n}$, can be written respectively as follows

$\displaystyle{\bf M}^{-1}$ $\displaystyle=$
$\displaystyle\exp{\left[-\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}\right]},$ (20) $\displaystyle{\bf M}^{T}$
$\displaystyle=$ $\displaystyle-\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)\,{\bf M}^{-1}\,\left(\begin{array}[]{cc}{\bf J}&0\\\
0&{\bf J}\end{array}\right),$ (25) $\displaystyle({\bf M})^{n}$
$\displaystyle=$ $\displaystyle\exp{\left[\left(\begin{array}[]{cc}{\bf
J}&0\\\ 0&{\bf J}\end{array}\right)(n{\bf L})\right]}.$ (28)

Thus, the Lie algebra multiplication in $sp(4,\mathbb{R})$ is given by the
matrix commutator $[,]_{m}$. When this multiplication acts on two arbitrary
elements ${\bf m}_{1}$ and ${\bf m}_{2}$ gives the element ${\bf m}_{3}$
defined as

${\bf m}_{3}:=\left[{\bf m}_{1},{\bf
m}_{2}\right]_{m}=\left[\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}_{1},\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right){\bf L}_{2}\right]=\left(\begin{array}[]{cc}{\bf J}&0\\\
0&{\bf J}\end{array}\right){\bf L}_{3},$ (29)

where the matrix ${\bf L}_{3}$ is also a real symmetric matrix with components
of the form

${\bf L}_{3}=\left(\begin{array}[]{cc}{\bf a}_{1}{\bf J}{\bf a}_{2}+{\bf
b}_{1}{\bf J}{\bf b}^{T}_{2}-{\bf a}_{2}{\bf J}{\bf a}_{1}-{\bf b}_{2}{\bf
J}{\bf b}^{T}_{1}&{\bf a}_{1}{\bf J}{\bf b}_{2}+{\bf b}_{1}{\bf J}{\bf
c}_{2}-{\bf a}_{2}{\bf J}{\bf b}_{1}-{\bf b}_{2}{\bf J}{\bf c}_{1}\\\ {\bf
b}^{T}_{1}{\bf J}{\bf a}_{2}+{\bf c}_{1}{\bf J}{\bf b}^{T}_{2}-{\bf
b}^{T}_{2}{\bf J}{\bf a}_{1}-{\bf c}_{2}{\bf J}{\bf b}^{T}_{1}&{\bf c}_{1}{\bf
J}{\bf c}_{2}+{\bf b}^{T}_{1}{\bf J}{\bf b}_{2}-{\bf c}_{2}{\bf J}{\bf
c}_{1}-{\bf b}^{T}_{2}{\bf J}{\bf b}_{1}\end{array}\right),$ (30)

hence, ${\bf m}_{3}$ is clearly an element in $sp(4,\mathbb{R})$.

Up to this point, we introduced the main concepts and notations required to
derive the relation between $sp(4,\mathbb{R})$ and its corresponding Lie group
$Sp(4,\mathbb{R})$. Let us proceed then to obtain the explicit relation
between the block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf D}$ and
the Lie algebra element ${\bf L}$. It is worth noting that the following
procedure can be applied to higher-order symplectic groups $Sp(2n,\mathbb{R})$
for $n\geq 3$, being this the main reason for its exposition in this section.

Let us collect the even and odd terms of the expansion in (15) as follows

$\displaystyle{\bf M}$ $\displaystyle=$ $\displaystyle\left[{\bf
1}+\frac{1}{2!}{\bf m}^{2}+\dots+\frac{1}{(2n)!}{\bf m}^{2n}+\dots\right]+{\bf
m}\left[{\bf 1}+\frac{1}{3!}{\bf m}^{2}+\dots+\frac{1}{(2n+1)!}{\bf
m}^{2n}+\dots\right],$ (31)

where ${\bf m}^{2}$ takes the form

${\bf m}^{2}=\left(\begin{array}[]{cc}-(\det{\bf a}+\det{\bf b}){\bf
1}_{2\times 2}&{\bf J}{\bf d}\\\ -{\bf J}{\bf d}^{T}&-(\det{\bf b}+\det{\bf
c}){\bf 1}_{2\times 2}\end{array}\right),$ (32)

and the matrix ${\bf d}$ is defined as

${\bf d}={\bf a}{\bf J}{\bf b}+{\bf b}{\bf J}{\bf c}.$ (33)

As can be seen from the expansion (31), to obtain the expression for ${\bf M}$
we need first to determine ${\bf m}^{2n}$. In Appendix VIII we obtain the
expression for ${\bf m}^{2n}$ given in Eq. (119). Let us replace this result
in the series expansion (31), which, after collecting the even and odd terms,
gives the following

$\displaystyle{\bf A}$ $\displaystyle=$
$\displaystyle\alpha^{(e)}+(\alpha^{(o)}-\beta^{(o)}\det{\bf b})\,{\bf
J}\,{\bf a}+\beta^{(o)}{\bf J}\,{\bf b}\,{\bf J}\,{\bf c}\,{\bf J}\,{\bf
b}^{T},$ (34) $\displaystyle{\bf B}$ $\displaystyle=$
$\displaystyle(\gamma^{(o)}-\beta^{(o)}\det{\bf a})\,{\bf J}\,{\bf
b}+\beta^{(e)}({\bf J}\,{\bf a}\,{\bf J}\,{\bf b}+{\bf J}\,{\bf b}\,{\bf
J}\,{\bf c})+\beta^{(o)}{\bf J}\,{\bf a}\,{\bf J}\,{\bf b}\,{\bf J}\,{\bf c},$
(35) $\displaystyle{\bf C}$ $\displaystyle=$
$\displaystyle(\alpha^{(o)}-\beta^{(o)}\det{\bf c})\,{\bf J}\,{\bf
b}^{T}+\beta^{(e)}({\bf J}\,{\bf b}^{T}\,{\bf J}\,{\bf a}+{\bf J}\,{\bf
c}\,{\bf J}\,{\bf b}^{T})+\beta^{(o)}{\bf J}\,{\bf c}\,{\bf J}\,{\bf
b}^{T}\,{\bf J}\,{\bf a},$ (36) $\displaystyle{\bf D}$ $\displaystyle=$
$\displaystyle\gamma^{(e)}+(\gamma^{(o)}-\beta^{(o)}\det{\bf b})\,{\bf
J}\,{\bf c}+\beta^{(o)}{\bf J}\,{\bf b}^{T}\,{\bf J}\,{\bf a}\,{\bf J}\,{\bf
b}.$ (37)

The coefficients $\alpha^{(e)}$, $\alpha^{(o)}$, $\beta^{(e)}$, $\beta^{(o)}$,
$\gamma^{(e)}$ and $\gamma^{(o)}$ were defined in the appendix IX.

These expressions for the matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf
D}$ link the components of the Lie algebra element ${\bf L}$ with the
corresponding symplectic matrix ${\bf M}$ and constitute the main result of
this section. Note also the “non-linear matrix relation” between the Lie
algebra elements and the group elements, particularly the role of the block
matrix ${\bf b}$.

A remarkable and direct application of this result is that it allows us to
compute the symplectic eigenvalues of the matrix ${\bf M}$. To do so, recall
that the characteristic polynomial for a $4\times 4$ matrix ${\bf M}$ with
$\det({\bf M})=1$ can be written in terms of the trace of its first three
powers by the expression

$\displaystyle\Lambda^{4}-(\mbox{Tr}({\bf
M}))\Lambda^{3}+\frac{\left[(\mbox{Tr}({\bf M}))^{2}-(\mbox{Tr}({\bf
M}^{2}))\right]}{2}\Lambda^{2}+\left[-\frac{(\mbox{Tr}({\bf
M}))^{3}}{6}+\frac{(\mbox{Tr}({\bf M}))(\mbox{Tr}({\bf M}^{2}))}{2}+\right.$
$\displaystyle\left.-\frac{(\mbox{Tr}({\bf M}^{3}))}{3}\right]\Lambda+1=0.$
(38)

where $\Lambda$ are the eigenvalues of the arbitrary matrix ${\bf M}$ and
$\mbox{Tr}({\bf M})$ is the trace of the matrix.

Using the relations (34)–(37) we obtain that the trace $\mbox{Tr}({\bf M})$ is
given by

$\mbox{Tr}({\bf
M})=2(\alpha^{(e)}+\gamma^{(e)})=\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right),$
(39)

where $\alpha^{(e)}$ and $\gamma^{(e)}$ are given in (157) and (161),
respectively, and eigenvalues $\lambda_{\pm}$ are given in (142). Moreover,
due to the linearity of the trace and the relation (28), we can verify that
$\mbox{Tr}({\bf M}^{n})$ is given by

$\mbox{Tr}({\bf
M}^{n})=\cosh\left(n\sqrt{\lambda_{+}}\right)+\cosh\left(n\sqrt{\lambda_{-}}\right),$
(40)

from which we obtain the expressions for $\mbox{Tr}({\bf M}^{2})$ and
$\mbox{Tr}({\bf M}^{3})$.

Inserting all these expressions in (38) for a symplectic matrix of the form
(15), and when calculating the roots of the polynomial, we obtain that their
eigenvalues are given by

$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\cosh\left(\sqrt{\lambda_{+}}\right)-\sinh\left(\sqrt{\lambda_{+}}\right)=e^{-\sqrt{\lambda_{+}}},$
(41) $\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\cosh\left(\sqrt{\lambda_{-}}\right)-\sinh\left(\sqrt{\lambda_{-}}\right)=e^{-\sqrt{\lambda_{-}}},$
(42) $\displaystyle\Lambda_{3}$ $\displaystyle=$
$\displaystyle\sinh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)=e^{\sqrt{\lambda_{-}}},$
(43) $\displaystyle\Lambda_{4}$ $\displaystyle=$
$\displaystyle\sinh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)=e^{\sqrt{\lambda_{+}}},$
(44)

Note that if $\lambda_{+}>\lambda_{-}$, then the eigenvalues are ordered as
$\Lambda_{1}<\Lambda_{2}<\Lambda_{3}<\Lambda_{4}$ which coincides with the
results of Williamson’s theorem Arnold ; Gosson2006 .

In milburn1984multimode ; castanos1987matrix  an alternative (and different)
formulation for each of the symplectic group generators was provided. Our
approach, however, not only reproduces to the same expressions for the
generators but also provides a direct relation with the Lie algebra matrix
generators ${\bf a}$, ${\bf b}$ and ${\bf c}$ a point that is absent in
milburn1984multimode ; castanos1987matrix . As a result, we can relate a
broader range of Lie algebra elements with their corresponding group elements.

Let us now show some of the relevant matrices and examples in the next section
in which this result can be applied.

##  III Quantum relations and examples

This section provides three examples where the relation between the Lie
algebra element and the group element is explicit. However, before proceeding,
let us introduce additional concepts and notations (see hall2018theory ;
adesso2014continuous  for more details), which will be relevant for the
quantum description.

###  III.1 Relation between $sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$

Consider the Lie algebra formed by second-order (operator) polynomials,
denoted by ${\cal P}(2,\mathbb{R})$. An arbitrary element $\widehat{s}$ is
given as

$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{a_{11}}{2}\widehat{q}^{2}_{1}+\frac{a_{12}}{2}(\widehat{q}_{1}\widehat{p}_{1}+\widehat{p}_{1}\widehat{q}_{1})+\frac{a_{22}}{2}\widehat{p}^{2}_{1}+b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{12}\widehat{q}_{1}\widehat{p}_{2}+b_{21}\widehat{p}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}+\right.$
(45)
$\displaystyle\left.+\frac{c_{11}}{2}\widehat{q}^{2}_{2}+\frac{c_{12}}{2}(\widehat{q}_{2}\widehat{p}_{2}+\widehat{p}_{2}\widehat{q}_{2})+\frac{c_{22}}{2}\widehat{p}^{2}_{2}\right].$

Here, $\widehat{q}_{j}$ and $\widehat{p}_{j}$, with $j=1,2$, are the position
and momenta operators satisfying the canonical commutation relations
$[\widehat{q}_{j},\widehat{p}_{k}]=i\hbar\delta_{j,k}$, and $a_{ij},\;b_{ij}$
and $c_{ij}$ are all real coefficients. The reason for this notation is that
formally $i\hbar\widehat{s}$ is a self-adjoint operator to be represented in a
Hilbert space ${\cal H}$, hence the exponential map $e^{\widehat{s}}$ gives
rise to a unitary operator in ${\cal H}$. Thus, in this sense, this notation
smoothes the way to the quantum representation analysis in section IV.

It is easy to check that $\widehat{s}$ can be written in the following form

$\widehat{s}=-\frac{i}{2\hbar}\widehat{\bf
R}^{T}\left(\begin{array}[]{cccc}a_{11}&a_{12}&b_{11}&b_{12}\\\
a_{12}&a_{22}&b_{21}&b_{22}\\\ b_{11}&b_{21}&c_{11}&c_{12}\\\
b_{12}&b_{22}&c_{12}&c_{22}\end{array}\right)\widehat{\bf R},$ (46)

that is a symmetric matrix, and where

$\widehat{\bf
R}^{T}=\left(\begin{array}[]{cccc}\widehat{q}_{1}&\widehat{p}_{1}&\widehat{q}_{2}&\widehat{p}_{2}\end{array}\right).$
(47)

Instead of the matrix commutator, the Lie algebra multiplication in ${\cal
P}(2,\mathbb{R})$ is given by the operator commutator $[,]$. Therefore, the
Lie algebra multiplication of two elements ${\widehat{s}_{1}}$ and
${\widehat{s}_{2}}$ gives a third element $\widehat{s}_{3}$ of the form

$\widehat{s}_{3}=\left[\widehat{s}_{1},\widehat{s}_{2}\right]=-\frac{i}{2\hbar}\widehat{\bf
R}^{T}{\bf L}_{3}\widehat{\bf R},$ (48)

where the matrix ${\bf L}_{3}$ is given by (30). Due to ${\bf L}_{3}$ is a
symmetric matrix, the operator ${\widehat{s}}_{3}$ is clearly in ${\cal
P}(2,\mathbb{R})$. Naturally, this result provides the isomorphism between
$sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$, i.e., the map

$\iota:sp(4,\mathbb{R})\rightarrow{\cal P}(2,\mathbb{R});{\bf
m}\mapsto\widehat{s}=\iota({\bf m})=-\frac{i}{2\hbar}\widehat{\bf R}^{T}\,{\bf
L}\,\widehat{\bf R},$ (49)

and this map preserves the linear properties of both Lie algebras, i.e., it is
a Lie algebra isomorphism.

An implication of this isomorphism is that due to ${\cal P}(2,\mathbb{R})$ is
a Lie algebra isomorphic to $sp(4,\mathbb{R})$, then the exponential map of
its elements ($\widehat{s}\mapsto e^{\widehat{s}}$) gives a (quantum) unitary
operator ($e^{\widehat{s}}$) which can be seen as the (quantum) unitary
representation of $Sp(4,\mathbb{R})$ as showed in the following diagram

$\begin{array}[]{ccc}\boxed{sp(4,\mathbb{R})}&\xLongleftrightarrow{\iota}&\boxed{{\cal
P}(2,\mathbb{R})}\\\ \big{\downarrow}&&\big{\downarrow}\\\
\boxed{Sp(4,\mathbb{R})}&\longrightarrow&\boxed{\widehat{Sp}(4,\mathbb{R})}\end{array}$
(50)

We can expect that if a representation of ${\cal P}(2,\mathbb{R})$ in a
Hilbert space ${\cal H}$ is known, then there is also a representation of
$\widehat{Sp}(4,\mathbb{R})$ in ${\cal H}$. However, in some scenarios like in
polymer quantum mechanics and LQC, it is not possible to obtain the
representation of $\widehat{S}$ out of the representation of $\widehat{s}$ in
${\cal H}$. The reason is that some elements of ${\cal P}(2,\mathbb{R})$
cannot be represented in the corresponding Hilbert space. This difficulty can
be overcome if we can represent directly the exponential $e^{\widehat{s}}$
instead of its infinitesimal generator $\widehat{s}$. This approach was done
for the case of polymer quantum mechanics in Garcia-Chung:2020cag .
Consequently, due to the operators in (45) can be used to describe the
dynamics of many physical systems ranging from two decoupled quantum harmonic
oscillators to the bipartite squeeze operators, a polymer representation of
these operators is possible, as we will show in section V. More details about
these aspects will be provided in section V.

In the next subsection, we show some of the explicit forms of ${\bf M}$.

###  III.2 Examples

####  III.2.1 Case ${\bf a},{\bf c}\neq{\bf 0}$ and ${\bf b}={\bf 0}$.

Let us consider the Lie algebra element with ${\bf b}={\bf 0}$ and ${\bf
a},{\bf c}\neq 0$, which, according to the expression (46), implies that there
is no interaction between the subsystems, that is, $\widehat{s}$ is of the
form

$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{2\hbar}\left[a_{11}\widehat{q}^{2}_{1}+a_{12}(\widehat{q}_{1}\widehat{p}_{1}+\widehat{p}_{1}\widehat{q}_{1})+a_{22}\widehat{p}^{2}_{1}+c_{11}\widehat{q}^{2}_{2}+c_{12}(\widehat{q}_{2}\widehat{p}_{2}+\widehat{p}_{2}\widehat{q}_{2})+c_{22}\widehat{p}^{2}_{2}\right].$
(51)

In this case, ${\bf d}={\bf 0}$ and $\lambda_{+}=-\det{\bf a}$ and
$\lambda_{-}=-\det{\bf c}$. After inserting ${\bf b}=0$ and the expressions
for $\lambda_{\pm}$ in (157)-(162) we obtain the following symplectic matrix

${\bf M}_{1}=\left(\begin{array}[]{cc}\cosh\left(\sqrt{-\det{\bf
a}}\right)+\frac{\sinh\left(\sqrt{-\det{\bf a}}\right)}{\sqrt{-\det{\bf
a}}}\,{\bf J}\,{\bf a}&{\bf 0}\\\ {\bf 0}&\cosh\left(\sqrt{-\det{\bf
c}}\right)+\frac{\sinh\left(\sqrt{-\det{\bf c}}\right)}{\sqrt{-\det{\bf
c}}}\,{\bf J}\,{\bf c}\end{array}\right).$ (52)

As can be seen, both block matrices in (52) are elements of $Sp(2,\mathbb{R})$
hence, the Lie algebra elements given by the parameters ${\bf a}$ and ${\bf
c}$ can be considered as the Lie algebra generators of
$Sp(2,\mathbb{R})\otimes Sp(2,\mathbb{R})\subset Sp(4,\mathbb{R})$. Moreover,
the matrix ${\bf M}_{1}$ is diagonal if and only if ${\bf a}$ and ${\bf c}$
are anti-diagonal matrices, i.e., only when there are no squared terms in
(45).

An important symplectic matrix of this type is

$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf J}\end{array}\right),$ (53)

which is often used to derive the transpose matrix as in (25). One can check
that this matrix can be obtained from (52) when ${\bf a}={\bf
c}=\mbox{diag}(\frac{\pi}{2},\frac{\pi}{2})$, i.e.,

$\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)=\exp{\left[\frac{\pi}{2}\left(\begin{array}[]{cc}{\bf
J}&0\\\ 0&{\bf J}\end{array}\right)\right]}.$ (54)

####  III.2.2  Case ${\bf c}={\bf a}=\mbox{diag}(a_{11},a_{22})$ and ${\bf
b}=$diag$(b_{11},b_{22})$.

In this case the operator $\widehat{s}$ is of the form

$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[a_{22}\left(\frac{1}{2}\widehat{p}^{2}_{1}+\frac{1}{2}\widehat{p}^{2}_{2}\right)+a_{11}\left(\frac{1}{2}\widehat{q}^{2}_{1}+\frac{1}{2}\widehat{q}^{2}_{2}\right)+b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(55)

i.e., the sub-systems interact via the matrix ${\bf b}$ but only with
couplings between coordinates $\widehat{q}_{1}\widehat{q}_{2}$ and momenta
operators $\widehat{p}_{1}\widehat{p}_{2}$. According to (142), the expression
for $\lambda_{\pm}$ for this case is

$\lambda_{\pm}=-(a_{11}\,a_{22}+b_{11}\,b_{22})\pm(a_{11}\,b_{22}+a_{22}\,b_{11})=-(a_{11}\mp
b_{11})(a_{22}\mp b_{22}).$ (56)

Note that when $b_{11}=\pm a_{11}$ or $b_{22}=\pm a_{22}$ the eigenvalues are
null and two particular systems emerge with their operators given by

$\displaystyle b_{11}=\pm a_{11}\rightarrow\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[a_{22}\left(\frac{1}{2}\widehat{p}^{2}_{1}+\frac{1}{2}\widehat{p}^{2}_{2}\right)+\frac{a_{11}}{2}\left(\widehat{q}_{1}\pm\widehat{q}_{2}\right)^{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(57) $\displaystyle b_{22}=\pm a_{22}\rightarrow\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{a_{22}}{2}\left(\widehat{p}_{1}+\widehat{p}_{2}\right)^{2}+a_{11}\left(\frac{1}{2}\widehat{q}^{2}_{1}+\frac{1}{2}\widehat{q}^{2}_{2}\right)+b_{11}\widehat{q}_{1}\widehat{q}_{2}\right].$
(58)

Both systems represent two interacting harmonic oscillators with a coupling
term in the momenta and the coordinates, respectively.

The symplectic matrix, denoted in this case as ${\bf M}_{2}$, is given by

${\bf
M}_{2}=\left(\begin{array}[]{cccc}\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(a_{22}-b_{22})S_{+}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(b_{22}-a_{22})S_{+}}{2}\\\
\frac{(b_{11}-a_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{11}-b_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}\\\
\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(b_{22}-a_{22})S_{+}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(a_{22}+b_{22})S_{-}+(a_{22}-b_{22})S_{+}}{2}\\\
\frac{(a_{11}-b_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)-\cosh\left(\sqrt{\lambda_{+}}\right)}{2}&\frac{(b_{11}-a_{11})S_{+}-(a_{11}+b_{11})S_{-}}{2}&\frac{\cosh\left(\sqrt{\lambda_{-}}\right)+\cosh\left(\sqrt{\lambda_{+}}\right)}{2}\\\
\end{array}\right),$ (59)

where we introduce the parameters $S_{\pm}$ as

$S_{\pm}:=\frac{\sinh\left(\sqrt{\lambda_{\pm}}\right)}{\sqrt{\lambda_{\pm}}}.$
(60)

####  III.2.3 Case ${\bf a}={\bf c}={\bf 0}$ and ${\bf b}\neq{\bf 0}$.

In this case, the operator $\widehat{s}$ is of the form

$\displaystyle\widehat{s}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[b_{11}\widehat{q}_{1}\widehat{q}_{2}+b_{12}\widehat{q}_{1}\widehat{p}_{2}+b_{21}\widehat{p}_{1}\widehat{q}_{2}+b_{22}\widehat{p}_{1}\widehat{p}_{2}\right],$
(61)

and this system corresponds, as we will see in the next section, to the
general case of the squeeze operator for a bi-partite system
adesso2014continuous .

Note that in this case, not only the matrices ${\bf a}={\bf c}$ are null, but
also the matrix ${\bf d}$, which implies that
$\lambda_{+}=\lambda_{-}=-\det{\bf b}$. Once we replace these expressions in
(157)-(162) the symplectic matrix takes the form

${\bf M}_{3}=\left(\begin{array}[]{cc}\cosh\sqrt{-\det{\bf
b}}&\frac{\sinh\sqrt{-\det{\bf b}}}{\sqrt{-\det{\bf b}}}\,{\bf J}\,{\bf b}\\\
\frac{\sinh\sqrt{-\det{\bf b}}}{\sqrt{-\det{\bf b}}}\,{\bf J}\,{\bf
b}^{T}&\cosh\sqrt{-\det{\bf b}}\end{array}\right),$ (62)

where the block matrices ${\bf A}$ and ${\bf D}$ are diagonal matrices.
Clearly, when $\det{\bf b}<0$ the coefficients of ${\bf M}_{3}$ will be given
by hyperbolic functions. In case $\det{\bf b}>0$ the coefficients are
described by trigonometric functions instead.

After giving some examples of symplectic matrices obtained through the
exponential map, we are ready to show some of their quantum mechanics
applications, both in the standard representation and the so-called polymer or
loop representation.

##  IV Quantum representation and its applications

The unitary representation of the group $Sp(2n,\mathbb{R})$ was given by
Moshinsky and Quesne in moshinsky1971linear . A review and a historical
analysis can be found in torre2005linear ; wolf2016development . However, to
be self-contained, we will show the main aspects of this group’s quantum
representation in standard quantum mechanics in the next subsection.

###  IV.1 Schrödinger representation of $Sp(2n,\mathbb{R})$

The symplectic group is a non-compact group which implies an infinite-
dimensional Hilbert space for its unitary representation. Consider the Hilbert
space ${\cal H}=L^{2}(\mathbb{R}^{n},d\vec{x})$ where $d\vec{x}$ is the
standard Lebesgue measure. The unitary representation of $Sp(2n,\mathbb{R})$
is the map

$\widehat{C}:Sp(2n,\mathbb{R})\rightarrow{\cal U}({\cal H});\;\widetilde{\bf
M}\mapsto\widehat{C}_{\widetilde{\bf M}},$ (63)

where $\widehat{C}_{\widetilde{\bf M}}$ is a unitary operator over ${\cal H}$,
i.e., formally $\widehat{C}^{\dagger}_{\widetilde{\bf
M}}=\widehat{C}^{-1}_{\widetilde{\bf M}}$. Note that the group action
considered in this map is $\widetilde{\bf M}$ instead of ${\bf M}$, i.e., we
used the “coordinatization” given by $\vec{X}$ introduced in section II.
Hence, in order to obtain a quantum (unitary) representation of a given
symplectic matrix ${\bf M}$ we first have to transform it into the other group
action $\widetilde{\bf M}$ using Eq. (9) with the corresponding matrix
${\bf\Gamma}(n)$ given by (10) or ${\bf\Gamma}(2)$ for $Sp(4,\mathbb{R})$
given in (11).

The map $\widehat{C}$ is given by the integral operator

$\widehat{C}_{\widetilde{\bf M}}\Psi(\vec{x})=\int
d\vec{x}^{\prime}C_{\widetilde{\bf
M}}(\vec{x},\vec{x}^{\prime})\Psi(\vec{x}^{\prime}),\qquad\quad\Psi(\vec{x})\in{\cal
H},$ (64)

and the kernel $C_{\widetilde{\bf M}}(\vec{x},\vec{x}^{\prime})$ of this
integral is

$C_{\widetilde{\bf
M}}(\vec{x},\vec{x}^{\prime})=\frac{e^{\frac{i}{2\hbar}\left[\vec{x}^{T}{\widetilde{\bf
D}}{\widetilde{\bf B}}^{-1}\vec{x}-2\vec{x}^{\prime T}{\widetilde{\bf
B}}^{-1}\vec{x}+\vec{x}^{\prime T}{\widetilde{\bf B}}^{-1}{\widetilde{\bf
A}}\vec{x}^{\prime}\right]}}{\sqrt{(2\pi i\hbar)^{n}\det{\widetilde{\bf
B}}}}.$ (65)

According to moshinsky1971linear , this representation results from imposing
two conditions on the operators $\widehat{C}_{\widetilde{\bf M}}$. The first
one is given by

$\widehat{C}_{\widetilde{\bf
M}}\left(\begin{array}[]{c}\vec{\widehat{q}}^{T}\\\
\vec{\widehat{p}}^{T}\end{array}\right)\widehat{C}^{-1}_{\widetilde{\bf
M}}=\widetilde{\bf M}^{-1}\left(\begin{array}[]{c}\vec{\widehat{q}}^{T}\\\
\vec{\widehat{p}}^{T}\end{array}\right),$ (66)

and relates the symplectic group elements $\widetilde{\bf M}$ with the
operators $\widehat{C}_{\widetilde{\bf M}}$. Here,
$\vec{\widehat{q}}:=(\widehat{q}_{1},\widehat{q}_{2},\dots,\widehat{q}_{n})$
and
$\vec{\widehat{p}}:=(\widehat{p}_{1},\widehat{p}_{2},\dots,\widehat{p}_{n})$
are the coordinate and momenta operators associated to the Heisenberg Lie
algebra of the system. The second condition is that

$\widehat{C}_{\widetilde{\bf M}}\cdot\left(\widehat{C}_{\widetilde{\bf
M}}\right)^{\dagger}=\widehat{1},$ (67)

where $\widehat{1}$ is the identity operator and this results in the unitarity
of $\widehat{C}_{\widetilde{\bf M}}$.

The factor $\det{\bf B}$ in (65) gives rise to a well define operator even in
the case where the matrix ${\bf B}$ is singular (for more details see
moshinsky1971linear ; wolf2016development ). Finally, it is worth mentioning
that this representation (64) is valid for the entire symplectic group and not
just for those elements close to the group identity.

Since the fundamental operators are unbounded the condition (66) only holds in
a subspace given by the domain of the operators $\widehat{q}_{j}$ and
$\widehat{p}_{j}$ in ${\cal H}$. To obtain a condition valid in the full
Hilbert space, we are forced to introduce the exponentiated version of
$\widehat{q}_{j}$ and $\widehat{p}_{j}$, that is to say, the Weyl algebra.
Briefly, the Weyl algebra is a $C^{*}$-unital algebra whose generators,
denoted by $\widehat{W}(\vec{a},\vec{b})$, are related with $\widehat{q}_{j}$
and $\widehat{p}_{j}$ with the following relation

$\widehat{W}(\vec{a},\vec{b}):=e^{\frac{i}{\hbar}\left(\vec{a}\;\vec{\widehat{q}}^{\,T}+\vec{b}\;\vec{\widehat{p}}^{\;T}\right)},$
(68)

and such that the real arrays $\vec{a}=(a_{1},a_{2},\dots,a_{n})$ and
$\vec{b}=(b_{1},b_{2},\dots,b_{n})$, which have dimensions
$[a_{j}]=\mbox{momentum}$ and $[b_{j}]=\mbox{position}$, label the Weyl
algebra generators.

The standard Schrödinger representation of $\widehat{q}_{j}$ and
$\widehat{p}_{j}$ is now used to obtain a representation for the generators
$\widehat{W}(\vec{a},\vec{b})$ in ${\cal H}$ given by

$\widehat{W}(\vec{a},\vec{b})\Psi(\vec{x})=e^{\frac{i}{2\hbar}\vec{a}\;\vec{b}^{\,T}}e^{\frac{i}{\hbar}\vec{a}\;\vec{x}^{\,T}}\Psi(\vec{x}+\vec{b}),$
(69)

and such that the canonical commutation relations give rise to the Weyl
algebra mutiplication

$\widehat{W}(\vec{a}_{1},\vec{b}_{1})\widehat{W}(\vec{a}_{2},\vec{b}_{2})=e^{-\frac{i}{2\hbar}\left(\vec{a}_{1}\;\vec{b}^{T}_{2}-\vec{b}_{1}\;\vec{a}^{T}_{2}\right)}\widehat{W}(\vec{a}_{1}+\vec{a}_{2},\vec{b}_{1}+\vec{b}_{2}).$
(70)

Combining (66) and (68) to obtain the exponentiated version of (66) yields

$\widehat{C}_{\widetilde{\bf
M}}\,\widehat{W}(\vec{a},\vec{b})\,(\widehat{C}_{\widetilde{\bf
M}})^{-1}=\widehat{W}(\vec{a}\;\widetilde{\bf D}^{T}-\vec{b}\;\widetilde{\bf
C}^{T},-\vec{a}\;\widetilde{\bf B}^{T}+\vec{b}\;\widetilde{\bf A}^{T}),$ (71)

where $\widetilde{\bf A}$, $\widetilde{\bf B}$, $\widetilde{\bf C}$ and
$\widetilde{\bf D}$ are the block matrices in $\widetilde{\bf M}$. This
relation allows us to obtain a representation of the symplectic group in the
Hilbert space used in polymer quantum mechanics and in loop quantum cosmology
Garcia-Chung:2020cag .

We are now ready to show, in the next subsections, some of the applications of
the representation of $Sp(4,\mathbb{R})$ given by (64) and (65).

###  IV.2 Schrödinger representation of the squeeze operator for a bi-partite
system.

The squeeze operator $\widehat{S}(\zeta)$ for a bi-partite system is given by
the exponential map

$\widehat{S}(\zeta)=e^{\widehat{s}_{\zeta}},$ (72)

where the operator $\widehat{s}_{\zeta}$, is given by

$\widehat{s}_{\zeta}:={\frac{1}{2}\left(\zeta^{*}\widehat{a}_{1}\widehat{a}_{2}-\zeta\widehat{a}^{\dagger}_{1}\widehat{a}^{\dagger}_{2}\right)}.$
(73)

Here, $\widehat{a}_{1}$ and $\widehat{a}_{2}$ are the annihilation operators
for each of the sub-systems, say, 1 and 2, of the bi-partite system,
$\widehat{a}^{\dagger}_{1}$ and $\widehat{a}^{\dagger}_{2}$ are their adjoint
operators respectively and $\zeta$ is a complex number labelling the amount of
squeezing. The operator $\widehat{S}(\zeta)$, when acting on the vacuum state
of the bi-partite quantum harmonic oscillators, gives a family of squeezed
states labelled by $\zeta$.

The operators in (73) are in the Fock representation, hence, let us transform
(73) to the Schrödinger representation described with operators
$\widehat{q}_{1}$, $\widehat{q}_{2}$, $\widehat{p}_{1}$ and $\widehat{p}_{2}$.
The relation between these representations is given by

$\displaystyle\widehat{a}_{j}=\frac{1}{\sqrt{2}}\frac{\widehat{q}_{j}}{l_{j}}+\frac{i}{\sqrt{2}}\frac{l_{j}\widehat{p}_{j}}{\hbar},\qquad\widehat{a}^{\dagger}_{j}=\frac{1}{\sqrt{2}}\frac{\widehat{q}_{j}}{l_{j}}-\frac{i}{\sqrt{2}}\frac{l_{j}\widehat{p}_{j}}{\hbar},$
(74)

for $j=1,2$ and $l_{j}:=\sqrt{\frac{\hbar}{m_{j}\omega_{j}}}$ where $m_{j}$
and $\omega_{j}$ stand for the masses and the frequencies of the oscillators.
Inserting these expressions for $\widehat{a}_{j}$ and
$\widehat{a}^{\dagger}_{j}$ in (73) the operator $\widehat{s}_{\zeta}$ takes
the following form

$\displaystyle\widehat{s}_{\zeta}$ $\displaystyle=$
$\displaystyle\frac{1}{2i\hbar}\left[\frac{\hbar\,\zeta_{y}}{l_{1}l_{2}}\;\widehat{q}_{1}\widehat{q}_{2}-\frac{l_{2}\,\zeta_{x}}{l_{1}}\;\widehat{q}_{1}\widehat{p}_{2}-\frac{l_{1}\,\zeta_{x}}{l_{2}}\;\widehat{p}_{1}\widehat{q}_{2}-\frac{l_{1}l_{2}\,\zeta_{y}}{\hbar}\widehat{p}_{1}\widehat{p}_{2}\right],$
(75)

where $\zeta_{x}$ and $\zeta_{y}$ are the real and imaginary parts of $\zeta$.

We now rewrite this operator in the form

$\widehat{s}_{\zeta}=-\frac{i}{4\hbar}(\vec{\widehat{R}}^{T}_{1},\vec{\widehat{R}}^{T}_{2})\left(\begin{array}[]{cc}0&{\bf
b}\\\ {\bf
b}^{T}&0\end{array}\right)\left(\begin{array}[]{c}\vec{\widehat{R}}_{1}\\\
\vec{\widehat{R}}_{2}\end{array}\right),$ (76)

where the matrix ${\bf b}$ is the following

${\bf
b}=\left(\begin{array}[]{cc}\frac{\hbar\,\zeta_{y}}{l_{1}l_{2}}&-\frac{l_{2}\,\zeta_{x}}{l_{1}}\\\
-\frac{l_{1}\,\zeta_{x}}{l_{2}}&-\frac{l_{1}l_{2}\,\zeta_{y}}{\hbar}\end{array}\right).$
(77)

Using the isomorphism $\iota^{-1}$ defined in (49) we obtain that the
corresponding Lie algebra element ${\bf
m}_{\zeta}=\iota^{-1}(\widehat{s}_{\zeta})$ is given by

${\bf m}_{\zeta}=\left(\begin{array}[]{cc}{\bf J}&0\\\ 0&{\bf
J}\end{array}\right)\left(\begin{array}[]{cc}0&{\bf b}\\\ {\bf
b}^{T}&0\end{array}\right).$ (78)

Note that the Lie algebra matrix ${\bf m}_{\zeta}$ isomorphic to the squeeze
operator $\widehat{s}_{\zeta}$, is of the type given in the third case
(III.2.3).

To obtain the symplectic matrix associated to this Lie algebra element, we
insert ${\bf m}_{\zeta}$ and its expressions for ${\bf a}$, ${\bf b}$ and
${\bf c}$ in (62). This results in the following symplectic matrix

${\bf M}_{\bf
s}(r,\phi)=\left(\begin{array}[]{cccc}\cosh(r)&0&-\sinh(r)\cos(\phi)\frac{l_{1}}{l_{2}}&-\sinh(r)\sin(\phi)\frac{l_{1}l_{2}}{\hbar}\\\
0&\cosh(r)&-\sinh(r)\sin(\phi)\frac{\hbar}{l_{1}l_{2}}&\sinh(r)\cos(\phi)\frac{l_{2}}{l_{1}}\\\
-\sinh(r)\cos(\phi)\frac{l_{2}}{l_{1}}&-\sinh(r)\sin(\phi)\frac{l_{1}l_{2}}{\hbar}&\cosh(r)&0\\\
-\sinh(r)\sin(\phi)\frac{\hbar}{l_{1}l_{2}}&\sinh(r)\cos(\phi)\frac{l_{1}}{l_{2}}&0&\cosh(r)\\\
\end{array}\right),$ (79)

where $r$ and $\phi$ are defined as $\zeta=re^{i\phi}$. Matrix ${\bf M}_{\bf
s}(r,\phi)$ can be considered as the classical symplectic transformation such
that when represented in $L^{2}(\mathbb{R}^{2},d^{2}\vec{x})$, gives rise to
the quantum operator $\widehat{S}(\zeta)$. Naturally, this means also that the
unitary representation of $\widehat{S}(\zeta)$ in the Schrödinger
representation is given by $\widehat{C}_{{\bf M}_{\bf s}}$, i.e.,
$\widehat{C}_{{\bf M}_{\bf s}}=e^{\widehat{s}_{\zeta}}$.

It is worth to mention that although the expression (79) depends on the proper
lengths $l_{1}$ and $l_{2}$, the matrix ${\bf M}_{\bf s}(r,\phi)$ is
$\hbar-$independent, i.e., it is entirely a classical object. Also, matrix
${\bf M}_{\bf s}$ produces classical squeezing but of course, adapted to the
classical phase space, which in this case is $(\mathbb{R}^{4},\\{,\\})$. To
illustrate the squeezing and the rotation properties of the matrix ${\bf
M}_{\bf s}$ as a canonical transformation for different values of $r$ and
$\phi$ we consider its action on a circular trajectory
$(q_{1}(t),p_{1}(t),q_{2}(t),p_{2}(t))$ where,
$q_{j}(t)=\cos(t)\;q_{j}+\sin(t)\;p_{j}$ and
$p_{j}(t)=-\sin(t)\;q_{j}+\cos(t)\;p_{j}$, for $j=1,2$.

(a) Squeezing a circular trajectory.

(b) Rotating a squeezed trajectory with $r=0.6$.

Figure 1:  In both figures, the solid, the dashed and the dotted lines
correspond to: (a) $r=0$, $r=0.3$ and $r=0.6$, respectively and (b) to
$\phi=0$, $\phi=\pi/4$ and $\phi=\pi/2$, respectively.

The action of ${\bf M}_{\bf s}$ on the trajectories is explicitly of the form

$\left(\begin{array}[]{c}q^{\prime}_{1}(t)\\\ p^{\prime}_{1}(t)\\\
q^{\prime}_{2}(t)\\\ p^{\prime}_{2}(t)\end{array}\right)={\bf M}_{\bf
s}\left(r,\phi\right)\left(\begin{array}[]{c}q_{1}(t)\\\ p_{1}(t)\\\
q_{2}(t)\\\ p_{2}(t)\end{array}\right).$ (80)

In figure 1 we showed the plot of $(q^{\prime}_{1}(t),p^{\prime}_{1}(t))$. As
expected, we note in Fig. (1(a)), that the amount of squeezing $r$ squeezes
the circular trajectory. Recall that symplectic transformations also preserve
the area, hence the trajectories are squeezed but the area is preserved. On
the other hand, the rotation angle $\phi$, as showed in (1(b)), rotates the
trajectories and also preserves the area.

Finally, observe that ${\bf M}_{\bf s}$ is given in the $\vec{Y}$
“coordinatization” which is not suitable for its quantum representation. To
make it suitable, let us provide the expression for the matrix $\widetilde{\bf
M}_{\bf s}$, which is given by

$\widetilde{\bf M}_{\bf
s}=\left(\begin{array}[]{cccc}\cosh(r)&-\frac{l_{1}\sinh(r)\cos(2\phi)}{l_{2}}&0&-\frac{l_{1}l_{2}\sinh(r)\sin(2\phi)}{\hbar}\\\
-\frac{l_{2}\sinh(r)\cos(2\phi)}{l_{1}}&\cosh(r)&-\frac{l_{1}l_{2}\sinh(r)\sin(2\phi)}{\hbar}&0\\\
0&-\frac{\hbar\sinh(r)\sin(2\phi)}{l_{1}l_{2}}&\cosh(r)&\frac{l_{2}\sinh(r)\cos(2\phi)}{l_{1}}\\\
-\frac{\hbar\sinh(r)\sin(2\phi)}{l_{1}l_{2}}&0&\frac{l_{1}\sinh(r)\cos(2\phi)}{l_{2}}&\cosh(r)\\\
\end{array}\right).$ (81)

This expression will be used to explore the analog of the bipartite squeeze
operator in polymer quantum mechanics in section V.

###  IV.3 Covariance matrix for squeezed states

Now we will show the relation between the covariance matrix, denoted by ${\bf
V}^{(2)}$, and the symplectic matrix $\widetilde{\bf M}$. Let us consider the
state $|\Psi_{\widetilde{\bf M}}\rangle\in L^{2}(\mathbb{R}^{n},d\vec{x})$
related with the symplectic matrix $\widetilde{\bf M}$ as

$|\Psi_{\widetilde{\bf M}}\rangle=\widehat{C}_{\widetilde{\bf M}}|0\rangle,$
(82)

where $|0\rangle=\int d\vec{x}\;\Psi_{0}(\vec{x})\,|\vec{x}\rangle$ is the
state $|0\rangle=|0\rangle_{1}\otimes|0\rangle_{2}\dots|0\rangle_{n}$, and the
ket $|0\rangle_{j}$ is the vacuum state of the $j$-th quantum harmonic
oscillator. Note that this construction can be extended to other states in
$L^{2}(\mathbb{R}^{n},d\vec{x})$ and not only for $|0\rangle$. However, for
simplicity in our exposition, let us consider the simplest example of the
covariance matrix for $\widehat{C}_{\widetilde{\bf M}}|0\rangle$.

To obtain the covariance matrix we first calculate the following amplitude

$\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf M}}\rangle=\langle
0|\widehat{C}^{\dagger}_{\widetilde{\bf
M}}\;\widehat{W}(\vec{a},\vec{b})\;\widehat{C}_{\widetilde{\bf M}}|0\rangle,$
(83)

where $\widehat{W}(\vec{a},\vec{b})$ is the Weyl-algebra generator introduced
in (68). Combining (64), (69) and the Gaussian form of the vacuum state of the
system given by $n$-decoupled harmonic oscillators, we obtain the following
expression for the amplitude in (83)

$\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle=\exp\left\\{-\frac{1}{4}\left(\begin{array}[]{cc}\vec{a}&\vec{b}\end{array}\right)^{T}{\bf\Lambda}\left(\begin{array}[]{c}\vec{a}\\\
\vec{b}\end{array}\right)\right\\},$ (84)

where the matrix ${\bf\Lambda}$ is given by

${\bf\Lambda}:={\widetilde{\bf
M}}\left(\begin{array}[]{cc}\frac{1}{\hbar^{2}}{\bf L}^{2}&{\bf 0}\\\ {\bf
0}&{\bf L}^{-2}\end{array}\right){\widetilde{\bf M}}^{T},$ (85)

and ${\bf L}=\mbox{diag}(l_{1},l_{2},\dots,l_{n})$, where $l_{j}$ was defined
earlier (74).

The covariance matrix ${\bf V}^{(2)}$ has components given by

${\bf V}^{(2)}=\left(\begin{array}[]{cc}\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}\;\widehat{x}_{k}|\Psi_{\widetilde{\bf
M}}\rangle&\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{x}_{j},\;\widehat{p}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle\\\ \frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{p}_{j},\;\widehat{x}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle&\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}\;\widehat{p}_{k}|\Psi_{\widetilde{\bf
M}}\rangle\end{array}\right),$ (86)

and these components can be obtained from (84) using the following relations

$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}\;\widehat{x}_{k}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{a_{j}a_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (87)
$\displaystyle\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{x}_{j},\;\widehat{p}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{a_{j}b_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (88)
$\displaystyle\frac{1}{2}\langle\Psi_{\widetilde{\bf
M}}|\left\\{\widehat{p}_{j},\;\widehat{x}_{k}\right\\}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{b_{j}a_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0},$ (89)
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}\;\widehat{p}_{k}|\Psi_{\widetilde{\bf
M}}\rangle=-\hbar^{2}\partial^{2}_{b_{j}b_{k}}\langle\Psi_{\widetilde{\bf
M}}|\widehat{W}(\vec{a},\vec{b})|\Psi_{\widetilde{\bf
M}}\rangle|_{\vec{a},\vec{b}=0}.$ (90)

Remarkably, the resulting expression for ${\bf V}^{(2)}$ in terms of the
symplectic matrix $\widetilde{\bf M}$ is

${\bf V}^{(2)}=\frac{1}{2}{\widetilde{\bf M}}\left(\begin{array}[]{cc}{\bf
L}^{2}&{\bf 0}\\\ {\bf 0}&\hbar^{2}{\bf
L}^{-2}\end{array}\right){\widetilde{\bf M}}^{T},$ (91)

and this shows the direct relation between the covariance matrix ${\bf
V}^{(2)}$ for the state $|\Psi_{\widetilde{\bf M}}\rangle$ and the symplectic
matrix ${\widetilde{\bf M}}$ associated with the unitary operator
$\widehat{C}_{\widetilde{\bf M}}$. Moreover, if we now consider the definition
(8), it can be shown that $\frac{2}{\hbar}{\bf V}^{(2)}$ is actually a
symplectic matrix. Let us apply this formula to some of the systems considered
before.

Consider the matrix ${\bf M}_{1}$ given in (52). Using (9) we obtain the
expression for ${\widetilde{\bf M}}_{1}$ which then is replaced in (91) giving
rise to the following covariance matrix

${\bf
V}^{(2)}_{1}=\frac{1}{2}\left(\begin{array}[]{cccc}{V}^{(2)}_{11}&0&{V}^{(2)}_{13}&0\\\
0&{V}^{(2)}_{22}&0&{V}^{(2)}_{24}\\\ {V}^{(2)}_{13}&0&{V}^{(2)}_{33}&0\\\
0&{V}^{(2)}_{24}&0&{V}^{(2)}_{44}\end{array}\right).$ (92)

Its components are given in the appendix (X) and in the particular case where
$a_{11}=a_{22}=c_{11}=c_{22}=0$, the covariance matrix ${\bf V}^{(2)}_{1}$
reduces to

${\bf
V}^{(2)}_{1}=\frac{1}{2}\left(\begin{array}[]{cccc}l^{2}_{1}e^{2a_{12}}&0&0&0\\\
0&l^{2}_{2}e^{2c_{12}}&0&0\\\ 0&0&\frac{\hbar^{2}}{l^{2}_{1}}e^{-2a_{12}}&0\\\
0&0&0&\frac{\hbar^{2}}{l^{2}_{2}}e^{-2c_{12}}\end{array}\right).$ (93)

We use this result to derive the uncertainties in the coordinates $\Delta
x_{j}$ for $j=1,2$ which according to (93) are given by

$\Delta x_{j}:=\sqrt{\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}^{2}_{j}|\Psi_{\widetilde{\bf
M}}\rangle-\langle\Psi_{\widetilde{\bf
M}}|\widehat{x}_{j}|\Psi_{\widetilde{\bf
M}}\rangle^{2}}=\frac{l_{j}e^{\alpha_{j}}}{\sqrt{2}},$ (94)

where due to the symmetry of the vacuum wavefunction we have
$\langle\Psi_{\widetilde{\bf M}}|\widehat{x}_{j}|\Psi_{\widetilde{\bf
M}}\rangle=0$. This can be verified calculating the first derivatives in (84).
Here, for simplicity we make $\alpha_{1}=a_{12}$ and $\alpha_{2}=c_{12}$. Note
the remarkably property of the squeezed states like in (82) which is that
$\Delta x_{j}$ can be smaller than the proper length of the vacuum state
$l_{j}$ when $\alpha_{j}<0$.

Similarly, the uncertainties in the momenta $\Delta p_{j}$, are

$\Delta p_{j}:=\sqrt{\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}^{2}_{j}|\Psi_{\widetilde{\bf
M}}\rangle-\langle\Psi_{\widetilde{\bf
M}}|\widehat{p}_{j}|\Psi_{\widetilde{\bf
M}}\rangle^{2}}=\frac{\hbar}{\sqrt{2}l_{j}e^{\alpha_{j}}},$ (95)

which can also be smaller than $\frac{\hbar}{l_{j}}$ when $\alpha_{j}>0$ and
also, $\langle\Psi_{\widetilde{\bf M}}|\widehat{p}_{j}|\Psi_{\widetilde{\bf
M}}\rangle=0$ as the previous case. Nevertheless, both uncertainties satisfy
Heisenberg’s uncertainty principle:

$\Delta x_{j}\,\Delta
p_{j}=\left(\frac{l_{j}e^{\alpha_{j}}}{\sqrt{2}}\right)\left(\frac{\hbar}{\sqrt{2}l_{j}e^{\alpha_{j}}}\right)=\hbar/2.$
(96)

Another interesting covariance matrix is the one related with the bipartite
squeeze operator (81) derived in the previous subsection. Inserting (81) in
(91) yields

${\bf
V}^{(2)}(r,\phi)=\left(\begin{array}[]{cccc}\frac{l_{1}^{2}\cosh(2r)}{2}&-\frac{l_{1}l_{2}\sinh(2r)\cos(2\phi)}{2}&0&-\frac{l_{1}\hbar\sinh(2r)\sin(2\phi)}{2l_{2}}\\\
-\frac{l_{1}l_{2}\sinh(2r)\cos(2\phi)}{2}&\frac{l_{2}^{2}\cosh(2r)}{2}&-\frac{l_{2}\hbar\sinh(2r)\sin(2\phi)}{2l_{1}}&0\\\
0&-\frac{l_{2}\hbar\sinh(2r)\sin(2\phi)}{2l_{1}}&\frac{\hbar^{2}\cosh(2r)}{2l_{1}^{2}}&\frac{\hbar^{2}\sinh(2r)\cos(2\phi)}{2l_{1}l_{2}}\\\
-\frac{l_{1}\hbar\sinh(2r)\sin(2\phi)}{2l_{2}}&0&\frac{\hbar^{2}\sinh(2r)\cos(2\phi)}{2l_{1}l_{2}}&\frac{\hbar^{2}\cosh(2r)}{2l_{2}^{2}}\end{array}\right),$
(97)

and this allows us to determine the correlation between the second moments of
the subsystem 1 and the subsystem 2

$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\left(\widehat{x}_{1}\pm\widehat{x}_{2}\right)^{2}|\Psi_{\widetilde{\bf
M}}\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{2r}}{4}\left[l^{2}_{1}+l^{2}_{2}\mp
2l_{1}\,l_{2}\cos(2\phi)\right]+\frac{e^{-2r}}{4}\left[l^{2}_{1}+l^{2}_{2}\pm
2l_{1}\,l_{2}\cos(2\phi)\right],$ (98)
$\displaystyle\langle\Psi_{\widetilde{\bf
M}}|\left(\widehat{p}_{1}\pm\widehat{p}_{2}\right)^{2}|\Psi_{\widetilde{\bf
M}}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}e^{2r}}{4l^{2}_{1}\,l^{2}_{2}}\left[l^{2}_{1}+l^{2}_{2}\pm
2l_{1}\,l_{2}\cos(2\phi)\right]+\frac{\hbar^{2}e^{-2r}}{4l^{2}_{1}\,l^{2}_{2}}\left[l^{2}_{1}+l^{2}_{2}\mp
2l_{1}\,l_{2}\cos(2\phi)\right].$ (99)

In the particular case where $l_{1}=l_{2}=l$ and $\phi=\frac{\pi}{2}$, the
uncertainties $(\Delta x_{1})_{\widetilde{\Psi}}$ and $(\Delta
x_{2})_{\widetilde{\Psi}}$ for the state $\widetilde{\Psi}$ are correlated as
follows

$(\Delta x_{1})^{2}_{\widetilde{\Psi}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}}=l^{2}\cosh(2r).$ (100)

These are the main results, at the standard quantum mechanics level, which we
want to show regarding the representation of the symplectic group in quantum
mechanics. There are others applications like the analysis of the Bohmian
trajectories for bipartite squeezed states, the analysis of the fidelity for
bipartite or tripartite squeezed states, and others which are currently in
preparation. For now, let us move to the analysis of the squeezed states in
polymer quantum mechanics given in the next section.

##  V Squeezed states in polymer quantum mechanics

Polymer quantum mechanics ashtekar2003quantum ; corichi2007polymer ;
velhinho2007quantum ; pawlowski2014separable , is a quantization scheme which
can be considered as a “toy model” looming from loop quantum cosmology. Hence,
exploring the nature and properties of squeezed states in polymer quantum
mechanics will help study those scenarios in loop quantum cosmology where such
states might play a significant role.

For this example, we will consider a system with two degrees of freedom and
both will be polymer quantized. Therefore, the Hilbert space of the entire
system is given by

${\cal H}_{poly}={\cal H}^{(1)}_{poly}\times{\cal H}^{(2)}_{poly},$ (101)

where the Hilbert spaces ${\cal H}^{(j)}_{poly}$ with $j=1,2$ are of the form

${\cal H}^{(j)}_{poly}=L^{2}(\overline{\mathbb{R}},dp^{(j)}_{Bohr}),$ (102)

where $\overline{\mathbb{R}}$ is the Bohr compactification of real line and
$dp^{(j)}_{Bohr}$ is the Bohr measure (see velhinho2007quantum  for more
details). This Hilbert space resembles the momentum representation used in the
standard quantum mechanics.

An arbitrary state in this Hilbert space ${\cal H}_{poly}$ is given by

$\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}},$
(103)

where $\\{\vec{x}_{j}\\}$ is a shorthand notation for the graph
$\\{(x^{(1)}_{j},x^{(2)}_{j})\\}^{j=n}_{j=1}$ associated with the state
$\Psi(\vec{p})$. In this notation, the array
$\vec{p}=\left(p_{1},p_{2}\right)$ denotes the momentum variables for the
system 1 and 2, respectively. The coefficients $\Psi_{\vec{x}_{j}}$ provide
the value for the norm of the state which is given by

$||\Psi(\vec{p})||=\sum_{\\{\vec{x}_{j}\\}}|\Psi_{\vec{x}_{j}}|^{2},$ (104)

hence these coefficients are different from zero and the sum converges (they
are non-null over countable number of points in the graph
$\\{\vec{x}_{j}\\}$). This norm arises from the inner product

$\langle\Psi|\Phi\rangle=\lim_{L_{1},L_{2}\rightarrow\infty}\frac{1}{4\,L_{1}L_{2}}\int^{L_{1}}_{-L_{1}}\int^{L_{2}}_{-L_{2}}\Psi^{*}(\vec{p})\,\Phi(\vec{p})dp_{1}dp_{2},$
(105)

which for the specific case of the plane waves takes the form of the Kronecker
delta

$\langle
e^{\frac{i}{\hbar}\vec{x}^{T}\vec{p}}|e^{\frac{i}{\hbar}\vec{x}^{\prime
T}\vec{p}}\rangle=\delta_{\vec{x},\vec{x}^{\prime}}.$ (106)

This inner product is the main signature of the polymer quantization as it
violates the Stone-von Neumann theorem. Consequently, polymer quantum
mechanics is not unitarily equivalent to the standard Schrödinger
representation. Moreover, in polymer quantum mechanics there is no momentum
operator hence infinitesimal spatial translations cannot be implemented.
Nevertheless, we can obtain a representation for the position operator, which
in the present case is given by

$\displaystyle\widehat{q}_{1}\Psi(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle i\hbar\frac{\partial}{\partial
p_{1}}\Psi(p_{1},p_{2})=-\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}x^{(1)}_{j}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}},$
(107) $\displaystyle\widehat{q}_{2}\Psi(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle i\hbar\frac{\partial}{\partial
p_{2}}\Psi(p_{1},p_{2})=-\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}x^{(2)}_{j}e^{\frac{i}{\hbar}\vec{x}^{T}_{j}\vec{p}}.$
(108)

Despite these peculiarities with the non-regularity of the polymer
representation, the representation of the symplectic group $Sp(2n,\mathbb{R})$
on the Hilbert space of polymer quantum mechanics was provided recently by one
of the authors in Ref. Garcia-Chung:2020cag . There, the representation is
given by the map $\widehat{C}^{(poly)}:Sp(2n,\mathbb{R})\rightarrow L({\cal
H}_{poly}),\;\widetilde{\bf M}\mapsto\widehat{C}^{(poly)}_{\widetilde{\bf
M}}$, where the linear operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}}$
acts on ${\cal H}_{poly}$ as

$\widehat{C}^{(poly)}_{\widetilde{\bf
M}}\Psi(\vec{p})=\lim_{L_{1},L_{2}\rightarrow\infty}\frac{1}{4\,L_{1}\,L_{2}}\int^{L_{1}}_{-L_{1}}\int^{L_{2}}_{-L_{2}}C^{(poly)}_{\widetilde{\bf
M}}(\vec{p},\vec{p}^{\prime})\Psi(\vec{p}^{\prime})d\vec{p}^{\prime}.$ (109)

The polymer kernel $C^{(poly)}_{\widetilde{\bf M}}(\vec{p},\vec{p}^{\prime})$
is given by

$C^{(poly)}_{\widetilde{\bf M}}(\vec{p},\vec{p}^{\prime})=\det(\widetilde{\bf
D}\widetilde{\bf
A}^{T})^{-\frac{1}{4}}e^{-\frac{i}{2\hbar}\vec{p}^{T}\widetilde{\bf
B}\widetilde{\bf
D}^{-1}\vec{p}}\sum_{\vec{x}}e^{\frac{i}{\hbar}\vec{p}^{T}\vec{x}-\frac{i}{\hbar}\vec{p}^{\prime
T}\widetilde{\bf D}^{T}\vec{x}+\frac{i}{2\hbar}\vec{x}^{T}{\widetilde{\bf
D}\widetilde{\bf C}^{T}}\vec{x}},$ (110)

and note that when the factor $\det(\widetilde{\bf D}\widetilde{\bf
A}^{T})^{-\frac{1}{4}}\neq 1$ it implies that this representation is not
unitary Garcia-Chung:2020cag .

Recall that one of the main features of loop quantum cosmology is its
intrinsic length scale given by the Planck length. As a toy model, polymer
quantum mechanics does not have an intrinsic length scale. However, it admits
a length scale that mimics some of the features of loop quantum cosmology.
This length scale is introduced at hand and is called polymer scale, usually
denoted by $\mu$. This polymer scale constitutes the analog of minimum length
for polymer quantum mechanics models, and therefore, it can be considered as a
lower bound for the uncertainties. In the present analysis, each system admits
a polymer scale $\mu_{1}$ and $\mu_{2}$ when the dynamics is considered.

Let us now consider the following questions: (1) is it possible to have
polymer states such that their uncertainties are lower than the polymer scale?
Furthermore, (2) do the correlations found in (98) have an analog in polymer
quantum mechanics?

To answer these questions let us consider the matrix $\widetilde{\bf M}_{1}$
with $a_{11}=a_{22}=c_{11}=c_{22}=0$, used to calculate the covariance matrix
${\bf V}^{(2)}_{1}$ in (93), but now with $a_{12}=-r_{1}$ and $c_{12}=-r_{2}$.
The explicit form is $\widetilde{\bf M}_{1}$

$\widetilde{\bf M}_{1}=\left(\begin{array}[]{cccc}e^{-r_{1}}&0&0&0\\\
0&e^{-r_{2}}&0&0\\\ 0&0&e^{r_{1}}&0\\\ 0&0&0&e^{r_{2}}\end{array}\right).$
(111)

The action of the group element $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$
on an arbitrary polymer state (103) gives the following state

$\widetilde{\Psi}_{\widetilde{\bf
M}_{1}}(p_{1},p_{2})=\widehat{C}^{(poly)}_{\widetilde{\bf
M}_{1}}\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x}_{j}}e^{\frac{i}{\hbar}\left(e^{-r_{1}}p_{1}x^{(1)}_{j}+e^{-r_{2}}p_{2}x^{(2)}_{j}\right)}.$
(112)

Using the representation of the position operators $\widehat{q}_{1}$ and
$\widehat{q}_{2}$ given in (107) and (108) we obtain the dispersion relations

$(\Delta x_{1})_{\widetilde{\Psi}}=e^{-r_{1}}(\Delta
x_{1})_{\Psi},\qquad(\Delta x_{2})_{\widetilde{\Psi}}=e^{-r_{2}}(\Delta
x_{2})_{\Psi},$ (113)

which show that the squeezed polymer state is indeed squeezed by a factor
$e^{-r_{1}}$ or $e^{-r_{2}}$. Consequently, if the initial dispersion of the
polymer state is given by $(\Delta x_{1})_{\Psi}$ or $(\Delta x_{2})_{\Psi}$,
then the squeeze operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$ gives
rise to a polymer state $\widetilde{\Psi}$ (112) whose dispersion is smaller
than that of the initial polymer state $\Psi$. Moreover, due to there is no
upper bound for the parameter $r$, these dispersion relations can be smaller
than the corresponding polymer scales $\mu_{1}$ and $\mu_{2}$.

Let us now consider the analog of the correlations (100) but for polymer
states. To do so, consider the polymer representation
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ of the symplectic matrix
(81) corresponding to a bi-partite system. The action of
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ on an arbitrary polymer
state (103) is given by

$\widetilde{\Psi}_{s}=\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf
s}}\Psi(p_{1},p_{2})=\sum_{\\{\vec{x}_{j}\\}}\Psi_{\vec{x_{j}}}e^{\frac{i}{\hbar}\left[\left(\cosh(r)\,x^{(1)}_{j}+\sinh(r)\,x^{(2)}_{j}\right)p_{1}+\left(\cosh(r)\,x^{(2)}_{j}+\sinh(r)\,x^{(1)}_{j}\right)p_{2}\right]},$
(114)

where again the representation in (109) was used.

We combine this result with the representation of the position operators in
(107) and (108) and obtain the following relations

$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}-(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=(\Delta x_{1})^{2}_{\Psi}-(\Delta
x_{2})^{2}_{\Psi},\qquad\forall\quad\Psi\in{\cal H}_{poly},$ (115)

where the conditions $l_{1}=l_{2}$ and $\phi=\\{0,\frac{\pi}{2},\pi\\}$ were
imposed. Remarkably, this result not only is independent of the parameter $r$
(which labels the amount of squeezing) but also applies to any polymer state
$\Psi\in{\cal H}_{poly}$. As can be seen, the difference of dispersions
squared is conserved, regardless of the amount of squeezing. Also, note that
$l_{1}$ and $l_{2}$ are considered group parameters and have no relation to
the dynamics, i.e., we are considering general states in ${\cal H}_{poly}$.
The same applies for $\phi$.

Let us now consider pure and symmetric polymeric states. The pure states are
those that can be written as the following product

$\Psi^{(p)}(p_{1},p_{2})=\left(\sum_{\\{x^{(1)}_{j}\\}}\Psi^{(1)}_{x^{(1)}_{j}}e^{\frac{i}{\hbar}p_{1}x^{(1)}_{j}}\right)\left(\sum_{\\{x^{(2)}_{j}\\}}\Psi^{(2)}_{x^{(2)}_{j}}e^{\frac{i}{\hbar}p_{2}x^{(2)}_{j}}\right).$
(116)

Secondly, both lattices $\\{x^{(1)}_{j}\\}$ and $\\{x^{(2)}_{j}\\}$ are
symmetric, i.e., for every positive point $0<x^{(s)}_{j}\in\\{x^{(s)}_{j}\\}$
there exist a negative point $0>x^{(s)}_{j^{\prime}}\in\\{x^{(s)}_{j}\\}$,
such that $x^{(s)}_{j}+x^{(s)}_{j^{\prime}}=0$, and the states are also
symmetric which implies that
$\Psi^{(s)}_{x^{(s)}_{j}}=\Psi^{(s)}_{x^{(s)}_{j^{\prime}}}$. These states are
the analog of the states described with even functions in the standard quantum
mechanics.

The dispersion relation for squeezed pure symmetric polymer states is given by

$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=\cosh(2r)\left((\Delta
x_{1})^{2}_{\Psi^{(p)}}+(\Delta x_{2})^{2}_{\Psi^{(p)}}\right),$ (117)

which takes the form

$(\Delta x_{1})^{2}_{\widetilde{\Psi}_{s}}+(\Delta
x_{2})^{2}_{\widetilde{\Psi}_{s}}=l^{2}\cosh(2r),$ (118)

when the dispersion of the pure states are $\left((\Delta
x_{1})_{\Psi^{(p)}}=(\Delta x_{2})_{\Psi^{(p)}}\right)=l/\sqrt{2}$. Notably,
Eq. (118) is the same as that obtained in (100) for the Schrödinger
representation. This shows that the correlations present in the standard
quantum mechanics using the $\widehat{C}_{\widetilde{\bf M}}$ operator for
both symplectic matrices $\widetilde{\bf M}_{1}$ and $\widetilde{\bf M}_{\bf
s}$ are the same to those obtained in polymer quantum mechanics using the
operator $\widehat{C}^{(poly)}_{\widetilde{\bf M}}$.

##  VI Conclusions

In this paper we provided the direct relation between the Lie algebra
$sp(4,\mathbb{R})$ and the symplectic group $Sp(4,\mathbb{R})$. The expression
shows the link between the block matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and
${\bf D}$ with those of the Lie algebra ${\bf a}$, ${\bf b}$ and ${\bf c}$
given in the Eqs. (34)-(37). This result has not been reported before and
applies to the full Lie algebra $sp(4,\mathbb{R})$ of the symplectic group
$Sp(4,\mathbb{R})$.

Such relation allows us to obtain some important symplectic matrices that were
used in subsequent sections. In the first example for ${\bf a},{\bf c}\neq{\bf
0}$ and ${\bf b}=0$, we show that the corresponding symplectic matrix ${\bf
M}_{1}$, given in (52), can be written as $\left({\bf
M}^{\prime}_{1}\otimes{\bf 1}\right)\cdot\left({\bf 1}\otimes{\bf
M}^{\prime}_{2}\right)$, where ${\bf 1},{\bf M}^{\prime}_{1},{\bf
M}^{\prime}_{2}\in Sp(2,\mathbb{R})$. Here, ${\bf M}^{\prime}_{1}$ and ${\bf
M}^{\prime}_{2}$ are symplectic matrices acting over each of the sub-systems
with coordinates $(q_{1},p_{1})$ and $(q_{2},p_{2})$ respectively. In the case
${\bf a}={\bf c}=\mbox{diag}(a_{11},a_{22})$ and ${\bf
b}=\mbox{diag}(b_{11},b_{22})$ the symplectic matrix ${\bf M}_{2}$ in (59),
describes two coupled harmonic oscillators with interaction terms labeled by
the coefficients of the matrix ${\bf b}$. Finally, in the equation (62) we
showed the symplectic matrix ${\bf M}_{3}$ for the case in which ${\bf a}={\bf
c}={\bf 0}$ and a general form of the matrix ${\bf b}$.

In section (IV) we analyzed the classical description of squeeze operators. We
showed that the symplectic matrix ${\bf M}_{\bf s}$ is the classical analog of
the squeeze operator
${\widehat{S}(\zeta)=e^{\widehat{s}_{\zeta}}}=\widehat{C}_{\widetilde{\bf
M}_{\bf s}}$. Also, we remarked the isomorphism between the Lie algebra
$sp(4,\mathbb{R})$ and ${\cal P}(2,\mathbb{R})$. Additionally, the general
form of the covariance matrix ${\bf V}^{(2)}$ for the squeezed vacuum state
$|\Psi_{\widetilde{\bf M}}\rangle$ was derived using the Weyl algebra
representation and the symplectic matrix ${\bf M}_{1}$. The components of this
covariant matrix were used to calculate the dispersion relations (94) and (95)
for the particular case where $a_{11}=a_{22}=c_{11}=c_{22}=0$. As is already
known, these dispersions can be smaller than the vacuum characteristic length
for the harmonic oscillators. They also satisfy the Heisenberg uncertainty
principle as was shown in (96). We then calculated the covariance matrix for
the symplectic matrix ${\bf M}_{\bf s}$ corresponding to the classical analog
of the bipartite squeeze operator. With this matrix we determined the
correlation (100). We also provided the general expressions for these
correlations in equations (98) and (99).

Applying the previous results, it is also possible to represent operators in
non-regular Hilbert spaces that are non-unitarily equivalent to the Fock-
Schrödinger representation, so in section (V) we analyze polymer quantized
systems. We calculated the dispersion relation for an arbitrary polymer state
using the representation of the symplectic group in polymer quantum mechanics.
We obtained that the polymer representation of the squeeze operator given by
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$, yields a dispersion relation
Eq. (113), which can be smaller than those of the initial state. This implies
that $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{1}}$ is indeed a polymer
squeeze operator and (112) describes a polymer squeezed state.

On the other hand, the polymer representation of the bipartite squeeze
operator given by $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ was used
to derive the polymer correlations (115) and (118). The first correlation
(115) shows that the difference of the dispersions’ square is preserved and is
independent of the initial polymer state. Clearly, this result only holds for
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ so a symplectic matrix
different than the one used in (111) will produce a different result. In the
case of (118), the result has the same form as the standard correlation in Eq.
(100), hence, the polymer representation of
$\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$ can be used to construct
correlated squeezed states for bipartite polymer systems. Naturally, this
brings some questions like whether there is any mechanism in nature, say loop
quantum cosmology or the interior of a black hole, from which a polymer
squeezed state can be created.

Moreover, establishing that squeezing is a property also present in non-
regular representations questions its role in the classicality of some
cosmological models. As we showed, it is possible to construct entangled
polymer states using $\widehat{C}^{(poly)}_{\widetilde{\bf M}_{\bf s}}$. Such
polymer entangled states correlations satisfy a relation identical to the one
obtained in the standard quantum mechanics. In this case, the states are
polymer bipartite squeezed states, similar to those used in the quantum
description of the inflaton field.

Finally, it is worth to mention that the polymer squeezed states obtained as a
result of the representation of $Sp(4,\mathbb{R})$ in ${\cal H}_{poly}$, given
in (112, 114), differ from those reported in the LQC literature
taveras2008corrections ; mielczarek2012gaussian ; gazeau2013quantum ;
diener2014numerical ; diener2014numerical2 ; corichi2011coherent . In these
references, a Gaussian form of the states is considered, whereas in our case,
the polymer state (112) is a general polymer state. In LQC, the squeezed
states are constructed by hand due after imposing some conditions to achieve
the squeezed nature of the dispersion relations. In our case, the squeezed
state results from the action of the squeeze operator.

The results of this work open the doors for studying the entangled states of
matter and geometry and the role that their correlations might play in some
physical scenarios.

##  VII Acknowledgments

I thank Academia de Matemáticas and Colegio de Física, UP, for the support and
enthusiasm.

##  VIII Appendix: calculation of ${\bf m}^{2n}$

In this appendix the expression for the matrix ${\bf m}^{2n}$ is obtained. To
do so, recall that the matrix ${\bf m}^{2}$ is formed by four $2\times 2$
block matrices where the upper left and the lower right matrices are multiples
of the identity matrix ${\bf 1}_{2\times 2}$. The upper right block is the
matrix ${\bf J}{\bf d}$ whereas the lower left is $-{\bf J}{\bf d}^{T}$.
Notably, we found that this block structure is preserved after exponentiating
the matrix ${\bf m}^{2}$ an integer number of times. That is to say, the
n-power of matrix ${\bf m}^{2}$ yields a new matrix $({\bf m}^{2})^{n}$ given
by

$({\bf m}^{2})^{n}=\left(\begin{array}[]{cc}\alpha_{n}{\bf 1}_{2\times
2}&\beta_{n}{\bf J}{\bf d}\\\ -\beta_{n}{\bf J}{\bf d}^{T}&\gamma_{n}{\bf
1}_{2\times 2}\end{array}\right).$ (119)

It is this pattern the one to be considered when this procedure is applied to
higher order symplectic groups.

The coefficients $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$, are to be
determined and depend on the values of the matrices ${\bf a}$, ${\bf b}$,
${\bf c}$ and ${\bf d}$. For $n=1$, these coefficients are given by the
factors in the block matrices of ${\bf m}^{2}$ given in (32) and can be
directly defined as

$\alpha_{1}:=-(\det{\bf a}+\det{\bf
b}),\qquad\beta_{1}:=+1,\qquad\gamma_{1}:=-(\det{\bf c}+\det{\bf b}).$ (120)

To calculate $\alpha_{n}$, $\beta_{n}$ and $\gamma_{n}$ for arbitrary $n$,
first note that they can be generated with the $(n-1)$-power of the matrix
${\bf T}$ as

$\displaystyle\left(\begin{array}[]{c}\alpha_{n}\\\ \beta_{n}\\\
\gamma_{n}\end{array}\right)={\bf
T}^{n-1}\left(\begin{array}[]{c}\alpha_{1}\\\ \beta_{1}\\\
\gamma_{1}\end{array}\right),$ (127)

where the matrix ${\bf T}$ is given by

${\bf T}=\left(\begin{array}[]{ccc}\alpha_{1}&\beta_{1}\det{\bf d}&0\\\
\beta_{1}&\gamma_{1}&0\\\ 0&\beta_{1}\det{\bf
d}&\gamma_{1}\end{array}\right).$ (128)

The calculation shows that ${\bf T}^{n-1}$ is a matrix of the form

${\bf T}^{n-1}=\left(\begin{array}[]{cc}{\bf U}^{n-1}&\vec{0}^{T}\\\
\vec{u}^{T}\gamma_{1}^{n-2}\sum^{n-2}_{j=0}\gamma^{-j}_{1}{\bf
U}^{j}&\gamma^{n-1}_{1}\end{array}\right),$ (129)

where $\vec{0}=(0,0)$ and $\vec{u}=(0,\beta_{1}\det{\bf d})$ and matrix ${\bf
U}$ is given by

$\displaystyle{\bf U}=\left(\begin{array}[]{cc}\alpha_{1}&\beta_{1}\det{\bf
d}\\\ \beta_{1}&\gamma_{1}\end{array}\right).$ (132)

Then, using (127) we have the following relation for the coefficients

$\displaystyle\left(\begin{array}[]{c}\alpha_{n}\\\
\beta_{n}\end{array}\right)={\bf U}^{n-1}\left(\begin{array}[]{c}\alpha_{1}\\\
\beta_{1}\end{array}\right),\qquad\gamma_{n}=\gamma^{n}_{1}+\vec{u}^{T}\gamma^{n-2}_{1}\sum^{n-2}_{j=0}\gamma^{-j}_{1}{\bf
U}^{j}\left(\begin{array}[]{c}\alpha_{1}\\\ \beta_{1}\end{array}\right).$
(139)

In order to calculate ${\bf U}^{n-1}$ we need to diagonalize the matrix ${\bf
U}$. Let ${\bf P}$ be the matrix diagonalizing ${\bf U}$, then

${\bf U}={\bf P}\,{\bf D}_{0}\,{\bf P}^{-1},$ (140)

where the matrix ${\bf P}$ is

${\bf
P}=\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right).$ (141)

The real arbitrary parameters $k_{1}$ and $k_{2}$ result from the
diagonalization procedure. Its values will be automatically cancelled as part
of the calculation of ${\bf U}^{n-1}$ further below. The eigenvalues of ${\bf
U}$, denoted by $\lambda_{\pm}$, have the following expression

$\displaystyle\lambda_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}+\gamma_{1}}{2}\pm\frac{1}{2}\sqrt{(\alpha_{1}-\gamma_{1})^{2}+4\beta^{2}_{1}\det{\bf
d}},$ (142) $\displaystyle=$ $\displaystyle-\frac{\det{\bf a}+\det{\bf
c}+2\det{\bf b}}{2}\pm\frac{1}{2}\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}.$

and the diagonal matrix ${\bf D}_{0}$ is

${\bf D}_{0}=\left(\begin{array}[]{cc}\lambda_{+}&0\\\
0&\lambda_{-}\end{array}\right).$ (143)

We now take the $n-1$ power of ${\bf U}$ given in (140) to obtain the
following result

$\displaystyle{\bf
U}^{n-1}=\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right)\left(\begin{array}[]{cc}\lambda^{n-1}_{+}&0\\\
0&\lambda^{n-1}_{-}\end{array}\right)\left(\begin{array}[]{cc}\frac{(\lambda_{+}-\gamma_{1})}{\beta_{1}}k_{1}&\frac{(\lambda_{-}-\gamma_{1})}{\beta_{1}}k_{2}\\\
k_{1}&k_{2}\end{array}\right)^{-1},$ (150)

which, when combined with the result in (139) together with the expression for
$\vec{u}$, gives

$\displaystyle\alpha_{n}$ $\displaystyle=$
$\displaystyle\frac{(\lambda_{+}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{+}-(\lambda_{-}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{-}}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\beta^{2}_{1}\det{\bf
d}}},$ (151) $\displaystyle\beta_{n}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{n}_{+}-\lambda^{n}_{-}}{\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\beta^{2}_{1}\det{\bf d}}},$ (152)
$\displaystyle\gamma_{n}$ $\displaystyle=$
$\displaystyle\frac{\left[(\lambda_{+}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{-}-(\lambda_{-}-\det{\bf b}-\det{\bf
c})\lambda^{n}_{+}\right]}{\sqrt{(\det{\bf a}-\det{\bf
c})^{2}+4\beta^{2}_{1}\det{\bf d}}}.$ (153)

These are the final expressions for the coefficients in $({\bf m}^{2})^{n}$.

##  IX Appendix: Series analysis

In this appendix we calculate the series expansion terms. To do so, note that
once the expression for $({\bf m}^{2})^{n}$ is inserted the expansion (31) and
the even and odd terms are collected, the matrix ${\bf M}({\bf a},{\bf b},{\bf
c})$ takes the form

${\bf M}({\bf a},{\bf b},{\bf c})=\left(\begin{array}[]{cc}\alpha^{(e)}{\bf
1}_{2\times 2}&\beta^{(e)}{\bf J}{\bf d}\\\ -\beta^{(e)}{\bf J}{\bf
d}^{T}&\gamma^{(e)}{\bf 1}_{2\times 2}\end{array}\right)+{\bf
m}\left(\begin{array}[]{cc}\alpha^{(o)}{\bf 1}_{2\times 2}&\beta^{(o)}{\bf
J}{\bf d}\\\ -\beta^{(o)}{\bf J}{\bf d}^{T}&\gamma^{(o)}{\bf 1}_{2\times
2}\end{array}\right),$ (154)

where the following coefficients are given by

$\alpha^{(e)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\alpha_{n},\qquad\beta^{(e)}:=\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\beta_{n},\qquad\gamma^{(e)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n)!}\gamma_{n},$
(155)
$\alpha^{(o)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\alpha_{n},\qquad\beta^{(o)}:=\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\beta_{n},\qquad\gamma^{(o)}:=1+\sum^{+\infty}_{n=1}\frac{1}{(2n+1)!}\gamma_{n}.$
(156)

We now insert (151), (152) and (153) in the relations (155) - (156) to obtain

$\alpha^{(e)}=\frac{1}{2}\left[\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)\right]+\frac{\left(\det{\bf
c}-\det{\bf
a}\right)\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right]}{2\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\det{\bf d}}},$ (157)

$\displaystyle\alpha^{(o)}=\frac{1}{2}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}+\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right]+\frac{\left(\det{\bf
c}-\det{\bf a}\right)}{2\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(158) $\beta^{(e)}=\frac{1}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right],$
(159) $\beta^{(o)}=\frac{1}{\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(160)

$\gamma^{(e)}=\frac{1}{2}\left[\cosh\left(\sqrt{\lambda_{+}}\right)+\cosh\left(\sqrt{\lambda_{-}}\right)\right]-\frac{\left(\det{\bf
c}-\det{\bf
a}\right)\left[\cosh\left(\sqrt{\lambda_{+}}\right)-\cosh\left(\sqrt{\lambda_{-}}\right)\right]}{2\sqrt{(\det{\bf
a}-\det{\bf c})^{2}+4\det{\bf d}}},$ (161)

$\displaystyle\gamma^{(o)}=\frac{1}{2}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}+\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right]-\frac{\left(\det{\bf
c}-\det{\bf a}\right)}{2\sqrt{(\det{\bf a}-\det{\bf c})^{2}+4\det{\bf
d}}}\left[\frac{\sinh\left(\sqrt{\lambda_{+}}\right)}{\sqrt{\lambda_{+}}}-\frac{\sinh\left(\sqrt{\lambda_{-}}\right)}{\sqrt{\lambda_{-}}}\right],$
(162)

where we have to recall the expression for the eigenvalues $\lambda_{\pm}$ in
(142).

##  X Appendix: Covariance matrix coefficients

In this appendix we show the explicit form of the coefficients of the
covariance matrix ${\bf V}^{(2)}$.

$\displaystyle{V}^{(2)}_{11}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{12}^{2}l^{4}_{1}+a_{22}^{2}\hbar^{2}\right)}{l^{2}_{1}\det{\bf
a}}+\frac{a_{12}l^{2}_{1}\sinh(2\sqrt{-\det{\bf a}})}{\sqrt{-\det{\bf
a}}}+l^{2}_{1}\cosh^{2}(\sqrt{-\det{\bf a}}),$ (163)
$\displaystyle{V}^{(2)}_{22}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{12}^{2}l^{4}_{2}+c_{22}^{2}\hbar^{2}\right)}{l^{2}_{2}\det{\bf
c}}+\frac{c_{12}l^{2}_{2}\sinh(2\sqrt{-\det{\bf c}})}{\sqrt{-\det{\bf
c}}}+l^{2}_{2}\cosh^{2}(\sqrt{-\det{\bf c}}),$ (164)
$\displaystyle{V}^{(2)}_{13}$ $\displaystyle=$
$\displaystyle\frac{\sinh(2\sqrt{-\det{\bf
a}})\left(a_{22}\hbar^{2}-a_{11}l^{4}_{1}\right)}{2l^{2}_{1}\sqrt{-\det{\bf
a}}}+\frac{a_{12}\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{11}l^{4}_{1}+a_{22}\hbar^{2}\right)}{l^{2}_{1}\det{\bf a}},$
(165) $\displaystyle{V}^{(2)}_{24}$ $\displaystyle=$
$\displaystyle\frac{\sinh(2\sqrt{-\det{\bf
c}})\left(c_{22}\hbar^{2}-c_{11}l^{4}_{2}\right)}{2l^{2}_{2}\sqrt{-\det{\bf
c}}}+\frac{c_{12}\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{11}l^{4}_{2}+c_{22}\hbar^{2}\right)}{l^{2}_{2}\det{\bf c}},$
(166) $\displaystyle{V}^{(2)}_{33}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
a}})\left(a_{11}^{2}l^{4}_{1}+a_{12}^{2}\hbar^{2}\right)}{l^{2}_{1}\det{\bf
a}}+\frac{a_{12}\hbar^{2}\sinh(2\sqrt{-\det{\bf a}})}{l^{2}_{1}\sqrt{-\det{\bf
a}}}+\frac{\hbar^{2}\cosh^{2}(\sqrt{-\det{\bf a}})}{l^{2}_{1}},$ (167)
$\displaystyle{V}^{(2)}_{44}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(\sqrt{-\det{\bf
c}})\left(c_{11}^{2}l^{4}_{2}+c_{12}^{2}\hbar^{2}\right)}{l^{2}_{2}\det{\bf
c}}+\frac{c_{12}\hbar^{2}\sinh(2\sqrt{-\det{\bf c}})}{l^{2}_{2}\sqrt{-\det{\bf
c}}}+\frac{\hbar^{2}\cosh^{2}(\sqrt{-\det{\bf c}})}{l^{2}_{2}}.$ (168)

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