# Fluctuations, uncertainty relations, and the geometry of quantum state
manifolds

Balázs Hetényi and Péter Lévay  MTA-BME Quantum Dynamics and Correlations
Research Group, Department of Physics, Budapest University of Technology and
Economics, H-1111 Budapest, Hungary

###### Abstract

The complete quantum metric of a parametrized quantum system has a real part
(usually known as the Provost-Vallee metric) and a symplectic imaginary part
(known as the Berry curvature). In this paper, we first investigate the
relation between the Riemann curvature tensor of the space described by the
metric, and the Berry curvature, by explicit parallel transport of a vector in
Hilbert space. Subsequently, we write a generating function from which the
complex metric, as well as higher order geometric tensors (affine connection,
Riemann curvature tensor) can be obtained in terms of gauge invariant
cumulants. The generating function explicitly relates the quantities which
characterize the geometry of the parameter space to quantum fluctuations. We
also show that for a mixed quantum-classical system both real and imaginary
parts of the quantum metric contribute to the dynamics, if the mass tensor is
Hermitian. A many operator generalization of the uncertainty principle results
from taking the determinant of the complex quantum metric. We also calculate
the quantum metric for a number of Lie group coherent states, including
several representations of the $SU(1,1)$ group. In our examples non-trivial
complex geometry results for generalized coherent states. A pair of oscillator
states corresponding to the $SU(1,1)$ group gives a double series for its
spectrum. The two minimal uncertainty coherent states show trivial geometry,
but, again, for generalized coherent states non-trivial geometry results.

††preprint: APS/123-QED

##  I Introduction

A parametrized quantum system exhibits interesting physics. In such systems a
metric structure is induced on the parameter space which is determined by the
type of quantum system in question Provost80 . If the metric is Riemannian, it
is then possible to derive an affine connection or a Riemann curvature,
quantities which further characterize the geometry of the parameter space.
Recently, Smith et al. Smith22  have considered the geodesic equation
associated with a quantum metric, and arrived at equations of motion similar
to those of general relativity. This led to the intriguing suggestion to use
parametrized quantum systems in the laboratory to mimic and study spacetime
physics.

The study of quantum metrics already has a long history. The complex quantum
metric of a parametrized quantum system was first derived Provost80  by
Provost and Vallee, and they showed that its real part corresponds to a gauge
invariant Riemannian metric (known as the Provost-Vallee metric (PVM)), while
its imaginary part is of symplectic structure. After the discovery Berry84  of
the quantum geometric phase characterizing adiabatic cycles (also known as the
Berry phase), it was quickly realized Shapere89  that the imaginary part of
the complex quantum metric is the Berry curvature (BC), whose area integral
within an adiabatic cycle gives the phase itself. In the study of quantum
phase transitions, a useful quantity is the fidelity Sachdev11 ; You07 ;
Zanardi07 ; Gu10 ; Dutta15 ; Carollo20 ; Austrich-Olivares22 , an overlap (Eq.
(31)) between quantum states, and the fidelity susceptibility, which is a
second derivative of the fidelity. This construction is entirely equivalent to
the Provost-Vallee one, which also starts from an overlap of wave functions,
and in fact, the fidelity susceptibility is the PVM (Eq. (38)). The quantum
metric is also relevant In the modern theory of polarization King-Smith93 ;
Resta94 ; Resta00 ; Vanderbilt18 , where the metric tensor expresses Souza00 ;
Ortiz00  the variance of the polarization in insulators. It can also be
connected to linear Kashihara23  and non-linear Liu23  response functions, and
can be used as a gauge to distinguish metals from insulators Resta05 ;
Marrazzo19 . More recently, the quantum metric has reappeared in the context
of topological physics, specifically, in the description of the fractional
quantum Hall effect Haldane11 ; Gromov17  and of topological insulators
Matsuura10 ; Varjas22 . Anandan and Aharonov Anandan90  have characterized the
geometric phase by the Fubini-Study metric, which was shown Pati92 ; Levay92 ;
Bohm03  to be related to the PVM. The PVM is a metric on the parameter space,
while the Fubini-Study metric is the metric of the projective Hilbert space of
the entire system. It follows that the PVM is the pullback of the Fubini-Study
metric with respect to the map defined by the parametrized family of spectral
projectors.  The Fubini-Study metric was also shown Levay04  to define a
measure of entanglement. Very recently Avdoshkin and Popov Avdoshkin23  have
derived quantum geometric tensors (including the Christoffel symbol) from a
three-point Bargmann invariant Bargmann64 . The Bargmann invariant can be
interpreted as a cumulant generating function Hetenyi22 , so the formalism of
Ref. Avdoshkin23  is related to ours.

In this work we investigate the full complex quantum metric. As a start we
investigate the relation between the two curvatures which both appear as the
characteristic of a parametrized quantum system. The BC appears as the second
derivative of the scalar product of two quantum states with respect to the
parameters, while the four-index Riemann curvature tensor appears as the
fourth derivative. We clarify the relation between these two quantities by
comparing the parallel transport of a vector in Hilbert space (which is
characterized by the BC) against the parallel transport of an ordinary vector
(characterized by the Riemann curvature tensor). Subsequently we write down
the generating function from which the PVM, as well as higher order geometric
quantities can be obtained. While it is known that the PVM is related to the
second cumulant (variance) of the fluctuations, we point out that higher order
geometric quantities correspond to higher order cumulants, for example, the
affine connection is related to the third cumulant (skew), while the Riemann
curvature tensor corresponds to the fourth cumulant (kurtosis). In the
fidelity language, this means that higher order geometric tensors are related
to ”nonlinear fidelity susceptibilities”.

Mixed quantum-classical systems obeying the Born-Oppenheimer approximation
were considered Mead79 ; Mead80  by Mead and Truhlar. They predicted the
“molecular Aharonov-Bohm effect” and introduced the notion of “molecular
magnetic field” a gauge field that modifies the force on the nuclei, and whose
origin is the BC. In addition, according to the analysis in Ref. Shapere89
the PVM gives rise to a “molecular electric” counterpart of the molecular
magnetic field. We revisit this issue, considering the possibility of a
complex (Hermitian) mass tensor, and find that the BC also contributes to the
molecular electric field (an extra potential term arises, since the anti-
symmetric part of the complex mass tensor couples to the BC, which is also
antisymmetric). Complex mass was first introduced in 1967 by Feinberg
Feinberg67 . Tachyons are quantum fields with imaginary mass which violate
physical principles and such particles are purely hypothetical. However,
recently, Hermitian mass matrices have been invoked to explain neutrino
oscillations and generalizations of the Dirac equation in this direction have
also been put forth Jones-Smith14 . By diagonalizing the inverse mass tensor,
we show that the resulting Born-Oppenheimer type system is like a ”usual”
system whose only possible unusual characteristic, in addition to the
molecular fields, is that the inverse masses are directionally dependent.

Requiring that the determinant of the complex quantum metric is greater than
or equal to zero leads to a many-operator generalization of the Schrödinger
uncertainty principle Schrodinger30 , which is a stronger statement than the
Heisenberg uncertainty principle (whose many operator generalization was done
by Robertson Robertson34 ). We also calculate the metric for three types of
Lie group coherent states Perelomov72 ; Gilmore72 ; Zhang90 : harmonic
oscillator, atomic ($SU(2)$), and hyperbolic ($SU(1,1)$). Non-trivial complex
geometry results from generalized Boiteux73 ; Roy82  coherent states, which
are generated from non-extremal starting states, hence they are no longer
minimum uncertainty states. This generalization of coherent states was first
proposed by Boiteux and Levelut Boiteux73 , who considered it a purely formal
development, but later, Roy and Singh Roy82  showed that the time-dependent
generalization of these generalized coherent states lead to dynamics in which
the probability distribution remains undistorted, and the averages perform
classical motion. For certain special points in the parameter space even the
quantum metric of generalized coherent states can become trivial, points where
the complex quantum metric tensor exhibits zero determinant. We analyze the
$SU(1,1)$ case extensively, where different representations of the group
correspond Alhassid83 ; Alhassid86 ; Mahran86 ; Gerry89 ; Kim89 ; Levay93  to
different quantum number series. A case of particular interest we treat is a
pair of coupled oscillators Mahran86 ; Gerry89 ; Kim89 ; Levay93 , which
corresponds to the direct sum of two projective representations of the
$SU(1,1)$ group, corresponding to a spectrum of two series (squeezed vacuum
state, squeezed one-photon state). In this case we find that the complex
quantum metric is zero for the extremal states of both series, in other words,
there are two cases where the geometry is trivial.

Our paper is organized as follows. In section II we give a brief overview of
the quantities used to characterize the geometry of curved spaces. In section
III we derive the Berry connection and curvature by explicit parallel
transport, and compare it to the affine connection and Riemann curvature. In
section IV we write down the cumulant generating function which generates
gauge invariant cumulants for a general parametrized quantum system. In
section V we investigate a mixed quantum-classical system with a complex mass
tensor. Then, in section VI we write the Christoffel symbol in terms of the
third gauge invariant cumulant in our formalism. In VII we relate geometric
tensors and quantum fluctuations to the uncertainty principle. Before
concluding in section IX, in section VIII we derive full quantum geometric
tensors for coherent states, and identify when the geometries described by the
tensors are trivial or not.

##  II Background: Geometric tensors and parallel transport

In this section we provide some mathematical background Arfken05  used in this
work. Given an $N$-dimensional metric space with coordinates
$s_{1},...,s_{N}$, the line element squared can be written as

$dl^{2}=g_{jk}ds^{j}ds^{k},$ (1)

where summation over doubly occurring indices is implied, and $g_{ij}$ denotes
the metric tensor. The metric tensor characterizes the geometry of the space
and can depend on the coordinates $\\{s_{i}\\}$.

The equation for geodesic curves can be obtained by requiring that the
integral of the variance of $dl$ along a curve is zero,

$\delta\int dl=0.$ (2)

This yields the second order differential equation

$g_{ki}\frac{d^{2}s^{i}}{dl^{2}}=-\frac{ds^{i}}{dl}\frac{ds^{j}}{dl}[ij,k],$
(3)

where $[ij,k]$ denotes the Christoffel symbol of the first kind,

$[ij,k]=\frac{1}{2}\left(\partial_{j}g_{ik}+\partial_{i}g_{kj}-\partial_{k}g_{ij}\right),$
(4)

where

$\partial_{i}g_{kj}=\frac{\partial g_{kj}}{\partial s^{i}}.$ (5)

The Christoffel symbol of the second kind is defined as

$\Gamma\indices{{}^{l}_{i}{}_{j}}=\frac{1}{2}g^{kl}(\partial_{j}g_{ik}+\partial_{i}g_{kj}-\partial_{k}g_{ij}),$
(6)

where the inverse (or dual) of the metric tensor, $g^{ij}$ is given by,

$g^{ij}g_{jk}=\delta\indices{{}^{i}_{k}}.$ (7)

The equation for parallel transporting a vector is usually given in terms of
the Christoffel symbol of the second kind. If we parallel transport a vector
along a curve $s^{k}(\lambda)$ (parametrized by $\lambda$) in a curved space
with some metric tensor $g_{ij}(\lambda)$, from which the Christoffel symbol
is obtained according to Eq. (6), then the change in the vector in the process
is given by

$\delta
V^{i}(\lambda)=-\Gamma\indices{{}^{i}_{j}{}_{k}}(\lambda)V^{j}(\lambda)\delta
s^{k}(\lambda).$ (8)

##  III Berry connection/Berry curvature and the Christoffel symbol/Riemann
curvature tensor

In the case of parametrized quantum systems there are two “connections” and
two “curvatures”. The Berry connection has one coordinate index, while the
affine connection associated with the parameter space has three. The Berry
curvature is a two-index quantity, whereas the Riemann curvature tensor has
four indices. The Berry curvature is the imaginary part of the complex quantum
metric, which is also a two-index quantity (defined below in Eqs. (35-38)).

The Berry connection and curvature correspond Simon83  to the gauge fields of
a fiber bundle. A fiber bundle is constructed by first considering the
projected Hilbert space of the parametrized quantum system, where the
projection occurs onto the space spanned by a chosen wave function or chosen
set of wave functions. The projection removes the phase indeterminacy (for
example, if a single state is used, then the projection,
$|\Psi(s)\rangle\langle\Psi(s)|$, has no arbitrary phase due to the
simultaneous presence of a bra and a ket). A fiber bundle restores the phase
by assigning a fiber to each point of the base space (which in this case is
the parameter space $s$). For example, in the case of a single state
projection, a typical fiber can be the space on which the phase is easily
represented, the $S^{1}$ unit circle. In this sense, there is a difference
between the Berry connection/curvature (which corresponds to the gauge fields
which live on the fibers) and the Christoffel and Riemann curvature tensors
(which live on the parameter space $s$), however, we demonstrate below a
useful parallel between these two sets of quantities.

We will argue below that the above discrepancy between the number of indices
between the two sets of quantities is due to the projection of the Hilbert
space necessary to obtain physically interesting cases (it is conventional to
suppress the indices of the projected states). It is in order to cite a few
examples. The Berry phase is defined via an integral of one quantum state over
an adiabatic cycle, and the state index is usually not explicitly indicated
when writing the Berry connection. The state under scrutiny is separated by an
energy gap from the other states. A sum over Berry phases over all the states
of a complete Hilbert space is zero Pacher89 ; Xiao10 . In the case of the
generalization of the Berry phase to a degenerate subset of states by Wilczek
and Zee Wilczek84 , it is the subset of states that is carried around an
adiabatic cycle, again, separated from the rest of the states by a gap. In the
modern theory of polarization of insulating systems King-Smith93 ; Resta94 ;
Resta00 ; Vanderbilt18 , a particular kind of Berry phase, a Zak phase Zak89 ,
is evaluated, which is an integral over the Brillouin zone over an occupied
band (or bands) of states. The projection here is justified by the fact that
only some bands are occupied, unoccupied bands do not contribute. In the case
of time-reversal invariant topological insulators Kane05a ; Kane05b  a
modified version of the Zak phase is calculated for occupied degenerate
Kramers bands. While the gap condition can be relaxed Aharonov87 , a non-
trivial value for a Berry phase or a Wilson loop still requires a projection
in Hilbert space.

We now turn to the main purpose of this section, which is to derive the Berry
connection and curvature by explicit parallel transport of a vector in Hilbert
space. Our aim is to place emphasis on the connections (no pun intended) and
the differences between the Berry connection and curvature on the one hand and
the Christoffel symbol and the Riemann curvature tensor on the other.

One way Arfken05  to arrive at the Christoffel symbol and the Riemann tensor
is to consider the change in a general vector ${\bf V}(s)$ defined on a curved
parameter space. ${\bf V}(s)$ is a mapping from each point of the parameter
space $s_{1},...,s_{N}$ to $V_{1},...,V_{M}$. One can expand the vector in a
local basis,

${\bf V}(s)=V^{j}(s){\bf e}_{j}(s),$ (9)

where $V^{j}(s)$ denote the components of the vector ${\bf V}(s)$ in the local
basis and ${\bf e}_{j}(s)$ denote the members of the basis themselves. In
general, a change in ${\bf V}(s)$ results from a change in the coefficients
themselves as well as a possible change in the basis,

$\frac{\partial{\bf V}(s)}{\partial s^{k}}=\frac{\partial{V^{j}(s)}}{\partial
s^{k}}{\bf e}_{j}(s)+V^{j}(s)\frac{\partial{\bf e}_{j}(s)}{\partial s^{k}}.$
(10)

We can expand the derivative in the second term as,

$\frac{\partial{\bf e}_{j}(s)}{\partial
s^{k}}=\Gamma\indices{{}^{l}_{j}{}_{k}}(s){\bf e}_{l}(s),$ (11)

where the Christoffel symbol, $\Gamma\indices{{}^{l}_{j}{}_{k}}(s)$ is the
expansion coefficient. Let us resolve the change in ${\bf V}(s)$ into
components as

${\bf e}^{l}(s)\cdot\frac{\partial{\bf V}(s)}{\partial
s^{k}}=\frac{\partial{V^{l}(s)}}{\partial
s^{k}}+V^{j}(s)\Gamma\indices{{}^{l}_{j}{}_{k}}(s).$ (12)

We define the covariant derivative in the $k$ direction as,

$D_{k}V^{l}(s)=\frac{\partial{V^{l}(s)}}{\partial
s^{k}}+\Gamma\indices{{}^{l}_{j}{}_{k}}(s)V^{j}(s).$ (13)

From the covariant derivative, it is possible to define the four-index Riemann
curvature tensor, as

$(D_{j}D_{k}-D_{k}D_{j})V^{l}(s)=R\indices{{}^{l}_{m}{}_{k}{}_{j}}(s)V^{m}(s),$
(14)

where

$\displaystyle R\indices{{}^{l}_{m}{}_{k}{}_{j}}(s)$ $\displaystyle=$
$\displaystyle\partial_{j}\Gamma\indices{{}^{l}_{k}{}_{m}}(s)-\partial_{k}\Gamma\indices{{}^{l}_{j}{}_{m}}(s)$
$\displaystyle+\Gamma\indices{{}^{l}_{j}{}_{n}}(s)\Gamma\indices{{}^{n}_{k}{}_{m}}(s)-\Gamma\indices{{}^{l}_{k}{}_{n}}(s)\Gamma\indices{{}^{n}_{j}{}_{m}}(s).$

Note, the first pair of indices in $R\indices{{}^{l}_{m}{}_{k}{}_{j}}$ ($l$
and $m$) rotate the vector components, while the second pair ($k$ and $j$) are
sensitive to the curvature of the surface on which the transport occurs.

In an exactly analogous manner, we can consider a quantum system which is
parametrized by a set of variables $s=s_{1},...,s_{N}$, and we can write a
general vector in the Hilbert space in some basis,

$|\Psi(s)\rangle=\sum_{\iota}|\iota(s)\rangle\langle\iota(s)|\Psi(s)\rangle=\sum_{\iota}|\iota(s)\rangle\Psi^{\iota}(s).$
(16)

We use Greek indices to label basis states. We can consider a change in this
state in the $k$-direction,

$\partial_{k}|\Psi(s)\rangle=\sum_{\iota}\left[|\iota(s)\rangle\partial_{k}\Psi^{\iota}(s)+\partial_{k}|\iota(s)\rangle\Psi^{\iota}(s)\right].$
(17)

For the directions in the parameter space we use Latin indices. Multiplying
from the left by another state $\langle\lambda(s)|$ we obtain the change of
$\Psi(s)$ in the basis,

$\displaystyle\tilde{D}_{k}\Psi^{\lambda}$ $\displaystyle=$
$\displaystyle\langle\lambda(s)|\partial_{k}|\Psi(s)\rangle$ $\displaystyle=$
$\displaystyle\partial_{k}\Psi^{\lambda}(s)+\sum_{\iota}\langle\lambda(s)|\partial_{k}|\iota(s)\rangle\Psi^{\iota}(s).$

Comparing Eqs. (12) and (III), we see that the two terms on the right hand
side of the equations correspond. The analog of the Christoffel symbol in Eq.
(III) is the expression

$\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}=\langle\lambda(s)|\partial_{k}|\iota(s)\rangle.$
(19)

The Berry connection is a special case of
$\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}$. The Berry phase arises
from transporting a single energy eigenstate around an adiabatic cycle. In
other words, the state $|\Psi(s)\rangle$ is itself some energy eigenstate, and
the basis used is also the energy eigenbasis, resulting in

$i\tilde{\Gamma}\indices{{}^{0}_{0}{}_{k}}=i\langle\Psi_{0}(s)|\partial_{k}|\Psi_{0}(s)\rangle.$
(20)

The general connection, $\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}$,
is a three-index quantity because it connects the states indexed by $\lambda$
and $\iota$ by a change in the coordinate $s_{k}$.
$\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}$ represents the
probability amplitude to go between quantum state $\lambda$ and $\iota$ when a
change in the coordinate $k$ is occurring. The Berry connection connects the
same states, and this state remains fixed during the entire adiabatic cycle,
and the only remaining index is the coordinate one, $k$. When
$\lambda\neq\iota$ , and the system is non-degenerate, the quantity
$\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}$ represents non-adiabatic
transitions between different quantum states under a change of the coordinate
$s_{k}$. As such, this quantity plays a central role in the field of non-
adiabatic molecular dynamics Tully90 ; Webster91 ; Coker93 : the non-adiabatic
coupling vector is closely related Coker95  to the Pechukas Pechukas69a ;
Pechukas69b  force, the force which acts on the nuclei of a molecular system
whose electrons undergo a transition between electronic quantum states, in
other words, when the Born-Oppenheimer approximation does not hold.

We can also derive the analog of the Riemann curvature tensor in this case,
via

$(\tilde{D}_{k}\tilde{D}_{l}-\tilde{D}_{l}\tilde{D}_{k})\Psi^{\lambda}=\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}\Psi^{\mu},$
(21)

resulting in

$\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}=\partial_{k}\tilde{\Gamma}\indices{{}^{\lambda}_{\mu}{}_{l}}-\partial_{l}\tilde{\Gamma}\indices{{}^{\lambda}_{\mu}{}_{k}}+\tilde{\Gamma}\indices{{}^{\lambda}_{\nu}{}_{k}}\tilde{\Gamma}\indices{{}^{\nu}_{\mu}{}_{l}}-\tilde{\Gamma}\indices{{}^{\lambda}_{\nu}{}_{l}}\tilde{\Gamma}\indices{{}^{\nu}_{\mu}{}_{k}}.$
(22)

Eq. (22) is identically zero if the indices run over the entire Hilbert space.
To show this, we write the $s$-dependent state $|\iota(s)\rangle$ using a
fixed basis, as,

$|\iota(s)=U(s)|\iota_{0}\rangle,$ (23)

where $|\iota_{0}\rangle$ denotes a member of the fixed basis, and $U(s)$ is a
unitary matrix spanning the entire Hilbert space. The quantum Christoffel
symbol becomes,

$\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}=\langle\lambda_{0}|U(s)^{\dagger}\partial_{k}U(s)|\iota_{0}\rangle.$
(24)

Using this definition of $\tilde{\Gamma}\indices{{}^{\lambda}_{\iota}{}_{k}}$
it is a relatively simple exercise that
$\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}$, as defined in Eq. (22),
is zero. But this only holds if the entire Hilbert space is considered, in
other words, a non-trivial quantum Riemann curvature results if the Hilbert
space is truncated, meaning that all the indices in Eq. (22) (including the
internal ones which are summed) are truncated Pacher89 . Projections of the
types discussed above are examples of such truncations. Comparing the sets of
Eqs. (14), (III) and (21), (22), we see that there is a correspondence between
the curvature tensor of the parametrized quantum system and the Riemann
curvature tensor known from differential geometry.

The four-index tensor $\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}$ is
not a curvature in the sense of the Riemann curvature tensor. To lower the
index $\lambda$ in $\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}$ one
uses not the metric tensor of the parameter space, but that of the Hilbert
space of states ($\delta_{\mu\nu}$ for complete orthonormal states). As for
contraction the Riemann curvature tensor can be contracted in three ways,
because the indices are equivalent, whereas
$\tilde{R}\indices{{}^{\lambda}_{\mu}{}_{l}{}_{k}}$ can only be contracted as

$\tilde{R}\indices{{}_{l}{}_{k}}=\tilde{R}\indices{{}^{\lambda}_{\lambda}{}_{l}{}_{k}}.$
(25)

The contraction over the entire Hilbert space leads to zero (for a proof of
this statement, see for example, the introduction of Ref. Xiao10, ), however
it is possible to sum over a part of the Hilbert space. The Berry curvature
involves only one state, its generalization Wilczek84 , the Wilson loop
involves a subset of the complete Hilbert space (usually a degenerate
subspace). In these cases $\tilde{R}\indices{{}_{l}{}_{k}}$ is not trivial.

If the metric tensor of the parameter space is invertible, one can raise one
of the indices of $\tilde{R}\indices{{}_{l}{}_{k}}$ as

$\tilde{R}\indices{{}^{j}_{k}}=g\indices{{}^{j}{}^{l}}\tilde{R}\indices{{}_{l}{}_{k}},$
(26)

and contract it to get the scalar curvature,

$\tilde{R}=\tilde{R}\indices{{}^{j}_{j}}.$ (27)

The scalar curvature obtained from the Berry curvature is zero on account of
the anti-symmetry in the two coordinate indices.

##  IV Cumulant generating function for quantum states

Before stating the cumulant generating function relevant to quantum state
manifolds, let us give the definitions of moments and cumulants from the
theory of probability. In this section, we use a notation that is closest to
the original derivation of Provost and Vallee Provost80 . In the Appendix we
develop a notation in which the relation between moments and cumulants in the
case of quantum state manifolds can be made manifest.

Moments and cumulants are quantities which characterize probability
distributions. Given a multi-variate normalized probability distribution:

$\displaystyle P(x_{1},...,x_{N})\geq 0,\forall x_{i};\hskip 85.35826pt$ (28)
$\displaystyle\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}dx_{1}...dx_{N}P(x_{1},...,x_{N})=1,$

the $M$th moment and the $M$th cumulant is obtained from the generating (or
characteristic) function, defined as,

$f(k_{1},...,k_{N})=\int
dx_{1}...dx_{N}e^{i\sum_{j=1}^{N}k_{j}x_{j}}P(x_{1},...,x_{N}),$ (29)

via the derivatives and logarithmic derivatives, respectively, as

$\displaystyle\mathcal{M}_{M}$ $\displaystyle=$
$\displaystyle\left.\prod_{j=1}^{N}\frac{1}{i^{m_{j}}}\frac{\partial^{m_{j}}}{\partial
k_{j}^{m_{j}}}f(k_{1},...,k_{N})\right|_{k_{1}=...=k_{N}=0},$ (30)
$\displaystyle\mathcal{C}_{M}$ $\displaystyle=$
$\displaystyle\left.\prod_{j=1}^{N}\frac{1}{i^{m_{j}}}\frac{\partial^{m_{j}}}{\partial
k_{j}^{m_{j}}}\ln f(k_{1},...,k_{N})\right|_{k_{1}=...=k_{N}=0},$

where $\sum_{j=1}^{N}m_{j}=M$. Cumulants can be written in terms of moments
(and vice versa).

We now turn to quantum state manifolds. Consider a family $\\{\Psi(s)\\}$ of
quantum state vectors parametrized smoothly by the $n$-dimensional variable
$s=s_{1},...,s_{n}$. We write the overlap between two wave functions with
different $s$ as,

$S(s^{\prime},s)=\langle\Psi(s^{\prime})|\Psi(s)\rangle.$ (31)

We define the generating function for gauge invariant cumulants as

$\displaystyle\mathcal{C}_{m_{1}^{\prime},...,m_{N}^{\prime};m_{1},...,m_{N}}=\hskip
71.13188pt$ (32)
$\displaystyle(-i)^{M^{\prime}}(i)^{M}(\partial_{1}^{\prime})^{m_{1}^{\prime}}...(\partial_{N}^{\prime})^{m_{N}^{\prime}}\partial^{m_{1}}_{1}...\partial^{m_{N}}_{N}\left.\ln
S(s^{\prime},s)\right|_{s^{\prime}=s}.$

In Eq. (32) $M^{\prime}=m_{1}^{\prime}+...+m_{N}^{\prime}$ and
$M=m_{1}+...+m_{N}$, and $\partial_{j}$($\partial_{j}^{\prime}$) denotes a
partial derivative with respect to $s_{j}$($s_{j}^{\prime}$). Since this
cumulant depends on the parameters of the vector in Hilbert space
$|\Psi(s)\rangle$ and its dual $\langle\Psi(s^{\prime})|$, there are two
numbers specifying the order of the cumulant, $M^{\prime}$, referring to the
dual vector $\langle\Psi(s^{\prime})|$ and $M$ referring to the vector
$|\Psi(s)\rangle$. In Section VII we show that in the case where the state
$|\Psi(s)\rangle$ is generated by simple translation operators,
$\mathcal{C}_{m_{1}^{\prime},...,m_{N}^{\prime};m_{1},...,m_{N}}$ correspond
exactly to the statistical cumulants given in Eq. (30).

Gauge invariance can be shown via substituting a wave function

$|\tilde{\Psi}(s)\rangle=\exp(-i\alpha(s))|\Psi(s)\rangle,$ (33)

a state with a modified phase, $\alpha(s)$ which depends on the parameter set
$s$. The overlap becomes,

$\tilde{S}(s^{\prime},s)=\ln\langle\tilde{\Psi}(s^{\prime})|\tilde{\Psi}(s)\rangle=\ln\langle\Psi(s^{\prime})|\Psi(s)\rangle+i(\alpha(s^{\prime})-\alpha(s)).$
(34)

While cumulants with either $M$ or $M^{\prime}$ zero are not gauge invariant,
all other cases (including cumulants which correspond to the geometric tensors
of interest) are independent of $\alpha(s)$.

The gauge invariant quantum metric tensor of Provost and Vallee Provost80  is
easily generated. We first relabel the cumulants of order $M=1$ and
$M^{\prime}=1$ as

$C_{2}(j;k)=C_{0,...,0,m_{j}^{\prime}=1,0,....,0;0,...,0,m_{k}=1,0,....,0}.$
(35)

Evaluating $C_{2}(j;k)$ using Eq. (32), we obtain

$C_{2}(j;k)=\gamma_{jk}-\beta_{j}\beta_{k},$ (36)

where

$\displaystyle\gamma_{jk}$ $\displaystyle=$
$\displaystyle\langle\partial_{j}\Psi(s)|\partial_{k}\Psi(s)\rangle,$ (37)
$\displaystyle\beta_{j}$ $\displaystyle=$
$\displaystyle-i\langle\Psi(s)|\partial_{j}\Psi(s)\rangle.$

$\beta_{j}$ is the Berry connection in the state $\Psi(s)$. The metric tensor
$g_{jk}$ and the Berry curvature $\sigma_{jk}$ are given by

$g_{jk}(s)=\mbox{Re}C_{2}(j;k);\hskip
28.45274pt\sigma_{jk}(s)=\mbox{Im}C_{2}(j;k).$ (38)

$g_{jk}$($\sigma_{jk}$) is symmetric(anti-symmetric) in its indices. The
second cumulant in ordinary statistics gives only a real correlation matrix.
In the quantum case the particular structure of the scalar product of the
probability amplitudes of quantum states ($S(s^{\prime},s)$) is what gives
rise to the additional imaginary anti-symmetric component.

##  V A Born-Oppenheimer system with Hermitian inverse mass tensor

One area in which the study of the PVM and the BC proved crucially important
was for mixed quantum-classical systems, in which the quantum subsystem
remains in one quantum state throughout the dynamics (Born-Oppenheimer
approximation). Mead and Truhlar showed that the BC gives rise to the
“molecular magnetic field”, which is measurable in several types of
experiments, for example, pseudorotation of triatomic molecules Delacetraz86 ;
Wolf89 .

In this section we show, on the one hand, that the metric tensor as well as
its derivative (a three-index quantity related to the Christoffel symbol)
appears in mixed quantum classical systems. In addition, we also show that a
complex inverse mass tensor leads to a “molecular electric field” which
depends not only on the PVM, but also the BC.

We consider a molecular system consisting of electrons and nuclei. Since
nuclei are at least three orders of magnitude heavier than electrons, the
dynamics of molecules are often studied by invoking the Born-Oppenheimer
approximation, which assumes that electrons remain in one electronic state
(the ground state). We first write the Hamiltonian (fully quantum) as

$\hat{H}_{tot}=\frac{1}{2}\sum_{jk}Q_{jk}^{*}\hat{P}_{j}\hat{P}_{k}+\hat{h}(\hat{\xi};\hat{s}),$
(39)

where $\hat{s}_{j}$ and $\hat{P}_{j}=i\partial_{s_{j}}$ represent the
positions and momenta of the nuclei, $\hat{\xi}$ denote the electronic
coordinates collectively ($s$ denotes the nuclear coordinates collectively),
and $h(\hat{\xi};\hat{s})$ is the Hamiltonian of the electrons. $Q_{ij}^{*}$
denotes the complex conjugate of the inverse mass tensor. We assume that the
inverse mass tensor has real and imaginary parts,

$Q_{jk}=Q^{\prime}_{jk}+iQ^{\prime\prime}_{jk}.$ (40)

From the hermiticity of the kinetic energy, it follows that
$Q^{\prime}_{jk}=Q^{\prime}_{kj}$, and
$Q^{\prime\prime}_{jk}=-Q^{\prime\prime}_{kj}$.

In the Born-Oppenheimer approximation one writes the total wave function as:

$\Psi_{tot}(\xi;s)=\psi_{nuc}(s)\phi_{0}(\xi;s),$ (41)

where $\psi_{nuc}(\hat{X})$ denotes the nuclear wave function, and
$\phi_{0}(\hat{\xi};\hat{X})$ denotes the ground state electronic wave
function, satisfying

$\hat{h}(\xi;s)\phi_{0}(\xi;s)=E_{0}(s)\phi_{0}(\xi;s).$ (42)

The effective Hamiltonian acting on the nuclei that results is

$\hat{H}_{eff}=\int d{\xi}\phi_{0}^{*}(\xi;s)\hat{H}_{tot}\phi_{0}(\xi;s).$
(43)

The effective Hamiltonian can be shown to be

$\displaystyle\hat{H}_{eff}$ $\displaystyle=$
$\displaystyle\sum_{jk}\frac{Q_{jk}^{*}}{2}(\hat{P}_{j}-A_{j}(s))(\hat{P}_{k}-A_{k}(s))$
(44) $\displaystyle+\Phi(s)+E_{0}(s),$

where, $A_{j}(s)$ and $\Phi(s)$ denote a vector potential and a scalar
potential, respectively, defined as:

$\displaystyle A_{j}(s)$ $\displaystyle=$ $\displaystyle-i\int
d{\bf\xi}\phi_{0}^{*}(\xi;s)\partial_{j}\phi_{0}(\xi;s)$ (45)
$\displaystyle\Phi(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{jk}Q_{jk}^{*}C_{2}(j;k)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{jk}(Q^{\prime}_{jk}g_{jk}(s)+Q^{\prime\prime}_{jk}\sigma_{jk}(s)).$

$g_{jk}(s)$ denotes the quantum metric tensor, derived in the previous
section. In Eq. (45) the real part of the mass tensor couples to the PVM, and
the imaginary part couples to the BC. Even though the mass tensor has an
imaginary component, the terms in the effective Hamiltonian obtained above are
all real, so the above potential corresponds to a classical system with a
modified potential (modified “molecular electric field”).

It is also possible to diagonalize the Hermitian mass tensor, and end up with
a more straightforward expression: $\hat{H}_{eff}$,

$\hat{H}_{eff}=\sum_{r}\frac{\hat{\Pi}_{r}^{2}}{2M_{r}}+\Phi(s)+E_{0}(s),$
(46)

where

$\hat{\Pi}_{r}=\sum_{j}U_{jr}(\hat{P}_{j}-A_{j}(s)),$ (47)

where $U$ diagonalizes $Q$, and $M_{r}$ are the inverse eigenvalues. The
potential adopts the form,

$\Phi(s)=\sum_{r}\frac{\tilde{g}_{r}(s)+\tilde{\sigma}_{r}(s)}{M_{r}},$ (48)

where

$\displaystyle\tilde{g}_{r}(s)$ $\displaystyle=$
$\displaystyle\sum_{jk}(U_{jr}U_{kr}^{*})^{\prime}g_{jk}(s)$ (49)
$\displaystyle\tilde{\sigma}_{r}(s)$ $\displaystyle=$
$\displaystyle\sum_{jk}(U_{jr}U_{kr}^{*})^{\prime\prime}\sigma_{jk}(s).$

In the form of Eq. (46) the effective Hamiltonian is not very different from
systems with an “effective mass”, since the kinetic energy has the usual from,
apart from exhibiting directionally dependent masses, and the BC term only
effects the potential in which the particle is moving. It is interesting that
in the original Hamiltonian (Eq. (39)), the imaginary part of the complex
Hermitian mass tensor appears in the kinetic energy, however, after invoking
the Born-Oppenheimer approximation, it only effects the potential energy.

##  VI Three index quantities: The quantum Christoffel symbol

We now consider the three-index quantity of the form

$C_{3}(j;kl)=(-i\partial_{j^{\prime}})(i\partial_{k})(i\partial_{l})\left.\ln
S(s^{\prime},s)\right|_{s^{\prime}=s}.$ (50)

$C_{3}(j;kl)$, like $C_{2}(j;k)$, is complex and takes the form,

$\displaystyle C_{3}(j;kl)$ $\displaystyle=$ $\displaystyle
i\langle\partial_{j}\Psi|\partial^{2}_{kl}\Psi\rangle-\langle\Psi|\partial^{2}_{kl}\Psi\rangle\beta_{j}$
$\displaystyle+\gamma_{jl}\beta_{k}+\gamma_{jk}\beta_{l}-2\beta_{j}\beta_{k}\beta_{l}.$

It is possible to show that

$\partial_{l}C_{2}(j;k)=\frac{1}{2i}\left(C_{3}(jl;k)-C_{3}(j;kl)\right),$
(52)

by directly taking the derivative of $C_{2}(j;k)$ (as defined in Eq. (36))
with respect to $s_{l}$, and substituting Eq. (VI) in the right hand side of
Eq. (52). Using this result, we now define the quantum extension of the
Christoffel symbol as,

$\displaystyle[jl;k]_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{j}C_{2}(l;k)+\partial_{l}C_{2}(k;j)-\partial_{k}C_{2}(j;l)\right),$
$\displaystyle=$
$\displaystyle\frac{1}{4i}\left(C_{3}(jl;k)-C_{3}(l;kj)+C_{3}(lk;j)-C_{3}(k;jl)\right.$
$\displaystyle\left.-C_{3}(kj;l)+C_{3}(j;lk)\right).$

The quantum Christoffel symbol is the analog of the expression in Eq. (4),
with the full quantum metric substituted in place of $g_{jk}$. In addition,
the quantum Christoffel symbol evaluates to a sum of various third order
cumulants. Eq. (VI) establishes a definite relation between quantum
fluctuations and the Christoffel symbol.

The usual Christoffel symbol of the first kind, which results from taking the
derivative of the PVM, is given by

$\mbox{Re}[jl;k]_{q}=[jl;k],$ (54)

and

$\mbox{Im}[jl;k]_{q}=\frac{1}{2}\left(\partial_{j}\sigma_{lk}+\partial_{l}\sigma_{kj}-\partial_{k}\sigma_{jl}\right).$
(55)

In the usual Christoffel symbol (Eq. (6)), the first and the last terms change
sign upon exchanging indices $j$ and $k$, and in $\mbox{Im}[jl;k]_{q}$ exactly
the opposite happens. For this reason, using $[jl;k]_{q}$, we can write the
force associated with the potential $\Phi(\hat{X})$ as

$\displaystyle F_{l}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{jk}Q_{jk}^{*}[jl;k]_{q},$
$\displaystyle-\frac{1}{2}\sum_{jk}\left(Q^{\prime}_{jk}\mbox{Re}[jl;k]_{q}+Q^{\prime\prime}_{jk}\mbox{Im}[jl;k]_{q}\right).$

The force is thus dependent on the extended quantum Christoffel symbol. The
broader conclusion is that in the most general case the parallel transport of
a vector in the parameter space $s$ proceeds via not only the derivative of
the PVM, but also of the BC. A quantum Christoffel symbol of the second kind
can only be defined if the complex quantum metric $C_{2}(j;k)$ is invertible.
In that case the definition in Eq. (6) can be used.

Since it is always possible to take further derivatives according to Eq. (52),
and the resulting higher order cumulants can then be combined to conform to
expressions of higher order geometric tensors (for example, the Riemann
curvature tensor), geometric tensors of any order can be expressed as quantum
fluctuations of the same order.

##  VII Geometric tensors, quantum fluctuations, and the uncertainty principle

In this section we investigate how quantum fluctuations are related to
geometric tensors. The most straightforward example Provost80  is states
generated by commuting operators $\\{\hat{A}_{j}\\}$, from some quantum state
$|\Psi_{0}\rangle$, as

$|\Psi\rangle=\exp\left(i\sum_{j}s_{j}\hat{A}_{j}\right)|\Psi_{0}\rangle.$
(57)

Apart from the fluctuations in $|\Psi_{0}\rangle$, this example can be
considered classical. Substituting Eq. (57) into Eq. (32), we obtain ordinary
statistical cumulants (Eq. 30). The metric tensor $g_{jk}$ is simply the
second statistical cumulant, $\sigma_{jk}$ is zero in this case. $C_{3}(jk;l)$
only has a real part and corresponds to the third order statistical cumulant
(the skew).

Further insight can be gained via considering a state generated by assuming
that the set of operators $\\{\hat{A}_{j}\\}$ are non-commuting (generators of
a Lie group). To start, we study the fluctuations near the origin of the
parameter space, by taking $s$ to zero in Eq. (32). In this case, the
cumulants will have imaginary parts. The second order cumulant looks like,

$\displaystyle g_{jk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle[\hat{A}_{j},\hat{A}_{k}]_{+}\rangle_{0}-\langle\hat{A}_{j}\rangle_{0}\langle\hat{A}_{k}\rangle_{0},$
$\displaystyle\sigma_{jk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle[\hat{A}_{j},\hat{A}_{k}]_{-}\rangle_{0},$
(58)

where $[,]_{+}$($[,]_{-}$) indicates an anti-commutator(commutator), and
$\langle\rangle_{0}$ indicates average over the state $|\Psi_{0}\rangle$. A
non-trivial Berry phase is a result of non-commuting generators.

It is instructive to consider the matrix $C_{2}(j;k)$ for a system with a two-
dimensional parameter space, where there are only two generators,
$\hat{A}_{1}$ and $\hat{A}_{2}$. In this case, the quantities $C_{2}(1;1)$ and
$C_{2}(2;2)$ are simply the variances of operators $\hat{A}_{1}$ and
$\hat{A}_{2}$. Since the quantity $C_{2}(j;k)$ is itself a variance, and
therefore has a determinant greater than or equal to zero, we obtain

$\sigma_{j}^{2}\sigma_{k}^{2}\geq\left|\frac{1}{2}\langle[\hat{A}_{j},\hat{A}_{k}]_{+}\rangle_{0}-\langle\hat{A}_{j}\rangle_{0}\langle\hat{A}_{k}\rangle_{0}\right|^{2}-\left|\frac{1}{2}\langle[\hat{A}_{j},\hat{A}_{k}]_{-}\rangle_{0}\right|^{2}.$
(59)

This equation is a form of the uncertainty relation, known as the Schrödinger
uncertainty relation Schrodinger30 , which is a stronger form of this
principle than the well-known one due to Heisenberg. Considering now the
general case of an arbitrary number of dimensions, a multi-dimensional
generalization of the uncertainty principle can be obtained via

$\mbox{det}[C_{2}(j;k)]\geq 0.$ (60)

This form of the uncertainty principle places a constraint on the variances
$\sigma_{1},...\sigma_{N}$ if the parameter space is $N$ dimensional. The
many-operator version of the Heisenberg uncertainty relation was first derived
by Robertson Robertson34 .

If the determinant of $C_{2}(j;k)$ is zero, it means that the complex metric
$C_{2}(j;k)$ can not be inverted. This also means that a Christoffel symbol of
the second kind can not be constructed, and that the parameter space, when
considered together with the quantum fluctuations, has a trivial geometry, the
entire parameter space represents the same point (all distances are zero). In
the next section we give examples of both cases.

##  VIII Coherent state examples

In this section we will analyze the geometry of quantum systems using coherent
states. Reviews of this subject are found in Refs. Zhang90, ; Gilmore72, ;
Perelomov86, . After deriving the uncertainty relation in a coherent state
context, we will consider three well-known examples of coherent states
associated with Lie groups. They are known as Glauber, $SU(2)$ and $SU(1,1)$
coherent states. Glauber coherent states are associated with the Weyl-
Heisenberg group Weyl50 . Glauber coherent states are minimum uncertainty
states of a quantum harmonic oscillator. $SU(2)$ coherent states are
constructed using angular momentum operators. The generators of the $SU(1,1)$
group can be obtained by modifications of the $SU(2)$ algebra. An interesting
connection Alhassid83 ; Alhassid86  exists between the $SU(2)$ and $SU(1,1)$
groups. The $SU(2)$ space corresponds to a sphere, which is a compact space.
The basis states of the $SU(2)$ correspond to the bound states of the Pöschl-
Teller potential. The modifications leading to the $SU(1,1)$ algebra lead to a
non-compact group space (a hyperboloid) and the modified basis states
correspond to the scattering states of the same potential. The $SU(1,1)$ group
space has several projections, which correspond to different series of quantum
numbers. In the case of $SU(1,1)$ we will consider several representations of
the group space below, including the case when the group space is the
universal covering space of the hyperboloid, as well as projections thereof.

Before turning to our concrete examples, we give a brief overview of coherent
states. Our purpose here is to calculate and analyze the quantum geometric
tensor associated with coherent states, but we believe it is in order to give
the general steps of their construction in a Lie group context.

Given a Lie group $G$ with a Lie algebra consisting of the identity $\hat{I}$
and two sets of generators, $\\{\hat{A_{i}}\\}$ and $\\{\hat{B_{i}}\\}$. The
original coherent states Zhang90  are minimum uncertainty states whose
starting point is an extremal state. The set $\\{\hat{A_{i}}\\}$ together with
the identity forms the stability subgroup of $G$, which we call $H$,
consisting of operators which leave the extremal state invariant. The set
$\\{\hat{B_{i}}\\}$, the complement of $H$ in $G$ consists of operators which
do not leave the extremal state invariant, and include at least one operator,
which eliminates the extremal state. In practice, this means that the extremal
state is an eigenstate of the operators included in $H$, with definite quantum
numbers, and the operators of the complement of $H$ in $G$ are ladder (raising
and lowering, or creation and annihilation) operators. Coherent states are
then constructed by applying the shift operator, $\hat{D}(s)$,

$|s\rangle=\hat{D}(s)|0\rangle,\hat{D}(s)=\exp\left(\sum_{j}s_{j}\hat{B_{j}}\right),$
(61)

to the extremal state. The argument of the exponential operator $D(s)$
consists of a sum over operators the complement of $H$ in $G$, and the
parameters $s_{j}$ are coordinates desginating points on the geometrical space
of the quotient group $G/H$. The complex metric can be written Chaturvedi87
as

$C_{2}(j;k)=\sum_{m\neq 0}\langle
0|[\hat{D}^{\dagger}(s)\partial_{j}\hat{D}(s)]^{\dagger}|m\rangle\langle
m|\hat{D}^{\dagger}(s)\partial_{k}\hat{D}(s)|0\rangle.$ (62)

It is also possible to construct generalized Boiteux73 ; Roy82  coherent
states via acting with the displacement operator on a state which is not
extremal, but still an eigenstate of the stability group. Such states are no
longer minimum uncertainty states, but they are interesting, because their
time-dependent generalization corresponds Roy82  to the averages moving
according to classical dynamics, while the full quantum distribution remains
rigid. Other generalizations of the original coherent state formalism include
a variety of squeezed states Zhang90 ; Mahran86 ; Gerry89 ; Kim89 .

To summarize, our construction of coherent states and the calculation of the
quantum geometric tensor in subsections VIII.2, VIII.3, and VIII.4 will
proceed via the following steps:

  * •

Identify the stability group and its complement.

  * •

Identify states which are eigenstates of the operators of the stability group.

  * •

Apply the shift operator to eigenstates of the stability group.

  * •

Calculate the quantum geometric tensor

For the case of $SU(1,1)$ we will first analyze (subsection VIII.4) single-
valued representations of the group which correspond to the universal covering
space of the hyperboloid. It is also possible to construct projective
representations in which the groups are multi-valued. In subsection VIII.5 we
give one example of this. The reader should note that the group theoretical
approach to potential problems is a very colorful subject, whose mathematical
intricacies are beyond the scope of this work, but excellent references exist
to quench unsatisfied further curiosity Bargmann47 ; Alhassid83 ; Alhassid86 .

###  VIII.1 Uncertainty principle in terms of coherent states

We can derive an uncertainty relation of the Schrödinger type for Lie group
coherent states of the form of Eqs. (61). These relations will include only
the operators of the coset space, and we show that it remains valid at any
point on the geometrical space $G/H$. We can write the operator

$\hat{D}^{\dagger}(s)\partial_{k}\hat{D}(s)=\sum_{l}A_{kl}(s)\hat{B}_{l},$
(63)

where $A_{kl}(s)$ is a matrix which depends on the coordinates $s$. This
linear combination includes only operators of the coset space. Therefore, we
can write $C_{2}(j;k)$ as

$C_{2}(j;k)=\langle
0|[\hat{D}^{\dagger}(s)\partial_{j}\hat{D}(s)]^{\dagger}\hat{D}^{\dagger}(s)\partial_{k}\hat{D}(s)|0\rangle.$
(64)

Using Eq. (63) we can write $C_{2}(j;k)$ as

$C_{2}(j;k)=\sum_{lm}A_{jl}^{\dagger}(s)A_{km}(s)\langle
0|\hat{B}_{l}^{\dagger}\hat{B}_{m}|0\rangle.$ (65)

The general complex variance can be written as

$C_{2}(j;k)=\sum_{lm}A_{jl}^{\dagger}(s)A_{km}(s)C_{2}^{(0)}(l;m).$ (66)

Taking the determinant leads to

$\mbox{Det}[C_{2}]=\mbox{Det}[A^{\dagger}(s)A(s)]\mbox{Det}[C_{2}^{(0)}].$
(67)

Since $\mbox{Det}[A^{\dagger}A]$ is greater than or equal to zero, and we
already know that this is so for $\mbox{Det}C_{2}^{(0)}$ from the uncertainty
relation, it follows that

$\mbox{Det}[C_{2}]\geq 0,$ (68)

alternatively, it follows that the usual uncertainty relation will hold
anywhere in the parameter space.

###  VIII.2 Glauber coherent states

For Glauber coherent states the relevant group is the Weyl-Heisenberg group
Weyl50 , which consists of elements
$\\{\hat{I},\hat{n},\hat{a}^{\dagger},\hat{a}\\}$, where
$\hat{a}$($\hat{a}^{\dagger}$) denotes the annihilation(creation) operator,
and $\hat{n}=\hat{a}^{\dagger}\hat{a}$. The stability group is
$\\{\hat{I},\hat{n}\\}$. Minimum uncertainty coherent states are generated
using the shift operator, as,

$|\alpha\rangle=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})|0\rangle,$
(69)

where $\alpha=\alpha_{1}+i\alpha_{2}$ (the complex plane) is the parameter
space. We will also consider generalized Glauber states, constructed from an
excited harmonic oscillator state,

$|\alpha\rangle_{m}=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})|m\rangle.$
(70)

Using Eq. (62), the result for the quantum geometric tensor is,

$C_{2}=\begin{pmatrix}2m+1&i\\\ -i&2m+1\end{pmatrix}.$ (71)

For $m>0$ the metric is invertible and there exists a non-trivial complex
metric, however, for $m=0$ the metric is not invertible, and the geometry is
trivial. Stated differently, coherent states constructed starting from
extremal states give rise to a trivial geometry, whereas generalized coherent
states are endowed with a non-trivial geometry.

###  VIII.3 $SU(2)$ (atomic) coherent states

The Lie group of angular momentum $G$ is generated by the angular momentum Lie
algebra, $\mathfrak{g}=\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z}\\}$, which obeys
the commutation relations,

$[\hat{J}_{i},\hat{J}_{j}]=i\hat{J}_{k},$ (72)

where $i,j,k$ denote a cyclic permutation of the coordinates $x,y,z$. For
later use we also define the raising and lowering operators,

$\hat{J}_{\pm}=\hat{J}_{x}\pm i\hat{J}_{y}.$ (73)

A convenient basis is formed by the states $|j,m\rangle$, which satisfy,

$\displaystyle\hat{J}^{2}|j,m\rangle$ $\displaystyle=$ $\displaystyle
j(j+1)|j,m\rangle$ (74) $\displaystyle\hat{J}_{z}|j,m\rangle$ $\displaystyle=$
$\displaystyle m|j,m\rangle,$

and for which

$\displaystyle j$ $\displaystyle=$
$\displaystyle\frac{1}{2},1,\frac{3}{2},2,...$ (75) $\displaystyle m$
$\displaystyle=$ $\displaystyle-j,-j+1,-j+2,...,j-2,j-1,j.$

The Casimir operator of the group $G$, $\hat{J}^{2}$, is defined as

$\hat{J}^{2}=\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}.$ (76)

The stability group $H$ of the group $G$, is the subgroup generated by
rotations about the $z$-axis (subalgebra $\mathfrak{h}=\\{\hat{J}_{z}\\}$).
The coset space of $G/H$ corresponds to a sphere (which we can parametrize by
angles $\theta,\phi$).

A generalized coherent state ($|\tau\rangle$) can be generated by the $SU(2)$
shift operator

$D(\tau)=\exp(\tau\hat{J}_{+})\exp(\beta\hat{J}_{z})\exp(-\tau^{*}\hat{J}_{-}),$
(77)

where

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\tan\frac{\theta}{2}e^{i\phi}$ (78) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle\ln(1+|\tau|^{2}),$

by acting on a basis state

$|\tau\rangle=D(\tau)|j,m\rangle.$ (79)

The usual atomic coherent states, which are minimum uncertainty states, are
generated by the above procedure from the extremal basis functions of which in
this case there are two the lower and upper bounds of the series of spherical
harmonic solutions ($m=-j$ or $m=j$).

After some algebra it is possible to show that

$\displaystyle D(\tau)^{\dagger}\partial_{\theta}D(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[e^{-i\phi}\hat{J}_{+}-e^{i\phi}\hat{J}_{-}\right],$
$\displaystyle D(\tau)^{\dagger}\partial_{\phi}D(\tau)$ $\displaystyle=$
$\displaystyle\frac{-i\sin\theta}{2}\left[e^{-i\phi}\hat{J}_{+}+e^{i\phi}\hat{J}_{-}\right].$
(80)

Using Eq. (62)), we obtain the complex metric for the atomic coherent states,

$C_{2}=\frac{1}{2}\begin{pmatrix}j(j+1)-m^{2}&-im\sin\theta\\\
im\sin\theta&(j(j+1)-m^{2})\sin^{2}\theta\end{pmatrix}.$ (81)

Taking the determinant results in,

$\mbox{Det}[C_{2}]=\frac{1}{4}[j(j+1)-m(m+1)][j(j+1)-m(m-1)]\sin^{2}\theta.$
(82)

The PVM together with the BC is recovered by setting $m=-j$ in Eq. (81). We
also find that there are four values of $m$ for which the determinant
vanishes, $m=-j-1,-j,j,j+1$, but two of these do not contribute, because they
are outside the range of $m$ for the representation of the $SU(2)$ group (Eq.
(75)). Hence, for arbitrary values of the coordinates, only the extremal
states correspond to a trivial complex quantum metric. For the remaining
states the metric is non-trivial, except if $\theta=0,\pi$, which are the
north and south poles of the sphere.

We can consider a mixed quantum classical system of the type chronicled in
Section V. A $2\times 2$ inverse mass tensor has the form,

$Q=\begin{pmatrix}Q^{\prime}_{11}&Q^{\prime}_{12}\\\
Q^{\prime}_{12}&Q^{\prime}_{22}\end{pmatrix}-i\begin{pmatrix}0&Q^{\prime\prime}_{12}\\\
-Q^{\prime\prime}_{12}&0\end{pmatrix}$ (83)

leading to a potential term of the form,

$\Phi(\theta)=\left(j(j+1)-m^{2}\right)(Q^{\prime}_{11}+Q^{\prime}_{22}\sin^{2}\theta)+2mQ^{\prime\prime}_{12}\sin\theta.$
(84)

The second term is the contribution due to the imaginary component of the
inverse mass tensor.

###  VIII.4 $SU(1,1)$ coherent states

$SU(1,1)$ coherent states are generated by the Lie algebra,
$\mathfrak{g}=\\{\hat{J}_{x},\hat{J}_{y},\hat{J}_{z}\\}$, whose members obey
the relations

$\displaystyle[\hat{J}_{x},\hat{J}_{y}]$ $\displaystyle=$
$\displaystyle-i\hat{J}_{z},$ (85) $\displaystyle[\hat{J}_{y},\hat{J}_{z}]$
$\displaystyle=$ $\displaystyle i\hat{J}_{x},$
$\displaystyle[\hat{J}_{z},\hat{J}_{x}]$ $\displaystyle=$ $\displaystyle
i\hat{J}_{y}.$

The raising and lowering operators are defined according to Eq. (73). A
convenient basis is formed by the states $|j,m\rangle$, which satisfy,

$\displaystyle\hat{J}^{2}|j,m\rangle$ $\displaystyle=$ $\displaystyle
j(j+1)|j,m\rangle$ (86) $\displaystyle\hat{J}_{z}|j,m\rangle$ $\displaystyle=$
$\displaystyle m|j,m\rangle,$

but here the Casimir operator of the group $G$, $\hat{J}$, is defined as

$\hat{J}^{2}=\hat{J}_{x}^{2}+\hat{J}_{y}^{2}-\hat{J}_{z}^{2}.$ (87)

The algebra of the stability group is $\mathfrak{h}=\\{\hat{J}_{z}\\}$, the
coset space of $G/H$ corresponds to a hyperboloid (parametrized below by
$\rho,\phi$).

The $SU(1,1)$ group has several possible series of quantum numbers, depending
on representationAlhassid86 . Using the notation used in Ref. Alhassid86 ,
here we list the ones corresponding to the single-valued representations of
the group. There are two discrete series, $D_{j}^{+}$ and $D_{j}^{-}$, and two
continuous series, $C_{k}^{0}$ and $C_{k}^{1/2}$. For $D_{j}^{+}$, the quantum
number $j$ can take negative integer or half-integer values, while, $m$ can
start from $-j$ and can increase in integral steps, but unlike the $SU(2)$
series, the $m$ quantum number is not bounded above. We have

$\displaystyle j$ $\displaystyle=$
$\displaystyle-\frac{1}{2},-1,-\frac{3}{2},-2,...$ (88) $\displaystyle m$
$\displaystyle=$ $\displaystyle-j,-j+1,-j+2,...$

The series of $m$ values is only bounded from below. For the series
$D_{j}^{-}$, the quantum number $j$ can take negative integer or half-integer
values, while, $m$ can start from $j$ and can decrease in integral steps,

$\displaystyle j$ $\displaystyle=$
$\displaystyle-\frac{1}{2},-1,-\frac{3}{2},-2,...$ (89) $\displaystyle m$
$\displaystyle=$ $\displaystyle j,j-1,j-2,...$

The series of $m$ values are only bounded from above. For the two continuous
series,

$\displaystyle j$ $\displaystyle=$ $\displaystyle-\frac{1}{2}+ik,$ (90)
$\displaystyle m$ $\displaystyle=$ $\displaystyle 0,\pm 1,\pm 2,...,\hskip
5.69046pt\mbox{for}\hskip 5.69046ptC_{k}^{0},$ $\displaystyle m$
$\displaystyle=$ $\displaystyle\pm\frac{1}{2},\pm\frac{3}{2},...,\hskip
5.69046pt\mbox{for}\hskip 5.69046ptC_{k}^{1/2},$

where $k$ is a real number, such that $k>0$.

A generalized $SU(1,1)$ coherent state ($|\tau\rangle$) is generated by the
$SU(1,1)$ shift operator,

$D(\tau)=\exp(\tau\hat{J}_{+})\exp(\beta\hat{J}_{z})\exp(-\tau^{*}\hat{J}_{-}),$
(91)

where

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle-\tanh\frac{\rho}{2}e^{-i\phi}$ (92) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle\ln(1-|\tau|^{2}),$

by acting on a basis state $|j,m\rangle$

$|\tau\rangle=D(\tau)|j,m\rangle.$ (93)

After some algebra it is possible to show that

$\displaystyle D(\tau)^{\dagger}\partial_{\rho}D(\tau)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[e^{-i\phi}\hat{J}_{+}-e^{i\phi}\hat{J}_{-}\right],$
$\displaystyle D(\tau)^{\dagger}\partial_{\phi}D(\tau)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sinh(\rho)\left[e^{-i\phi}\hat{J}_{+}+e^{i\phi}\hat{J}_{-}\right].$
(94)

Using Eq. (62)), we obtain the complex metric for the hyperbolic coherent
states,

$C_{2}=\frac{1}{2}\begin{pmatrix}\left[-j(j+1)+m^{2}\right]&-im\sinh\rho\\\
im\sinh\rho&\left[-j(j+1)+m^{2}\right]\sinh^{2}\rho\end{pmatrix}.$ (95)

Taking the determinant results in,

$\mbox{Det}[C_{2}]=\frac{1}{4}[j(j+1)-m(m+1)][j(j+1)-m(m-1)]\sinh^{2}\rho.$
(96)

This determinant is zero if $\rho=0$. It is also possible to get zero for
particular values of the quantum numbers. The determinant of the complex
quantum metric is zero for the extremal states in each of the series
$D_{j}^{+}$ ($m=-j$) and $D_{j}^{-}$ ($m=j$). For the two continuous series
there is no way for the determinant of the complex quantum metric to be zero.
We note in passing that there exists Alhassid86  also a supplementary series
for the $SU(1,1)$ group. This series does not contribute to the solution of
the relevant Laplace equation (defined on a hyperboloid), since the series
$D_{j}^{+}$, $D_{j}^{-}$, $C_{k}^{0}$, and $C_{k}^{1/2}$ together form a
complete set in which any function can be expanded. For this supplementary
series,

$\displaystyle-\frac{1}{2}<j<0,$ (97) $\displaystyle m=0,\pm 1,\pm 2,...$

The determinant of the complex quantum metric is never zero.

We also consider a mixed quantum classical system with inverse mass tensor of
the form given in Eq. (83), the potential term will be of the form,

$\Phi(\rho)=(-j(j+1)+m^{2})(Q^{\prime}_{11}+Q^{\prime}_{22}\sinh^{2}\rho)+2mQ^{\prime\prime}_{12}\sinh\rho.$
(98)

Again, the second term is the contribution due to the imaginary component of
the inverse mass tensor.

###  VIII.5 Projective representations of $SU(1,1)$ and two coupled
oscillators as an example

Group spaces can have single-valued and multi-valued representations. The
former correspond to the universal covering space of the group, whereas the
latter to projective representations. An intuitive example is the case of the
one-dimensional translation group, which is single-valued, and its group space
is the one-dimensional line. The group of uniaxial rotations correspond to a
projective representation of the one-dimensional translation group, but in
this case the representation is no longer single valued, and the group space
will be the circle. The one-dimensional translation group is known as the
universal covering group of the group of uniaxial rotations. Analogously, the
one-dimensional line is the universal covering space of the circle. Exactly
this situation occurs for $SU(1,1)$. In addition to the single-valued
representations studied in the previous subsection, $SU(1,1)$ also has
projective representations, which correspond to multi-valued group
representations.

The Lie algebras are the same (Eq. (85)), because locally the universal
covering space and the group space of the multivalued represesntations is the
same. The form of the quantum metric is also the same as of the single-valued
representations, given by Eq. (95), what changes are the series of quantum
numbers. We state without proof that the projective discrete representation
corresponds to the the series,

$m=-j,-j+1,...,$ (99)

for $j<0$, $j$ real. In this case, $\mbox{Det}[C_{2}]$ is zero for $m=-j$. The
projective continuous representation is characterized by two numbers,

$\displaystyle j=-\frac{1}{2}+i\delta\hskip 14.22636pt\delta=\mbox{real}>0,$
(100) $\displaystyle m_{0}=\mbox{real},\hskip 14.22636pt0\leq m_{0}<1.$

The spectrum of $J_{z}$ in this case is

$m=m_{0},m_{0}\pm 1,m_{0}\pm 2,...$ (101)

As an example we can consider the case of a pair of coupled oscillators, a
system which has been studied in Ref. Mahran86, ; Gerry89, ; Kim89, ; Levay93,
. We define

$\hat{J}_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger},\hat{J}_{-}=\frac{1}{2}\hat{a}\hat{a},\hat{J}_{z}=\frac{1}{2}\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right),$
(102)

where $\hat{a}^{\dagger}$($\hat{a}$) are bosonic creation(annihilation)
operators. It can easily be shown that the operators $\hat{J}_{+}$,
$\hat{J}_{-}$, and $\hat{J}_{z}$ satisfy the $SU(1,1)$ commutation rules,
showing that the Lie algebra is the same as before. The Casimir operator is
also unchanged, meaning that Eq. (86) is still valid. After some algebra we
find, though, that the Casimir operator, using the definitions in Eq. (102)
becomes,

$\hat{J}^{2}=-\frac{3}{16}\hat{I},$ (103)

where $\hat{I}$ denotes the identity. Solving Eq. (86) results in two possible
$j$ values,

$j=-\frac{1}{4},-\frac{3}{4}.$ (104)

The essential point is that Eq. (103) does not fit into any of the sequences
given in subsection VIII.4 ($D_{j}^{+}$, $D_{j}^{-}$, $C_{k}^{0}$, and
$C_{k}^{1/2}$ ) even though the Lie algebra of the generators is unchanged.
However, it can be accommodated by the sequence given in Eq. (99).

$m=\frac{1}{4},\frac{5}{4},...\hskip 14.22636ptm=\frac{3}{4},\frac{7}{4},...$
(105)

The two series are each distinct projective discrete representations, and
their direct sum is the representation of the two-oscillator system. The
series corresponding to $j=-\frac{1}{4}$($j=-\frac{3}{4}$) are the odd(even)
photon squeezed vacuum states Gerry89  (see also Refs. Mahran86, ; Gerry89, ;
Kim89, ; Levay93, ). In such a system, characterized by a direct sum, there
are two instances where the geometry becomes trivial, $m=\frac{1}{4}$ and
$m=\frac{3}{4}$.

##  IX Conclusion

In this paper we investigated the complex geometric tensor of quantum state
manifolds. We showed that the imaginary part of this tensor, the Berry
curvature, gives rise to a potential term in mixed quantum-classical systems
obeying the Born-Oppenheimer approximation, if the inverse mass tensor has an
imaginary component.

From the scalar product of two quantum states a generating function can be
written, which generates, in principle, all orders of quantum fluctuations.
The second cumulant (variance) corresponds to the complex quantum metric,
higher order cumulants correspond to higher order geometric quantities, the
skew is related to the affine connection, the kurtosis gives the complex
analog of the four-index Riemann curvature tensor. Requiring the determinant
of the complex quantum metric to be positive definite gives generalized
uncertainty relations. Our calculations for Lie group coherent states led to
trivial geometries for the usual coherent states, when they are generated from
an extremal state, however for other cases, the determinant of the complex
metric can be nontrivial. Of particular interest is the representation of the
$SU(1,1)$ group formed by coupling a pair of harmonic oscillator creation and
annihilation operators, whose representation consists of the direct sum of two
projective representations of the group. In this case the extremal states of
both direct sums give zero for the determinant of the complex quantum metric,
meaning the geometry is trivial. Again, for other states the geometry is non-
trivial.

We envision a variety of interesting further studies based on the formalism we
presented. In quantum optics, the properties of squeezed coherent states are
measured via various correlation functions of the number operator Mahran86 ;
Gerry89 ; Kim89 , which are four operator products of creation and
annihilation operators. Another interesting direction would be to study time-
dependent coherent states to connect the underlying geometry with the
dynamics.

## Acknowledgments

BH was supported by the National Research, Development and Innovation Fund of
Hungary within the Quantum Technology National Excellence Program (Project Nr.
2017-1.2.1-NKP-2017-00001) and by the BME-Nanotechnology FIKP grant (BME FIKP-
NAT). PL was supported by the Ministry of Culture and Innovation and the
National Research, Development and Innovation Office within the Quantum
Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004).

##  Appendix A Moments and cumulants

In this appendix, we derive the moments and cumulants from the scalar product
of two quantum states and compare them to relations known for ordinary moments
and cumulants used in statistics. First we review the definition of ordinary
moments and cumulants.

Given a multi-variate normalized probability distribution:

$\displaystyle P(x_{1},...,x_{N})\geq 0,\forall x_{i};\hskip 85.35826pt$ (106)
$\displaystyle\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}dx_{1}...dx_{N}P(x_{1},...,x_{N})=1,$

The generating (or characteristic) function is defined as,

$f(k_{1},...,k_{N})=\int
dx_{1}...dx_{N}e^{i\sum_{j=1}^{N}k_{j}x_{j}}P(x_{1},...,x_{N}).$ (107)

Moments are obtained by taking derivatives of $f(k_{1},...,k_{N})$. The first
three moments can be written, Cumulants are logarithmic derivatives of the
characteristic function.

$\displaystyle\mathcal{M}_{1}(l)$ $\displaystyle=$
$\displaystyle\frac{1}{i}\frac{\partial}{\partial
k_{l}}f(k_{1},...,k_{N})|_{\bf k=0},$ $\displaystyle=$ $\displaystyle\langle
x_{l}\rangle,$ $\displaystyle\mathcal{M}_{2}(l,m)$ $\displaystyle=$
$\displaystyle\frac{1}{i^{2}}\frac{\partial^{2}}{\partial k_{l}\partial
k_{m}}f(k_{1},...,k_{N})|_{\bf k=0},$ $\displaystyle=$ $\displaystyle\langle
x_{l}x_{m}\rangle,$ $\displaystyle\mathcal{M}_{3}(l,m,n)$ $\displaystyle=$
$\displaystyle\frac{1}{i^{3}}\frac{\partial^{3}}{\partial k_{l}\partial
k_{m}\partial k_{n}}f(k_{1},...,k_{N})|_{\bf k=0},$ $\displaystyle=$
$\displaystyle\langle x_{l}x_{m}x_{n}\rangle.$

The first three cumulants are defined as

$\displaystyle\mathcal{C}_{1}(l)$ $\displaystyle=$
$\displaystyle\frac{1}{i}\frac{\partial}{\partial k_{l}}\ln
f(k_{1},...,k_{N})|_{\bf k=0},$ (109) $\displaystyle\mathcal{C}_{2}(l,m)$
$\displaystyle=$ $\displaystyle\frac{1}{i^{2}}\frac{\partial^{2}}{\partial
k_{l}\partial k_{m}}\ln f(k_{1},...,k_{N})|_{\bf k=0},$
$\displaystyle\mathcal{C}_{3}(l,m,n)$ $\displaystyle=$
$\displaystyle\frac{1}{i^{3}}\frac{\partial^{3}}{\partial k_{l}\partial
k_{m}\partial k_{n}}\ln f(k_{1},...,k_{N})|_{\bf k=0},$

or in terms of moments, they can be written,

$\displaystyle\mathcal{C}_{1}(l)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{1}(l),$ (110) $\displaystyle\mathcal{C}_{2}(l,m)$
$\displaystyle=$
$\displaystyle\mathcal{M}_{2}(l,m)-\mathcal{M}_{1}(l)\mathcal{M}_{1}(m),$
$\displaystyle\mathcal{C}_{3}(l,m,n)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{3}(l,m,n)-\mathcal{M}_{2}(l,m)\mathcal{M}_{1}(n)$
$\displaystyle-\mathcal{M}_{2}(l,n)\mathcal{M}_{1}(m)-\mathcal{M}_{2}(m,n)\mathcal{M}_{1}(l)$
$\displaystyle+2\mathcal{M}_{1}(l)\mathcal{M}_{1}(m)\mathcal{M}_{1}(n).$

In the formalism used in our work the role of the characteristic function is
played by the scalar product
$S(s^{\prime},s)=\langle\Psi(s^{\prime})|\Psi(s)\rangle$ (Eq. (31)). There are
two sets of variables, the primed ones,
$s^{\prime}=s^{\prime}_{1},...,s^{\prime}_{n}$, associated with the bra vector
of the scalar product, and the unprimed ones, $s=s_{1},...,s_{n}$, associated
with the ket. Derivatives of either one can be used to define the analog of a
moment. There are two types of first moments, for which we introduce the
following notation,

$\displaystyle\mathcal{M}_{1}(\\_;l)$ $\displaystyle=$
$\displaystyle(i\partial_{l})S(s^{\prime},s)|_{s^{\prime}=s}$ (111)
$\displaystyle\mathcal{M}_{1}(l;\\_)$ $\displaystyle=$
$\displaystyle(-i\partial_{l}^{\prime})S(s^{\prime},s)|_{s^{\prime}=s}.$

The underscore $\\_$ is used when no derivatives are taken for a particular
set of variables. For the first moments, it holds that

$\displaystyle\mathcal{M}_{1}(\\_;l)=\mathcal{M}_{1}(l;\\_)=\mathcal{M}_{1}(l)$
(112) $\displaystyle=-\beta_{l},$

see Eq. (37) for the definition of $\beta_{l}$. Depending on whether
derivatives are applied to the primed or unprimed variables, there are three
different types of second moments. We write them as,

$\displaystyle\mathcal{M}_{2}(\\_;kl)$ $\displaystyle=$
$\displaystyle(i\partial_{k})(i\partial_{l})S(s^{\prime},s)|_{s^{\prime}=s}$
(113) $\displaystyle\mathcal{M}_{2}(k;l)$ $\displaystyle=$
$\displaystyle(-i\partial_{k}^{\prime})(i\partial_{l})S(s^{\prime},s)|_{s^{\prime}=s}$
$\displaystyle\mathcal{M}_{2}(kl;\\_)$ $\displaystyle=$
$\displaystyle(-i\partial_{k}^{\prime})(-i\partial_{l}^{\prime})S(s^{\prime},s)|_{s^{\prime}=s}.$

$\mathcal{M}_{2}(k;l)$ corresponds to $\gamma_{kl}$ in Eq. (37). As for third
moments, there are four distinct types, but we will only write the two that
are useful in establishing the third order Christoffel symbol (section VI),

$\displaystyle\mathcal{M}_{3}(j;kl)$ $\displaystyle=$
$\displaystyle(-i\partial_{j}^{\prime})(i\partial_{k})(i\partial_{l})S(s^{\prime},s)|_{s^{\prime}=s}$
(114) $\displaystyle\mathcal{M}_{3}(jk;l)$ $\displaystyle=$
$\displaystyle(-i\partial_{j}^{\prime})(-i\partial_{k}^{\prime})(i\partial_{l})S(s^{\prime},s)|_{s^{\prime}=s}$

Cumulants are obtained in a similar way, by applying derivatives to $\ln
S(s^{\prime},s)$. The first order cumulants can be written,

$\displaystyle\mathcal{C}_{1}(\\_;l)$ $\displaystyle=$
$\displaystyle(i\partial_{l})\ln S(s^{\prime},s)|_{s^{\prime}=s}$ (115)
$\displaystyle\mathcal{C}_{1}(l;\\_)$ $\displaystyle=$
$\displaystyle(-i\partial_{l}^{\prime})\ln S(s^{\prime},s)|_{s^{\prime}=s}.$

the second ones as,

$\displaystyle\mathcal{C}_{2}(\\_;kl)$ $\displaystyle=$
$\displaystyle(i\partial_{k})(i\partial_{l})\ln
S(s^{\prime},s)|_{s^{\prime}=s}$ (116) $\displaystyle\mathcal{C}_{2}(k;l)$
$\displaystyle=$ $\displaystyle(-i\partial_{k}^{\prime})(i\partial_{l})\ln
S(s^{\prime},s)|_{s^{\prime}=s}$ $\displaystyle\mathcal{C}_{2}(kl;\\_)$
$\displaystyle=$
$\displaystyle(-i\partial_{k}^{\prime})(-i\partial_{l}^{\prime})\ln
S(s^{\prime},s)|_{s^{\prime}=s}.$

and the third order ones we use here as,

$\displaystyle\mathcal{C}_{3}(j;kl)$ $\displaystyle=$
$\displaystyle(-i\partial_{j}^{\prime})(i\partial_{k})(i\partial_{l})\ln
S(s^{\prime},s)|_{s^{\prime}=s}$ (117) $\displaystyle\mathcal{C}_{3}(jk;l)$
$\displaystyle=$
$\displaystyle(-i\partial_{j}^{\prime})(-i\partial_{k}^{\prime})(i\partial_{l})\ln
S(s^{\prime},s)|_{s^{\prime}=s}.$

In the text it was shown that cumulants which have at least one derivative as
a function of each set of parameters (the primed and unprimed) are gauge
invariant. Moments are not gauge invariant. It can also be shown that
cumulants can be expressed in terms moments, and the expressions look similar
to Eqs. (110), with appropriate modifications. For first order cumulants, it
holds that,

$\mathcal{C}_{1}(\\_;l)=\mathcal{C}_{1}(l;\\_)=\mathcal{M}_{1}(l).$ (118)

For the second order cumulants, we have,

$\displaystyle\mathcal{C}_{2}(\\_;kl)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{2}(\\_;kl)-\mathcal{M}_{1}(\\_;k)\mathcal{M}_{1}(\\_;l)$
(119) $\displaystyle\mathcal{C}_{2}(k;l)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{2}(k;l)-\mathcal{M}_{1}(k;\\_)\mathcal{M}_{1}(\\_;l)$
$\displaystyle\mathcal{C}_{2}(kl;\\_)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{2}(kl;\\_)-\mathcal{M}_{1}(k;\\_)\mathcal{M}_{1}(l;\\_).$

The indices on both sides of the equation “keep their sides of the semi-
colon”, otherwise the second cumulants obey the same relation as for ordinary
statistical cumulants (Eq. (110)). For the third order cumulants, we have,

$\displaystyle\mathcal{C}_{3}(j;kl)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{3}(j;kl)-\mathcal{M}_{2}(j;k)\mathcal{M}_{1}(\\_;l)$
$\displaystyle-\mathcal{M}_{2}(j;l)\mathcal{M}_{1}(\\_;k)-\mathcal{M}_{1}(j;\\_)\mathcal{M}_{2}(\\_;kl)$
$\displaystyle+2\mathcal{M}_{1}(j;\\_)\mathcal{M}_{1}(\\_;k)\mathcal{M}_{1}(\\_;l),$
$\displaystyle\mathcal{C}_{3}(jk;l)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{3}(jk;l)-\mathcal{M}_{1}(j;\\_)\mathcal{M}_{2}(k;l)$
$\displaystyle-\mathcal{M}_{1}(k;\\_)\mathcal{M}_{2}(j;l)-\mathcal{M}_{2}(jk;\\_)\mathcal{M}_{1}(\\_;l)$
$\displaystyle+2\mathcal{M}_{1}(j;\\_)\mathcal{M}_{1}(k;\\_)\mathcal{M}_{1}(\\_;l).$

Again, the indices maintain “their sides of the semicolon”. Further
simplifications can be made, by using $\mathcal{M}_{1}(l)$, but we wanted to
emphasize the pattern of the placement of indices.

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