# Duality of quantum geometries

Jan Naudts
Universiteit Antwerpen
Physics Department, Universiteitsplein 1, 2610 Antwerpen, Belgium
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-4646-1190

###### Abstract

Quantum connections are defined by parallel transport operators acting on a
Hilbert space. They transport tangent operators along paths in parameter
space. The metric tensor of a Riemannian manifold is replaced by an inner
product of pairs of operator fields, similar to the inner product of the Kubo-
Mori formalism of Linear Response Theory. The metric is used to define the
dual of a quantum connection. Directional derivatives of the parallel
transport operators are the elements of the quantum vector potential. They
define the covariant derivatives of operator fields. The covariant derivatives
are used to quantify the holonomy of the quantum connection. It is shown that
a quantum connection is holonomic if and only if its dual is holonomic. If the
parallel transport operators are unitary then an alpha-family of quantum
connections can be defined in a way similar to Amari’s alpha family of
connections in Information Geometry. The minus alpha connection is the dual of
the alpha connection. In particular, the alpha equal zero connection is self-
dual. An operator field can be combined with a path in parameter space to
produce a path in operator space. A definition is given for such a path in
operator space to be autoparallel. The path in parameter space is then a
geodesic for the induced connection. If the quantum connection is self-dual
then the induced connection is the metric connection.

##  1 Introduction

This research is part of an effort to generalize results of Information
Geometry [10, 14, 16, 17] to the non-commutative context. Several approaches
have been tried out already.

Chapter 7 of [14] starts from a quantum state space formed by density matrices
on a Hilbert space. Hasegawa [12] introduced an alpha family of quantum
divergences for pairs of density matrices. More generally, the density
matrices are replaced by states on a $C^{*}$-algebra. These states form a
differentiable manifold that can be studied by the techniques of Information
Geometry. Many authors have followed this approach [19, 20, 21, 22, 25, 26].
The approach has two drawbacks. Based on works of Araki [5] it is argued in
[24, 25] that the definition of quantum exponential families is highly non-
unique because of the non-uniqueness of the Radon–Nikodym derivative in a non-
commutative context. The other drawback is that the approach with a
differentiable manifold of quantum states is far away of the current practice
in both Solid State Physics and in Quantum Field Theory.

The present work shifts the emphasis from quantum states to tangent operators.
Intuitively, tangent operators are directional derivatives of smooth paths in
the space of bounded operators on the Hilbert space. This yields a different
starting point than when one considers paths in the space of quantum states.
The latter point of view is worked out in [23] for a relativistic field of
quantum harmonic oscillators. The quantum states are parameterized by
positions in spacetime. The tangent vectors are linear functionals on a von
Neumann algebra. Gauge transformations implement parallel transport along
paths in spacetime. Much of this approach is taken over by the present work.
However, spacetime is replaced by an arbitrary finite-dimensional space of
parameters. The usual identification of a vector of parameters with a point in
a differentiable manifold is replaced by an identification with a tangent
space of wave functions, subspace of the Hilbert space. After introduction of
an inner product in the way it is done in the Kubo-Mori formalism [3, 4, 6, 7]
this tangent space becomes a tangent space of operators.

Parameterized statistical models belonging to an exponential family are the
central study object of Information Geometry. Their geometry is described by a
connection which is the dual of the Euclidean connection on the space of
canonical parameters. An important result (see Theorem 3.6 of [14] or Theorem
2 of [27]) states that if a connection is flat then the dual connection is
flat as well. Hence, the geometry of a statistical model belonging to an
exponential family is flat. In the present work the analogue of the Euclidean
connection is a quantum connection with unitary parallel transport operators.
It is shown in Proposition 11.2 below that if a unitary connection is
holonomic then the dual connection is holonomic as well. The notion of
holonomy [2] is used here as a substitute for the Riemannian curvature.

The structure of the paper is as follows. Sections 2 to 6 introduce the notion
of a quantum connection and of a metric operator. The dual of a quantum
connection is defined in an obvious manner. Unitary connections are discussed
in Section 4. Starting from Section 7 the parallel transport operators are
considered as gauge transformations, the directional derivatives of which
define the quantum vector potential. Sections 9 and 10 elucidate the relation
that exists between the antisymmetric force tensor and the holonomy of the
quantum connection. Sections 12 and 13 show how the quantum geometry makes the
parameter space into a Riemannian manifold.

Section 14 contains a discussion of various issues that arise in the previous
sections. Finally, Section 15 evaluates the results obtained so far.

##  2 Tangent operators

In Solid State Physics the Kubo-Mori formalism of Linear Response Theory [6]
replaces the tangent vectors of a Riemannian manifold by operators acting on a
Hilbert space. For a mathematical foundation see [7, 8]. The Riemannian metric
is replaced by a complex inner product/scalar product defined on pairs of
operators. It is of the form

$(X,Y)_{\sim}=(T\,X\,\Omega,T\,Y\Omega),\qquad X,Y\in\mathfrak{M}.$ (1)

with $X,Y$ bounded operators on $\mathscr{H}$ belonging to some von Neumann
algebra $\mathfrak{M}$. The vector $\Omega$ is a normalized element of the
Hilbert space $\mathscr{H}$. Its is called the wave vector in what follows.
The operator $T$ is a strictly positive self-adjoint operator with domain
including $\mathfrak{M}\Omega$. It is called the metric operator in what
follows,

The situation described above can depend on a vector of parameters
$\theta_{p}$, $p=1,2,\cdots,n$, belonging to some open connected domain
${\mathbb{M}}$ in ${\mathbb{R}}^{n}$. For the sake of simplicity the Hilbert
space $\mathscr{H}$ is assumed to be independent of $\theta$. The von Neumann
algebra $\mathfrak{M}$, the metric operator $T$, the wave function $\Omega$
and the inner product receive a lower index to stress there dependence on the
parameters $\theta_{p}$.

In the Literature the inner product $(\cdot,\cdot)_{\sim}$ is interpreted as a
quantum statistical correlation between pairs $X,Y$ of observable operators.
Here, the operators belonging to $\mathfrak{M}$ are called tangent operators
and are considered to be the analogues of the tangent vectors of a Riemannian
manifold. For convenience, also unbounded operators affiliated with
$\mathfrak{M}$ are considered to be tangent operators. The inner product
$(\cdot,\cdot)_{\theta}$ is then the inner product defined in the tangent
space of the quantum manifold at the point labeled $\theta$, i.e.

$(X,Y)_{\theta}=(T_{\theta}X\Omega_{\theta},T_{\theta}Y\Omega_{\theta}),\qquad
X,Y\in\mathfrak{M}_{\theta}.$ (2)

The points of the quantum manifold are the quantum states. However, they are
not further specified in the present paper. The quantum manifold is assumed to
be fully specified by its tangent spaces. The conventional identification
between the set of parameters ${\mathbb{M}}$ and the manifold is replaced by
an identification between a vector of parameters $\theta$ and the
corresponding tangent space.

For convenience, it is assumed throughout the paper that the metric operators
$T_{\theta}$ are bounded operators with bounded inverse. This avoids some of
the technical discussions about domains of unbounded operators. In addition,
they satisfy

$T_{\theta}\Omega_{\theta}=\Omega_{\theta},\qquad\theta\in{\mathbb{M}}.$

It is also assumed that $\mathfrak{M}_{\theta}\Omega_{\theta}$ is dense in the
Hilbert space $\mathscr{H}$. This implies that the inner product
$(\cdot,\cdot)_{\theta}$ is non-degenerate. It makes the linear space
$\mathfrak{M}$ into a pre-Hilbert space the completion of which is denoted
$\mathscr{H}_{\theta}$. The adjoint of an operator $X$ on the Hilbert space
$\mathscr{H}$ is denoted $X^{\dagger}$. On the other hand, the adjoint of an
operator $\hat{X}$ on $\mathscr{H}_{\theta}$ is denoted $\hat{X}^{*}$.

##  3 Quantum connections

###### Definition 3.1

A quantum connection $\hat{\Pi}$ is determined by a collection of parallel
transport operators $\Pi(\gamma)^{t}_{s}$. These are bounded linear operators
acting on the Hilbert space $\mathscr{H}$. They describe the transport of
tangent operators from $\gamma_{s}$ to $\gamma_{t}$ along a smooth path
$\gamma:\,t\mapsto\gamma_{t}$ in ${\mathbb{M}}$.

The following properties are assumed.

  * A1

$\Pi(\gamma)^{t}_{t}$ is the identity operator;

  * A2

The inverse of $\Pi(\gamma)^{t}_{s}$ exists and equals $\Pi(\gamma)^{s}_{t}$;

  * A3

If $X$ is a tangent operator at $\gamma_{s}$ then
$\Pi(\gamma)^{t}_{s}\,X\,\Pi(\gamma)^{s}_{t}$ is a tangent operator at
$\gamma_{t}$, i.e.

$\Pi(\gamma)^{t}_{s}\,\mathfrak{M}_{s}\,\Pi(\gamma)^{s}_{t}=\mathfrak{M}_{t};$

  * A4

The wave vector $\Omega_{\theta}$ appearing in the definition (1) of the
quantum metric is covariant in the sense that

$\Pi(\gamma)^{t}_{s}\,\Omega_{s}=[\Pi(\gamma)^{s}_{t}]^{\dagger}\,\Omega_{s}=\Omega_{t}.$

Note that $\mathfrak{M}_{s}$ is written instead of $\mathfrak{M}_{\gamma_{s}}$
if it is clear which path $\gamma$ is intended. A similar remark holds for
other quantities.

From the transport operators one can derive covariant derivatives and
connection coefficients [1]. This will be done further on.

Introduce operators $\hat{\Pi}(\gamma)^{t}_{s}$ acting on $\mathscr{H}_{s}$,
defined by

$\hat{\Pi}(\gamma)^{t}_{s}X=\Pi(\gamma)^{t}_{s}\,X\,\Pi(\gamma)^{s}_{t},\qquad
X\in\mathfrak{M}_{s}.$ (3)

The operator $\hat{\Pi}(\gamma)^{t}_{s}X$ belongs to $\mathfrak{M}_{t}$ by
assumption A3.

###### Proposition 3.2

Assume that $T_{t}$ is a bounded operator with bounded inverse. Then
$\hat{\Pi}(\gamma)^{t}_{s}$ is a bounded operator. Its adjoint is given by

$[\hat{\Pi}(\gamma)^{t}_{s}]^{*}=\hat{Z}^{s}_{t}\quad\mbox{ with
}Z^{s}_{t}=T^{-2}_{s}\,[\Pi(\gamma)^{t}_{s}]^{\dagger}\,T^{2}_{t}.$ (4)

Proof

Calculate

$\displaystyle(\hat{\Pi}(\gamma)^{t}_{s}X,\hat{\Pi}(\gamma)^{t}_{s}X)_{t}$
$\displaystyle=\bigg{(}T_{t}\,\Pi(\gamma)^{t}_{s}\,X\,\Omega_{t},T_{t}\,\Pi(\gamma)^{t}_{s}\,X\,\Omega_{t}\bigg{)}$
$\displaystyle\leq||T_{t}||^{2}\,||\Pi(\gamma)^{t}_{s}||^{2}\,||X\,\Omega_{t}||^{2}$
$\displaystyle\leq||T_{t}||^{2}\,||\Pi(\gamma)^{t}_{s}||^{2}\,||T^{-1}_{t}||^{2}\,(X,X)_{t}.$

This shows that $\hat{\Pi}(\gamma)^{t}_{s}$ is bounded by

$||\hat{\Pi}(\gamma)^{t}_{s}||_{t}\leq||T_{t}||\,||\Pi(\gamma)^{t}_{s}||\,||T^{-1}_{t}||.$

Let us finally show (4). One has

$\displaystyle[Z^{s}_{t}]^{-1}\Omega_{s}$
$\displaystyle=T^{-2}_{t}\,[\Pi(\gamma)^{s}_{t}]^{\dagger}\,T^{2}_{s}\Omega_{s}$
$\displaystyle=Z^{t}_{s}\Omega_{s}$ $\displaystyle=\Omega_{t}.$

Use this to calculate for $X$ in $\mathfrak{M}_{s}$ and $Y$ in
$\mathfrak{M}_{t}$

$\displaystyle(X,\hat{Z}^{s}_{t}Y)_{s}$
$\displaystyle=(T_{s}\,X\Omega_{s},T_{s}Z^{s}_{t}\,Y\,[Z^{s}_{t}]^{-1}\,\Omega_{s})$
$\displaystyle=(T_{s}\,X\Omega_{s},T_{s}Z^{s}_{t}Y\,\Omega_{t})$
$\displaystyle=(X\Omega_{s},[\Pi(\gamma)^{t}_{s}]^{\dagger}\,T^{2}_{t}\,Y\,\Omega_{t})$
$\displaystyle=(T_{t}\,\Pi(\gamma)^{t}_{s}\,X\Omega_{s},T_{t}\,Y\,\Omega_{t})$
$\displaystyle=(T_{t}\,\hat{\Pi}(\gamma)^{t}_{s}\,X\Omega_{t},T_{t}\,Y\,\Omega_{t})$
$\displaystyle=(\hat{\Pi}(\gamma)^{t}_{s}\,X,T_{t}\,Y)_{t}.$

This shows that the adjoint of $\hat{\Pi}(\gamma)^{t}_{s}$ is
$\hat{Z}^{s}_{t}$.

$\square$

##  4 Dual connections

In the context of Differential Geometry dual connections are defined relative
to the metric in the tangent space and relative to a path $t\mapsto\gamma_{t}$
in the manifold. In an analoguous way one can define the dual of a quantum
connection.

###### Definition 4.1

A dual quantum connection $\hat{\Pi}^{\star}$, if it exists, is defined by
transport operators $\Pi^{*}(\gamma)^{t}_{s}$ which satisfy

$\bigg{(}\hat{\Pi}(\gamma)^{t}_{s}\,X,\hat{\Pi}^{\star}(\gamma)^{t}_{s}\,Y\bigg{)}_{t}=\bigg{(}X,Y\bigg{)}_{s},\qquad
X,Y\in\mathfrak{M}_{s}.$ (5)

The proof of the following proposition is straightforward.

###### Proposition 4.2

If the assumptions of Proposition 3.2 are satisfied and the dual quantum
connection $\hat{\Pi}^{\star}$ exists then it is unique and it is given by

$\hat{\Pi}^{\star}(\gamma)^{t}_{s}=[\hat{\Pi}(\gamma)^{s}_{t}]^{*}.$ (6)

The problem of the existence of the dual quantum connection is that a dual
defined by (6) should satisfy the assumptions A1, A2, A3, A4 formulated in
Section 3.

###### Proposition 4.3

Assume that the metric operators $T_{\theta}$ are bounded operators with
bounded inverse. Assume in addition that $X$ in $\mathfrak{M}_{\theta}$
implies that $T^{-2}_{\theta}X^{\dagger}T^{2}_{\theta}$ belongs to
$\mathfrak{M}_{\theta}$ as well. Then each quantum connection $\hat{\Pi}$ has
a unique dual w.r.t. the metric operators $T_{\theta}$. It is given by

$\Pi^{\star}(\gamma)^{t}_{s}=T^{-2}_{t}\,[\Pi(\gamma)^{s}_{t}]^{\dagger}\,T^{2}_{s}$
(7)

and it satisfies (6).

Proof

Let $\Pi^{\star}(\gamma)^{t}_{s}$ be defined by (7), i.e.

$\Pi^{\star}(\gamma)^{t}_{s}=Z^{t}_{s}\quad\mbox{ with }\quad
Z^{t}_{s}=T^{-2}_{t}\,[\Pi(\gamma)^{s}_{t}]^{\dagger}\,T^{2}_{s}.$

These operators clearly satisfy assumptions A1, A2 and A4 of Section 3.

One has for an arbitrary $X$ in $\mathfrak{M}_{s}$

$\displaystyle\hat{Z}^{t}_{s}\,X$ $\displaystyle=Z^{t}_{s}\,X\,Z^{s}_{t}$
$\displaystyle=T^{-2}_{t}\,[\Pi(\gamma)^{s}_{t}]^{\dagger}\,T^{2}_{s}\,X\,T^{-2}_{s}\,[\Pi(\gamma)^{t}_{s}]^{\dagger}\,T^{2}_{t}$
$\displaystyle=T^{-2}_{t}\,\bigg{(}\Pi(\gamma)^{t}_{s}\,T^{-2}_{s}X^{\dagger}\,T^{2}_{s}\,\Pi(\gamma)^{s}_{t}\bigg{)}^{\dagger}\,T^{2}_{t}.$

It is easy to see using twice the additional assumption of the proposition
that this operator belongs to $\mathfrak{M}_{t}$. Hence, assumption A3 is also
satisfied.

One concludes that $\hat{\Pi}^{\star}$, as defined by (7) satisfies the
assumptions A1 to A4 of Section 3. Hence, it is a quantum connection. It then
follows from Proposition 3.2 that (6) is satisfied.

$\square$

Another way of writing (7) is

$T^{2}_{s}=\Pi^{\dagger}(\gamma)^{t}_{s}\,T^{2}_{t}\Pi^{\star}(\gamma)^{t}_{s}.$
(8)

Hence, if the connection $\hat{\Pi}$ and its dual $\hat{\Pi}^{\star}$ are
known and the metric operator $T_{\gamma_{t}}$ is known at a single site
$\gamma_{t}$ in ${\mathbb{M}}$ then it can be calculated all along the path
$\gamma$ by the above expression.

##  5 Unitary transport

A special role is played by the quantum connections $\hat{\Pi}$ for which the
parallel transport operators $\Pi(\gamma)^{t}_{s}$ are unitary operators. Such
a connection is said to be unitary in what follows. Unitary connections have
an additional symmetry which is exploited to introduce alpha-families of
connections.

From a Physics point of view it is obvious to require that the transport
operators are unitary because then they conserve the normalization of the wave
vectors and conserve the probabilistic interpretation of Quantum Mechanics.

If the connection is unitary and the operators $T_{m}$ appearing in the
definition (1) of the metric are bounded with bounded inverse then (7) becomes

$\Pi^{\star}(\gamma)^{t}_{s}=T^{-2}(\gamma_{t})\,\Pi(\gamma)^{t}_{s}\,T^{2}(\gamma_{s}).$
(9)

This suggests the introduction of a family of connections $\hat{\Pi}_{\alpha}$
labeled by a parameter $\alpha$ and defined by the following parallel
transport operators

$\Pi_{\alpha}(\gamma)^{t}_{s}=T^{-(1-\alpha)}(\gamma_{t})\,\,\Pi_{1}(\gamma)^{t}_{s}\,\,T^{1-\alpha}(\gamma_{s}),$
(10)

where $\hat{\Pi}_{1}$ is a unitary connection. Clearly is
$\hat{\Pi}_{-1}=\hat{\Pi}_{1}^{\star}$.

The folllowing proposition requires a sharpening of the assumptions made in
Proposition 4.3.

###### Proposition 5.1

Assume that $X\in\mathfrak{M}_{\theta}$ implies $Y\in\mathfrak{M}_{\theta}$
for any $Y$ of the form

$Y=T^{\lambda}(\theta)\,X^{\dagger}\,T^{-\lambda}(\theta)$ (11)

with $\lambda\in{\mathbb{R}}$. Then the parallel transport operators
$\Pi_{\alpha}(\gamma)^{t}_{s}$ defined by (10) satisfy the axioms A1 to A4 of
Section 3 and hence determine a quantum connection $\hat{\Pi}_{\alpha}$.

Proof

Assumptions A1 and A2 are satisfied in a rather trivial manner.

Take $X$ in $\mathfrak{M}_{s}$ and calculate using unitarity of the connection

$\displaystyle\Pi_{\alpha}(\gamma)^{t}_{s}\,X\,\Pi_{\alpha}(\gamma)^{s}_{t}$
$\displaystyle=T^{-(1-\alpha)}(\gamma_{t})\,\,\Pi_{1}(\gamma)^{t}_{s}\,\,\bigg{[}T^{1-\alpha}(\gamma_{s})\,X\,T^{-(1-\alpha)}(\gamma_{s})\bigg{]}$
$\displaystyle\quad\times\Pi_{1}(\gamma)^{s}_{t}\,\,T^{1-\alpha}(\gamma_{t})$
$\displaystyle=T^{-(1-\alpha)}(\gamma_{t})\,\,\Pi_{1}(\gamma)^{t}_{s}\,\,Y^{\dagger}\,\Pi_{1}(\gamma)^{s}_{t}\,\,T^{1-\alpha}(\gamma_{t})$
$\displaystyle=T^{-(1-\alpha)}(\gamma_{t})\,\bigg{[}\Pi_{1}(\gamma)^{t}_{s}\,Y\Pi_{1}(\gamma)^{s}_{t}\,\bigg{]}^{\dagger}\,T^{1-\alpha}(\gamma_{t}),$

with

$Y=T^{-(1-\alpha)}(\gamma_{s})\,X^{\dagger}\,T^{(1-\alpha)}(\gamma_{s}).$

Use now the assumption of the proposition to see that $Y$ belongs to
$\mathfrak{M}_{s}$ so that $\Pi_{1}(\gamma)^{t}_{s}\,Y\Pi_{1}(\gamma)^{s}_{t}$
belongs to $\mathfrak{M}_{t}$. Apply again the assumption to conclude that the
whole expression belongs to $\mathfrak{M}_{t}$. This proves that assumption A3
is satisfied.

Finally consider assumption A4. A calculation gives

$\displaystyle[\Pi_{\alpha}(\gamma)^{s}_{t}]^{\dagger}\,\Omega_{s}=\Pi_{\alpha}(\gamma)^{t}_{s}\,\Omega_{s}$
$\displaystyle=T^{-(1-\alpha)}(\gamma_{t})\,\,\Pi_{1}(\gamma)^{t}_{s}\,\,T^{1-\alpha}(\gamma_{s})\,\Omega_{s}$
$\displaystyle=\Omega_{t}.$

This proves A4.

$\square$

A short calculation shows that

$\left[\Pi_{\alpha}(\gamma)^{t}_{s}\right]^{\dagger}=\Pi_{2-\alpha}(\gamma)^{s}_{t}.$
(12)

Hence, the connection $\hat{\Pi}_{\alpha}$ is unitary when $\alpha$ is such
that $\Pi_{\alpha}=\Pi_{2-\alpha}$. This is of course the case for $\alpha=1$.

An important property is the following.

###### Proposition 5.2

The dual connection of $\hat{\Pi}_{\alpha}$ is $\hat{\Pi}_{-\alpha}$. In
particular, $\hat{\Pi}_{0}$ is self-dual.

Proof

A short calculation using (9) gives

$\displaystyle T^{2}(\gamma_{t})\,\Pi^{\star}_{\alpha}(\gamma)^{t}_{s}$
$\displaystyle=\Pi^{\dagger}_{\alpha}(\gamma)^{s}_{t}\,T^{2}(\gamma_{s})$
$\displaystyle=\bigg{[}T^{-(1-\alpha)}(\gamma_{s})\,\Pi(\gamma)^{s}_{t}\,T^{1-\alpha}(\gamma_{t})\bigg{]}^{\dagger}\,T^{2}(\gamma_{s})$
$\displaystyle=T^{1-\alpha}(\gamma_{t})\,\Pi^{\dagger}(\gamma)^{s}_{t}\,T^{1+\alpha}(\gamma_{s})$
$\displaystyle=T^{2}(\gamma_{t})\,T^{-(1+\alpha)}(\gamma_{t})\,\Pi(\gamma)^{t}_{s}\,T^{1+\alpha}(\gamma_{s})$
$\displaystyle=T^{2}(\gamma_{t})\Pi_{-\alpha}(\gamma)^{t}_{s}.$

This implies $\hat{\Pi}_{\alpha}^{\star}=\hat{\Pi}_{-\alpha}$.

$\square$

##  6 Connections of the product form

A subclass of connections is formed by those for which a composition law
holds.

A connection $\hat{\Pi}$ is of the product form if there exists a non-
degenerate operator field $V(\theta)$ such that for any smooth path $\gamma$
one has

$\Pi(\gamma)^{t}_{s}=V(\gamma_{t})\,V^{-1}(\gamma_{s});$ (13)

Such a connection satisfies the composition law

$\Pi(\gamma)^{t}_{s}\,\Pi(\gamma)^{s}_{p}=\Pi(\gamma)^{t}_{p}$ (14)

for any path $\gamma$ in ${\mathbb{M}}$.

If a unitary connection $\hat{\Pi}$ is of the product form then all members of
the corresponding alpha-family are of the product form as well.

##  7 The quantum vector potential

In Differential Geometry connection coefficients $\Gamma^{r}_{pq}$ can be
obtained from the parallel transport operators by taking partial derivatives.
The story here is slightly different due to the appearance of non-commuting
operators.

Consider a path in parameter space ${\mathbb{M}}$ of the form

$\gamma_{p}^{\theta}:t\mapsto\theta+tg_{p}$

where $g_{p}$ is the vector in ${\mathbb{R}}^{n}$ with components $g_{p}^{q}$,
which equal $1$ if $p=q$ and zero otherwise. The directional derivative of the
parallel transport operator $\Pi$ along the path $\gamma_{p}^{\theta}$ at
$t=0$ is denoted $A_{p}(\theta)$. It is given by

$A_{p}(\theta)=i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Pi(\gamma_{p}^{\theta})^{t}_{0}\bigg{|}_{t=0}=-i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Pi(\gamma_{p}^{\theta})^{0}_{t}\bigg{|}_{t=0}.$
(15)

The imaginary number $i$ is added in the definition of $A_{p}$ to guarantee
that the latter is a Hermitian operator on $\mathscr{H}$ in the case of a
unitary connection. The constant of Planck $\hbar$ is added for agreement with
applications in Physics.

The assumptions needed in what follows are the following.

  * A5

The directional derivatives $A_{p}(\theta)$ exist and are linear so that for
any smooth path $\gamma$ in ${\mathbb{M}}$ one has

$i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Pi(\gamma)^{t}_{s}\bigg{|}_{s=t}=\dot{\gamma}^{p}\,A_{p}(\gamma_{t});$
(16)

  * A6

The domains of the operators $A_{p}(\theta)$, of the commutators

$\frac{i}{\hbar}[A_{p},A_{q}]_{\hskip-1.5pt\mbox{\tiny--}}$

and of the directional derivatives

$\frac{\partial\,}{\partial\theta^{q}}A_{p}(\theta)=\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\theta+tg_{q})\bigg{|}_{t=0}.$

include $\mathfrak{M}_{\theta}\Omega_{\theta}$; These operators belong to
$\mathfrak{M}_{\theta}$ or are affiliated with it.

As a consequence of (16) the field of wave functions $\Omega_{\theta}$
satisfies

$i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Omega_{t}=\dot{\gamma}^{p}A_{p}(\gamma_{t})\Omega_{t}$
(17)

along any smooth path $\gamma$ in the parameter space ${\mathbb{M}}$. This is
the Schrödinger equation with Hamiltonian

$H_{\gamma}(t)=\dot{\gamma}^{p}A_{p}(\gamma_{t}).$

##  8 Covariant derivatives

The vector of operators $A_{p}(\theta)$, $p=1,2,\cdots,n$, is known as the
quantum vector potential. It is used to define an operator-valued covariant
derivative $\nabla_{\dot{\gamma}}$ by

$[\nabla_{\dot{\gamma}}X]_{t}=\dot{\gamma}^{p}\,[\nabla_{p}X]_{t}\quad\mbox{
with }\quad[\nabla_{p}X]_{\theta}=\frac{\partial
X}{\partial\theta^{p}}+\frac{i}{\hbar}\bigg{[}A_{p}(\theta),X(\theta)\bigg{]}_{\hskip-1.5pt\mbox{\tiny--}},$
(18)

where $\gamma$ is any smooth path in ${\mathbb{M}}$ and $X$ is an operator-
valued field over ${\mathbb{M}}$.

The justification for this definition is that for any smooth path $\gamma$ in
the parameter space ${\mathbb{M}}$ and any smooth operator-valued field
$X(\theta)$ one has

$\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{[}\Pi(\gamma)^{s}_{t}\,X(\gamma_{t})\,\Pi(\gamma)^{t}_{s}\bigg{]}_{s=t}=[\nabla_{\dot{\gamma}}X]_{t}.$
(19)

Defined in this way $\nabla_{\dot{\gamma}}$ has the properties one usually
requests a covariant derivative to satisfy:

  * 1)

The map

$n\mapsto\nabla_{n}X=n^{p}\nabla_{p}X$

is linear by construction;

  * 2)

$\nabla_{p}X$ is additive in $X$;

  * 3)

The product rule holds; Indeed, one has for any function $f(\theta)$

$\displaystyle[\nabla_{p}fX]_{\theta}$ $\displaystyle=\frac{\partial
fX}{\partial\theta^{p}}+\frac{i}{\hbar}\bigg{[}A_{p}(\theta),f(\theta)\,X(\theta)\bigg{]}_{\hskip-1.5pt\mbox{\tiny--}}$
$\displaystyle=\frac{\partial
f}{\partial\theta^{p}}\,X(\theta)+f(\theta)\,\frac{\partial
X}{\partial\theta^{p}}+f(\theta)\,\frac{i}{\hbar}\bigg{[}A_{p}(\theta),X(\theta)\bigg{]}_{\hskip-1.5pt\mbox{\tiny--}}$
$\displaystyle=\frac{\partial
f}{\partial\theta^{p}}\,X(\theta)+f(\theta)\,[\nabla_{p}X]_{\theta}.$

The obvious definition of connection coefficients $\Gamma_{qp,r}$ is

$\Gamma_{qp,r}(\theta)=(\nabla_{q}A_{p},A_{r})_{\theta},$ (20)

where $(\cdot,\cdot)_{\theta}$ is the inner product defined by (1). Introduce
a metric tensor $g(\theta)$ defined by

$g_{pq}(\theta)=(A_{p},A_{q})_{\theta}$ (21)

Assume that $g$ is non-degenerate. Then one has

$\nabla_{q}A_{p}=\Gamma^{r}_{qp}A_{r}+\cdots,$ (22)

where the missing term is orthogonal to the tangent operators $A_{p}$ and
where

$\Gamma^{r}_{qp}:=g^{rs}\Gamma_{qp,s}.$

##  9 The force field tensor

Figure 1: Infinitesimal paths considered in Section 9.

Let $\gamma^{\theta}_{p}$ denote the path $t\mapsto\theta+tg_{p}$, as before.
Consider two composed paths which both go from $\theta$ to
$\theta+sg_{p}+tg_{q}$:

$\displaystyle\theta\rightarrow\theta+sg_{p}\rightarrow\theta+sg_{p}+tg_{q},$
(23)
$\displaystyle\theta\rightarrow\theta+tg_{q}\rightarrow\theta+sg_{p}+tg_{q}.$
(24)

See the figure. Introduce operators $H_{pq}(\theta)$ which are formally
defined by

$H_{pq}(\theta)=i\hbar\,\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{(}\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}-\Pi(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}\bigg{)}\bigg{|}_{s=t=0}.$
(25)

They are called the holonomy operators in what follows.

###### Proposition 9.1

The holonomy operators satisfy

$H_{pq}(\theta)=F_{pq}(\theta)-\frac{i}{\hbar}\,[A_{p}(\theta),A_{q}(\theta)]_{\hskip-1.5pt\mbox{\tiny--}},$
(26)

with the antisymmetric force field tensor $F$ defined by

$F_{pq}(\theta)=\frac{\partial\,}{\partial\theta^{q}}A_{p}(\theta)-\frac{\partial\,}{\partial\theta^{p}}A_{q}(\theta).$
(27)

Proof

One has

$\displaystyle\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\left(\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}\right)\bigg{|}_{s=t=0}$
$\displaystyle=-\frac{i}{\hbar}\frac{{\rm\,d}\,}{{\rm\,d}t}\,A_{p}(\theta+tg_{q})\,\Pi(\gamma^{\theta}_{q})^{t}_{0}\bigg{|}_{t=0}$
$\displaystyle=-\frac{i}{\hbar}\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\theta+tg_{q})\bigg{|}_{t=0}-\frac{1}{\hbar^{2}}A_{p}(\theta)\,A_{q}(\theta)$
$\displaystyle=-\frac{i}{\hbar}\frac{\partial\,}{\partial\theta^{q}}A_{p}(\theta)-\frac{1}{\hbar^{2}}\,A_{p}(\theta)\,A_{q}(\theta).$

This result implies (26).

$\square$

Remember that the partial derivative $\partial
A_{p}(\theta)/\partial\theta^{q}$ is a shortcut notation for the directional
derivative

$\frac{\partial\,}{\partial\theta^{q}}A_{p}(\theta)=\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\theta+tg_{q})\bigg{|}_{t=0}.$

In particular, consecutive partial derivatives of operators do not necessarily
commute with each other.

If $H_{pq}$ vanishes then one has

$F_{pq}=\frac{i}{\hbar}\,[A_{p},A_{q}]_{\hskip-1.5pt\mbox{\tiny--}},$ (28)

which is a well-known expression for the anti-symmetric force tensor.

Note that (26) implies that the operator-valued tensor $H_{pq}$ is anti-
symmetric.

Some further properties follow now.

###### Proposition 9.2

The covariant derivatives of the elements $A_{p}$ of the vector potential
satisfy

$[\nabla_{q}A_{p}]_{\theta}=\frac{\partial\,}{\partial\theta^{p}}A_{q}-H_{qp}.$
(29)

Proof

From the definition (18) of $\nabla_{q}$ one obtains

$[\nabla_{q}A_{p}]_{\theta}=\frac{\partial
A_{p}}{\partial\theta^{q}}+\frac{i}{\hbar}\bigg{[}A_{q}(\theta),A_{p}(\theta)\bigg{]}_{\hskip-1.5pt\mbox{\tiny--}}.$

Next use (26) and (27) to obtain (29).

$\square$

###### Proposition 9.3

If the connection $\hat{\Pi}$ is of the product form (13) then $H_{pq}$
vanishes.

Proof

A short calculation gives

$\displaystyle H_{pq}(\theta)$
$\displaystyle=i\hbar\,\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{(}\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}-\Pi(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}\bigg{)}\bigg{|}_{s=t=0}$
$\displaystyle=i\hbar\,\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{(}V(\theta+tg_{q}+sg_{p})\,V^{-1}(\theta+tg_{q})\,V(\theta+tg_{q})\,V^{-1}(\theta)$
$\displaystyle\quad-V(\theta+tg_{q}+sg_{p})\,V^{-1}(\theta+sg_{p})\,V(\theta+sg_{p})\,V^{-1}(\theta)\bigg{)}\bigg{|}_{s=t=0}$
$\displaystyle=0.$

$\square$

##  10 Holonomy

The previous section introduces the holonomy operators $H_{pq}$ without
elucidating the connection with holonomy, which is the property that parallel
transport of an operator along a closed loop in parameter space ${\mathbb{M}}$
is an identity operation. This connection is made here.

###### Proposition 10.1

If the holonomy operators $H_{pq}(\theta)$ of the previous section exist then
the difference of parallel transport along the two paths (23) can be expanded
as follows

$\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}-\Pi(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}=st\,H_{pq}(\theta)+\mbox{
o}(s)\,+\mbox{ o}(t).$ (30)

Proof

The l.h.s. vanishes for $s=t=0$. Hence, there is no term of order
$s^{0}t^{0}$. The term in $st$ follows from the definition of $H_{pq}$.

If $s=0$ then

$\displaystyle\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}-\Pi(\gamma^{m_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}$
$\displaystyle=\Pi(\gamma^{\theta}_{q})^{t}_{0}-\Pi(\gamma^{m}_{q})^{t}_{0}$
$\displaystyle=0.$

Hence, the term in $t$ times $\mbox{ O}(s^{0})$. A similar argument holds for
the absence of a term proportional to $s$ times $\mbox{ O}(t^{0})$.

$\square$

The expansion (30) can be used to calculate the transport along the closed
loop

$(\gamma^{\theta}_{p})^{0}_{s}\,\circ\,(\gamma_{q}^{\theta+sg_{p}})^{0}_{t}\,\circ\,(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\circ\,(\gamma^{\theta}_{q})^{t}_{0}.$

One finds

$\Pi(\gamma^{\theta}_{p})^{0}_{s}\,\Pi(\gamma_{q}^{\theta+sg_{p}})^{0}_{t}\,\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}={\mathbb{I}}+stH_{pq}(\theta)+\mbox{
o}(s)\,+\mbox{ o}(t).$ (31)

Hence, the holonomy operators $H_{pq}$ quantify the holonomy of the quantum
connection $\hat{\Pi}$.

##  11 The dual field operators

The quantum vector potential of the dual quantum connection
$\hat{\Pi}^{\star}$ is denoted $A^{\star}_{p}$. The definition (15) implies

$A^{\star}_{p}(\theta)=i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Pi^{\star}(\gamma_{p}^{\theta})^{t}_{0}\bigg{|}_{t=0}=-i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}\Pi^{\star}(\gamma_{p}^{\theta})^{0}_{t}\bigg{|}_{t=0}.$
(32)

###### Proposition 11.1

Assume that the metric operators $T_{\theta}$ satisfy the conditions of
Proposition 4.3. Then the quantum vector potential $A_{p}$ and its dual
$A^{\star}_{p}$ satisfy the following relation

$\displaystyle
T_{\theta}\,A^{\star}_{p}(\theta)T^{-1}_{\theta}-T^{-1}_{\theta}A^{\dagger}_{p}(\theta)\,T_{\theta}=-i\hbar\,T^{-1}_{\theta}\left(\frac{\partial\,}{\partial\theta^{p}}T^{2}_{\theta}\right)\,T^{-1}_{\theta}.$
(33)

Proof

Expression (7) for $\gamma$ equal to $\gamma^{\theta}_{p}$ with

$\gamma^{\theta}_{p}(t)=\theta+tg_{p}$

implies

$T^{2}_{t}\,\Pi^{\star}(\gamma^{\theta}_{p})^{t}_{0}=[\Pi(\gamma^{\theta}_{p})^{0}_{t}]^{\dagger}\,T^{2}_{0}(\theta).$

Take a derivative w.r.t. $t$ and put $t=0$. This gives

$\displaystyle\frac{\partial}{\partial\theta^{p}}T^{2}_{\theta}-\frac{i}{\hbar}T^{2}_{\theta}\,A^{\star}_{p}(\theta)$
$\displaystyle=-\frac{i}{\hbar}A^{\dagger}_{p}(\theta)\,T^{2}_{\theta}.$

This can be written as (33).

$\square$

###### Proposition 11.2

If the connection $\hat{\Pi}$ is unitary then the holonomy operators $H_{pq}$
are related to their duals $H^{\star}_{pq}$ by

$T_{\theta}\,H^{\star}_{pq}\,T^{-1}_{\theta}=T^{-1}_{\theta}\,H_{pq}\,T_{\theta}.$
(34)

In particular, $H$ vanishes if and only if $H^{\star}$ vanishes.

Proof

Use (30) and (7) to obtain

$\displaystyle H^{\star}_{pq}(\theta)$
$\displaystyle=\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{[}\Pi^{\star}(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi^{\star}(\gamma^{\theta}_{q})^{t}_{0}-\Pi^{\star}(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi^{\star}(\gamma^{\theta}_{p})^{s}_{0}\bigg{]}\bigg{|}_{s=t=0}$
$\displaystyle=\frac{{\rm\,d}\,}{{\rm\,d}s}\frac{{\rm\,d}\,}{{\rm\,d}t}\bigg{[}T^{-2}_{\theta+tg_{q}+sg_{p}}$
$\displaystyle\times\bigg{(}\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}\,-\Pi(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}\bigg{)}\,T^{2}_{\theta}\bigg{]}\bigg{|}_{s=t=0}$
$\displaystyle=T^{-2}_{\theta}\,H_{pq}(\theta)\,T^{2}_{\theta}$

because if $s=0$ or $t=0$ then

$\bigg{(}\Pi(\gamma^{\theta+tg_{q}}_{p})^{s}_{0}\,\Pi(\gamma^{\theta}_{q})^{t}_{0}\,-\Pi(\gamma^{\theta+sg_{p}}_{q})^{t}_{0}\,\Pi(\gamma^{\theta}_{p})^{s}_{0}\bigg{)}$

vanishes.

$\square$

##  12 Autoparallel operator fields

Consider now a path $t\mapsto\gamma_{t}$ in parameter space ${\mathbb{M}}$ and
an operator field $X(\theta)$. The induced path $t\mapsto X(\gamma_{t})$
satisfies

$\nabla_{\dot{\gamma}}\,X(\gamma_{t})=\frac{{\rm\,d}\,}{{\rm\,d}t}X(\gamma_{t})+\frac{i}{\hbar}\dot{\gamma}^{p}\,\bigg{[}A_{p}(\gamma_{t}),X(\gamma_{t})\bigg{]}_{\hskip-1.5pt\mbox{\tiny--}}.$
(35)

One can say that the operator field $X$ is autoparallel w.r.t. the path
$\gamma$ if the covariant derivative $\nabla_{\dot{\gamma}}\,X(\gamma_{t})$
vanishes. The physical meaning of this property is that the quantum
measurement of an operator autoparallel w.r.t. the path $\gamma$ reveals an
intrinsic property of a test particle following the path.

More general is the following definition.

###### Definition 12.1

The operator field $X(\theta)$ defined for $\theta$ on the path
$t\mapsto\gamma_{t}\in{\mathbb{M}}$ is autoparallel w.r.t. the path $\gamma$
for the quantum connection $\hat{\Pi}$ if it satisfies

$\Pi(\gamma)^{t}_{s}\,X(\gamma_{s})=X(\gamma_{t})\,\Pi(\gamma)^{t}_{s}$ (36)

for all $s,t$ in the domain of definition of the path.

If $X(\theta)$ is defined all over ${\mathbb{M}}$ then one can take the
derivative of (36) w.r.t. $t$ and use (35) to see that the covariant
derivative $\nabla_{\dot{\gamma}}\,X(\gamma_{t})$ of an autoparallel operator
field $X$ vanishes.

The set of fields $X(\theta)$ autoparallel w.r.t. the given path $\gamma$
forms an algebra. It contains the identity, which is the constant field
$X(\theta)={\mathbb{I}}$. If the connection $\hat{\Pi}$ is unitary then the
adjoint $X^{\dagger}(\theta)$ of any element of the algebra does also belong
to the algebra. In that case it is an involutive algebra.

If the connection $\hat{\Pi}$ is of the product form with transport operators
given by (13) then the operator field $X(\theta)$ belongs to the algebra of
autoparallel fields if and only if the operator

$V^{-1}(\gamma_{t})\,X(\gamma_{t})\,V(\gamma_{t})$

is constant along the path. In particular, given a connection $\hat{\Pi}$ of
the product form (13) then any operator $A$ on $\mathscr{H}$ defines an
autoparallel operator field $X(\theta)$ by

$X(\gamma_{t})=V(\gamma_{t})\,A\,V^{-1}(\gamma_{t}).$

The proof of the following result is straightforward.

###### Proposition 12.2

If the operator field $X(\theta)$ is autoparallel w.r.t. the path $\gamma$ for
the unitary connection $\hat{\Pi}_{1}$ then

$T^{-(1-\alpha)}\,X\,T^{1-\alpha}$

is autoparallel w.r.t. the path $\gamma$ for the connection
$\hat{\Pi}_{\alpha}$.

##  13 Geodesics

The quantum connection $\hat{\Pi}$ induces a geometry on the parameter space
${\mathbb{M}}$.

###### Definition 13.1

A path $\gamma$ in ${\mathbb{M}}$ is a geodesic for the quantum connection
$\hat{\Pi}$ if the operator field $\dot{\gamma}_{t}^{p}\,A_{p}$ is
autoparallel w.r.t. $\gamma$.

The well-known geodesic equation of Riemannian geometry reads

$\ddot{\gamma}^{s}+\Gamma^{s}_{pq}\dot{\gamma}^{p}\dot{\gamma}^{q}=0.$ (37)

###### Proposition 13.2

If the path $\gamma$ is a geodesic for the quantum connection $\hat{\Pi}$
according to Definition 13.1 then it satisfies (37) with connection
coefficients $\Gamma^{s}_{pq}$ given by (20).

Proof

From Definition 12.1 it follows that if $\gamma$ is a geodesic then one has

$\Pi(\gamma)^{t}_{s}\,\dot{\gamma}^{p}(s)\,A_{p}(\gamma_{s})=\dot{\gamma}^{p}(t)\,A_{p}(\gamma_{t})\,\Pi(\gamma)^{t}_{s}.$
(38)

Take a derivative of (38) w.r.t. $s$ and put $t=s$. Then one obtains

lhs
$\displaystyle=\frac{{\rm\,d}\,}{{\rm\,d}s}\bigg{[}\Pi(\gamma)^{t}_{s}\,\dot{\gamma}^{p}(s)\,A_{p}(\gamma_{s})\bigg{]}_{s=t}$
$\displaystyle=\frac{i}{\hbar}\dot{\gamma}^{q}A_{q}\,\dot{\gamma}^{p}\,A_{p}+\ddot{\gamma}^{p}\,A_{p}+\dot{\gamma}^{p}\,\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\gamma_{t})$

and

rhs
$\displaystyle=\frac{{\rm\,d}\,}{{\rm\,d}s}\bigg{[}\dot{\gamma}^{p}(t)\,A_{p}(\gamma_{t})\,\Pi(\gamma)^{t}_{s}\bigg{]}_{s=t}$
$\displaystyle=\frac{i}{\hbar}\dot{\gamma}^{p}\,A_{p}\,\dot{\gamma}^{q}A_{q}.$

Equating lhs and rhs gives

$\displaystyle\ddot{\gamma}^{p}\,A_{p}+\dot{\gamma}^{p}\,\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\gamma_{t})=0$
(39)

Write

$\displaystyle\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\gamma_{t})$
$\displaystyle=\dot{\gamma}^{q}\left[\frac{\partial
A_{p}}{\partial\theta^{q}}\right]_{m=\gamma_{t}}$
$\displaystyle=H_{pq}+\left[\nabla_{p}A_{q}\right]_{\gamma_{t}}$

and use the expansion (22) to obtain

$\frac{{\rm\,d}\,}{{\rm\,d}t}A_{p}(\gamma_{t})=H_{pq}+\Gamma^{r}_{pq}A_{r}+\cdots,$

where the missing term is orthogonal to the operators $A_{r}$. An inner
product of (39) with $A_{s}$ yields

$\ddot{\gamma}^{p}g_{ps}+\Gamma^{r}_{pq}\dot{\gamma}^{p}\dot{\gamma}^{q}g_{rs}=0$

where use is made of the anti-symmetry of $H_{pq}$. The result can be written
as (37).

$\square$

###### Proposition 13.3

If $\gamma$ is a geodesic for a self-dual quantum connection $\hat{\Pi}$ then
the quantity $g_{pq}\dot{\gamma}^{p}_{t}\dot{\gamma}^{q}_{t}$ does not depend
on $t$.

Proof

One has using self-duality followed by (38)

$\displaystyle g_{pq}(\gamma_{s})\,\dot{\gamma}^{p}_{s}\dot{\gamma}^{q}_{s}$
$\displaystyle=\bigg{(}\dot{\gamma}^{p}_{s}A_{p}(\gamma_{s}),\dot{\gamma}^{q}_{s}A_{q}(\gamma_{s})\bigg{)}_{\gamma_{s}}$
$\displaystyle=\bigg{(}\Pi(\gamma)^{t}_{s}\,\dot{\gamma}^{p}_{s}A_{p}(\gamma_{s})\,\Pi(\gamma)^{s}_{t},\Pi(\gamma)^{t}_{s}\,\dot{\gamma}^{q}_{s}A_{q}(\gamma_{s})\,\Pi(\gamma)^{s}_{t}\bigg{)}_{\gamma_{s}}$
$\displaystyle=\bigg{(}\dot{\gamma}^{p}_{s}A_{p}(\gamma_{s}),\dot{\gamma}^{q}_{s}A_{q}(\gamma_{s})\bigg{)}_{\gamma_{t}}$
$\displaystyle=\bigg{(}\dot{\gamma}^{p}_{t}A_{p}(\gamma_{t}),\dot{\gamma}^{q}_{t}A_{q}(\gamma_{t})\bigg{)}_{\gamma_{t}}$
$\displaystyle=g_{pq}(\gamma_{t})\,\dot{\gamma}^{p}_{s}\dot{\gamma}^{q}_{s}.$

$\square$

This proposition shows that the connection coefficients $\Gamma^{s}_{pq}$ of a
self-dual quantum connection $\hat{\Pi}$ are those of the Levi-Civita
connection for the metric tensor $g$.

##  14 Discussion

### The metric

The inner product (1) used in the Kubo-Mori formalism is that of Bogoliubov
[13]. It describes the statistical correlations between pairs of operators as
observed in equilibrium systems of Quantum Statistical Mechanics. There is an
extensive Literature about such systems and the inner product of Bogoliubov is
an important ingredient of it. See for instance [15].

In applications of the present formalism to quantum field theories other inner
products may be appropriate. In the present paper no assumption is made about
the existence of quantum states. The manifold ${\mathbb{M}}$ is treated as a
space of parameters which label the state of the system whatever this state
may be.

### Coordinate independence

A change of coordinates in the space of parameters
${\mathbb{M}}\subset{\mathbb{R}}^{n}$ results in a change of many of the
quantities studied in the present work. The parallel transport operators
$\Pi(\gamma)^{t}_{s}$ depend on the chosen path $\gamma$ in ${\mathbb{M}}$ but
not on the choice of basis in ${\mathbb{R}}^{n}$. However, the element $A_{p}$
of the quantum vector potential is a directional derivative in the direction
$p$ and hence depends on the choice of basis. It is assumed that the quantum
vector potential transforms as a vector and that the anti-symmetric force
tensor $F_{pq}$ and the holonomy operators $H_{pq}$ transform as an operator-
valued tensor.

### Tangent operators

The definition of tangent operators is kept intentionally vague. In a more
concrete context one can define paths in Hilbert space and in operator space
in a more explicit way. See for instance the exponential arcs of [24, 25]. The
tangent operators are then directional derivatives along these exponential
arcs.

### Quantum connections

The assumptions A1 to A4 made in Section 3 about the properties that a quantum
connection should posses are open for debate. The assumption A3 requires that
tangent vectors are transported onto tangent vectors by the parallel transport
operators. The assumption is non-trivial as is demonstrated by the proof of
Proposition 4.3.

Assumption A4 is used at many places. The parallel transport
$\hat{\Pi}(\gamma)^{t}_{s}$ of an operator $X$ from $\gamma_{s}$ to
$\gamma_{t}$ along a path $\gamma$ is given by (3). However, when acting on
the wave vector $\Omega_{s}$ one has

$\displaystyle\hat{\Pi}(\gamma)^{t}_{s}X\Omega_{s}$
$\displaystyle=\Pi(\gamma)^{t}_{s}\,X\,\Pi(\gamma)^{s}_{t}\Omega_{s}$
$\displaystyle=\Pi(\gamma)^{t}_{s}\,X\,\Omega_{t}.$

Assumption A4 is essential for this step. The assumption that

$[\Pi(\gamma)^{s}_{t}]^{\dagger}\,\Omega_{s}=\Omega_{t}$

is used in Proposition 4.3 to obtain that the adjoint $\hat{\Pi}^{*}(\gamma)$
of a quantum connection $\hat{\Pi}(\gamma)$ satisfies assumption A4 as well.

### Gauge theories

Parallel transport along a geodesic in parameter space can be considered as a
gauge transformation [11]. In principle, one can choose a different Hilbert
space $\mathscr{H}$ for each value of the vector of parameters $\theta$. It is
assumed right from the start that all these Hilbert spaces can be identified
with one single copy. This identification of two Hilbert spaces, of the same
dimension or separable, is unique up to a unitary transformation. This non-
uniqueness introduces a gauge freedom in the definition of the parallel
transport operators. Assumptions A3 and A4 partly restrict this gauge freedom.

In the present terminology the connection used in [23] is unitary and of the
product form. In particular, it has a vanishing holonomy operators. This is
all right for a model of free-propagating electromagnetic waves. A more
complete description of Quantum Electrodynamics requires a less-trivial
geometry.

### Holonomy

The operators in the l.h.s. of (31) belong to what one can call the holonomy
group of the quantum connection at the point $\theta$ of the parameter space
${\mathbb{M}}$. A study of this group is out of scope of the present paper. It
requires a more detailed setup and other analytic tools.

### Operator topologies

The existence of the directional derivatives in Section 7 and following depend
on the operator topology one chooses. If the Hilbert space $\mathscr{H}$ is
finite-dimensional then one can try to replace these directional derivatives
by Fréchet derivatives using the operator norm. In the general case Gateaux
derivatives of paths in Hilbert space is the best one can hope for. Then the
vector $A_{p}(\theta)$ in the Hilbert space $\mathscr{H}_{\theta}$ is defined
by

$\displaystyle T_{\theta}\,A_{p}(\theta)\,\Omega_{\theta}$
$\displaystyle=i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}T_{\theta}\,\Pi(\gamma^{\theta}_{p})^{t}_{0}\,\Omega\bigg{|}_{t=0}$
$\displaystyle=i\hbar\frac{{\rm\,d}\,}{{\rm\,d}t}T_{\theta}\,\Omega_{\theta+tg_{p}}\bigg{|}_{t=0}.$

The convergence of such derivatives is out of scope of the present paper and
should be studied in a more explicit context such as that of [7].

### Berry phase and curvature

An obvious definition of an operator-valued curvature tensor $R_{pq}$ is

$R_{pq}=\nabla_{p}\nabla_{q}-\nabla_{q}\nabla_{p}$

These operators are not considered in the present work because they require an
additional level of directional derivatives.

The phenomenon on non-holonomy of the electromagnetic field is known in Solid
State Physics as the Berry phase[9, 18]. It is the phase factor acquired by an
eigenvector of the Hamiltonian when transported along a closed path. This is
covered by Proposition 10.1 and its consequence (31).

### Geodesics

The Definition 13.1 of a geodesic of a quantum connection is rather
restrictive because auto-parallelism of vector fields (Definition 12.1)
demands equality of operators. This raises the question whether quantum
geodesics exist for arbitrary quantum connections.

##  15 Conclusion

The present work starts from the paradigm that in Quantum Information Geometry
the tangent vectors of a Riemannian manifold are replaced by operator fields
over a parameter space ${\mathbb{M}}$. An inner product for pairs of tangent
operators is introduced and quantum connections are defined by parallel
transport operators. The inner product is used to introduce dual connections.
If the parallel transport operators are unitary then an alpha-family of
connections can be defined. It has the expected property that the minus alpha
connection is the dual of the alpha connection. In particular, the $\alpha=0$
connection is self-dual. This is the first result of the paper.

The directional derivatives of parallel transport along a path in parameter
space define the quantum vector potential $A_{p}$. It generalizes the vector
potential of Quantum Electrodynamics. It is used to introduce covariant
derivatives and the anti-symmetric force tensor $F_{pq}$. In a natural way
operators $H_{pq}$ show up which add up to the forces $F_{pq}$. They quantify
the holonomy when transporting the quantum vector potential along a closed
path in parameter space.

The second result of the paper is that if the operators $H_{pq}$ vanish for a
quantum connection then they vanish also for the dual connection. This result
mimicks one of the corner stones of Information Geometry, which is that a
connection is flat if and only if its dual is flat. The analogue of the
m-connection of Information Geometry is the unitary quantum connection. If it
has vanishing holonomy operators then its dual has vanishing holonomy
operators as well and is the analogue of the e-connection, which characterizes
the exponential families of statistical models. The vanishing of the holonomy
operators is important because it implies that the two well-known expressions
for the anti-symmetric force tensor coincide.

The final part of the paper considers the geometry of the parameter space
${\mathbb{M}}$ induced by the quantum geometry. It is shown that if the
quantum connection is self-dual then the induced connection is that of Levi-
Civita.

An application of the present work to Quantum Field Theory in curved spaces is
forthcoming.

##

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# On the nontrivial extremal eigenvalues of graphs

Wenbo Li<EMAIL_ADDRESS>Shiping Liu<EMAIL_ADDRESS>
###### Abstract

We present a finer quantitative version of an observation due to Breuillard,
Green, Guralnick and Tao which tells that for finite non-bipartite Cayley
graphs, once the nontrivial eigenvalues of their normalized adjacency matrices
are uniformly bounded away from $1$, then they are also uniformly bounded away
from $-1$. Unlike previous works which depend heavily on combinatorial
arguments, we rely more on analysis of eigenfunctions. We establish a new
explicit lower bound for the gap between $-1$ and the smallest normalized
adjacency eigenvalue, which improves previous lower bounds in terms of edge-
expansion, and is comparable to the best known lower bound in terms of vertex-
expansion.

##  1 Introduction

One of the main topics of spectral graph theory is to explore the relationship
between structural properties of a graph and eigenvalues of associated
matrices. Let $G=(V,E)$ be a finite graph with $n$ vertices. Denote by

$\mu_{n}\leq\mu_{n-1}\leq...\leq\mu_{2}\leq\mu_{1}$

the eigenvalues of its normalized adjacency matrix. We call an eigenvalue
trivial if it equals $1$ or $-1$. Recall that $\mu_{1}$ is always equal to $1$
and $\mu_{n}=-1$ if and only if the graph has a bipartite connected component.
The well-known Cheeger inequality [2, 1, 11] $h^{2}/2\leq 1-\mu_{2}\leq 2h$
relates the spectral gap $1-\mu_{2}$ and the edge-expansion (also called
Cheeger constant) $h$. In this article we show that the following inequality
involving $\mu_{2}$, $\mu_{n-1}$ and $h$ holds.

######  Theorem 1.

Let $G=(V,E)$ be a finite connected graph. Then we have

$1+\mu_{n-1}\geq\frac{(1-\mu_{2})^{2}}{2h^{2}}\left(\sqrt{1+\frac{h^{2}}{1-\mu_{2}}}-1\right)^{2}.$
(1)

As an application, we show the following estimates for the spectral gap
$1+\mu_{n}$ of a non-bipartite vertex-transitive graph.

######  Corollary 1.

Let $G$ be a finite, non-bipartite, vertex-transitive graph. There hold

$1+\mu_{n}\geq{\rm
min}\left\\{\frac{2}{d},\frac{(\sqrt{3}-1)^{2}}{8}h^{2}\right\\},$ (2)

and

$1+\mu_{n}\geq{\rm
min}\left\\{\frac{2}{d},2\left(\sqrt{1+\frac{1-\mu_{2}}{4}}-1\right)^{2}\right\\}.$
(3)

This provides a finer version of an observation due to Breuillard-Green-
Guralnick-Tao [8, Proposition E.1], which tells that for finite non-bipartite
Cayley graphs, combinatorial expansion implies spectral expansion. More
precisely, they show that for such a connected graph, there exits $\delta>0$
depending only on its degree $d$ and vertex-expansion $h_{out}$, such that the
following holds

$1+\mu_{n}\geq\delta(d,h_{out}).$ (4)

Here the vertex-expansion $h_{out}$ is closely related to the edge-expansion
$h$. Indeed, for $d$-regular graphs there holds $h_{out}/d\leq h\leq h_{out}.$

By Cheeger inequality, their observation simply tells that, for finite non-
bipartite Cayley graphs, once the gap $1-\mu_{2}$ is uniformly bounded away
from $0$, so does the gap $1+\mu_{n}$.

It is natural to seek for an explicit formula for the lower bound of
$1+\mu_{n}$ in terms of $d$, $h_{out}$ or other closely related constants like
the edge-expansion $h$ for Cayley, or more generally, vertex-transitive
graphs. Various works have been done on this topic, see, for example, [4, 5,
6, 14, 15, 12]. A recent result of Saha [15] states that for non-bipartite
vertex-transitive graphs there holds

$1+\mu_{n}\geq C\frac{h^{2}}{d^{2}}.$ (5)

We use $C$ to denote an absolute constant which may change from line to line.
Indeed, we have by the dual Cheeger inequality [16, 3]
$1+\mu_{n}\geq\beta^{2}/2$, where $\beta$ is the bipartiteness constant of the
graph $G$. Saha [15] proves for non-bipartite vertex-transitive graphs that
$\beta\geq Ch/d$, solving an open question of Moorman-Ralli-Tetali [14].
Extending the work of Bobkov-Houdré-Tetali [7] on vertex-expansion to the
setting of signed graphs, Hu-Liu [12] establish that for non-bipartite vertex-
transitive graphs

$1+\mu_{n}\geq C\frac{h_{out}^{2}}{d}.$ (6)

Recalling $h_{out}\geq h$, the estimate $(\ref{21})$ improves (5). Notice that
our estimates (2) and (3) cannot be derived from (6) and vice versa.

We achieve our estimates via a very different strategy from previous works [4,
5, 6, 14, 15, 12]. We rely more on spectral graph theoretic methods than
combinatorial arguments. We are motivated by the Remark in [8, Appendix E] to
consider the multiplicity of the eigenvalue $\mu_{n}$ and the product of two
eigenfunctions, and find [10] inspiring in obtaining Lemma 3.

Our strategy can be described as follows. By Theorem 1, all eigenvalues of a
finite connected graph with multiplicity greater than $1$ is bounded away from
$-1$. On the other hand, for non-bipartite vertex-transitive graphs there is a
natural gap of at least $2/d$ between any simple eigenvalue and $-1$, due to
the symmetry of the graph. Combining the above two cases leads to the bounds
in (2) and (3).

##  2 Preliminaries

Let $G=(V,E)$ be a finite graph. Recall that the degree matrix $D$ and
adjacency matrix $A$ of $G$ is defined as follows:

$D_{ij}=\begin{cases}d_{i}&i=j,\\\ 0&i\neq j,\end{cases}$

where $d_{i}$ is the degree of $i\in V$ and

$A_{ij}=\begin{cases}1&\\{i,j\\}\in E,\\\ 0&\\{i,j\\}\notin E.\end{cases}$

The _normalized adjacency matrix_ of a graph is defined as $D^{-1}A$.

Let $\mathbb{R}^{V}$ be the set of real-valued functions on $V$. For any
$f,g\in\mathbb{R}^{V}$, we define their inner product as

$\langle f,g\rangle:=\sum_{u\in V}f(u)g(u)d_{u}.$

The corresponding $\ell^{2}$-norm of a function $f$ is defined as
$\|f\|_{2}:=\langle f,f\rangle^{\frac{1}{2}}$. The following two identities
are straightforward.

######  Lemma 1.

Let $G=(V,E)$ be a finite graph and $f$ be a function on $V$. We have

$\sum_{\\{u,v\\}\in E}(f(u)-f(v))^{2}=\ \langle f,(I-D^{-1}A)f\rangle$ (7)

and

$\sum_{\\{u,v\\}\in E}(f(u)+f(v))^{2}=\ \langle f,(I+D^{-1}A)f\rangle,$ (8)

where $I$ is the $n\times n$ identity matrix.

We list the $n:=|V|$ eigenvalues of $D^{-1}A$ counting multiplicity as

$\mu_{n}\leq\mu_{n-1}\leq...\leq\mu_{2}\leq\mu_{1}.$

In fact, $\mu_{2}=1$ if and only if $G$ is disconnected, and $\mu_{n}=-1$ if
and only if G has a bipartite connected component.

For any subset $S\subset V$, we define its volume vol$(S)$ as

${\rm vol}(S)=\sum_{i\in S}d_{i},$

and its boundary $\partial S$ as

$\partial S=\left\\{\\{i,j\\}\in E:i\in S,j\notin S\right\\}.$

The edge-expansion of $S$ is then defined to be

$h(S)=\frac{|\partial S|}{{\rm min}\left({\rm vol}(S),{\rm vol}(V-S)\right)}.$
(9)

The edge-expansion (also called Cheeger constant) of $G$, denoted by $h$, is
defined as

$h:=\mathop{\rm min}\limits_{S\subset V}h(S).$ (10)

Note that $0\leq h\leq 1$, and $h=0$ if and only if $G$ is disconnected. Let
us recall two classic results involving $h$.

######  Proposition 1 ([9, Corollary 2.10]).

Let $G=(V,E)$ be a finite graph. For any nonzero function $f$ on $V$
satisfying $\sum_{u\in V}f(u)d_{u}=0$, we have

$\frac{\sum_{\\{u,v\\}\in E}|f(u)-f(v)|}{\sum_{u\in
V}|f(u)|d_{u}}\geq\frac{1}{2}h.$ (11)

######  Theorem 2 (Cheeger inequality [2, 1, 11]).

Let $G=(V,E)$ be a finite graph. We have

$2h\geq 1-\mu_{2}\geq\frac{1}{2}h^{2}.$ (12)

##  3 Proof of Theorem 1

First, we prepare the following lemmas.

######  Lemma 2.

Let $G=(V,E)$ be a finite graph and $f,g$ be two eigenfunctions of the
eigenvalues $\mu$ and $\nu$ of the normalized adjacency matrix $D^{-1}A$,
respectively. We have the following estimate

$\sum_{\\{u,v\\}\in
E}|f(u)g(u)-f(v)g(v)|\leq\frac{\sqrt{2}}{2}\left(\sqrt{1+\mu}+\sqrt{1+\nu}\right)\|f\|_{2}\|g\|_{2}.$

###### Proof.

By Cauchy-Schwarz inequality and Lemma 1, we calculate

$\displaystyle\sum_{\\{u,v\\}\in E}2|f(u)g(u)-f(v)g(v)|$ $\displaystyle=$
$\displaystyle\sum_{\\{u,v\\}\in
E}|(f(u)-f(v))(g(u)+g(v))+(f(u)+f(v))(g(u)-g(v))|$ $\displaystyle\leq$
$\displaystyle\sum_{\\{u,v\\}\in E}|(f(u)-f(v))(g(u)+g(v))|+\sum_{\\{u,v\\}\in
E}|(f(u)+f(v))(g(u)-g(v))|$ $\displaystyle\leq$
$\displaystyle\sqrt{\sum_{\\{u,v\\}\in
E}(f(u)-f(v))^{2}}\sqrt{\sum_{\\{u,v\\}\in E}(g(u)+g(v))^{2}}$
$\displaystyle\hskip 99.58464pt+\sqrt{\sum_{\\{u,v\\}\in
E}(f(u)+f(v))^{2}}\sqrt{\sum_{\\{u,v\\}\in E}(g(u)-g(v))^{2}}$
$\displaystyle=$
$\displaystyle\left(\sqrt{(1-\mu)(1+\nu)}+\sqrt{(1+\mu)(1-\nu)}\right)\|f\|_{2}\|g\|_{2}$
$\displaystyle\leq$
$\displaystyle\sqrt{2}(\sqrt{1+\mu}+\sqrt{1+\nu})\|f\|_{2}\|g\|_{2}.$

This completes the proof. ∎

Lemma 2 tells that the $\ell^{1}$-energy of the product $fg$ of two
eigenfunctions is close to $0$, whenever the corresponding two eigenvalues are
close to $-1$.

The next lemma shows that if an eigenvalue is close to $-1$, then the absolute
value of its eigenfunction is close to being a constant.

######  Lemma 3.

Let $G=(V,E)$ be a finite connected graph. Assume that $f$ is an eigenfunction
of the eigenvalue $\mu$ of the normalized adjacency matrix $D^{-1}A$ with
$\|f\|_{2}=1$. Let $\boldsymbol{c}$ is the constant function such that
$\|c\|_{2}=1$ and $f_{1}:=|f|-\langle|f|,\boldsymbol{c}\rangle\boldsymbol{c}$.
Then we have

$\|f_{1}\|_{2}^{2}\leq\frac{1+\mu}{1-\mu_{2}}$

and

$\langle|f|,\boldsymbol{c}\rangle^{2}\geq 1-\frac{1+\mu}{1-\mu_{2}}.$

###### Proof.

Oberserving that $\langle f_{1},\boldsymbol{c}\rangle=0$, we have

$(1-\mu_{2})\langle f_{1},f_{1}\rangle\leq\langle
f_{1},(I-D^{-1}A)f_{1}\rangle.$

Therefore, we estimate by Lemma 1

$\begin{split}\|f_{1}\|_{2}^{2}&\leq\frac{1}{1-\mu_{2}}\langle
f_{1},(I-D^{-1}A)f_{1}\rangle=\frac{1}{1-\mu_{2}}\langle|f|,(I-D^{-1}A)|f|\rangle\\\
&=\frac{1}{1-\mu_{2}}\sum_{\\{u,v\\}\in E}(|f(u)|-|f(v)|)^{2}\\\
&\leq\frac{1}{1-\mu_{2}}\sum_{\\{u,v\\}\in E}(f(u)+f(v))^{2}\\\
&=\frac{1}{1-\mu_{2}}\langle f,(I+D^{-1}A)f\rangle\\\
&=\frac{1+\mu}{1-\mu_{2}}.\end{split}$

The other inequality holds since

$1=\|f\|_{2}^{2}=\langle|f|,\boldsymbol{c}\rangle^{2}+\|f_{1}\|_{2}^{2}.$

This concludes the proof. ∎

The following lemma is a useful consequence of Lemma 3. It bounds from below
the $\ell^{1}$-norm of the product of two eigenfunctions.

######  Lemma 4.

Let $G=(V,E)$ be a finite connected graph and $f,g$ be eigenfunctions of the
eigenvalues $\mu$ and $\nu$ of $D^{-1}A$, respectively, such that
$\|f\|_{2}=\|g\|_{2}=1$. Then

$\langle|f|,|g|\rangle\geq\sqrt{1-\frac{1+\mu}{1-\mu_{2}}}\sqrt{1-\frac{1+\nu}{1-\mu_{2}}}-\sqrt{\frac{1+\mu}{1-\mu_{2}}}\sqrt{\frac{1+\nu}{1-\mu_{2}}}.$
(13)

###### Proof.

Let $\boldsymbol{c}$ be a positive constant function with $\|c\|_{2}=1$.
Decompose $|f|$ and $|g|$ such that

$|f|=\langle|f|,\boldsymbol{c}\rangle\boldsymbol{c}+f_{1},\,\,\,\,\text{and}\,\,\,\,|g|=\langle|g|,\boldsymbol{c}\rangle\boldsymbol{c}+g_{1}.$

Then we compute

$\begin{split}\langle|f|,|g|\rangle&=\langle|f|,\boldsymbol{c}\rangle\langle|g|,\boldsymbol{c}\rangle+\langle
f_{1},g_{1}\rangle\\\
&\geq\langle|f|,\boldsymbol{c}\rangle\langle|g|,\boldsymbol{c}\rangle-\|f_{1}\|_{2}\|g_{1}\|_{2}\\\
&\geq\sqrt{1-\frac{1+\mu}{1-\mu_{2}}}\sqrt{1-\frac{1+\nu}{1-\mu_{2}}}-\sqrt{\frac{1+\mu}{1-\mu_{2}}}\sqrt{\frac{1+\nu}{1-\mu_{2}}}\end{split}$

where the last inequality comes from Lemma 3. ∎

We are now prepared to prove Theorem 1.

###### Proof of Theorem 1.

Let $f,g$ be eigenfunctions of the eigenvalues $\mu_{n}$ and $\mu_{n-1}$ such
that $\|f\|_{2}=\|g\|_{2}=1$ and $\langle f,g\rangle=0$, i.e., $\sum_{u\in
V}f(u)g(u)d_{u}=0$. Applying Proposition 1 to the function $fg$ and inserting
the estimates from Lemma 2 and Lemma 4, we derive the following inequality

$\frac{\sqrt{2}}{2}h\leq\frac{\sqrt{1+\mu_{n}}+\sqrt{1+\mu_{n-1}}}{\sqrt{1-\frac{1+\mu_{n}}{1-\mu_{2}}}\sqrt{1-\frac{1+\mu_{n-1}}{1-\mu_{2}}}-\sqrt{\frac{1+\mu_{n}}{1-\mu_{2}}}\sqrt{\frac{1+\mu_{n-1}}{1-\mu_{2}}}},$

when $1+\mu_{n-1}<(1-\mu_{2})/2$. Notice that the following function

$f(x,y)=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{1-\frac{1}{1-\mu_{2}}x}\sqrt{1-\frac{1}{1-\mu_{2}}y}-\sqrt{\frac{1}{1-\mu_{2}}x}\sqrt{\frac{1}{1-\mu_{2}}y}}$
(14)

is continuous on $[0,\frac{1}{2}(1-\mu_{2}))\times[0,\frac{1}{2}(1-\mu_{2}))$,
monotonically increasing with respect to either $x$ or $y$ and $f(0,0)=0$. Let
$c=c(h,1-\mu_{2})$ be the number such that

$f(c,c)=\frac{\sqrt{2}}{2}h.$

Then we have

$1+\mu_{n-1}\geq c,$

since, otherwise,

$f(1+\mu_{n},1+\mu_{n-1})<f(c,c)=\frac{\sqrt{2}}{2}h$

is a contradiction. A direct computation shows that

$\sqrt{c}=\frac{1-\mu_{2}}{\sqrt{2}h}\left(\sqrt{1+\frac{h^{2}}{1-\mu_{2}}}-1\right).$

This proves the inequality (1) when $1+\mu_{n-1}<(1-\mu_{2})/2$.

For the case that $1+\mu_{n-1}\geq(1-\mu_{2})/2$, the inequality (1) still
holds since

$1+\mu_{n-1}\geq\frac{1-\mu_{2}}{2}\geq\frac{(1-\mu_{2})^{2}}{2h^{2}}\left(\sqrt{1+\frac{h^{2}}{1-\mu_{2}}}-1\right)^{2},$

while the last inequality comes from the fact that $1-\mu_{2}\geq 0$. ∎

It is direct to check that $c=c(h,1-\mu_{2})$ is monotonically increasing with
respect to either $h$ or $1-\mu_{2}$. Replacing $1-\mu_{2}$ with powers of $h$
or doing the opposite via the Cheeger inequality (12) yields the following
result.

######  Corollary 2.

Let $G=(V,E)$ be a finite graph. Let $n$, $\mu_{2}$, $\mu_{n-1}$ and $h$ be as
in Theorem 1. Then, we have

$1+\mu_{n-1}\geq 2\left(\sqrt{1+\frac{1-\mu_{2}}{4}}-1\right)^{2},$ (15)

$1+\mu_{n-1}\geq\frac{(\sqrt{3}-1)^{2}}{8}h^{2}.$ (16)

######  Remark 1.

If we do not care about the constant, the inequality (16) can also be derived
from the higher order dual Cheeger inequalities [13, Theorem 1.2]. Indeed, we
have $1+\mu_{n-1}\geq C(1-\overline{h}(2))^{2}$, where $\overline{h}(2)$
stands for the two-way dual Cheeger constant. Then the inequality (16) follows
directly from the observation that $1-\overline{h}(2)\geq h$. The proof of
higher order dual Cheeger inequalities involves applying deep results from the
random partition theory. Our proof here is much simpler and more elementary.
We also obtain a better constant here.

##  4 On non-bipartite vertex-transitive graphs

For graphs such that $\mu_{n}=\mu_{n-1}$, Theorem 1 provides a bound for the
smallest eigenvalue of the normalized adjacency matrix. For vertex-transitive
graphs we have the following gap phenomenon.

######  Theorem 3.

Let $G$ be a finite connected vertex-transitive graph and $\mu$ a simple
eigenvalue of its normalized adjacency matrix. Then

$\mu=\frac{2k}{d}-1$ (17)

where $0\leq k\leq d$ is an integer and $d$ is the degree of $G$.

###### Proof.

Fix $a\in V$. Suppose that $\mu$ is a simple eigenvalue of $D^{-1}A$ with $f$
being its eigenfunction. Assume further that $f(a)=-1$ or $1$. For every
$g\in\mathrm{Aut}(G)$, the function $f_{g}$ defined as
$f_{g}(\cdot)=f(g(\cdot))$ is still an eigenfunction of the same eigenvalue
$\mu$. Since $\mu$ is simple, there exits $\lambda_{g}$ such that
$f_{g}=\lambda_{g}f$. Since $\|f\|_{2}=\|f_{g}\|_{2}$, we have
$\lambda_{g}\in\\{-1,1\\}$. From the fact that $G$ is vertex transitive,
$\mathrm{Aut}(G)$ acts transitively on $G$, we have for any $u\in V$ that
$f(u)\in\\{-1,1\\}$. Calculating $D^{-1}Af(a)$ yields the result. ∎

Combining Corollary 2 and the above Theorem 3, we prove Corollary 1.

For graphs with no simple eigenvalues, we can get rid of the term $2/d$ in (2)
and (3).

######  Corollary 3.

Let $(G,S)$ be a finite connected Cayley graph with a finite group $G$ and a
gernerating set $S$. Assume that $G$ is a simple group or the size $|G|$ is
odd. Let

$-1<\mu_{n}\leq\mu_{n-1}\leq...\leq\mu_{2}\leq\mu_{1}=1$

be the eigenvalues of its normalized adjacency matrix where $n=|G|$. Denote by
$h$ its edge-expansion. Then we have

$1+\mu_{n}\geq\frac{(\sqrt{3}-1)^{2}}{8}h^{2}$ (18)

and

$1+\mu_{n}\geq 2\left(\sqrt{1+\frac{1-\mu_{2}}{4}}-1\right)^{2}.$ (19)

###### Proof.

The map $\lambda_{g}:G\rightarrow\\{\pm 1\\}$ in the proof of Theorem 3 is in
fact a homomorphism of groups since for any $g_{1},g_{2}\in G$, it holds that

$\lambda_{g_{1}g_{2}}f(e)=f_{g_{1}g_{2}}(e)=f(g_{1}g_{2})=f_{g_{1}}(g_{2})=\lambda_{g_{1}}f(g_{2})=\lambda_{g_{1}}\lambda_{g_{2}}f(e).$

Here we denote by $e$ the identity element of the group $G$. If $G$ is a
simple group or the size $|G|$ is odd, then such a homomorphism must be
trivial otherwise there will be a subgroup of $G$ of index 2. As a result all
non-trivial eigenvalues of $(G,S)$ will be of multiplicity great than 1 and
the above estimates hold. ∎

## Acknowledgement

SL is very grateful to Paul Horn and Matthias Keller for very inspiring
discussions on related topics. This work is supported by the National Key R
and D Program of China 2020YFA0713100, the National Natural Science Foundation
of China (No. 12031017), and Innovation Program for Quantum Science and
Technology 2021ZD0302902.

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