UNIVERSITÄT LEIPZIG
FAKULTÄT FÜR PHYSIK UND GEOWISSENSCHAFTEN

M.Sc. Mathematical Physics

RELATIVE ENTROPY FOR
FERMIONIC QUANTUM FIELD THEORY

Stefano Galanda

Supervisor:
Dr. Albert Much

Second Assessor:
Prof. Dr. Rainer Verch

October 2022

Abstract

We study the relative entropy, in the sense of Araki, for the representation
of a self-dual CAR algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. We
notice, for a specific choice of $f\in\mathcal{H}$, that the associated
element in $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is unitary. As a
consequence, we explicitly compute the relative entropy between a quasifree
state over $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ and an excitation of it
with respect to the abovely mentioned unitary element. The generality of the
approach, allows us to consider $\mathcal{H}$ as the Hilbert space of
solutions of the classical Dirac equation over globally hyperbolic spacetimes,
making our result, a computation of relative entropy for a Fermionic Quantum
Field Theory.
Our result, extends those of Longo [1], Casini et al. [2] for the relative
entropy between a quasifree state and a coherent excitation of it for a free
Scalar Quantum Field Theory, to the case of fermions.
As a first application, we computed such a relative entropy for a Majorana
field on an ultrastatic spacetime.

Acknowledgements

I would like to thank Dr. Albert Much for supervising this thesis, especially
for his great quality of creating an amicable and professional working
enviroment and for the large amount of time he dedicated to me. While
completing my thesis, I was given the opportunity to deepen my understanding
in a variety of fascinating topics which I greatly appreaciate and which will
profoundly influence my future. I also thank Prof. Dr. Rainer Verch for his
help and the discussions we had. Indeed, his guidance during the last months
presents a considerable part behind the realization of this work.
In addition, I have benefited from discussions with Dr. Markus Fröb (on the
definition of fermionic field algebras and Tomita-Takesaki modular theory),
whom I want to thank for this reason.

With this work, I am closing an important chapter of my professional as well
as personal life. It is for this reason, that I must thank the people that
made these years special and supported me during any difficult time. Above
all, and from the bottom of my heart, my family: my parents Stefania and
Paolo, my elder brother Francesco and my grandparents Giuliana and Benito.
Your love and the awareness of having you on my side were the biggest source
of motivation.
Distinct words are required for Leonardo. Not only did we share this
experience abroad together, but we especially share the same passion for
mathematics and physics. The discussions with you are a central part of this
thesis and are, in general, among the things I enjoy most. You are one of my
best friends and these years together will remain unforgettable.
Furthermore, I want to thank Filippo for his support, especially in the most
difficult period, my lifelong friends Madiara, Enes, Alessandro, Matthias and
also the people that helped me just for part of these two years.

Finally, among the people I met here, I am especially grateful to Tim for his
motivational support and contagious enthusiasm and to Paula for her virtue of
being able to say the right words in the right moment.

I am indebted to you all.

###### Contents

  1. 0.1 Introduction
  2. 0.2 Conventions
  3. 1 Algebraic Quantum Field Theory
    1. 1.1 Preliminary definitions and results
      1. 1.1.1 Operator algebras
      2. 1.1.2 State functionals
      3. 1.1.3 GNS theorem and Fock space representation
      4. 1.1.4 KMS and ground states
    2. 1.2 Axioms of AQFT
    3. 1.3 Dirac and/or Majorana fields
      1. 1.3.1 Spin bundle
      2. 1.3.2 Dirac spinors and cospinors
      3. 1.3.3 The classical Dirac equation
      4. 1.3.4 Quantisation
  4. 2 The role of entropy and information in modern physics
    1. 2.1 Quantum information theory
      1. 2.1.1 The Von Neumann relative entropy
      2. 2.1.2 Introduction to von Neumann factors I
    2. 2.2 Relative entropy in QFT
      1. 2.2.1 Introduction to von Neumann factors II
      2. 2.2.2 The necessity of Type $III$ factors in QFT
      3. 2.2.3 Tomita-Takesaki modular theory
      4. 2.2.4 Araki’s relative entropy
      5. 2.2.5 Coherent states and relative entropy
  5. 3 Relative entropy for fermionic fields
    1. 3.1 Single Unitary Fermionic Excitation of the Vacuum
    2. 3.2 Multiple Unitary Fermion Excitation of the Vacuum
    3. 3.3 Extension to more general types of excitations
      1. 3.3.1 Standard subspaces for a fermionic QFT
      2. 3.3.2 Unitary exponential excitation
    4. 3.4 Comparison with Von Neumann relative entropy
    5. 3.5 Relative entropy for Majorana fields on ultrastatic spacetimes
      1. 3.5.1 Majorana fields in 1+1 dimensions
    6. 3.6 Conclusion
  6. A Appendix
    1. A.1 Elements of Lorentzian geometry
    2. A.2 The Reeh-Schlieder theorem
    3. A.3 The Bisognano-Wichmann theorem
    4. A.4 Clifford algebra

## Introduction and conventions

###  0.1 Introduction

The first half of the past century was marked by the search for a fundamental
theory describing nature, incorporating both the classical understanding of
empty space as well as electromagnetism. During the same period, the particle-
wave ”duality” was one of the most debated concepts. Attempting to clarify the
latter, de Broglie was the first to assume, in his work of 1926 [3], ”the
existence of a certain periodic phenomenon of a yet to be determined
character, which is to be attributed to each and every isolated energy
particle”. Looking back, we know he was referring to the existence of what we
call a Quantum field. The first to introduce the notion of such a field was
Dirac in [4], who attempted to formulate a relativistically invariant theory
describing a charged particle interacting with an electromagnetic field. Up to
this date, enormous contributions to the formulation and understanding of the
theory have led to what we nowadays call Quantum Field Theory. However,
referring to it as a theory, however, may be slightly misleading. Instead, we
should rather call it a framework, in which the ”physical theories” are
established. Quantum Electrodynamics (QED), for instance, presents one of the
most precisely tested theories ever formulated and is nothing but a a specific
type of Quantum field Theory obtained by incorporating electromagnetism in
such a framework.
However, during its developement, certain mathematical problems arose (see for
example [5]) and agreement between the predictions and experimental outcomes
was prioritized at the expense of sacrificing mathematical rigor. For this
reason, the subject has increasingly gained interest among both mathematicians
as well as mathematical physicists. An attempt to a more mathematically
rigorous formulation of QFT, in the spirit of Heisenberg matrix mechanics, was
given in the pioneering work of Haag and Kastler ([6]). In their work, they
define QFT in an axiomatic way, where the focus is on the study of the
properties of observables, that’s why is called Algebraic Quantum Field
Theory. One of the great innovations in their work, was the freedom regarding
the choice of the underlying spacetime, allowing gravitational effects to be
included in the theory. In this framework, a QFT is formulated on a spacetime
where gravity is treated classically (in the sense that the evolution of the
quantum field, does not have any backreaction on the spacetime geometry
itself), and is hence called semiclassical gravity.
The study of semiclassical gravity, using the tools of AQFT led, during the
last $50$ years, to a deeper understanding of gravity itself, in particular in
the context of Black Holes. In fact, in 1974, Hawking ([7]) showed that
despite classically, crossing the event horizon is a point of no return, Black
Holes possess a grey-body spectrum of emission of radiation in the form of
quantum fields. Together with the subsequent assignment of a temperature, this
observation made concrete, the previously existing analogy between the laws of
thermodynamics and Black Hole dynamics.
Once the area of a Black Hole horizon had been identified with its entropy,
the interest in entropy measures in this context became greater. However,
aiming for mathematical rigor, the entropy measures in QFT need to be reviewed
as, for instance, the von Neumann formula is no longer well defined in this
context. The reason for this mainly stems from the fact that when the number
of degrees of freedom of the theory become uncountably many, certain
mathematical properties of the observable algebra pertain. This led to a focus
on the study of von Neumann algebras and their classification, at least from
the point of view of a mathematical physicists. The first to solve the problem
were Araki [8] and Uhlmann [9], who introduced a reformulated notion of
relative entropy, generalizing the one by von Neumann.
However, such a result remains very abstract. In order to make it more
concretely applicable to contexts of physical relevance, Longo [1] and Casini
et al. [2] recently carried out the analysis for a particular type of unitary
excitation of the vacuum of a free scalar QFT, for which the relative entropy
is expressed in a form much more easily interpreted. Such a result was used,
for instance, in the context of semiclassical gravity by Kurpicz, Pinamonti
and Verch [10] as well as by D’Angelo [11] to investigate thermodynamical
properties of Black Holes that are not stationary (i.e. that incorporate a
dynamically changing mass). By doing so, attempts were made of extending the
thermodynamical analogy to the case of Black Holes away from equilibrium
(stationary case).

This seeks to present a generalization of the mentioned result, regarding the
computation of the relative entropy for the free scalar case, to the case of a
fermionic quantum field theory. The text is organized in three main chapters.
The first one aims at introducing the mathematical framework of AQFT and at
defining Fermi quantum fields on a curved spacetime background. The second
chapter is devoted to the discussion of entropy measures that, as we are going
to see, need to be reviewed in the context of Quantum Field Theory due to
ultraviolet divergences, arising from the type of observables algebras. In
chapter two still, we we present the results for the bosonic case, allowing
for the computation of the relative entropy for specific types of excitations.
Lastly, in chapter three, we present our work regarding the computation of the
relative entropy in the fermionic case.

###  0.2 Conventions

I here list the convention we adopt:

  * •

The four vectors are denoted in the abstract index notation. Namely $x^{a}$
denotes a vector field over the spacetime manifold and $x_{a}$ the
corresponding covector with indices lowered and raised using the Lorentzian
metric tensor $g_{ab}$ over $M$. In a specific coordinate chart over $M$ the
four vectors are denoted with greek indices, e.g. $x^{\mu}$, while its spatial
components either with latin indices or with bold font:
$x^{\mu}=(x^{0},x^{i})=(x^{0},\mathbf{x})$

  * •

We adopt the Einstein summation convention for repeated indices

  * •

The adjoint of an operator over an Hilbert space, is denoted as ∗. While, if
that operator is in particular a matrix over $\mathbb{C}^{n}$ with the
standard inner product, we will use the more familiar notation †.

  * •

We work with natural units, namely:

$\hbar=k_{B}=c=G=1$

  * •

The set of $k$-continuous functions over the spacetime $M$, are denoted by
$\mathcal{C}^{k}(M)$ while the smooth functions as $\mathcal{C}^{\infty}(M)$.
Analogously, a smooth $k$-continuous (resp. smooth) function over $M$ with
compact support are denoted $\mathcal{C}^{k}_{0}(M)$ (resp.
$\mathcal{C}^{\infty}_{0}(M)$). Finally, we denote by $\mathcal{S}(M)$ the
Schwartz space of functions.

  * •

We will denote by $[\cdot,\cdot]$ the commutator and by $[\cdot,\cdot]_{+}$
the anticommutator.

  * •

The fundamental group of a manifold $M$ is denoted by $\pi_{1}(M)$.

  * •

A spacetime is a Lorentzian Manifold $M$ with metric $g$ of signature $(+---)$

  * •

We denote the causal future/past of a closed subset $C\subset M$ by
$J^{\pm}(C)$. For a closed achronal set $S\subset M$ we denote the future/past
causal developement by $D^{\pm}(S)$. Finally the set of all causally complete
regions on $M$ is denoted as $\mathcal{K}$.

A brief review of the fundamentals of Lorentzian geometry is reported in
Appendix A.1, where we explain better the concepts mentioned in the last
point.

##  Chapter 1 Algebraic Quantum Field Theory

Quantum Field Theory (QFT) is the theory that combines Quantum Mechanics (QM)
and Special Relativity, namely it aims to incorporate the relativity principle
into the theory describing the microscopic scale. This implies that the
fundamental constituents of nature (the Quantum fields) as measured by
different inertial observers, must transform according to the transformation
connecting the two inertial observers. That is, there must exist a unitary
representation of the Lorentz group, encoding the transformation properties of
the quantum fields. One of the biggest achievements of the theory is its
capability to give meaning to the concept of Spin: a consequence of the
symmetry group of the spacetime that we encounter, once we ask, what the
elementary properties are, that classify the fields and that further both
observers agree on.
Algebraic Quantum Field Theory (AQFT) is an approach to QFT that differs from
the standard one but that incorporates it as a particular case, such that we
may see it as a generalization. The term Algebraic refers to the way in which
the Quantum theory is treated, in this sense we are formulating an Algebraic
approach to Quantum Mechanics that allows us to incorporate special relativity
with minimal effort.
The mathematical description of the standard approach to QM, starts with a
Hilbert space $\mathcal{H}$ and the set of bounded linear operators defined
over it $\mathfrak{B}(\mathcal{H})$. The subset
$\mathfrak{B}_{sa}(\mathcal{H})\subset\mathfrak{B}(\mathcal{H})$ of self
adjoint operators is the set of all operators representing quantities that can
be measured (as their point spectrum is real) on the system and for that
reason is called the set of observables. Finally a state is described by a so
called density matrix, i.e. an element of:

$\mathfrak{D}(\mathcal{H}):=\\{\rho\in\mathfrak{B}_{sa}(\mathcal{H}):\rho\geq
0\,\,\mathrm{and}\,\,\Tr(\rho)=1\\}$

Any density matrix gives rise to a state functional over the set of bounded
operators defined as: $\omega_{\rho}:\mathfrak{B}(\mathcal{H})\to\mathbb{C}$
via the relation:

$\omega_{\rho}(A):=\Tr(A\rho)\hskip 15.0pt\forall
A\in\mathfrak{B}(\mathcal{H})$

######  Remark 1.0.1.

In only considering bounded operators as observables, in this first brief
discussion about the motivation behind AQFT, we are not giving up generality.
In fact, from the spectral theorem, it follows that we can decompose any self
adjoint operator $A$ as:

$A=\int_{\mathbb{R}}\lambda dE_{A}(\lambda)$

where $E_{A}(\Delta)$ are the spectral measures corresponding to the
measurable set $\Delta\in\mathbb{R}$ and in particular
$E_{A}(\Delta)\in\mathfrak{B}(\mathcal{H})$ whenever $\Delta\in\mathbb{R}$ is
bounded. Now, whenever we want to measure some quantity associated with a
possibly unbounded operator $A$, we employ a measurement device which always
has limited capacities, i.e. there are a highest and lowest values that is
capable of registering. This means, that what we measure is not the unbounded
operator $A$, but rather:

$\tilde{A}=\int_{a_{min}}^{a_{max}}\lambda dE_{A}(\lambda)$

which is bounded.
Therefore, unbounded operators, that we will consider and study in what
follows, are in fact just ”idealized observables” and for the purpose of
outlining the standard approach to QM, considering only bounded observables is
perfectly fine.

Given the ideas of the standard approach, two questions may arise:

  * 1)

How do we know that we are considering and/or able to measure all possible
elements in $\mathfrak{B}_{sa}(\mathcal{H})$?

  * 2)

How does our results depend on the choice of the initial Hilbert space and can
we somehow relate the same physically measurable quantity in terms of bounded
operators on two different such Hilbert spaces?

The answer to the first question is that in general we do not have access to
the all set of observables. Therefore we need to investigate the consequences
of knowing only a restrict subset of them. A concrete example in which we have
access to just a subset of the observables, arises when introducing causality,
for instance in QFT. As a consequence, we will be able to measure just what
lies in our causal future, any other observable of the general theory defined
over the entire spacetime is practically unaccessible.
For what concerns the second question, as long as we deal with finitely
generated algebras, the Stone- von Neumann theorem ensures that there is a
unitary transformation mapping the representation in one Hilbert space into
that of another Hilbert space whenever the representations are irreducible.
However, the same theorem tells us that when the algebras have infinitely many
generators, there are unitarily inequivalent representations and thus the
choice of the initial Hilbert space might lead to unwanted consequences, like
preventing us from describing any physically relevant state of the system as a
vector in that single Hilbert space. It is worth noting that, as we are
interested in QFT, the algebras in this context, as we will see, have
infinitely many generators: the solutions of the field equations.
AQFT solves both these two issues as it takes as primary object the algebra of
”accessible” observables and as states the positive, normal functionals
defined over it. In this approach, the notion of Hilbert space is derived via
the so called GNS construction (Gelfand-Naimark-Segal) once a state is chosen.
In this way, one is able to treat all states equally: just as functionals.

The chapter starts off with a section that aims at inroducing all the
necessary mathematical theorems and definitions required for the formulation
of AQFT. We will continue on in the second section with an axiomatic
definition of AQFT. Finally, we will discuss an example of an AQFT presenting
the Dirac field.

###  1.1 Preliminary definitions and results

As the algebraic approach starts off by specifying as primary data the
Algebras of observables (i.e. those sets of opertors that are relevant for the
study of the physical system), the first thing we will introduce is the notion
of algebras of operators. Then, in order to specify a configuration of a
system, we will introduce the notion of functionals over the Algebra and
correspondingly the notion of a state. Finally, combining these two we will
derive the Hilbert space and thus the standard formulation of QFT.

####  1.1.1 Operator algebras

The first core concept is that of an algebra, which is nothing but a vector
space with an additional structure:

######  Definition 1.1.1.

An algebra $\mathfrak{A}$ is a vector space $V$ on a field $\mathbb{K}$ (if
$\mathbb{K}=\mathbb{R},\mathbb{C}$ we speak of a real resp. complex algebra)
on which a product operation is defined $\circ:V\times V\to V$ which is
associative:

$A\circ(B\circ C)=(A\circ B)\circ C\hskip 15.0pt\forall A,B,C\in\mathfrak{A}$

and fulfills distributivity over the vector space structures.

######  Example 1.1.1.

The usual example is that of $\mathbb{R}^{3}$ with respect to the vector
product $\times:\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$, defined
for any $\mathbf{v},\mathbf{w}\in\mathbb{R}^{3}$ as:

$\mathbf{v}\times\mathbf{w}=\det\begin{pmatrix}\hat{x}&\hat{y}&\hat{z}\\\
v_{x}&v_{y}&v_{z}\\\ w_{x}&w_{y}&w_{z}\end{pmatrix}$

Where we have denoted with $x,y,z$ the cartesian directions and
$\hat{x},\hat{y},\hat{z}$ the corresponding versors.

######  Example 1.1.2.

The set of bounded linear operators $\mathfrak{B}(\mathcal{H})$ over an
Hilbert space $\mathcal{H}$ is an algebra with respect to composition. The
simplest example holds when $\dim\mathcal{H}=n<\infty$, then the bounded
operators $\mathfrak{B}(\mathcal{H})=\mathrm{Mat}(n,\mathbb{C})$ and the
composition simply becomes the matrix multiplication.

Algebras that contain the neutral element of the multiplication operation, are
called unital algebras. Namely, by calling $\mathfrak{A}$ the unital algebra,
there exist an element $\mathbb{1}\in\mathfrak{A}$ such that:

$\mathbb{1}\cdot A=A\cdot\mathbb{1}=A$

for any $A\in\mathfrak{A}$. As far as we are concerned, all algebras that we
are going to consider in the following will be unital.
It is possible to introduce further structures on an algebra, one of which is
the $*$-operation:

######  Definition 1.1.2.

Let $\mathfrak{A}$ be a complex algebra. An antilinear map:

$\displaystyle*:\mathfrak{A}$ $\displaystyle\to\mathfrak{A}$ $\displaystyle A$
$\displaystyle\mapsto A^{*}$

is called an involution if $A=(A^{*})^{*}$ and $(AB)^{*}=B^{*}A^{*}$ for all
$A,B\in\mathfrak{A}$. An algebra with an involution is called a $*$-algebra.

We can also introduce a topology over an algebra and, taking advantage of the
vector space structure, this can be done by introducing a norm:

######  Definition 1.1.3.

Let $\mathfrak{A}$ be a complex algebra. We can define a norm:

$\displaystyle\|\cdot\|:\mathfrak{A}$ $\displaystyle\to\mathbb{R}^{+}$
$\displaystyle A$ $\displaystyle\mapsto\|A\|$

With the usual properties:

  1. 1.

$\|A\|=0$ if and only if $A=0$

  2. 2.

$\|kA\|=|k|\,\|A\|$

  3. 3.

$\|A+B\|\leq\|A\|+\|B\|$

  4. 4.

$\|AB\|\leq\|A\|\,\|B\|$

for any $A,B\in\mathfrak{A}$ and all $k\in\mathbb{C}$. An algebra with a norm
is called a normed algebra.

######  Definition 1.1.4.

A normed algebra $(\mathfrak{A},\|\cdot\|)$ that is complete in the topology
induced by $\|\cdot\|$, is called a Banach algebra

The next presents the type of algebra, we will be interested in the most:

######  Definition 1.1.5.

Let $(\mathfrak{A},\|\cdot\|)$ be a $*$-Banach algebra.
$(\mathfrak{A},\|\cdot\|)$ is called a $C^{*}$-algebra if the norm satisfies:

$\|AA^{*}\|=\|A\|^{2}$

######  Example 1.1.3.

Consider the algebra of bounded operators over a Hilbert space
$\mathfrak{B}(\mathcal{H})$. We can make it a $C^{*}$-algebra by considering
the operator norm and as involution the adjoint with respect to the inner
product on $\mathcal{H}$. The corresponding induced topology will be called
operator topology or uniform topology. The operator norm is defined as
follows:

$\|A\|_{op}:=\sup_{v\in\mathcal{H}}\frac{\|Av\|}{\|v\|}$

for all $A\in\mathfrak{B}(\mathcal{H})$, and the norms on the right hand side
are those induced by the scalar product on $\mathcal{H}$. We can always write
a vector $v\in\mathcal{H}$ as:

$v=\|v\|\frac{v}{\|v\|}$

And clearly $\hat{v}:=v/\|v\|$ is a normalized vector. For this reason we can
rewrite:

$\|A\|_{op}=\sup_{v\in\mathcal{H}\,,\,\|v\|=1}\|Av\|$

The operator norm fulfills all the properties we listed before such that
$(\mathfrak{B}(\mathcal{H}),\|\cdot\|_{op})$ is a normed algebra:

  1. 1.

$\|A\|_{op}=0\Leftrightarrow Av=0\,\,\,\,\forall v\in\mathcal{H}$ but this
implies that $A=0$

  2. 2.

It easily follows from the properties of the norm over the Hilbert space:

$\displaystyle\|kA\|_{op}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\|kAv\|$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}|k|\,\|Av\|$
$\displaystyle=|k|\,\|A\|_{op}$

  3. 3.

Again, one has from the Hilbert space norm:

$\displaystyle\|A+B\|_{op}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\|(A+B)v\|$
$\displaystyle\leq\sup_{v\in\mathcal{H},\|v\|=1}(\|Av\|+\|Bv\|)$
$\displaystyle=\|A\|_{op}+\|B\|_{op}$

  4. 4.

To prove the last property, notice that from the definition one has:

$\|Av\|\leq\|A\|_{op}\|v\|$

Then, if we look at:

$\displaystyle\|AB\|_{op}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\|ABv\|$
$\displaystyle\leq\sup_{v\in\mathcal{H},\|v\|=1}\|A\|_{op}\|Bv\|$
$\displaystyle=\|A\|_{op}\sup_{v\in\mathcal{H},\|v\|=1}\|Bv\|=\|A\|_{op}\|B\|_{op}$

$(\mathfrak{B}(\mathcal{H}),\|\cdot\|_{op})$ is also a Banach algebra as can
be derived from the general result:

######  Proposition 1.1.1.

Let $\mathcal{X},\mathcal{Y}$ be normed vector spaces. If $\mathcal{Y}$ is a
Banach space, then $(\mathfrak{B}(\mathcal{X},\mathcal{Y}),\|\cdot\|_{op})$ is
a Banach algebra

###### Proof.

Consider a Cauchy sequence
$\\{A_{n}\\}\subset\mathfrak{B}(\mathcal{X},\mathcal{Y})$ for the operator
norm. By definition of an operator norm we have:

$\|A_{n}v-A_{m}v\|\leq\|A_{n}-A_{m}\|_{op}\,\|v\|$

As $\\{A_{n}\\}$ is a Cauchy sequence, $\\{A_{n}v\\}$ must be as well. But we
know that $\mathcal{Y}$ is complete so there is a vector in $\mathcal{Y}$ such
that:

$Av:=\lim_{n\to\infty}A_{n}v$

As all the $A_{n}$ are linear operators, so must be the operator $A$.
Now, as $\\{A_{n}\\}$ is Cauchy, for any choice of $\varepsilon$ if we take
$m,n$ sufficiently large we have:

$\|A_{m}-A_{n}\|_{op}<\varepsilon$

Then we can estimate:

$\|A_{n}v-A_{m}v\|\leq\|A_{n}-A_{m}\|_{op}\|v\|\leq\varepsilon\|v\|$

As we are taking the norm over $\mathcal{Y}$:

$\|Av-A_{m}v\|=\lim_{n\to\infty}\|A_{n}v-A_{m}v\|\leq\varepsilon\|v\|$

where in the inequality, we implicitly assume that $m$ is sufficiently large.
Now since $\|Av\|\leq\|Av-A_{n}v\|+\|A_{n}v\|$, we have:

$\|Av\|\leq(\varepsilon+\|A_{n}\|_{op})\|v\|$

The above proves that $A$ is bounded. Finally we need to show that as
$n\to\infty$, $\|A-A_{n}\|_{op}\to 0$. To see this, just notice that from the
above:

$\|A-A_{n}\|_{op}=\sup_{v\in\mathcal{H},\|v\|=1}\|Av-A_{n}v\|\leq\varepsilon$

The result follows from the freedom of choice of $\varepsilon$ for
sufficiently large $n$. ∎

Since in our case, we are considering bounded operators over a Hilbert space,
the fact that $\mathfrak{B}(\mathcal{H})$ is a Banach algebra, is an immediate
consequence of this proposition.

Finally, let us show that the $C^{*}$-algebra condition holds. First of all
let us show that $\|A^{*}\|_{op}=\|A\|_{op}$. We have for $v,x\in\mathcal{H}$:

$\innerproduct{A^{*}v}{x}=\innerproduct{v}{Ax}\leq\|v\|\,\|A\|_{op}\|x\|$

But, if we choose $x=A^{*}v$ we get:

$\|A^{*}v\|^{2}\leq\|v\|\,\|A\|_{op}\|A^{*}v\|$

This implies: $\|A^{*}\|_{op}\leq\|A\|_{op}$. Now, as the adjoint is
involutive:

$\innerproduct{v}{A^{*}x}=\overline{\innerproduct{A^{*}x}{v}}=\overline{\innerproduct{x}{Av}}=\innerproduct{Av}{x}$

We also have: $\|A\|_{op}=\|(A^{*})^{*}\|_{op}\leq\|A^{*}\|_{op}$ which
implies:

$\|A^{*}\|_{op}=\|A\|_{op}$

Further, we have from the multiplicative property of the operator norm, proven
above: $\|A^{*}A\|_{op}\leq\|A^{*}\|_{op}\|A\|_{op}=\|A\|_{op}^{2}$. For the
converse inequality, observe that:

$\displaystyle\|A\|_{op}^{2}$
$\displaystyle=\bigg{(}\sup_{v\in\mathcal{H},\|v\|=1}\|Av\|\bigg{)}^{2}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\|Av\|^{2}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\innerproduct{Av}{Av}$
$\displaystyle=\sup_{v\in\mathcal{H},\|v\|=1}\innerproduct{v}{A^{*}Av}$
$\displaystyle\leq\sup_{v\in\mathcal{H},\|v\|=1}\|A^{*}Av\|=\|A^{*}A\|_{op}$

Therefore, $\|A^{*}A\|_{op}=\|A\|_{op}^{2}$ and we have proven that
$(\mathfrak{B}(\mathcal{H}),\|\cdot\|_{op})$ is a $C^{*}$-algebra.

Another important concept is that of a $C^{*}$-subalgebra:

######  Definition 1.1.6.

Let $\mathfrak{A}$ be a $C^{*}$-algebra, and let
$\mathfrak{B}\subset\mathfrak{A}$ be a $*$-subalgebra in the sense that it is
closed under the product and involution of its elements and let it be closed
in the topology induced by the norm. Then, we say that $\mathfrak{B}$ is a
$C^{*}$-subalgebra of $\mathfrak{A}$, and clearly, it is a $C^{*}$-algebra on
its own.

We now introduce specific type of $C^{*}$-algebras that are going to be widely
used and mentioned in what follows.

#####  1.1.1.1 CCR $*$-algebra and Weyl $C^{*}$-algebra

We start by abstractly defining what a $CCR$ algebra is, the motivation for it
will be explained in an incoming example:

######  Definition 1.1.7.

Let $\mathbf{P}$ be a symplectic space with symplectic form
$\sigma(\cdot,\cdot)$. A CCR (Canonical Commutation Relation) algebra
$\mathfrak{A}_{CCR}(\mathbf{P},\sigma)$, is the quotient of the $*$-algebra,
generated by $A^{*}(f)$, $A(f)$ and the identity $\mathbb{1}$ (where
$f\in\mathbf{P}$), by the following relations:

$\displaystyle(A^{*}(f))^{*}$ $\displaystyle=A(f)$ $\displaystyle[A(f),A(g)]$
$\displaystyle=i\sigma(f,g)\mathbb{1}$ $\displaystyle
A(c_{1}f_{1}+c_{2}f_{2})$ $\displaystyle=c_{1}A(f_{1})+c_{2}A(f_{2})$
$\displaystyle A^{*}(h)$ $\displaystyle=A(\overline{h})$

for $c_{1},c_{2}\in\mathbb{C}$ and $\overline{h}$ denotes the complex
conjugate of $h\in\mathbf{P}$.

An example of such an abstract algebra, is the algebra of free scalar field
operators, i.e. the operator valued distributions, as the fundamental
solutions of the Klein-Gordon equation, smeared with test functions.
Unfortunately, on the $CCR$ $*$-algebra one cannot define a $C^{*}$-norm. This
is due to the following Proposition:

######  Proposition 1.1.2.

There are no self-adjoint operators $q$ and $p$ such that, on a common
invariant subspace, $[q,p]=ic\mathbb{1}$ (for $c\in\mathbb{C}$) and at the
same time $q$ and $p$ are bounded.

###### Proof.

Suppose that $[q,p]=ic\mathbb{1}$ on a common invariant space $D$ where both
$q$ and $p$ are bounded. Restrict to the closure $\overline{D}$, extending
$q$, $p$ as self adjoint operators over it, and consider it as the Hilbert
space. From the commutation relations we get:

$[p,q^{n}]=-ic(nq^{n-1})$

Using that, as a self-adjoint, bounded operator, $q^{k}=(q^{k})^{*}$ for any
$k\in\mathbb{N}$ and assuming $n$ odd:

$\displaystyle n\|q\|_{op}^{n-1}$ $\displaystyle=n\|q^{n-1}\|_{op}$
$\displaystyle\leq 2\|p\|_{op}\|q^{n}\|_{op}$ $\displaystyle\leq
2\|p\|_{op}\|q\|_{op}\|q^{n-1}\|_{op}$
$\displaystyle=2\|p\|_{op}\|q\|_{op}\|q\|_{op}^{n-1}$

Noticing now that, substituting $q^{\prime}=q+c_{1}\mathbb{1}$ for
$c_{1}\in\mathbb{C}$, will not alter the assumptions of the proposition, we
always have $\|q\|_{op}\neq 0$. Therefore, we have obtained for any
$n=1,3,\dots$:

$n\leq 2\|p\|_{op}\|q\|_{op}<+\infty$

that contraddicts the boundedness. ∎

So, the elements of a finitely generated $CCR$ algebra do not admit
representations as buonded operators, forbidding the definition of a norm. As
a consequence, we cannot even define a norm over the abstract $*$-algebra.
Finally, being a general $CCR$ algebra the union of the finitely generated
ones, this prevents the definition of a norm in general.
The $CCR$ not being a $C^{*}$-algebra leads inevitably, as we will state when
introducing the GNS theorem, to a representation of its elements as unbounded
operators over a Hilbert space. As a consequence, domain issues in defining
products of the represented elements in the algebra may arise. Therefore, in
order to avoid them, the idea is to ”turn” the $CCR$ into a $C^{*}$-algebra,
for which the same GNS theorem ensures that the elements are represented as
bounded operators. For this reason, let us define the so called Weyl
$C^{*}$-Algebra that are a specific type of $C^{*}$-algebras that we can
construct once over the same symplectic space:

######  Definition 1.1.8.

Let $(\mathbf{P},\sigma)$ be a non-trivial real symplectic space, with
$\sigma:\mathbf{P}\times\mathbf{P}\to\mathbb{R}$ non degenerate. The Weyl
$C^{*}$-algebra $C\mathcal{W}(\mathbf{P},\sigma)$ associated to
$(\mathbf{P},\sigma)$ is a $C^{*}$-algebra generated by elements $W(f)$ with
$f\in\mathbf{P}$ that satisfy the following relations:

$W(f)W(f^{\prime})=e^{-\frac{i}{2}\sigma(f,f^{\prime})}W(f+f^{\prime})\hskip
20.0ptW(f)^{*}=W(-f)$

From the given definition however, is just clear that
$C\mathcal{W}(\mathbf{P},\sigma)$ is a $*$-algebra. We still need to argue
that any Weyl $*$-algebra admits a norm with the $C^{*}$-property. Let me here
only quote the procedure and refer to Remark $11.49$ number $\mathbf{(5)}$ in
[12] for a rigorous explanation. For any two Weyl $*$-algebras on the same
underlying symplectic space, one can show that there always exist a unique
$*$-isomorphism between them. Then, one shows that given $(\mathbf{P},\sigma)$
one can always construct a Weyl $*$-algebra of bounded operators
$\mathfrak{B}(\mathcal{H})$ over a Hilbert space
$\mathcal{H}=L^{2}(\mathbf{P},\mu)$, where $\mu$ is the counting measure
defined on $(\mathbf{P},\Sigma)$ with $\Sigma$ the power set of $\mathbf{P}$
that we take as our sigma algebra. In particular, for any $A\in\Sigma$ the
counting measure is:

$\mu(A):=\left\\{\begin{aligned} &\sharp A\hskip
20.0pt\mathrm{if}\,\,A\,\,\mathrm{is}\,\,\mathrm{finite}\\\ &+\infty\hskip
20.0pt\mathrm{if}\,\,A\,\,\mathrm{is}\,\,\mathrm{infinite}\end{aligned}\right.$

Where $\sharp\cdot$ denotes the cardinality of the set. However, since we know
that any $(\mathfrak{B}(\mathcal{H}),\|\cdot\|_{op})$ is a $C^{*}$-algebra, we
can define a $C^{*}$-norm on any other Weyl algebra on $(\mathbf{P},\sigma)$
starting from $\|\cdot\|_{op}$ using the existing $*$-isomorphism between any
Weyl algebra on the same underlying symplectic space.

Finally, to give some motivation and make these abstract definitions more
concrete we present an example, that we are familiar with, from standard
Quantum Mechanics.

######  Example 1.1.4 (CCR in QM).

In Schrödinger mechanics, the main operators are position and momentum. The
Canonical Commutation Relation algebra (CCR) for finitely many degrees of
freedom is defined as a $*$-algebra generated by self-adjoint
$q_{1},\dots,q_{n},p_{1},\dots,p_{n}$, with the relations:

$\displaystyle[q_{j},p_{k}]=i\delta_{jk}\mathbb{1}$ $\displaystyle\hskip
15.0pt[q_{j},q_{k}]=[p_{j},p_{k}]=0$ $\displaystyle q_{k}=q_{k}^{*}\,\,\,\,$
$\displaystyle\mathrm{and}\,\,\,\,p_{k}=p_{k}^{*}$

Such an abstract $*$-algebra does not admit a representation in terms of
bounded operators over a Hilbert space, as we have seen in Prop. 1.1.2,
preventing the definition of a norm. The canonical way to bypass this issue is
to transform such self-adjoint operators into unitaries, namely defining a
Weyl algebra for it.
In this case, we have an underlying symplectic space stemmin from the
classical theory that describes the positions and momenta of our collection of
classical pointlike particles: $(\mathbb{R}^{2n},\sigma)$. Therefore,
$\mathbf{P}=\mathbb{R}^{2n}=\\{(Q,P)|Q,P\in\mathbb{R}^{n}\\}$ is the classical
phase space and (denoting with $\cdot$ the standard inner product):

$\sigma((Q_{1},P_{1}),(Q_{2},P_{2})):=Q_{1}\cdot P_{2}-Q_{2}\cdot P_{1}$

This expression is compatible with the Poisson bracket as follows: given
$f=(Q,P)\in\mathbf{P}$, the map $\sigma_{f}:h\mapsto\sigma(f,h)$ is a linear
observable and, taking advantage of the non-degeneracy of the symplectic form,
we can write any linear map $\mathbf{P}\to\mathbb{R}$ in this way. For
instance, we can express the position and momentum operators
$q_{i}(Q,P):=Q_{i}$ and $p_{i}(Q,P):=P_{i}$ as:

$q_{i}=\sigma_{(0,-e_{i})}\hskip 20.0ptp_{i}=\sigma_{(e_{i},0)}$ (1.1)

where $e_{i}$ is the $i$-th canonical vector in $\mathbb{R}^{n}$. The
compatibility with the Poisson bracket then follows:

$\displaystyle\\{\sigma_{(Q_{1},P_{1})},\sigma_{(Q_{2},P_{2})}\\}_{P.B.}$
$\displaystyle=\frac{\partial\sigma_{(Q_{1},P_{1})}}{\partial
Q}\cdot\frac{\partial\sigma_{(Q_{2},P_{2})}}{\partial
P}-\frac{\partial\sigma_{(Q_{1},P_{1})}}{\partial
P}\cdot\frac{\partial\sigma_{(Q_{2},P_{2})}}{\partial Q}$
$\displaystyle=(-P_{1})\cdot Q_{2}-(-P_{2})\cdot Q_{1}$
$\displaystyle=\sigma((Q_{1},P_{1}),(Q_{2},P_{2}))$

We can rewrite the CCR $*$-algebra relations using Eq. (1.1) noticing that:

$\sigma((0,-e_{j}),(0,-e_{k}))=\sigma((e_{j},0),(e_{k},0))=0\hskip
15.0pt\sigma((0,-e_{j}),(e_{k},0))=\delta_{jk}$

Then, by quantizing the classical theory:

$\displaystyle[\sigma_{(0,-e_{j})},\sigma_{(e_{k},0)}]=i\delta_{jk}\mathbb{1}\hskip
15.0pt$
$\displaystyle[\sigma_{(0,-e_{j})},\sigma_{(0,-e_{k})}]=[\sigma_{(e_{j},0)},\sigma_{(e_{k},0)}]=0$
$\displaystyle\sigma_{(0,-e_{k})}=\sigma_{(0,-e_{k})}^{*}\,\,\,\,\,\,$
$\displaystyle\mathrm{and}\,\,\,\,\,\,\sigma_{(e_{k},0)}=\sigma_{(e_{k},0)}^{*}$

where we have used the fact that $e_{i}\in\mathbb{R}^{n}$. In this
formulation, for any $f\in\mathbf{P}$ (e.g. $f=(0,-e_{i})$), we define the
associated Weyl algebra as generated by elements $e^{i\sigma_{f}}$, checked to
fulfill:

$e^{i\sigma_{f}}e^{i\sigma_{h}}=e^{-\frac{i}{2}\sigma(f,h)}e^{i\sigma_{f+h}}\hskip
20.0pt(e^{i\sigma_{f}})^{*}=e^{i\sigma_{-f}}$

#####  1.1.1.2 Self-dual CAR algebra

Similarly with the way we defined an abstract $CCR$ algebra, we can define an
abstract $*$-algebra that has Canonical Anticommutation Relations (CAR)
implemented in it as done in [13]. In this case, the underlying structure is
that of a Hilbert space:

######  Definition 1.1.9.

Let $\mathcal{H}$ be a complex Hilbert space with inner product
$(\cdot,\cdot)_{\mathcal{H}}$, and let $\Gamma$ be an antiunitary operator
satisfying:

$\displaystyle\Gamma^{2}=\mathbb{1}$ $\displaystyle\hskip 20.0pt\Gamma
i=-i\Gamma$ $\displaystyle(\Gamma f_{1},\Gamma f_{2})_{\mathcal{H}}$
$\displaystyle=(f_{2},f_{1})_{\mathcal{H}}$

A selfdual CAR algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is the
quotient of the $*$-algebra, generated by $B^{*}(f)$, $B(f)$ and the identity
$\mathbb{1}$ (where $f\in\mathcal{H}$), by the following relations:

$\displaystyle(B^{*}(f))^{*}$ $\displaystyle=B(f)$
$\displaystyle[B^{*}(f),B(g)]_{+}$
$\displaystyle=(g,f)_{\mathcal{H}}\mathbb{1}$ $\displaystyle
B(c_{1}f_{1}+c_{2}f_{2})$ $\displaystyle=c_{1}B(f_{1})+c_{2}B(f_{2})$
$\displaystyle B^{*}(h)$ $\displaystyle=B(\Gamma h)$

for $c_{1},c_{2}\in\mathbb{C}$.

Contrary to the $CCR$ case, one can define a $C^{*}$-norm over a self-dual
$CAR$ algebra, making it a $C^{*}$-algebra after taking the completion of the
previous $*$-algebra with respect to this norm. Let me just quote the main
argument, and refer to Prop. $12.50$ in [14] for a rigorous proof. Note first
of all that $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is an infinite
dimensional Clifford algebra, due to the anticommutation relations. Moreover,
each finite dimensional Clifford algebra admits an injective reppresentation,
as an algebra of bounded operators over some Hilbert space, see [15] Section
$4$. Then, as bounded operators, they admit a $C^{*}$-norm. Follows, that each
finite dimensional Clifford algebra admits a $C^{*}$-norm as well. Finally, as
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is the completion of the union of
these finite dimensional $C^{*}$-subalgebras, we must have that
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is also a $C^{*}$-algebra.
As a consequence of this, one obtains that the corresponding operators in
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$, are reppresented, by the GNS
theorem, as bounded operators on a Hilbert space.

In what follows, $\mathfrak{A}_{(\mathbf{K},s)}$ will denote an algebra built
on $(\mathbf{K},s)$ with $s$ a bilinear form on $\mathbf{K}$, i.e. the
elements in $\mathfrak{A}_{(\mathbf{K},s)}$ are maps $(\mathbf{K},s)\ni f\to
T(f)\in\mathfrak{A}_{(\mathbf{K},s)}$. In particular, if $(\mathbf{K},s)$ is a
symplectic space, $\mathfrak{A}_{(\mathbf{K},s)}$ denotes either a Weyl
algebra or the $CCR$ algebra. While, if $(\mathbf{K},s)$ is a Hilbert space,
$\mathfrak{A}_{(\mathbf{K},s)}$ denotes a self-dual CAR algebra.

#####  1.1.1.3 Von Neumann algebras

The last important type of $C^{*}$-algebra that we will study are the so
called von Neumann algebras (named after John von Neumann) that are the types
of algebras we will be interested in, once we represent the abstract algebras
on a concrete Hilbert space. These are unital algebras that we will encounter
when studying the Tomita-Takesaki modular theory in the next chapter.
The first thing we need to introduce, is the notion of commutant algebra.

######  Definition 1.1.10.

Let $\mathfrak{M}\subset\mathfrak{B}(\mathcal{H})$ be a subset of the bounded
operators over the Hilbert space $\mathcal{H}$. The commutant algebra of
$\mathfrak{M}$ is defined:

$\mathfrak{M}^{\prime}:=\\{T\in\mathfrak{B}(\mathcal{H}):TA-
AT=0\,\,\,\,\mathrm{for}\,\,\,\,\mathrm{any}\,\,\,\,A\in\mathfrak{M}\\}$

######  Remark 1.1.1.

If $\mathfrak{M}$ is closed under hermitian conjugation, then
$\mathfrak{M}^{\prime}$ is a $*$-algebra with unit. One can also convince
himself that $\mathfrak{M}_{1}\subset\mathfrak{M}_{2}$ then
$\mathfrak{M}^{\prime}_{2}\subset\mathfrak{M}^{\prime}_{1}$ and that
$\mathfrak{M}\subset(\mathfrak{M}^{\prime})^{\prime}$. This implies that one
cannot reach more than the first order of commutant as
$\mathfrak{M}^{\prime}=((\mathfrak{M}^{\prime})^{\prime})^{\prime}:=(\mathfrak{M}^{\prime})^{\prime\prime}$.
This is true as, from the second statement applied on $\mathfrak{M}^{\prime}$,
we get:

$\mathfrak{M}^{\prime}\subset(\mathfrak{M}^{\prime})^{\prime\prime}$

but at the same time since
$\mathfrak{M}\subset(\mathfrak{M}^{\prime})^{\prime}$ we must have, from the
first statement above, that for the commutant algebras:

$((\mathfrak{M}^{\prime})^{\prime})^{\prime}\subset\mathfrak{M}^{\prime}$

The next theorem proves that the second commutant of
$\mathfrak{M}\subset\mathfrak{B}(\mathcal{H})$ corresponds to its closure in
the weak topology and, as we will see, this is required for defining von
Neumann algebras:

######  Theorem 1.1.1 (Double commutant theorem).

If $\mathcal{H}$ is a complex Hilbert space and $\mathfrak{A}$ is a unital
$*$-subalgebra of $\mathfrak{B}(\mathcal{H})$ the following statements are
equivalent:

  * i)

$\mathfrak{A}^{\prime\prime}=\mathfrak{A}$

  * ii)

$\mathfrak{A}$ is closed in the weak topology111A sequence of operators
$T_{i}\in\mathfrak{B}(\mathcal{H})$ converges in the weak operator topology to
$T$ if for any $x\in\mathcal{H}$ and $y\in\mathcal{H}^{*}$ (the dual space of
$\mathcal{H}$ with respect to the inner product):
$\innerproduct{y}{(T_{i}-T)x}\to 0$

  * iii)

$\mathfrak{A}$ is closed in the strong topology222A sequence of operators
$T_{i}\in\mathfrak{B}(\mathcal{H})$ converges in the strong operator topology
to $T$ if for any $x\in\mathcal{H}$: $\|T_{i}x-Tx\|\to 0$

Then:

$\mathfrak{A}^{\prime\prime}=\overline{\mathfrak{A}}^{s}=\overline{\mathfrak{A}}^{w}$

Where $\overline{(\cdot)}^{s}$ denotes the closure w.r.t. the strong topology
and $\overline{(\cdot)}^{w}$ the closure in the weak topology.

###### Proof.

See Theorem $3.88$ in [12] ∎

######  Remark 1.1.2.

Notice that if $\mathfrak{A}$, as introduced above, is closed in the strong
topology it must be closed also in the operator (uniform) topology. Indeed, if
$\|T_{i}-T\|_{op}\to 0$ in the operator topology, it must also converge in the
strong one as:

$\|T_{i}x-Tx\|\leq\|T_{i}-T\|_{op}\|x\|$

The converse is not true since, by taking as operator
$\mathbb{1}_{N}\in\mathfrak{B}(l^{2}(\mathbb{N}))$ defined via:

$\mathbb{1}_{N}=\sum_{n=0}^{N}\outerproduct{n}{n}$

its action on any $x\in l^{2}(\mathbb{N})$ is:

$\mathbb{1}_{N}=\sum_{n=0}^{N}\ket{n}\innerproduct{n}{x}=\sum_{n=0}^{N}\ket{n}x_{n}$

As $\ket{n}\in l^{2}(\mathbb{N})$ is nothing but a sequence of zeros except a
single $1$ at the $n$-th place. But then in the strong operator topology,
assuming w.l.o.g that $M>N$, we have that for all $x\in l^{2}(\mathbb{N})$:

$\|\mathbb{1}_{N}x-\mathbb{1}_{M}x\|=\|\sum_{n=N+1}^{M}\ket{n}x_{n}\|\to 0$

As, by assumption $x\in l^{2}(\mathbb{N})$, which means that only a finite
number of $x_{n}$ must be non-zero, so by taking $N$ sufficiently large we can
ascertain that no other $x_{n}$ for $n\geq N$ are nonvanishing. That means
that $\mathbb{1}_{N}\to\mathbb{1}$ in the strong topology. On the other hand:

$\|\mathbb{1}_{N}-\mathbb{1}_{N+1}\|_{op}=\|\ket{N+1}\innerproduct{N+1}{N+1}\|=1$

proves that this sequence is not Cauchy, i.e. in particular $\mathbb{1}_{N}$
cannot converge in the operator topology.
So, we have proven:

$\mathrm{uniform}\Rightarrow\mathrm{strong}$

But then, if we take the closure of a set $C$ in the uniform topology, this
closure must itself be contained in the closure of the strong topology of that
set, as the convergence in the uniform topology of each sequence implies that
all the accumulation points must be contained also in the closure w.r.t. the
strong topology:

$\overline{C}^{op}\subseteq\overline{C}^{s}$

showing that if a set is closed in the strong topology sense, it must be
closed in the uniform topology as well.

The main implication of this result is that, if $\mathfrak{A}^{\prime\prime}$
is $*$-closed, then it is a $C^{*}$-subalgebra of $\mathfrak{B}(\mathcal{H})$,
as it is also closed in the operator topology.

######  Remark 1.1.3.

$\mathfrak{M}^{\prime}$ is always closed in the uniform topology. This can be
seen by noticing that
$\mathfrak{M}^{\prime}=(\mathfrak{M}^{\prime})^{\prime\prime}$ and as such it
is closed in the strong topology from the above theorem.

Now we have introduced all the necessary formal notions to define a von
Neumann algebra:

######  Definition 1.1.11.

Let $\mathcal{H}$ be a complex Hilbert space. A von Neumann algebra over
$\mathcal{H}$ is a unital $*$-subalgebra of $\mathfrak{B}(\mathcal{H})$ that
fulfills one of the three equivalent statements $i);ii);iii)$ of the double
commutant theorem.
Given a von Neumann algebra $\mathfrak{M}$, its center is the subset
$\mathfrak{M}\cap\mathfrak{M}^{\prime}$.

We see then, from the above remarks, that any von Neumann algebra is in
particular a $C^{*}$-subalgebra of
$(\mathfrak{B}(\mathcal{H}),\|\cdot\|_{op})$.
Consequently, in general, we can obtain a von Neumann algebra whenever we are
given $\mathfrak{M}\subset\mathfrak{B}(\mathcal{H})$ that is closed under
Hermitian conjugation, by simply taking its double commutant
$\mathfrak{M}^{\prime\prime}$. It turns out that this is the smallest von
Neumann algebra that one can obtain starting from $\mathfrak{M}$ as, whenever
we take $\mathfrak{M}\subset\mathfrak{A}$ with $\mathfrak{A}$ a von Neumann
algebra containing $\mathfrak{M}$, we have
$\mathfrak{A}^{\prime}\subset\mathfrak{M}^{\prime}$. But then:
$\mathfrak{M}^{\prime\prime}\subset\mathfrak{A}^{\prime\prime}=\mathfrak{A}$
showing that $\mathfrak{M}^{\prime\prime}$ is the smallest von Neumann algebra
containing $\mathfrak{M}$.

####  1.1.2 State functionals

As mentioned at the beginning, now that we have introduced the concept of what
an algebra of operators is, we need to define the notion of a state if we want
to specify the configuration of the system. These are linear functionals over
the algebra $\mathfrak{A}$, namely $\omega$ is a linear functional over
$\mathfrak{A}$ if $\omega:\mathfrak{A}\to\mathbb{C}$ and:

$\omega(\alpha A+\beta B)=\alpha\omega(A)+\beta\omega(B)$

For $\alpha,\beta\in\mathbb{C}$ and any $A,B\in\mathfrak{A}$. The space of all
linear functionals corresponds to the dual space $\mathfrak{A}^{*}$ of
$\mathfrak{A}$.
Let us start by introducing some additional structure over this space of
functionals. As usual, we want a notion of convergence and of normalization
for the functionals, in order to be able to tell how the configuration of the
system can change continuously, and to recast the probabilistic interpretation
of Quantum Mechanics into our new formalism. We thus start by defining a norm
over the space of functionals $\omega\in\mathfrak{A}^{*}$ as:

$\|\omega\|:=\sup\\{|\omega(A)|\,:\,\|A\|=1\\}$

In fact, with the above norm, one can show that $(\mathfrak{A}^{*},\|\cdot\|)$
is a Banach space.
The functionals we will be interested in, for physical reasons, are listed in
the following definition:

######  Definition 1.1.12.

Let $\mathfrak{A}$ be a $C^{*}$-algebra. A linear functional
$\omega\in\mathfrak{A}^{*}$ is defined to be:

  * •

positive if for all $A\in\mathfrak{A}$:

$\omega(A^{*}A)\geq 0$

  * •

normalized if $\|\omega\|=1$

  * •

a state if it is positive and normalized

We denote the space of states by $\mathfrak{G}(\mathfrak{A})$.
A state is called faithful if it is strictly positive, i.e. if
$\omega(A^{*}A)=0$ implies that $A=0$.

######  Remark 1.1.4.

For a state, one can in fact drop the assumption of it being continuous, as
being a positive and linear functional over a $C^{*}$-algebra also implies
continuity. To see it, assume by absurd that $\omega$ is a positive linear but
discontinuous functional. Then, let us pick a sequence of operators $A_{n}$ in
the $C^{*}$-algebra, such that:

$\|A_{n}\|\leq\frac{1}{2^{n}}$

But then by the discontinuity, we can always find a state $\omega$ such that:

$\omega(A^{*}_{n}A_{n})\geq 1$

As we are on a $C^{*}$-algebra, we know that
$\|A_{n}^{*}A_{n}\|=\|A_{n}\|^{2}$, so:

$\displaystyle\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|$
$\displaystyle\leq\sum_{n=0}^{\infty}\|A_{n}\|^{2}$
$\displaystyle\leq\sum_{n=0}^{\infty}\frac{1}{4^{n}}$
$\displaystyle=\frac{4}{3}$

where we have used the geometric series. This implies that:

$\omega\bigg{(}\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\bigg{)}<\infty$

since:

$\displaystyle\omega\bigg{(}\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\bigg{)}$
$\displaystyle\leq\bigg{|}\omega\bigg{(}\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\bigg{)}\bigg{|}$
$\displaystyle=\bigg{|}\omega\bigg{(}\frac{\sum_{n=0}^{\infty}A_{n}^{*}A_{n}}{\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|}\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|\bigg{)}\bigg{|}$
$\displaystyle=\bigg{|}\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|\omega\bigg{(}\frac{\sum_{n=0}^{\infty}A_{n}^{*}A_{n}}{\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|}\bigg{)}\bigg{|}$
$\displaystyle\leq\|\omega\|\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|$
$\displaystyle=\|\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\|<\infty$

But then, by linearity and the discontinuity of the state $\omega$, we will
also have:

$\omega\bigg{(}\sum_{n=0}^{N}A_{n}^{*}A_{n}\bigg{)}\geq N$

for all finite $N\in\mathbb{N}$. So, by positivity of the state, it follows:

$\infty>\omega\bigg{(}\sum_{n=0}^{\infty}A_{n}^{*}A_{n}\bigg{)}\geq\omega\bigg{(}\sum_{n=0}^{N}A_{n}^{*}A_{n}\bigg{)}\geq
N$

which is a contraddiction, i.e. $\omega$ must be continuous

######  Remark 1.1.5.

Any positive functional is automatically Hermitian, i.e.
$\omega(A)\in\mathbb{R}$ if $A$ is self-adjoint. To see it, consider
$B=A+\mathbb{1}$, then from the positivity of the functional and linearity we
have:

$\displaystyle 0$ $\displaystyle\leq\omega((A^{*}+\mathbb{1})(A+\mathbb{1}))$
$\displaystyle=\omega(A^{*}A)+\omega(A^{*})+\omega(\mathbb{1})+\omega(A)$

But then, by taking the imaginary part of this, we must get a vanishing
expression as the whole expression must be positive:

$0=\imaginary(\omega(A^{*}A)+\omega(A^{*})+\omega(\mathbb{1})+\omega(A))=\imaginary(\omega(A^{*})+\omega(A))$

This implies $\imaginary(\omega(A^{*}))=-\imaginary(\omega(A))$. On the other
hand, if we consider $B=A+i\mathbb{1}$ and we take into account positivity:

$0\leq\omega(A^{*}A)+i\omega(A^{*})+\omega(\mathbb{1})-i\omega(A)$

Then, by again taking the imaginary part of this we obtain:
$\real(\omega(A))=\real(\omega(A^{*}))$. From which it follows that:

$\omega(A^{*})=\overline{\omega(A)}$

######  Remark 1.1.6.

From the assumption of positivity, one can prove that
$\|\omega\|=\omega(\mathbb{1})$. Then the normalization property is equivalent
to $\omega(\mathbb{1})=1$.

From the point of view of a physicist, $\omega(A)$ is the expectation value of
the operator $A$, if $A$ is symmetric, with respect to the system described by
$\omega$. Furthermore, to give the intuition behind the definition of state,
we notice:

  * •

The positivity assumption implies that we can consider $\omega(B^{*}A)$ as
defining a degenerate inner product on the elements in $\mathfrak{A}$ and this
will imply, as we will see, that we can get a Hilbert space once we specify a
state on the algebra.

  * •

The normalization assumption is needed in order to give a probabilistic
interpretation to the expectation values on the states. To see it, consider
any $A\in\mathfrak{A}$ with $\|A\|=1$, while we also have $\|A^{*}A\|=1$ from
the $C^{*}$-algebra condition. But since for any normalized element $A$ of the
algebra we also have: $|\omega(A)|\leq\|\omega\|$ and in this case
$\|\omega\|=1$, we have:

$0\leq\omega(A^{*}A)\leq 1$

#####  1.1.2.1 Pure and mixed states

Following the goal of building a bridge to the standard approach to QM, we
want to be able to describe with state functionals also systems corresponding
to ensembles. In the standard approach to quantum theory, we can talk of pure
and mixed states by investigating the corresponding form of the density
matrix. Before defining pure and mixed state functionals, let us state and
prove the following:

######  Lemma 1.1.1.

Let $\mathfrak{A}$ be a $C^{*}$-algebra. The set of states is convex, i.e.
whenever we take $\omega_{1},\omega_{2}\in\mathfrak{G(A)}$ as states and
$0\leq\lambda\leq 1$ we have that:

$\omega:=\lambda\omega_{1}+(1-\lambda)\omega_{2}$

is still a state.

###### Proof.

We need to prove positivity and normalization as linearity is manifestly true.
For what concerns normalization let us compute:

$\displaystyle\omega(\mathbb{1})$
$\displaystyle=\lambda\omega_{1}(\mathbb{1})+(1-\lambda)\omega_{2}(\mathbb{1})$
$\displaystyle=\lambda+(1-\lambda)=1$

For what concerns positivity, consider any $A\in\mathfrak{A}$, then:

$\displaystyle\omega(A^{*}A)$
$\displaystyle=\lambda\omega_{1}(A^{*}A)+(1-\lambda)\omega_{2}(A^{*}A)$
$\displaystyle\geq 0$

as $0\leq\lambda\leq 1$ and both $\omega_{2}(A^{*}A),\omega_{1}(A^{*}A)\geq
0$, so the above is just a sum of positive terms ∎

Given this geometric aspect of $\mathfrak{G(A)}$, we define pure states:

######  Definition 1.1.13.

A pure state, is an extremal element of $\mathfrak{G(A)}$. That means that it
cannot be written as a non trivial convex linear combination of any two other
states. Moreover, any state that is not pure is called a mixed state.

#####  1.1.2.2 Quasi-free states

The last important class of states that I want to discuss is that of quasi-
free states. The definition is motivated by physics, as these are those states
for which any correlation between operators is either zero, if the number of
operators is odd or, if the number is even, can be decomposed in a product of
two point correlation functions. This is precisely what happens for Gaussians
distributions, where each moment of the distribution can be expressed via the
variance (the moment of order one) recursively; for this reason such states
are also called Gaussian states. The physical motivation comes, for instance,
from the quantum harmonic oscillator as the wavefunction resulting by solving
the Schrödinger equation for the ground state is a Gaussian: the corresponding
state functional is quasi-free. Moreover, as one can prove, the ground and the
states describing thermal equilibrium (called KMS) of a free QFT are quasi-
free. This, in a very heuristic way, can be justified for the free sclar
field, remembering that is nothing but a infinite set of decoupled harmonic
oscillators.
We will say more about ground and KMS states in a later section. For the
moment, we start defining the quasi-free property.

######  Definition 1.1.14.

A state $\omega$ over an algebra $\mathfrak{A}_{(\mathbf{K},s)}$, where
$(\mathbf{K},s)$ is either a Hilbert space with inner product $s(\cdot,\cdot)$
or a symplectic space with $s(\cdot,\cdot)$ denoting a symplectic form, is
quasifree if for any $A(f_{n})\in\mathfrak{A}_{(\mathbf{K},s)}$
($f_{n}\in(\mathbf{K},s)$) we have:

$\omega(A(f_{1})\cdots A(f_{n}))=\left\\{\begin{aligned} &0\hskip
20.0pt\mathrm{for}\,\,n\,\,\mathrm{odd}\\\
&\sum_{\pi\in\mathcal{P}_{n}}\omega(A(f_{\pi_{1}})A(f_{\pi_{2}}))\cdots\omega(A(f_{\pi_{n-1}})A(f_{\pi_{n}}))\hskip
20.0ptn\,\,\mathrm{even}\,\,\mathrm{and}\,\,(\mathbf{K},s)\,\,\mathrm{symplectic}\\\
&\sum_{\pi\in\mathcal{P}_{n}}(-1)^{|\pi|}\omega(A(f_{\pi_{1}})A(f_{\pi_{2}}))\cdots\omega(A(f_{\pi_{n-1}})A(f_{\pi_{n}}))\hskip
20.0ptn\,\,\mathrm{even}\,\,\mathrm{and}\,\,(\mathbf{K},s)\,\,\mathrm{Hilbert}\end{aligned}\right.$

Where $\mathcal{P}_{n}$ denotes the group of ordered permutations of $n$
elements, for which it holds:

$\displaystyle\pi_{m-1}<\pi_{m}\,\,$ $\displaystyle\mathrm{with}\,\,1\leq
m\leq n$ $\displaystyle\pi_{m-1}<\pi_{m+1}\,\,$
$\displaystyle\mathrm{with}\,\,1\leq m<n$

and $|\pi|$ denotes the order of the permutation with respect to the initial
order $\\{1,2,\dots,n\\}$.

######  Example 1.1.5.

In the case in which we are considering a CCR algebra over a sympllectic space
$(\mathbf{P},\sigma)$, the condition of being quasi-free is formulated
equivalently at the Weyl algebra level by the condition:

$\omega\big{(}e^{i\sigma_{f}}\big{)}=e^{-\mu(f,f)/2}\hskip 15.0pt\forall
f\in\mathbf{P}$

Where:

$\mu(f,h):=\frac{1}{2}(\omega(\sigma_{h}\sigma_{f})+\omega(\sigma_{f}\sigma_{h}))$

and $\sigma_{f}$ are the generators of the CCR algebra for $f\in\mathbf{P}$.
To show the equivalence, consider a CCR algebra defined via the relations:

$\displaystyle[\sigma_{f},\sigma_{h}]$ $\displaystyle=i\sigma(f,h)\mathbb{1}$
$\displaystyle\sigma_{\lambda f+h}$
$\displaystyle=\lambda\sigma_{f}+\sigma_{h}$ $\displaystyle\sigma_{f}^{*}$
$\displaystyle=\sigma_{\overline{f}}$

for any $f,h\in\mathbf{P}^{\mathbb{C}}:=\\{f+ih|f,h\in\mathbf{P}\\}$ where we
are assuming $\mathbf{P}$ real. From the definition of $\mu(\cdot,\cdot)$ we
have:

$\omega(\sigma_{f}\sigma_{h}):=w(f,h)=\mu(f,h)+\frac{i}{2}\sigma(f,h)$

Then for $f_{1},f_{2},f_{3},f_{4}\in\mathbf{P}^{\mathbb{C}}$, if the state is
assumed to be quasi-free (introducing the shorthand notation
$\sigma_{1}:=\sigma_{f_{1}}$):

$\displaystyle\omega(\sigma_{1},\dots,\sigma_{4})$
$\displaystyle=\frac{1}{2}\omega\big{(}\\{\sigma_{1},\sigma_{2}\\}\sigma_{3}\sigma_{4}\big{)}+\frac{1}{2}\omega\big{(}[\sigma_{1},\sigma_{2}]\sigma_{3}\sigma_{4}\big{)}$
$\displaystyle=\frac{1}{4}\omega\big{(}\\{\sigma_{1},\sigma_{2}\\}\\{\sigma_{3},\sigma_{4}\\}\big{)}+\frac{1}{4}\omega\big{(}\\{\sigma_{1},\sigma_{2}\\}[\sigma_{3},\sigma_{4}]\big{)}+\frac{1}{4}\omega\big{(}[\sigma_{1},\sigma_{2}]\\{\sigma_{3},\sigma_{4}\\}\big{)}+\frac{1}{4}\omega\big{(}[\sigma_{1},\sigma_{2}][\sigma_{3},\sigma_{4}]\big{)}$
$\displaystyle=\frac{1}{4}\frac{d}{d\lambda_{4}}\dots\frac{d}{d\lambda_{1}}\omega\big{(}e^{i\sigma_{\lambda_{1}f_{1}+\lambda_{2}f_{2}}}e^{i\sigma_{\lambda_{3}f_{3}+\lambda_{4}f_{4}}}\big{)}\bigg{|}_{\lambda_{1}=\dots=\lambda_{4}=0}-\frac{i}{2}\sigma(f_{3},f_{4})\frac{d}{d\lambda_{2}}\frac{d}{d\lambda_{1}}\omega\big{(}e^{i\sigma_{\lambda_{1}f_{1}+\lambda_{2}f_{2}}}\big{)}\bigg{|}_{\lambda_{1}=\lambda_{2}=0}$
$\displaystyle\,\,\,\,\,\,-\frac{i}{2}\sigma(f_{1},f_{2})\frac{d}{d\lambda_{4}}\frac{d}{d\lambda_{3}}\omega\big{(}e^{i\sigma_{\lambda_{3}f_{3}+\lambda_{4}f_{4}}}\big{)}\bigg{|}_{\lambda_{3}=\lambda_{4}=0}-\frac{1}{4}\sigma(f_{1},f_{2})\sigma(f_{3},f_{4})$

Using now the quasi-free condition:

$\omega\big{(}e^{i\sigma_{\lambda_{1}f_{1}+\lambda_{2}f_{2}}}\big{)}=e^{-\mu(\lambda_{1}f_{1}+\lambda_{2}f_{2},\lambda_{1}f_{1}+\lambda_{2}f_{2})/2}$

Computing all the derivatives one gets:

$\omega(\sigma_{1},\dots,\sigma_{4})=w(f_{1},f_{2})w(f_{3},f_{4})+w(f_{1},f_{3})w(f_{2},f_{4})+w(f_{1},f_{4})w(f_{2},f_{3})$

That is an example showing the equivalence.

As mentioned at the beginning, together with a notion of normalization, we
also want a notion of convergence over the space of linear functionals.
Therefore, to end the section, let us discuss how to define a topology on
$\mathfrak{A}^{*}$ called the weak $*$-topology. We will need it in the next
chapter to argue that the physics is encoded in the algebra and not in the
Hilbert space on which the algebra is represented, giving in this way the main
arguments in favour of the Algebraic approach.

######  Definition 1.1.15.

For any set of operators $A_{1},\dots,A_{n}\in\mathfrak{A}$, define a
seminorm333A seminorm has all the properties of a norm, except that
$\sigma_{A_{1},\dots,A_{n}}(\phi)=0$ does not imply $\phi=0$
$\sigma_{A_{1},\dots,A_{n}}$ on $\mathfrak{A}^{*}$ by:

$\sigma_{A_{1},\dots,A_{n}}(\phi)=\sup\\{|\phi(A_{k})|:k=1,\dots,n\\}$

######  Remark 1.1.7.

This is a seminorm, as the states are not assumed to be faithful

Then, the weak $*$-topology on $\mathfrak{A}^{*}$ is defined as the topology
generated by the open neighborhoods:

$\mathcal{U}(\phi;A_{1},\dots,A_{n};\epsilon)=\\{\phi^{\prime}\in\mathfrak{A}^{*}:\sigma_{A_{1},\dots,A_{n}}(\phi-\phi^{\prime})<\epsilon\\}$

for all $\phi\in\mathfrak{A}^{*}$, $A_{1},\dots,A_{n}\in\mathfrak{A}$ and
$\epsilon>0$. In particular a sequence $\phi_{k}\in\mathfrak{A}^{*}$
($k\in\mathbb{N}$) converges to $\phi\in\mathfrak{A}^{*}$ in the weak
$*$-topology iff:

$\phi_{k}(A)\to\phi(A)\hskip 15.0pt\forall A\in\mathfrak{A}$

Such a topology has a strikingly concrete interpretation. Suppose that our
system is initially in the configuration
$\omega\in\mathfrak{G}(\mathfrak{A})$. As the accuracy of experiments is
limited and as the number of experiments we can perform is finite, we can
practically determine the system to be in the state $\omega$ just up to a
small neighborhood. This neighborhood is exactly
$\mathcal{U}(\omega;A_{1},\dots,A_{n};\epsilon)$ if we interpet $A_{i}$ for
$i=1,\dots,n$ as the finite measures that we perform and $\epsilon$ the
limited accuracy in the knowledge of the measurement to agree with the value
of $\omega(A_{i})$ for all the $i=1,\dots,n$. This means that $\omega$ is
physically equivalent to all other states belonging to the $*$-weak
neighborhood as, by performing measurements on the system, we are not able to
distinguish states belonging to
$\mathcal{U}(\omega;A_{1},\dots,A_{n};\epsilon)$.

####  1.1.3 GNS theorem and Fock space representation

The algebraic approach gives a way to abstractly investigate the theory, but
for practical computations we need to represent these algebras on some Hilbert
space, recasting the standard approach. This section aims at presenting the
fundamental theorems leading to such a representation.

The following is the theorem ensuring the existence of a Hilbert space
representation that, by the name of the auhors, we will refer to as GNS
theorem.

######  Theorem 1.1.2 (Gelfand-Naimark-Segal).

Let $\mathfrak{A}$ be a $C^{*}$-algebra with unit and let
$\omega\in\mathfrak{G}(\mathfrak{A})$. Then:

  * a)

There exist a triple $(\mathcal{H}_{\omega},\pi_{\omega},\Omega_{\omega})$,
where: $\mathcal{H}_{\omega}$ is a Hilbert space,
$\pi_{\omega}:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{\omega})$ a
$\mathfrak{A}$-representation on $\mathcal{H}_{\omega}$ and
$\Omega_{\omega}\in\mathcal{H}_{\omega}$, such that:

    1. 1.

$\Omega_{\omega}$ is cyclic for $\pi_{\omega}$, that means that
$\mathcal{D}_{\omega}:=\pi_{\omega}(\mathfrak{A})\Omega_{\omega}$ is a dense
subset of $\mathcal{H}_{\omega}$

    2. 2.

For every $A\in\mathfrak{A}$ we have:
$\omega(A)=\innerproduct{\Omega_{\omega}}{\pi_{\omega}(A)\Omega_{\omega}}$

  * b)

If
$((\mathcal{H}^{\prime}_{\omega},\pi^{\prime}_{\omega},\Omega^{\prime}_{\omega}))$
satisfies $1.$ and $2.$, then there must exist a unitary operator
$U:\mathcal{H}_{\omega}\to\mathcal{H}^{\prime}_{\omega}$, such that
$\Omega^{\prime}_{\omega}=U\Omega_{\omega}$ and for any $A\in\mathfrak{A}$:

$\pi^{\prime}_{\omega}(A)=U\pi_{\omega}(A)U^{-1}$

###### Proof.

See Theorem $14.4$ in [12] ∎

######  Remark 1.1.8.

There’s a version of the GNS theorem also for unital $*$-algebras
$\mathfrak{C}$, in which case the elements of the algebra are represented as
unbounded operators over the Hilbert space with same domain of definition
$D_{1}:=\pi_{\omega}(\mathfrak{C})\Omega_{\omega}$. The explicit statement and
proof of it can be found in [12] Theorem $14.24$. In particular, this version
of the theorem, provides a representation of elements of a $CCR$ algebra (e.g.
the one of Bosonic fields) as unbounded operators over a Hilbert space
$\mathcal{H}_{\omega}$.

######  Remark 1.1.9.

Starting from an abstract algebra and performing a GNS construction, one can
always get a von Neumann algebra on a Hilbert space. Let $\mathfrak{A}$ be an
abstract $C^{*}$-algebra and consider $\omega\in\mathfrak{G}(\mathfrak{A})$.
Performing the GNS, we get a subset
$\pi_{\omega}(\mathfrak{A})\subset\mathfrak{B}(\mathcal{H}_{\omega})$ that is
a $*$-subalgebra of bounded operators. However, we can always take its double
commutant $\pi_{\omega}(\mathfrak{A})^{\prime\prime}$ making it closed in the
uniform topology and thus a von Neumann algebra. In the following, without
mentionig explicitly these steps every time, we will say that any abstract
$C^{*}$-algebra is represented as a von Neumann algebra via its GNS
construction.

######  Remark 1.1.10.

For our purposes, all representations in what follows are considered to be
faithful i.e. the map $\pi_{\omega}$ is injective. This is not a loss of
generality, as also in the case of non faithful representation $\pi_{\omega}$
we can always define
$\tilde{\mathfrak{A}}:=\mathfrak{A}\backslash\ker(\pi_{\omega})$ and start
with this other algebra instead.

######  Remark 1.1.11.

One can show that a state $\omega$ is pure iff the corresponding GNS
repesentation is irreducible, i.e., $\mathcal{H}_{\omega}$ does not have
subspaces that are invariant under $\pi_{\omega}(\mathfrak{A})$ except for
$\mathcal{H}_{\omega}$ itself and $\\{0\\}$.

To abbreviate the notation, when referring to the GNS construction of a state
$\omega$ on an algebra $\mathfrak{A}$ we refer to the uniquely determined, up
to unitary equivalence, triple
$(\mathcal{H}_{\omega},\pi_{\omega},\Omega_{\omega})$.

#####  1.1.3.1 Equivalence of representations

Once a representation is specified, we ask how different ones are related. For
that purpose, we start identifying a specific subset of $\mathfrak{G(A)}$

######  Definition 1.1.16.

Let $\mathfrak{A}$ be a $C^{*}$-algebra and
$\pi:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{\pi})$ a representation of it
over a Hilbert space $\mathcal{H}_{\pi}$. Then, a state
$\omega\in\mathfrak{G(A)}$ is said to be normal with respect to the
representation $\pi$ or $\pi$-normal if there’s a density matrix
$\rho_{\omega}\in\mathfrak{B}(\mathcal{H}_{\pi})$ such that
$\omega(\cdot)=\Tr(\rho_{\omega}\pi(\cdot))$. The set of all $\pi$-normal
states, denoted $\mathfrak{G}^{(\pi)}\mathfrak{(A)}$ is a convex subset of
$\mathfrak{G(A)}$ and is called the folium of $\pi$.
Finally, any normal state of the form:

$\omega(\cdot)=\Tr(\outerproduct{\Psi}{\Psi}\pi(\cdot))$

with unit vector $\ket{\Psi}\in\mathcal{H}_{\pi}$, is called a vector state of
the representation $\pi$.

As proven in Section III$.2.2$ of [16], any state in $\mathfrak{G(A)}$ can be
approximated with arbitrary accuracy in the weak $*$\- topology by a sequence
of normal states.

With this notion we can now define different levels of equivalence between
representations:

######  Definition 1.1.17.

Let $\mathfrak{A}$ be a $C^{*}$-algebra, and let
$\pi_{1}:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{1})$ and
$\pi_{2}:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{2})$ be two different
representations. We say that:

  * •

$\pi_{1}$ is uinitarily equivalent to $\pi_{2}$ ($\pi_{1}\simeq\pi_{2}$) if it
exist a unitary $U:\mathcal{H}_{1}\to\mathcal{H}_{2}$ such that:

$U\pi_{1}(A)U^{*}=\pi_{2}(A)\hskip 15.0pt\forall A\in\mathfrak{A}$

  * •

$\pi_{1}$ is quasi-equivalent to $\pi_{2}$ if
$\mathfrak{G}^{(\pi_{1})}\mathfrak{(A)}=\mathfrak{G}^{(\pi_{2})}\mathfrak{(A)}$

  * •

$\pi_{1}$ is physically-equivalent to $\pi_{2}$ if for every state
$\omega_{1}\in\mathfrak{G}^{(\pi_{1})}\mathfrak{(A)}$ and every weak
$*$-neighborhood $\mathcal{U}(\omega_{1};A_{1},\dots,A_{n};\epsilon)$ of
$\omega_{1}$ there exist a state
$\omega_{2}\in\mathfrak{G}^{(\pi_{2})}(\mathfrak{A})$ such that
$\omega_{2}\in\mathcal{U}(\omega_{1};A_{1},\dots,A_{n};\epsilon)$.
Equivalently, the folium of $\pi_{1}$ is physically indistinguishable from
that of $\pi_{2}$.

The natural question that arises now is: which equivalence do we have for GNS
representations of different states over the same algebra? The most important
result, in finite dimensional QM, is the Stone-von Neumann theorem:

######  Theorem 1.1.3 (Stone-von Neumann).

Let $\mathcal{H}$ be a complex Hilbert space and let $(\mathbf{P},\sigma)$ be
a symplectic vector space of real dimension $2n$. Suppose that we have on
$\mathcal{H}$ a representation of a Weyl $C^{*}$-algebra as a subset of
bounded operators
$\pi(C\mathcal{W}(\mathbf{P},\sigma))\subset\mathfrak{B}(\mathcal{H})$ such
that:

  * •

$\mathcal{H}$ is irreducible under $\pi(C\mathcal{W}(\mathbf{P},\sigma))$

  * •

For every $\mathbf{x}\in\mathbf{P}$ we have that the representation is
strongly continuous:

$s-\lim_{s\to 0}\pi(W(s\mathbf{x}))=\pi(W(0))$

Then, in a given standard symplectic basis of $\mathbf{P}$ for which
$\mathbf{x}\in\mathbf{P}$ is determined by
$(\mathbf{t}^{\mathbf{x}},\mathbf{u}^{\mathbf{x}})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$,
there exist a Hilbert space isomorphism $S:\mathcal{H}\to
L^{2}(\mathbb{R}^{n},dx)$ (where $dx$ denotes the Lebesgue measure) such that
for any $\mathbf{x}$:

$S\pi(W(\mathbf{x}))S^{-1}:=\exp{i\sum_{k=1}^{n}t_{k}^{(\mathbf{x})}q_{k}+u_{k}^{(\mathbf{x})}p_{k}}$

Where $q_{k},p_{k}:\mathcal{S}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n},dx)$
are:

$\displaystyle(q_{k}\psi)(\mathbf{x})$ $\displaystyle=x_{k}\psi(\mathbf{x})$
$\displaystyle(p_{k}\psi)(\mathbf{x})$
$\displaystyle=-i\frac{\partial\psi}{\partial x_{k}}(\mathbf{x})$

As a consequence, $\mathcal{H}$ must be separable as
$L^{2}(\mathbb{R}^{n},dx)$ is.

###### Proof.

See Theorem $11.43$ in [12] ∎

######  Remark 1.1.12.

This theorem shows that any irreducible representation on different Hilbert
spaces $\mathcal{H}$, i.e. with respect to different pure states in the
algebraic formalism, of the Weyl algebra on a finite dimensional symplectic
space is isomorphic to a Weyl algebra on $L^{2}(\mathbb{R}^{n},dx)$. This
means, that the different representations themselves are isomorphic.
Therefore, the representation is unique up to unitary equivalence that, in the
category of equivalences introduced above, means that all the irreducible
representations are unitarily equivalent.

However, the Stone-von Neumann theorem holds just for finite dimensional
symplectic spaces and, when we deal with QFT, the symplectic space is that of
classical solutions of field equations: that is infinite dimensional. So, we
expect the existence of unitarily inequivalent representations. This might
give as a consequence that some state, representing a physically accessible
configuration for the system, might not be represented by a vector in the
chosen Hilbert space associated to another state via a GNS.
Of course this is an issue of the standard approach, where one starts with
just a fixed Hilbert space i.e. one works in a specified GNS representation.
Although, one of the strengths of AQFT is that we do not choose a
representation in the first place and thus we can treat all states on equal
footing and thus study this issue in $\mathfrak{A}^{*}$. In fact, one can show
that, despite the absence of unitary equivalence of different representations,
we always have physical equivalence, i.e. the physics is independent from the
choice of the representation. The following theorem holds:

######  Theorem 1.1.4.

Any two faithful representations of the algebra of observables are physically
equivalent

###### Proof.

I refer to Theorem $2.2.13$ in [6] for further discussion and the, there
cited, original paper [17] for a proof. ∎

The conclusion is that, as one should expect, the representation is just a
convenient choice for practical computations, but it does not have any real
physical implication. We will make use of this in a concrete example in
Chapter $4$. For this reason, if we want to study some process related to a
state $\omega\in\mathfrak{G}(\mathfrak{A})$ is more convenient to choose that
representation where such a state is normal (as they are dense in
$\mathfrak{G}(\mathfrak{A})$), i.e. is represented by a density matrix. What
we have presented, shows that the physically relevant informations are encoded
in the algebra $\mathfrak{A}$ rather than in the Hilbert space representation.

#####  1.1.3.2 Fock representation

Finally, we want to study under which conditions the GNS construction leads to
a Fock space representation:

######  Theorem 1.1.5.

We distinguish:

  1. 1.

For $\omega$ quasifree, defined on the CCR algebra
$\mathfrak{A}_{(\mathbf{P},\sigma)}$, there’s a unique (up to unitary
equivalence) Hilbert space $\mathfrak{h}$ and a real linear map
$K:\mathbf{P}\to\mathfrak{h}$ such that:

$\displaystyle\overline{K\mathbf{P}+iK\mathbf{P}}$
$\displaystyle=\mathfrak{h}$ $\displaystyle\mu(f,h)$
$\displaystyle=\real(\innerproduct{Kf}{Kh}_{\mathfrak{h}})$
$\displaystyle\sigma(f,h)$
$\displaystyle=2\imaginary(\innerproduct{Kf}{Kh}_{\mathfrak{h}})$

The GNS construction applied to $\omega$ yields the symmetric Fock space
$\mathcal{H}$ built upon $\mathfrak{h}$, with the GNS vector given by the Fock
space vacuum $\Omega$ and:

$\pi_{\omega}(A(f))=a_{B}^{*}(Kf)+a_{B}(K\overline{f})$

With $a_{B}^{*}(\psi),a_{B}(\psi)$ the creation annihilation operators for
$\psi\in\mathfrak{h}$, fulfilling:

$\displaystyle a_{B}(\psi)\Omega$ $\displaystyle=0$
$\displaystyle[a_{B}(\psi),a_{B}^{*}(\phi)]$
$\displaystyle=\innerproduct{\psi}{\phi}_{\mathfrak{h}}$

  2. 2.

Let $\mathfrak{A}_{(\mathbf{K},s)}$ be a self-dual CAR algebra. If
$\mathbf{K}$ has even finite dimension or is infinite dimensional, there is a
basis projection444Such a basis projection $P$ is motivated by the physical
model we consider. Usually is the projection onto the positive energy
eigenmodes of the one particle Hamiltonian of the system. $P$ on $\mathbf{K}$
such that $\Gamma P\Gamma=1-P$. We have
$P:\mathbf{K}\to\mathfrak{h}:=P\mathbf{K}$ and we equip $\mathfrak{h}$ with
the restriction of the inner product $s(\cdot,\cdot)$. Then, we can define a
quasifree state $\omega_{P}$ from $P$ which GNS construction yields the
antisymmetric Fock space $\mathcal{H}$ built upon $\mathfrak{h}$, with the GNS
vector given by the Fock space vacuum $\Omega$. Moreover:

$\pi_{P}(B(f))=a_{F}^{*}(Pf)+a_{F}(P\Gamma f)$

Where $f\in\mathbf{K}$ and $B(f)\in\mathfrak{A}_{(\mathbf{K},s)}$ and the
creation and annihilation operators are such that:

$\displaystyle a_{F}(\psi)\ket{\Omega}$ $\displaystyle=0$
$\displaystyle[a_{F}(\psi),a_{F}^{*}(\phi)]_{+}$
$\displaystyle=\innerproduct{\psi}{\phi}_{\mathfrak{h}}$

###### Proof.

For details, see Lemma $3.3$ in [13] together with Lemma $4.3.$ in [18] for
the CAR case and see [19] Prop. $3.2$ and the discussion thereafter for the
bosonic case. Moreover, for details regarding the construction of the state
$\omega_{P}$ see Lemma 3.1.1. ∎

######  Remark 1.1.13.

The Fock space $\mathcal{H}$ mentioned in the above theorem is built as usual:

$\mathcal{H}=\mathbb{C}\oplus\bigoplus_{n>0}E_{n}\mathfrak{h}^{\otimes n}$

Where $E_{n}$ is either the projector onto the totally symmetric subspace (CCR
case) or onto the totally antisymmetric subspace (CAR case). The vacuum vector
corresponds to $\Omega=(1,0,0,\dots)$. Finally, the creation and annihilation
operators mentioned above are defined for $\chi\in\mathfrak{h}$ and for any
$\ket{\Psi_{n}}=E_{n}(\ket{\psi_{1}}\otimes\dots\otimes\ket{\psi_{n}})\in
E_{n}\mathcal{H}^{\otimes n}$ as follows:

$\displaystyle a_{B}^{*}(\chi)\ket{\Psi_{n}}$
$\displaystyle=\sqrt{(n+1)}E_{n+1}\ket{\chi\otimes\Psi_{n}}$ $\displaystyle
a_{B}(\chi)\ket{\Psi_{n}}$
$\displaystyle=\sqrt{\frac{1}{n}}\sum_{j=1}^{n}\innerproduct{\chi}{\psi_{j}}_{\mathfrak{h}}E_{n-1}\ket{\psi_{1}}\otimes\dots\hat{\ket{\psi_{j}}}\otimes\dots\otimes\ket{\psi_{n}}$

for the Bosonic case and as:

$\displaystyle a_{F}^{*}(\chi)\ket{\Psi_{n}}$
$\displaystyle=\sqrt{(n+1)}E_{n+1}\ket{\chi\otimes\Psi_{n}}$ $\displaystyle
a_{F}(\chi)\ket{\Psi_{n}}$
$\displaystyle=\sqrt{\frac{1}{n}}\sum_{j=1}^{n}\innerproduct{\chi}{\psi_{j}}_{\mathfrak{h}}(-1)^{j-1}E_{n-1}\ket{\psi_{1}}\otimes\dots\hat{\ket{\psi_{j}}}\otimes\dots\otimes\ket{\psi_{n}}$

for the Fermionic case. Where the hat means that the vector is removed. In
particular, we see consistently that, e.g. in the $CAR$ case:

$\displaystyle[a_{F}^{*}(\chi),a_{F}(\chi^{\prime})]_{+}\ket{\Psi_{n}}$
$\displaystyle=a_{F}^{*}(\chi)\bigg{(}\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\innerproduct{\chi^{\prime}}{\psi_{j}}E_{n-1}\ket{\psi_{1}}\otimes\dots\hat{\ket{\psi_{j}}}\dots\otimes\ket{\psi_{n}}\bigg{)}+a_{F}(\chi^{\prime})\bigg{(}\sqrt{n+1}E_{n+1}\ket{\chi\otimes\Psi_{n}}\bigg{)}$
$\displaystyle=\bigg{(}\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\innerproduct{\chi^{\prime}}{\psi_{j}}\sqrt{n}E_{n}\ket{\chi}\ket{\psi_{1}}\otimes\dots\hat{\ket{\psi_{j}}}\dots\otimes\ket{\psi_{n}}\bigg{)}$
$\displaystyle\,\,\,\,\,\,+\sqrt{n+1}\frac{1}{\sqrt{n+1}}\bigg{(}\innerproduct{\chi^{\prime}}{\chi}\ket{\Psi_{n}}-\sum_{j=1}^{n}\innerproduct{\chi^{\prime}}{\psi_{j}}E_{n}\ket{\chi}\otimes\ket{\psi_{1}}\otimes\dots\hat{\ket{\psi_{j}}}\dots\otimes\ket{\psi_{n}}\bigg{)}$
$\displaystyle=\innerproduct{\chi^{\prime}}{\chi}\ket{\Psi_{n}}$

Where the minus in the braket at the second step, follows from the fact that
$E_{n}$ is the antisymmetrization projection and thus by putting the
annihilation operator through the $\ket{\chi}$ we need to account for a minus
sign. From this follows that:

$[a_{F}^{*}(\chi),a_{F}(\chi^{\prime})]_{+}=\innerproduct{\chi^{\prime}}{\chi}\mathbb{1}$

In the same way one can show that in the $CCR$ case:

$[a_{B}(\chi^{\prime}),a_{B}^{*}(\chi)]=\innerproduct{\chi^{\prime}}{\chi}\mathbb{1}$

In the following we will drop the $B$ and $F$ subscripts, when the context is
clear.

####  1.1.4 KMS and ground states

In this section, we describe a particular class of states of high physical
relevance: the ground and thermal equilibrium or KMS (Kubo-Martin-Schwinger)
states.
Whenever we have a $C^{*}$-algebra $\mathfrak{A}_{(\mathbf{K},s)}$ and a
transformation $U$ over $(\mathbf{K},s)$, preserving $s(\cdot,\cdot)$, we can
lift it to a $*$-automorphism over the abstract algebra
$\mathfrak{A}_{(\mathbf{K},s)}$555In order $U$ to give a $*$-automorphism in
the case of fermionic fields, one further needs to require $[\Gamma,U]=0$,
with $\Gamma$ the involution over the Hilbert space $(\mathbf{K},s)$.. Namely,
for any $A(f)\in\mathfrak{A}_{(\mathbf{K},s)}$ we have:

$\alpha_{U}A(f):=A(U^{-1}f)\hskip 20.0pt\forall f\in\mathbf{K}$

One particular example is that of a time-evolution. Namely, when the structure
preserving transformation is a one parameter group $V_{t}$ (either of
symplectomorphisms or unitaries). Then, if $t\to V_{t}$ is strongly continuos,
by Stone’s theorem:

######  Theorem 1.1.6 (Stones’s Theorem).

Let $(V_{t})_{t\in\mathbb{R}}$ be a strongly continuous one-parameter unitary
group. Then, there exist a unique, possibly unbounded, operator
$A:\mathcal{D}_{A}\to\mathcal{H}$, that is self-adjoint and such that:

$V_{t}=e^{itA}\hskip 20.0pt\forall t\in\mathbb{R}$

Where:

$\mathcal{D}_{A}=\bigg{\\{}\psi\in\mathcal{H}|\lim_{\varepsilon\to
0}\frac{-i}{\varepsilon}(U_{\epsilon}\psi-\psi)\,\,\mathrm{exists}\bigg{\\}}$

###### Proof.

See Theorem $9.33$ in [12]. ∎

We have existence of a ”Hamiltonian” $\mathbf{h}$, generating the dynamics
$V_{t}$. In this specific case, we shall denote the induced $*$-automorphism
over the algebra as:

$\alpha_{t}(A(f))=A(V_{-t}f)$

Finally, one may further raise this to an automorphism over the states as:

$\alpha_{t}^{*}\omega(A(f)):=\omega(\alpha_{t}(A(f)))$

For any $\omega$ state over $\mathfrak{A}_{(\mathbf{K},s)}$ and
$A(f)\in\mathfrak{A}_{(\mathbf{K},s)}$.

Now that we know how states evolve, we define those that are stationary with
respect to the time evolution:

$\alpha_{t}^{*}\omega=\omega$

Two such examples are the ground and the KMS (Kubo-Martin-Schwinger) states.
We start introducing the first, proving that is stationary.

######  Definition 1.1.18.

A state $\omega$ on $\mathfrak{A}_{(\mathbf{K},s)}$ is called a ground state,
with respect to the time evolution $\alpha_{t}$, if:

$-i\partial_{t}\omega(A^{*}\alpha_{t}(A))|_{t=0}\geq 0\hskip 20.0pt\forall
A\in\mathfrak{A}_{(\mathbf{K},s)}$

######  Proposition 1.1.3.

A ground state $\omega$ over $\mathfrak{A}_{(\mathbf{K},s)}$ is stationary

###### Proof.

Note that to prove the stationarity, is enough to show that
$\partial_{t}\alpha_{t}^{*}\omega|_{t=0}=0$. Choose an $A$ self-adjoint666An
element $A\in\mathfrak{A}_{(\mathbf{K},s)}$ is self-adjoint if $A^{*}=A$:

$\displaystyle\partial_{t}\omega(\alpha_{t}(A^{2}))|_{t=0}$
$\displaystyle=\partial_{t}\omega(\alpha_{t}(A)A)|_{t=0}+\partial_{t}\omega(A\alpha_{t}(A))|_{b=0}$
$\displaystyle=2\real\partial_{t}\omega(A\alpha_{t}(A))|_{t=0}$
$\displaystyle=0$

Proving the invariance when we evaluate squares of self-adjoint operators. But
now, any element $A\in\mathfrak{A}_{(\mathbf{K},s)}$ can be written as a
combination of two sel-adjoint elements
$A_{1/2}\in\mathfrak{A}_{(\mathbf{K},s)}$ as $A=A_{1}+iA_{2}$. Moreover, we
can write any of the self-adjoints:

$A_{1/2}=\frac{1}{2}(\mathbb{1}+A_{1/2})(\mathbb{1}+A_{1/2})-\frac{1}{2}\mathbb{1}-\frac{1}{2}A_{1/2}A_{1/2}$

where all the terms involved are squares of self adjoint elements. So, by
linearity of the state functional and the abovely proven result for squares of
self-adjoint operators, follows the claim. ∎

######  Remark 1.1.14.

The physical motivation behind the definition of ground state is the
following. By the just proven stationarity of the ground state, we have that
the automorphism $\alpha_{t}$ is implemented over the GNS by a unitary
$U_{t}=e^{-itH}$, which leaves $\ket{\Omega}$ invariant:

$\displaystyle-i\partial_{t}\omega(A^{*}\alpha_{t}(A))|_{t=0}$
$\displaystyle=-i\partial_{t}\innerproduct{\Omega}{\pi_{\omega}(A)^{*}e^{itH}\pi_{\omega}(A)e^{-itH}\Omega}$
$\displaystyle=\innerproduct{\pi_{\omega}(A)\Omega}{H\pi_{\omega}(A)\Omega}$

Hence, $\omega$ ground state is equivalent to demand $H\geq 0$ with
$H\ket{\Omega}=0$. Namely, choosing a ground state, corresponds to pick that
representation in which the ”Hamiltonian” generating the time evolution, is
bounded below by $0$.

Let us move on to the definition of KMS state:

######  Definition 1.1.19.

A state $\omega$ over $\mathfrak{A}_{(\mathbf{K},s)}$ is called a Kubo-Martin-
Schwinger (KMS) state, with inverse temperature $\beta$ (with respect to the
time evolution $\alpha_{t}$) if for all $A,B\in\mathfrak{A}_{(\mathbf{K},s)}$:

  * •

The funciton

$F_{A,B}(z):=\omega(A\alpha_{z}(B))$

of complex variable $z$, is bounded for $0\leq\imaginary(z)\leq\beta$ and
analytic in the complex strip $0<\imaginary(z)<\beta$

  * •

The function $F_{A,B}(z)$ and the funciton
$G_{A,B}(z):=\omega(\alpha_{z}(A)B)$ are related:

$F_{A,B}(t)=G_{A,B}(t+i\beta)\hskip 20.0pt\forall t\in\mathbb{R}$

######  Proposition 1.1.4.

A KMS state $\omega$ over $\mathfrak{A}_{(\mathbf{K},s)}$ is stationary

###### Proof.

Take in the KMS condition $A=\mathbb{1}$, then for $z\in\mathbb{C}$ it
becomes:

$F_{B}(z)=\omega(\alpha_{z}(B))$

This function is analytic in $0<\imaginary z<\beta$ and bounded in
$0\leq\imaginary z\leq\beta$, with boundary values which are periodic,
$F_{B}(t+i\beta)=F_{B}(t)$. But then, as an analytic function on a strip with
periodic boundary conditions, it must be constant. We have proven that for all
$B\in\mathfrak{A}_{(\mathbf{K},s)}$ and $t\in\mathbb{R}$,
$\omega(\alpha_{t}(B))=\omega(B)$ that is equivalent to:
$\alpha_{t}^{*}\omega(B)=\omega(B)$ ∎

######  Remark 1.1.15.

The appaently very abstract definition of KMS state, is actually just the
generalization of the condition that one gets in QM on finite dimensional
Hilbert spaces in defining thermal equilibrium or Gibbs states. To see it
explicitly, consider on a system of finitely many degrees of freeedom the
Gibbs state of inverse temperature $\beta$:

$\omega_{\beta}(A):=\frac{\Tr(Ae^{-\beta H})}{Z_{\beta}}\hskip 20.0pt\forall
A\in\mathfrak{A}_{(\mathbf{K},s)}$

where $Z_{\beta}$ is the partition function associated to the thermal state.
The time evolution is given by the Heisenberg equation:

$\alpha_{t}A=e^{itH}Ae^{-itH}$

with Hamiltonian $H$ bounded from below. Therefore:

$\omega_{\beta}(A\alpha_{t}(B))=\frac{\Tr(Ae^{itH}Be^{-(\beta+it)H})}{Z_{\beta}}$

As $H$ is semibounded, the map $z\to e^{izH}$ into bounded operators is
bounded for $\imaginary(z)\geq 0$ and analytic $\imaginary(z)>0$. As a
consequence, the map $z\to e^{-(\beta+iz)H}$ is bounded for
$\imaginary(z)\leq\beta$ and analytic for $\imaginary(z)<\beta$. It follows,
that map $z\to\omega_{\beta}(A\alpha_{z}(B))$ is bounded for
$0\leq\imaginary(z)\leq\beta$ and has domain of analyticity
$0<\imaginary(z)<\beta$. To prove also the second condition of a $KMS$ state,
notice:

$\displaystyle\omega_{\beta}(A\alpha_{t+i\beta}(B))$
$\displaystyle=\frac{\Tr(Ae^{-(\beta-it)H}Be^{-itH})}{Z_{\beta}}$
$\displaystyle=\frac{\Tr(e^{itH}Be^{-itH}Ae^{-\beta H})}{Z_{\beta}}$
$\displaystyle=\omega_{\beta}(\alpha_{t}(B)A)$

Therefore, the Gibbs state at inverse temperature $\beta$ is KMS at inverse
temperature $\beta$.
In this sense, the KMS condition generalizes the notion of thermal equilibrium
in the context of field theories where, the lacking of a notion of trace on
local operator algebras (we discuss this more in Chapter $2$ when we introduce
von Neumann factors), prevents us from defining a density matrix.
Therefore, the motivation behind the KMS condition, is to keep the same
analyticity conditions of the finite dimensional case that are independent
from the fact that $e^{-\beta H}$ must be of trace class. These same
conditions, become analyticity conditions on the state functional with respect
to a given time evolution.

######  Remark 1.1.16.

The existence of a time evlution is crucial, from the physical point of view,
in order to define the notions of ground and KMS states. In fact, is the
general definition of time evolution that allows to assign temperatures in an
observer independent way. In the same way, is the existence of a global time
evolution that allows to define a global notion of energy and as such fix the
scale of the lowest energy level to zero.

We close this section, noticing that a ground state is also $KMS$. Defining:

$F_{A,B}(t)=\innerproduct{\pi_{\omega}(A^{*})\Omega}{e^{itH}\pi_{\omega}(B)\Omega}$

if $\omega$ is a ground state, we have $H\geq 0$ and therefore the function
$z\to F_{A,B}(z)$ is bounded for $\imaginary(z)\geq 0$ and analytic for
$\imaginary(z)>0$. One can even show that the converse holds, i.e. a KMS state
with associated function $F_{A,B}(z)$ bounded for $\imaginary(z)\geq 0$ and
analytic for $\imaginary(z)>0$, defines a ground state (see Prop. $5.3.19$ in
[20]). In this way, a ground state can be understood as the $\beta\to\infty$
limit of a KMS state.

###  1.2 Axioms of AQFT

Now that we have introduced all the necessary mathematical background and
explanied most of the reasons in favour of the algebraic approach to QM, we
define what is a QFT in the algebraic sense. This was done in an axiomatic
way, first by Wightman and Garding in [21] and later to nets of local algebras
by Haag and Kastler in [6]. Here we adopt the Haag-Kastler axioms, and present
them in a more modern fashion following Section $2.3$ of [22].
In algebraic quantum field theory, the algebraic relations between the quantum
fields are encoded in nets of $C^{*}$-algebras associated with spacetime
regions, with partial ordering given by the usual inclusion of sets on the
spacetime. The formalism is then perfect to define Quantum Fields even on
curved backgrounds (in a semiclassical way), generalizing already the standard
approach. The structure and type of such nets depends of course on the type of
field theory and on the spacetime background.
To start and establish the connection with the standard approach, consider
Minkowski spacetime $(\mathbb{R}^{4},\eta)$. Consider on it the set of
causally complete regions $\mathcal{K}$ and the subset of causal diamonds
$\mathcal{O}$. The Poincaré group acts freely on Minkowski, i.e. for any
$g\in\mathcal{P}^{\uparrow}_{+}=\mathcal{L}_{+}^{\uparrow}\ltimes\mathbb{R}^{4}$,
that we write as $g=(\Lambda,a)$, we have:

$g\cdot x=\Lambda x+a\hskip 15.0pt\forall x\in\mathbb{R}^{4}$

Now, since Poincaré transformations are isometries of Minkowski, they map
causal diamonds in other causal diamonds as they are causally complete
regions.

######  Definition 1.2.1 (Algebraic Quantum Field Theory).

An algebraic quantum field theory (AQFT) is defined by a $C^{*}$-algebra
$\mathfrak{A}$, called the algebra of observables777The name is a bit
misleading, as elements of $A\in\mathfrak{A}$ are not all self adjoint, i.e.
not all of them are proper observables in the sense of being actually
measurable., and an assignement to every causal diamond $\mathcal{O}$ of a
$C^{*}$-subalgebra of $\mathfrak{A}$:

$\mathcal{K}\ni\mathcal{O}\mapsto\mathfrak{A}(\mathcal{O})\subset\mathfrak{A}$

which are called the local algebras of observables. This net of algebras
$\mathfrak{A}(\mathcal{O})$ must satisfy the following set of axioms:

  * A1

(Isotony) If $\mathcal{O}_{1}\subset\mathcal{O}_{2}$ we have
$\mathfrak{A}(\mathcal{O}_{1})\subset\mathfrak{A}(\mathcal{O}_{2})$. Moreover:

$\mathfrak{A}=\overline{\bigcup_{\mathcal{O}}\mathfrak{A}(\mathcal{O})}$

With the completion taken with respect to the $C^{*}$-norm

  * A2

(Causality) If $\mathcal{O}_{1}$ is spacelike separated from $\mathcal{O}_{2}$
we must have:
$[\mathfrak{A}(\mathcal{O}_{1}),\mathfrak{A}(\mathcal{O}_{2})]=0$. That means
that algebras at spacelike separation must commute. This can also be rewritten
as:

$\mathfrak{A}(O^{\prime})\subset\mathfrak{A}(O)^{\prime}$

  * A3

(Relativistic Covariance) For each transformation
$g\in\tilde{\mathcal{P}}^{\uparrow}_{+}$ (the universal covering group of
$\mathcal{P}^{\uparrow}_{+}$), there’s an automorphism $\alpha_{g}$ on
$\mathfrak{A}$ such that:

$\alpha_{g}\mathfrak{A}(\mathcal{O})=\mathfrak{A}(\Lambda\mathcal{O}+a)$

for all causal diamonds $\mathcal{O}$ and such that:
$\alpha_{g}\alpha_{g^{\prime}}=\alpha_{gg^{\prime}}$,
$\alpha_{(\mathbb{1},0)}=\mathbb{1}$

  * A4

(Vacuum) There’s a unique state $\omega_{0}$ over $\mathfrak{A}$ that is
invariant under $\alpha_{g}$. On its GNS representation
$(\pi_{0},\mathcal{H}_{0},\ket{0})$, $\alpha_{g}$ is implemented by a
projective representation $U$, with positive energy, of the universal covering
$\tilde{\mathcal{P}}^{\uparrow}_{+}$. This means that
$\pi_{0}(\alpha_{g}(A))=U(g)\pi_{0}(A)U(g)^{*}$ for all $A\in\mathfrak{A}$ and
$g\in\tilde{\mathcal{P}}^{\uparrow}_{+}$. Positive energy means that the
representation is strongly continuous and if we pick $x\in\mathbb{R}^{4}$ and
the corresponding translation operator
$(\mathbb{1},x)\in\tilde{\mathcal{P}}^{\uparrow}_{+}$ we have:

$U(x)=\exp(-iP^{\mu}x_{\mu})$

And the vector generator $P=(P^{\mu})$ has spectrum $p=p^{\mu}$ in the forward
lightcone $p\in\overline{V}^{+}=\\{p|p^{2}\geq 0,p^{0}>0\\}$

######  Remark 1.2.1.

The axioms are motivated by our expectation. The first reflects the intuition
that, if $\mathfrak{A}(\mathcal{O}_{2})$ represents everything that can be
observed in the region $\mathcal{O}_{2}$, whenever we consider a smaller
region $\mathcal{O}_{1}\subset\mathcal{O}_{2}$ all that can be observed here
is for sure contained in what was observable in the wider region. The second
axiom, as the name suggests, is just the reflection of the classical idea of
causality translated in the language of Quantum Mechanics: as the operators
commute, we can in an abstract sense measure something in
$\mathfrak{\mathcal{O}}$ without affecting the measure in
$\mathfrak{A}(O^{\prime})$ as these are compatible observables. The third
axiom reflects the attachement of the net of algebras to the spacetime regions
and as such they should transform according to the way in which the spacetime
regions transform. The fourth is motivated by the fact that each inertial
observer should measure the same physics in his system of reference.

If one considers a more general spacetime background, one needs to modify
these axioms. Clearly axioms $\mathbf{A1},\mathbf{A2}$ can be generalized
straightforwardly to the case of curved backgrounds as they do not involve any
specific feature of $(\mathbb{R}^{4},\eta)$. At the same time, if the
spacetime has a specific symmetry group $G$, one can generalize also
$\mathbf{A3}$ requiring the same to hold after replacing
$\tilde{\mathcal{P}}^{\uparrow}_{+}$ with $G$. What il less easy to generalize
is $\mathbf{A4}$ as, in general spacetimes, there’s no unique vacuum. This is
due to the general absence of a timelike Killing vector field. If we would
have rather had such a symmetry (Minkowski spacetime or more generally on
static spacetimes), we can impose a globally defined notion of positive and
negative energy modes giving a globally defined notion of vacuum once we
consider any two different observers moving along the flow lines of the
timelike Killing field.

However, in the axiomatic definition, there’s still a caveat: from standard
QFT and statistical physics we know that fermions have anticommutation
relations. This translates in fermionic field opeators that do not commute at
spacelike separation but rather anticommute. This violates the Causality axiom
of an AQFT. There are two way out for this problem:

  1. 1.

Modify the axioms by requiring a weaker version of locality called graded
locality

  2. 2.

Consider a subset of the fermionic algebra of fields corresponding to field
binomial such that causality holds on that subset

A good reason in favour of the second choice, is that we are just able to
measure quantities composed by fermionic binomials, as one can see also from
the form of the Dirac Lagrangian. However, as it is easier to work with the
full algebra, we will here adopt the first approach and consider the fermionic
algebras to be an example of an AQFT with respect to graded locality.
So, let me introduce this concept:

######  Definition 1.2.2.

Let $(\mathfrak{I}(\mathcal{O}))_{\mathcal{O}}$ be the net of local
$C^{*}$-algebras acting on a Hilbert space $\mathcal{H}$, i.e.
$\mathfrak{I}(\mathcal{O})\subset\mathfrak{B}(\mathcal{H})$ and let
$\ket{\Omega}$ be the cyclic vector for $\mathcal{H}$. A
$\mathbb{Z}_{2}$-grading on $\mathfrak{I}(\mathcal{O})$ is defined by
$\Gamma\in\mathfrak{B}(\mathcal{H})$ such that $\Gamma=\Gamma^{-1}=\Gamma^{*}$
and:

$\displaystyle\Gamma\ket{\Omega}$ $\displaystyle=\ket{\Omega}$
$\displaystyle\Gamma\mathfrak{I}(\mathcal{O})\Gamma$
$\displaystyle=\mathfrak{I}(\mathcal{O})\hskip
15.0pt\forall\mathcal{O}\in\mathcal{K}$ $\displaystyle[\Gamma,U(\tilde{g})]$
$\displaystyle=0\hskip 15.0pt\forall\tilde{g}\in\tilde{P}^{\uparrow}_{+}$

An operator $A\in\mathfrak{I}$ such that $\Gamma A\Gamma=\pm A$ is called
homogeneous Bose (even) or Fermi (odd) depending on the alternative $\pm$. For
this reason, we introduce a degree $I_{A}$ for the Homogeneous operator $A$
that is $0$ if it is Bose and $1$ if it is Fermi

We will call a net of local algebra with a $\mathbb{Z}_{2}$-grading a
$\mathbb{Z}_{2}$-graded net of local algebras. Moreover, any
$A\in\mathfrak{I}$ can be decomposed as a sum $A=A_{+}+A_{-}$ of a Bose and a
Fermi operator:

$A_{\pm}:=\frac{A\pm\Gamma A\Gamma}{2}$

Where of course $A_{\pm}\in\mathfrak{I}(\mathcal{O})$.
Now that we have introduced the notion of grading, we define the graded
commutator

######  Definition 1.2.3.

Let $(\mathfrak{I}(\mathcal{O}))_{\mathcal{O}}$ be a $\mathbb{Z}_{2}$-graded
net of local algebras. We define the graded commutator for
$A,B\in\mathfrak{I}$ homogenous as:

$[A,B]_{\Gamma}:=AB-(-1)^{I_{A}\cdot I_{B}}BA$

and we extend this to general $A,B\in\mathfrak{I}$ by linearity.

From the definition follows immediately that, for both homogeneous operators
of the Fermi type, the graded commutator reduces to an anticommutator, while
is a standard commutator for Bose homogeneous operators.
We can now replace the axiom $\mathbf{A2}$ with a generalized version:

  * A2’

Let $(\mathfrak{I}(\mathcal{O}))_{\mathcal{O}}$ be a $\mathbb{Z}_{2}$-graded
net of local algebras. We say that the local algebra satisfies graded locality
if, for any pair of spacelike separated regions
$\mathcal{O}_{1},\mathcal{O}_{2}$, we have that
$[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}=0$

In this manner Fermionic Quantum Field Theories fit in the formalism of AQFT.
The notion of $\mathbb{Z}_{2}$-grading is often replaced in litterature by
twisted locality [23],[24] and [25]:

######  Definition 1.2.4.

Let $(\mathfrak{I}(\mathcal{O}))_{\mathcal{O}}$ be a $\mathbb{Z}_{2}$-graded
net of local algebras and let:

$Z:=\frac{1-i\Gamma}{1-i}$

We define the twisted commutant of a local algebra as
$\mathfrak{I}(\mathcal{O})^{t^{\prime}}:=Z\mathfrak{I}(\mathcal{O})^{\prime}Z^{*}$.
We say that the net satisfies twisted locality if, for any two spacelike
separated regions $\mathcal{O}_{1},\mathcal{O}_{2}$, we have
$\mathfrak{I}(\mathcal{O}_{1})\subset\mathfrak{I}(\mathcal{O}_{2})^{t^{\prime}}$

One can prove that the two are equivalent:

######  Lemma 1.2.1.

Let $\mathfrak{I}(\mathcal{O})$ be a $\mathbb{Z}_{2}$-graded net of local
algebras. Then, it satisfies graded locality if and only if it satisfies
twisted locality

###### Proof.

$(\Rightarrow)$ By assumption
$[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}=0$. We
want to show that $\mathfrak{I}(\mathcal{O}_{1})\subset
Z\mathfrak{I}(\mathcal{O}_{2})Z^{*}=:\mathfrak{I}(\mathcal{O}_{2})^{t^{\prime}}$.
Start noticing:

$\displaystyle ZZ^{*}$ $\displaystyle=\frac{1}{2}(1-i\Gamma)(1+i\Gamma)$
$\displaystyle=\frac{1}{2}(1+i\Gamma-i\Gamma+\Gamma^{2})$ $\displaystyle=1$

Let us first look at the case in which the $\mathfrak{I}(\mathcal{O}_{i})$ is
homogeneous for all $i$. For Bose operators, we know:

$\displaystyle[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}$
$\displaystyle=\mathfrak{I}(\mathcal{O}_{1})\mathfrak{I}(\mathcal{O}_{2})-\mathfrak{I}(\mathcal{O}_{2})\mathfrak{I}(\mathcal{O}_{1})=0$
$\displaystyle Z\mathfrak{I}(\mathcal{O}_{2})^{\prime}Z^{*}$
$\displaystyle=\mathfrak{I}(\mathcal{O}_{2})^{\prime}$

But the first of these equations implies that:

$\mathfrak{I}(\mathcal{O}_{1})\subset\mathfrak{I}(\mathcal{O}_{2})^{\prime}=\mathfrak{I}(\mathcal{O}_{2})^{t^{\prime}}$

For what concerns the case of Fermi homogeneous operators we know:

$[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}=\mathfrak{I}(\mathcal{O}_{1})\mathfrak{I}(\mathcal{O}_{2})+\mathfrak{I}(\mathcal{O}_{2})\mathfrak{I}(\mathcal{O}_{1})=0$

But then, if we look at:

$\displaystyle\mathfrak{I}(\mathcal{O}_{1})Z\mathfrak{I}(\mathcal{O}_{2})Z^{*}-Z\mathfrak{I}(\mathcal{O}_{2})Z^{*}\mathfrak{I}(\mathcal{O}_{1})$
$\displaystyle=\frac{1}{2}\mathfrak{I}(\mathcal{O}_{1})(\mathfrak{I}(\mathcal{O}_{2})+i\mathfrak{I}(\mathcal{O}_{2})\Gamma-i\Gamma\mathfrak{I}(\mathcal{O}_{2})+\Gamma\mathfrak{I}(\mathcal{O}_{2})\Gamma)$
$\displaystyle\,\,\,\,\,\,-\frac{1}{2}(\mathfrak{I}(\mathcal{O}_{2})+i\mathfrak{I}(\mathcal{O}_{2})\Gamma-i\Gamma\mathfrak{I}(\mathcal{O}_{2})+\Gamma\mathfrak{I}(\mathcal{O}_{2})\Gamma)\mathfrak{I}(\mathcal{O}_{1})$
$\displaystyle=\frac{1}{2}\mathfrak{I}(\mathcal{O}_{1})(+i\mathfrak{I}(\mathcal{O}_{2})\Gamma-i\Gamma\mathfrak{I}(\mathcal{O}_{2}))-\frac{1}{2}(+i\mathfrak{I}(\mathcal{O}_{2})\Gamma-i\Gamma\mathfrak{I}(\mathcal{O}_{2}))\mathfrak{I}(\mathcal{O}_{1})$
$\displaystyle=\frac{1}{2}(i\mathfrak{I}(\mathcal{O}_{1})\mathfrak{I}(\mathcal{O}_{2})\Gamma+i\Gamma\mathfrak{I}(\mathcal{O}_{1})\mathfrak{I}(\mathcal{O}_{2}))-\frac{1}{2}(-i\mathfrak{I}(\mathcal{O}_{2})\mathfrak{I}(\mathcal{O}_{1})\Gamma-i\Gamma\mathfrak{I}(\mathcal{O}_{2})\mathfrak{I}(\mathcal{O}_{1}))$
$\displaystyle=\frac{i}{2}\Gamma[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}+\frac{i}{2}[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}\Gamma=0$

Where we have first used the fact that for homogenous Fermi operators
$\Gamma\mathfrak{I}(\mathcal{O}_{i})\Gamma=-\mathfrak{I}(\mathcal{O}_{i})$ and
at the third step that $\Gamma^{2}=1$.
The general case follows from the fact that any
$A\in\mathfrak{I}(\mathcal{O}_{i})$ can be written as $A=A_{+}+A_{-}$.

$(\Leftarrow)$ From the above computations, we have seen for homogeneous Bose:
$\mathfrak{I}(\mathcal{O}_{i})^{\prime}=\mathfrak{I}(\mathcal{O}_{i})^{t^{\prime}}$,
so if
$\mathfrak{I}(\mathcal{O}_{1})\subset\mathfrak{I}(\mathcal{O}_{2})^{t^{\prime}}$
we have:

$[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}=0$

While, for the Fermi case we have seen above that:

$\mathfrak{I}(\mathcal{O}_{1})Z\mathfrak{I}(\mathcal{O}_{2})Z^{*}-Z\mathfrak{I}(\mathcal{O}_{2})Z^{*}\mathfrak{I}(\mathcal{O}_{1})=\frac{i}{2}\Gamma[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}+\frac{i}{2}[\mathfrak{I}(\mathcal{O}_{1}),\mathfrak{I}(\mathcal{O}_{2})]_{\Gamma}\Gamma$

But if
$\mathfrak{I}(\mathcal{O}_{1})\subset\mathfrak{I}(\mathcal{O}_{2})^{t^{\prime}}$,
the left hand side must vanish, from which it follows the claim. ∎

We mentioned this because we are going to use the notion of graded locality
pointing although at litterature where the notion of twisted locality is used.

From what we said in the previous sections, formulating a Quantum field theory
just requires the construction of an algebra, corresponding to the operators
representing the quantum fields, fulfilling the above axioms. Once that is
done we can formulate the standard approach to a free QFT by picking a
specific Fock representation using a quasi-free state. In the following
sections, I will present an example of QFT that fulfills the above listed
axioms: the Dirac-Majorana field. From what we have just mentioned, we need to
show how to construct the algebras of observables starting from the classical
field equations. This process is often called quantisation.
Moreover, as mentioned above, AQFT is the perfect framework to generalize to
aribtrary spacetimes backgrounds, therefore we will work on curved spacetimes.
This will require some technicalities in the classical definition of Dirac
fields, that we are going to discuss at the beginning of the next chapter.
The choice of presenting just the Dirac field, despite it is more complicated
than the free scalar field, is due to the results of this thesis that are
presented in Chapter $3$ and that regard fermionic fields. Morevoer, the free
scalar field is treated in most of the literature so I refer the interested
reader to [26], [16], [12] and the literature cited there, for the original
papers.

###  1.3 Dirac and/or Majorana fields

The free scalar fields on arbitrary curved spacetime $(M,g)$, are just
sections of a trivial bundle $M\times\mathbb{C}$. As such, they are always
well defined on each manifold, provided that we have existence and uniqueness
of solutions of fields equations, i.e. if the spacetime is globally
hyperbolic. In contrast the formulation of the classical Dirac equation on a
spacetime manifold $(M,g)$ requires the spacetime manifold to have additional
properties and the Dirac fields are going to be sections of a non-trivial
bundle.
We will start the chapter by discussing which additional properties the
Manifold must have in order to be able to formulate the classical Dirac
equation.
I refer to Chapter $13$ of [27] for the physical motivation and applications
behind the introduction of Spin manifolds, to the Lecture notes [28] for a
modern mathematically rigorous approach to Spin geometry and to [29], [30] and
[31] for literature related to the quantization of Dirac fields.

####  1.3.1 Spin bundle

The concpet of Spin on Minkowski spacetime, relies on the existing projective
representation of its symmetry group (the Poincaré group or to be more precise
the proper orthocronus part of its universal covering) over the Hilbert space
of the QFT. Following Wigner’s approach [32], the ”elementary” fields are
classified according to the irreducible representations of the symmetry group.
These are labelled by the eigenvalues of the Casimir operators $m^{2}$ and
$S^{2}$, respectively the mass $m$ a positive real number and a discrete888In
the massless case is known the existence of irreducible representations with
non-trivial translations that admit continuous spin representations. However,
such fields have, so far, never been of physical relevance. For further
details on it and additional literature I refer to [33] integer or semiinteger
that labels the so called spin or, in the massless case, helicity.
The strict connection with the underlying spacetime geometry is manifest and
one should expect that notions like spin and spinors (vectors that transform
according to the two fold covering of the proper orthocronus Lorentz group)
have to be reviewed on arbitrary spacetime geometries where the isometry group
can also be trivial.
The idea is to rely on Einstein’s Equivalence Principle (EEP): each spacetime
is, at least locally, Minkowski. Namely, if we consider an observer $O$
specified at each spacetime point $x\in M$ by a tetrad of orthonormal vectors
$\\{e_{a}\\}_{a=0,1,2,3}\in T_{x}M$, where $e_{0}$ is the normalized vector
tangent to the worldline of $O$, another observer $\tilde{O}$ at the same
spacetime point $x\in M$ has its tetrad of vectors that must be related to
that of $O$ by a Lorentz transformation $\phi_{g}$. Moreover, each measurement
done by $O$ on a system $\mathcal{S}$ at $x$ must agree with a measurement
done by $\tilde{O}$ at the same spacetime point $x$ on $\tilde{\mathcal{S}}$
that is: $\tilde{\mathcal{S}}=\tilde{\phi}_{g}\mathcal{S}$ where the
transformation $\tilde{\phi}_{g}$ is the induced action on physical
observables existing by EEP.

Formally, the above means that at each $x\in M$, for each time oriented
tetrads, we can find Lorentz transformations mapping one tetrad into another.
This transformation must induce, by EEP, a transormation over the elements of
a theory satisfying the EEP. This, holding for tetrads at the same spacetime
point $x$, is a fiberwise statement. Our aim, is to extend the relation
between the tetrads in the fiber globally, in a continuous way, as a relation
between sections of the fiber bundle.
So, start defining the principal fiber bundle as the bundle of oriented, time
oriented and orthonormal tetrads on which we have a free action of the proper
orthocronus Lorentz group $\mathcal{L}_{+}^{\uparrow}$. A section of this
bundle is an assignement of a frame at each spacetime point. In this way the
fibers are diffeomorphic to the proper orthocronus Lorentz group:

  * •

The time orientation of the orthonormal basis gives that each observer has the
same time orientation, thus they must be related by an orthochronous Lorentz
transformation

  * •

The orientation assumption tells us that the observers must be related to each
other by a transformation in the same connected component of the Lorentz group
and since the identity transformation maps ”two” such observers, we are
considering the proper part of the Lorentz group

Therefore, starting from the standard tetrad coming from the orientation and
time orientation assumptions, we have a one to one correspondence between
elements in $\mathcal{L}_{+}^{\uparrow}$ and tetrads
$\\{e^{a}\\}_{a=0,1,2,3}$.
We call such a fiber bundle a Frame bundle and denote it as $FM$ and we will
denote the principal frame bundle, to emphasize the existence of a free right
action of the proper orthocronus Lorentz group, by
$(FM,\mathcal{L}^{\uparrow}_{+},M,\phi)$ with
$\phi:FM\times\mathcal{L}^{\uparrow}_{+}\to FM$ the free right action defined
as:

$\phi:(v,A)\mapsto v\cdot A$

Where $\cdot$ denotes the matrix multiplication between the element in the
Lorentz group associated to the section $v(x)$ at each spacetime point $x\in
M$ and the element $A\in\mathcal{L}^{\uparrow}_{+}.$

However, as we know from the flat spacetime case, in order to define spinors
with non trivial behavior under a $2\pi$ rotation, we need the universal
covering of $\mathcal{L}_{+}^{\uparrow}$. Therefore we ”duplicate” each fiber
to produce a principal $Spin_{1,3}^{0}$ bundle over $M$, where
$Spin_{1,3}^{0}$ denotes the universal covering999The explicit definition of
the Spin group is reported in Appendix A.4 of $\mathcal{L}_{+}^{\uparrow}$.

######  Definition 1.3.1.

A spin structure on $(M,g)$ is a pair $(SM,p)$, where $SM$ is a principal
$Spin_{1,3}^{0}$-bundle over $M$, the spin frame bundle, which carries a right
action $R_{S}$ with respect to $S\in Spin_{1,3}^{0}$. While, $p:SM\to FM$ is a
base-point preserving bundle homomorphism such that:

$p\circ R_{S}=R_{\Lambda(S)}\circ p$

Where $S\mapsto\Lambda(S)$ is the canonical universal covering map of the
Lorentz group:

$\displaystyle\Lambda:Pin_{1,3}$ $\displaystyle\to\mathcal{L}$ $\displaystyle
S$ $\displaystyle\mapsto\Lambda^{a}_{\,\,b}(S)$

see Appendix A.4.

The just outlined construction of spin structures, may be prevented by
topological obstructions of the manifold $M$. To understand why, consider
first the simple case of $M$ simply connected and thus orientable and time
orientable101010Suppose M is not orientable by absurd. Consider then its
oriented double covering $p:X\to M$ and pick $x_{0}\in X$. Let $\lambda$ be a
path connecting $x_{0}$ and some other point in the different fiber of the
double covering, call this point $x_{1}$ (here we are using that the double
cover of a non-orientable $M$ is connected - since it is a manifold, it is
also path-connected). Now, $p\circ\lambda$ is a loop in $M$, which lifts to
$\lambda$. Since $M$ is simply connected, $p\circ\lambda$ is homotopically
trivial. But this is an absurd, since $\lambda$ is not a loop and this will
imply $x_{0}=x_{1}$. Construct on it the frame bundle
$(FM,\mathcal{L}^{\uparrow}_{+},M,\phi)$ of oriented and time oriented frames.
Consider a curve $\gamma\in FM$, from the simply connectedness of $M$, we know
that $\mathfrak{p}\circ\gamma$ (the projection of the path $\gamma\in FM$ to
the base manifold $M$) is contractible to the trivial closed curve at a point
$x$ lying on $\mathfrak{p}\circ\gamma$. This means that we can continuously
deform $\gamma$ in $FM$ to a loop restricted to
$\mathfrak{p}^{-1}(x)=(\mathcal{L}^{\uparrow}_{+})_{x}$ (the subscript denotes
the fiber in $FM$ at the point $x$). But we know that
$\pi_{1}(\mathcal{L}^{\uparrow}_{+})=\mathbb{Z}_{2}$, therefore this loop is
not contractible within $\mathcal{L}^{\uparrow}_{+}$. The fact that such a
loop is not contractible is, from standard QFT, crucial for the definition of
spinors on Minkowski spacetime, as it assigns a sign to a spinor depending on
their behaviour under a $2\pi$ rotation. In this case,
$(\mathcal{L}^{\uparrow}_{+})_{x}$ is the fiber of $FM$ and thus if we want a
global notion of spin we need this loop to be incontractible also throughout
$FM$, otherwise we may start with Spin half field on a fiber move along a
closed curve in $FM$ and end up having a spin zero field. Therefore, we need
to make sure that such loops, not just fiberwise, but also by deforming them
throughout $FM$, are not all contractible: if $FM$ is simply connected, the
notion of spinors on M cannot be defined.111111 Actually, it exists a more
general result: $FM$ is not simply connected if and only if its second
Stiefel-Whitney class of $M$ vanishes. I refer to [27] for further literature
on it, where the Stiefel-Whitney class is defined..
In the case in which $M$ is not simply connected, the manifold may first of
all fail to be orientable or time orientable. In that case, even $FM$ as the
frame bundle of oriented and time oriented basis cannot be defined. Therefore,
we need to asume $M$ to be at least orientable and time orientable if we want
to define a notion of Spin.
Then, $M$ is assumed orientable and time orientable but not simply connected.
In constructing $SM$, we cannot simply take the universal covering of $FM$ as
now also $M$ is not simply connected. If we were to do that, the universal
covering will also have a modified base manifold $M$ while we want to
”duplicate” just the fibers. We can still demand, just the fibers to be
”unwrapped”, if the fundamental group is canonically splitted:

$\pi_{1}(FM)=\pi_{1}(\mathcal{L}^{\uparrow}_{+})\times\pi_{1}(M)$ (1.2)

In this case, we can split the task of finding the universal covering of the
fibers and of the base manifold, as each loop in the bundle can be splitted in
a loop in $M$ and another in the fiber. Moreover, if $M$ is simply connected
we get back the previously discussed case.
Therefore, Eq. (1.2) is a necessary condition to have spinors121212Actually
the above splitting might not be unique, leading to different notions of
spinors introducing an ambiguity that results in the so called exotic spin
structures and the number of them equals that of generators of the first
cohomology group $H^{1}(M;\mathbb{Z}_{2})$. A possible physical interpretation
of them was discussed in [34].

Now that we have discussed the conditions for existence of Spinors, it is
natural to ask whether the manifolds we are interested in admit such
structures. In particular, as we want to formulate field theories, to have
existence and uniqueness of solutions of the Cauchy problem for differential
equations we need to assume the spacetime to be globally hyperbolic 131313If
we are dealing with spacetimes with boundaries, like the one of the Casimir
effect, $I\times\mathbb{R}$ for $I\subset\mathbb{R}$ some compact real
interval, then we can still have existence and uniqueness of solutions for
boundary value problems by assigning Dirichlet boundary condition, despite
such spacetimes are not globally hyperbolic. However, in what follows, we will
study Cauchy problems for the field equation, therefore the spacetimes will
always be assumed to be globally hyperbolic.. For this reason, it’s natural to
ask whether a globally hyperbolic, orientable spacetime admits spinors. Let me
start quoting a general theorem proved by Geroch [35]:

######  Theorem 1.3.1.

Let $M$ be an open manifold with a Lorentzian metric $g$, then $M$ admits a
spinor structure if and only if there exists on $M$ a global system of
orthonormal tetrads

The condition of admitting a global orthonormal tetrad, is equivalent for the
spacetime to be parallelizable141414A Manifold $M$ is parallelizable, if there
exist smooth vector fields $V_{1},\dots,V_{n}$ on the manifold, such that at
every point $p\in M$ the tangent vectors $V_{1}(p),\dots,V_{n}(p)$ provide a
basis of the tangent space at $p$. Equivalently a manifold is parallelizable
if the tangent bundle is trivial, i.e. if the frame bundle has a global
section on $M$. A globally hyperbolic spacetime is parallelizable, by the
following two theorems ([26], [36]):

######  Theorem 1.3.2.

A globally hyperbolic spacetime is diffeomorphic to $\mathbb{R}\times\Sigma$
and isometric to $\mathbb{R}\times\Sigma$, equipped with metric:

$g=-\beta(\tau,\mathbf{x})d\tau^{2}+\gamma_{ij}(\tau,\mathbf{x})dx^{i}dx^{j}$

with $\tau\in\mathbb{R}$ and $\mathbf{x}=\\{x^{i}\\}$ coordinates on $\Sigma$,
such that $\beta>0$, $\Sigma_{\tau}=\\{\tau\\}\times\Sigma$ is a Cauchy
surface and $(\Sigma_{\tau},\gamma_{ij}(\tau,\dot{)})$ is a Riemannian
Manifold

######  Theorem 1.3.3 (Stiefel).

An orientable $3$ dimensional manifold $\Sigma$ is parallelizable

Therefore, an orientable globally hyperbolic spacetime151515Because time
orientability is implied by global hyperbolicity from the previous theorem, by
the first theorem, is diffeomorphic to $M\simeq\mathbb{R}\times\Sigma$ and
since, by assumption, it was orientable also $\Sigma$ must be orientable.
Therefore, by the Stiefel theorem, $\Sigma$ is parallelizable. So, since the
cartesian product of parallelizabe manifolds is still parallelizable, we
conclude from Geroch’s result that each globally hyperbolic manifold admits a
spinor structure.

####  1.3.2 Dirac spinors and cospinors

During the next section we will use, especially for proofs, concepts and
results related to the Dirac Clifford algebra that are reported in the
Appendix A.4. However, leaving aside proofs, the section can be followed
without all the details reported in Appendix and we will always cite
definitions and statements necessary to follow the discussion. Moreover, from
now on, $(M,g)$ is assumed to be globally hyperbolic admitting thus a spin
structure.
We turn to the construction of spinors. Let’s start by choosing a complex
irreducible representation $\pi$ of the Dirac algebra $D$ and matrices $A,C\in
GL(4,\mathbb{C})$ reppresenting the charge and hermitian conjugation (see
Appenix A.4 and Definition A.4.2).
Consider a globally hyperbolic spin spacetime $SM$, we define the assocaited
vector bundle:

$DM:=SM\times_{Spin_{1,3}^{0}}\mathbb{C}^{4}$

Where $Spin_{1,3}^{0}$ acts on $SM$ from the right and on $\mathbb{C}^{4}$
from the left via the representation $\pi$. In other words, $DM$ is obtained
from the product bundle $SM\times\mathbb{C}^{4}$ by identifying:

$[E,z]=[R_{S}E,\pi(S^{-1})z]$

Where we think of $z\in\mathbb{C}^{4}$ as a column vector. A spinor is thus an
equivalence class of tetrads or of a $2\pi$ rotation of it (due to the double
covering definition of $SM$) and an ordered quadrupole of comlpex numbers
transforming according to a representation of the $Spin^{0}_{1,3}$. The
equivalence relation identifies what observers, differing by a Lorentz
transformation, see. A spinor, is thus Lorentz invariant by definition. In
this way, if we let $M_{2\pi}$ denote the $2\pi$ rotation in $Spin_{1,3}^{0}$
with respect to a frame element $e_{3}$ of the section $E\in SM$, we see that
$[E,z]$ and $[R_{2\pi}E,z]=[E,\pi(M_{2\pi})z]$ describe different spinors,
allowing to distinguish them depending on their sign under a $2\pi$ rotation.
If we have worked with $FM$ instead of $SM$ this distinction would not have
been possible.
We can analogously introduce the dual bundle $D^{*}M$, or use the usual
isomorphism from $\mathbb{C}^{4}\to(\mathbb{C}^{4})^{*}$ induced by the
standard inner product over $\mathbb{C}^{4}$, to define elements of $D^{*}M$:

$[E,w^{*}]=[R_{S}E,w^{*}\pi(S)]$

In this way, we can pointwise on the base manifold (fiberwise) interpret
$D^{*}M$ as linear functionals over $DM$ via the following notation:

$\langle[E,w^{*}],[E,z]\rangle:=w^{*}(z)=\langle w,z\rangle$

Where we need both elements having the same base ”point” $E$, in order this
inner product to be well defined. The right most expression above, denotes the
standard inner product of $\mathbb{C}^{4}$.

######  Definition 1.3.2.

The vector bundle $DM$ is called the Dirac spinor bundle, its elements are
called Dirac spinors and a section of $DM$ is called a Dirac spinor field. The
space of all smooth spinor fields is denoted as $C^{\infty}(DM)$, and the
compactly supported ones as $C^{\infty}_{0}(DM)$.
The vector bundle $D^{*}M$ is called the Dirac cospinor bundle with relatively
analogous definitions and notations.

For notational convenience we indicate the canonical pairing of a spinor field
$u$ with a cospinor $v$ by:

$vu(x):=\langle v(x),u(x)\rangle$

that defines a sesquilinear map (by the sesquilinearity of the standard inner
product on $\mathbb{C}^{4}$): $C^{\infty}(D^{*}M)\times C^{\infty}(DM)\to
C^{\infty}(M)$ as the image is a function of the spacetime point $x$.
Let us now introduce the charge conjugation and Dirac adjoint maps, for spinor
and cospinors (later we will generalize this for spinor and cospinor fields)
as we need these notions to formulate the Dirac equation. As we’ll see, we
need the chosen matrices $A,C\in GL(4,\mathbb{C})$ mentioned at the beginning
and whose properties are discussed in Appendix A.4:

######  Definition 1.3.3.

We define the maps:

$\displaystyle(\cdot)^{\sharp}$ $\displaystyle:DM\to D^{*}M$
$\displaystyle(\cdot)^{\sharp}$ $\displaystyle:D^{*}M\to DM$
$\displaystyle(\cdot)^{c}$ $\displaystyle:DM\to DM$ $\displaystyle(\cdot)^{c}$
$\displaystyle:D^{*}M\to D^{*}M$

as follows:

$\displaystyle[E,z]^{\sharp}:=[E,z^{*}A]\hskip
15.0pt[E,z^{*}]^{\sharp}:=[E,A^{-1}z]$
$\displaystyle[E,z]^{c}:=[E,C^{-1}\overline{z}]\hskip
15.0pt[E,z^{*}]^{c}:=[E,\overline{z}^{*}C]$

######  Remark 1.3.1.

These maps are base-point preserving and act just on the spinorial component.

These are well defined maps, compatibly with the definitions of $DM$ and
$D^{*}M$, i.e. compatibly with the identifications of frames and vectors under
the free action of $Spin_{1,3}^{0}$. To see it one needs Lemma A.4.2 and:

$\displaystyle[R_{S}E,\pi(S^{-1})z]^{\sharp}$
$\displaystyle=[R_{S}E,z^{*}\pi(S^{-1})^{*}A]$
$\displaystyle=[R_{S}E,z^{*}A\pi(S)]$
$\displaystyle=[E,z^{*}A]=[E,z]^{\sharp}$

Moreover, the abovely defined maps, are vector bundle anti-isomorphisms:

######  Lemma 1.3.1.

For $q=[E,z]\in DM$ and $p=[E,w^{*}]\in D^{*}M$ we have:

$\displaystyle q^{\sharp\sharp}=q=q^{cc}$ $\displaystyle\hskip
20.0ptp^{\sharp\sharp}=p=p^{cc}$ $\displaystyle q^{\sharp c}=-q^{c\sharp}$
$\displaystyle\hskip 20.0ptp^{\sharp c}=-p^{c\sharp}$ $\displaystyle\langle
q^{\sharp},p^{\sharp}\rangle=$ $\displaystyle\overline{\langle
p,q\rangle}=\langle p^{c},q^{c}\rangle$

###### Proof.

Recalling that, from Definition A.4.2, we have $A=A^{*}$ and
$\overline{C}C=\mathbb{1}$, we can compute for $q$:

$\displaystyle q^{\sharp\sharp}$
$\displaystyle=[E,z]^{\sharp\sharp}=[E,z^{*}A]^{\sharp}=[E,A^{-1}A^{*}z]=[E,z]=q$
$\displaystyle q^{cc}$
$\displaystyle=[E,z]^{cc}=[E,C^{-1}\overline{z}]^{c}=[E,C^{-1}\overline{C^{-1}\overline{z}}]=[E,z]=q$

While for $p$ we have:

$\displaystyle p^{\sharp\sharp}$
$\displaystyle=[E,w^{*}]^{\sharp\sharp}=[E,A^{-1}w]^{\sharp}=[E,w^{*}(A^{-1})^{*}A]=[E,w^{*}]=p$
$\displaystyle p^{cc}$
$\displaystyle=[E,w^{*}]^{cc}=[E,\overline{w^{*}}C]^{c}=[E,\overline{\big{(}\overline{w^{*}}C\big{)}}C]=[E,w^{*}]=p$

For what concerns the second set of equalities, compute using Theorem A.4.2:

$\displaystyle q^{c\sharp}$
$\displaystyle=[E,z]^{c\sharp}=[E,C^{-1}\overline{z}]^{\sharp}=[E,\overline{z}^{*}(C^{-1})^{*}A]$
$\displaystyle=-[E,\overline{z^{*}A}C]=-[E,z^{*}A]^{c}=-[E,z]^{\sharp
c}=-q^{\sharp c}$ $\displaystyle p^{c\sharp}$
$\displaystyle=[E,w^{*}]^{c\sharp}=[E,\overline{w}^{*}C]^{\sharp}=[E,A^{-1}C^{*}\overline{w}]$
$\displaystyle=-[E,C^{-1}\overline{A^{-1}w}]=-[E,A^{-1}w]^{c}=-[E,w^{*}]^{\sharp
c}=-p^{\sharp c}$

Finally for what concerns the last result:

$\displaystyle\langle q^{\sharp},p^{\sharp}\rangle$
$\displaystyle=\langle[E,z]^{\sharp},[E,w^{*}]^{\sharp}\rangle=\langle[E,z^{*}A],[E,A^{-1}w]\rangle$
$\displaystyle=z^{*}w=\overline{w^{*}z}=\overline{\langle p,q\rangle}$
$\displaystyle\langle p^{c},q^{c}\rangle$
$\displaystyle=\langle[E,\overline{w}^{*}C],[E,C^{-1}\overline{z}]\rangle=\overline{w}^{*}\overline{z}$
$\displaystyle=\overline{w^{*}z}=\overline{\langle p,q\rangle}$

∎

######  Definition 1.3.4.

The maps $(\cdot)^{\sharp}$ are called Dirac adjoint, while $(\cdot)^{c}$
Dirac charge conjugation.

For spinor and cospinor fields these maps are defined pointwise and extended
gobally after. This means that for $u\in C^{\infty}(DM)$ we have
$u^{\sharp}(x)=u(x)^{\sharp}$, manifesting that the charge conjugation and
adjoint preserve the support.
Now that we have introduced the spinor and cospinor bundles, defining
pointwise on the manifold vector spaces, we can construct tensors and tensor
fields starting from them. Therefore, on $M$, we have now the following bundle
structures: $DM$, $D^{*}M$, $TM$, $T^{*}M$. Therefore, we can construct
tensors combining both the spin and the standard tangent bundles, forming in
this way a mixed spinor-tensor algebra. In presenting it, I will adopt the
”physicist” notation working in local coordinates but, in order to do that, we
need to introduce a proper index notation to distinguish spinor and vector
indices.
Let us start from a local section $E$ of $SM$ and a real basis $b_{A}$ of
$\mathbb{C}^{4}$. Then, we get local coordinates $E_{A}$ on $DM$ and, as a
consequence, also on $TM$:

$E_{A}=[E,b_{A}]\,\,\,\,\Rightarrow\,\,\,\,e=p\circ E$

Where we have used the canonical projection map $p:SM\to FM$. Now $(E_{A})$
represents a local frame for $DM$ while $\\{e_{a}\\}_{a=0,\dots,3}$ is a frame
for $TM$. One can further assume these to be normalized:

$e^{b}(e_{a})=\delta^{b}_{\,\,a}\hskip 15.0ptE^{B}E_{A}=\delta^{B}_{\,\,A}$

Where we have introduced the dual basis $e^{b}$, $E^{B}$ canonically obtained
from the inner product structure that we have. The difference, with standard
tensors over a manifold, stems from the way they transform under change of
coordinates. In order to explain it, let $E^{\prime}$ be a different section
of $SM$ on the same spacetime region $\mathcal{O}\subset M$. Then, calling
$S:\mathcal{O}\to Spin_{1,3}^{0}$ the map that associates to spacetime points
transformations in the universal covering of the fibers of $SM$, we know that:
$E^{\prime}=R_{S^{-1}}E$. As a consequence:

$E^{\prime}_{A}=[E^{\prime},b_{A}]=[R_{S^{-1}}E,b_{A}]=[E,b_{B}\pi(S^{-1})^{B}_{\,\,A}]=E_{B}\pi(S^{-1})^{B}_{\,\,A}$

So we have derived, the following transformation rules for frames of $DM$:

$E^{\prime}_{A}=E_{B}\pi(S^{-1})^{B}_{\,\,A}\hskip
15.0pt(E^{\prime})^{A}=\pi(S)^{A}_{\,\,B}E^{B}$

and the standard ones for frames of $TM$ ($\Lambda$ is the double covering map
of the proper orthocronus Lorentz group):

$e^{\prime}_{a}=e_{b}\Lambda(S^{-1})^{b}_{\,\,a}\hskip
20.0pt(e^{\prime})^{a}=\Lambda(S)^{a}_{\,\,b}e^{b}$

From which, for a general spinor-tensor $T=T^{Aa}_{\,\,\,\,\,\,Bb}E_{A}\otimes
e_{a}\otimes E^{B}\otimes e^{b}$, the transformation rule of its components
becomes:

$(T^{\prime})^{Aa}_{\,\,\,\,\,\,Bb}=\pi(S)^{A}_{\,\,F}\pi(S^{-1})^{D}_{\,\,B}\Lambda(S^{-1})^{d}_{\,\,b}\Lambda(S)^{a}_{f}T^{Ff}_{\,\,\,\,\,\,Dd}$

To better understand the local coordinate notation, we express charge
conjugation and hermitian adjointeness using local frames. In particular, from
the transformation rules of frames:

$E_{A}^{\sharp}=\delta_{AB}A^{B}_{\,\,C}E^{C}\hskip
15.0ptE_{A}^{c}=E_{B}(C^{-1})^{B}_{\,\,A}$

we find, the following general transformation rules:

$\displaystyle u^{\sharp}$
$\displaystyle=(u^{A}E_{A})^{\sharp}=\overline{u^{A}}\delta_{AB}A^{B}_{\,\,C}E^{C}=(u^{\sharp})_{C}E^{C}$
$\displaystyle u^{c}$
$\displaystyle=(u^{A}E_{A})^{c}=\overline{u^{A}}E_{B}(C^{-1})^{B}_{\,\,A}=(u^{c})^{B}E_{B}$
$\displaystyle v^{\sharp}$
$\displaystyle=(v_{A}E^{A})^{\sharp}=\overline{v_{A}}E_{C}\delta^{AB}(A^{-1})^{C}_{\,\,B}=(v^{\sharp})^{C}E_{C}$
$\displaystyle v^{c}$
$\displaystyle=(v_{A}E^{A})^{c}=\overline{v_{A}}E_{C}C^{C}_{\,\,B}\delta^{BA}=(v^{c})^{C}E_{C}$

From which we extract:

$\displaystyle(u^{\sharp})_{C}$
$\displaystyle=\overline{u}^{A}\delta_{AB}A^{B}_{\,\,C}\hskip
20.0pt(u^{c})^{A}=(C^{-1})^{A}_{\,\,B}\overline{u}^{B}$
$\displaystyle(v^{\sharp})^{A}$
$\displaystyle=(A^{-1})^{A}_{\,\,B}\delta^{BC}\overline{v_{C}}\hskip
20.0ptv^{c}_{A}=\overline{v_{B}}C^{B}_{\,\,A}$

Finally, let us introduce for vector fields $v$ and covector fields $k$ the
Feynman slash notation:

$\cancel{v}:=v^{a}\gamma^{A}_{\,\,\,\,aB}\hskip
20.0pt\cancel{k}:=k_{a}\gamma^{Aa}_{\,\,\,\,\,\,B}$

Before moving on to the next section, where we present and study the Dirac
equation, we still owe a proof of independence, of the described spin bundles,
from the choice of representation of the Dirac algebra $D$ and matrices
$A,C\in GL(4,\mathbb{C})$ that we did at the beginning:

######  Proposition 1.3.1.