Download \chapter{Introduction}

\chapter{Introduction}
To write a thesis on the vacuum is, from a naive point of view, to write a thesis on
nothing at all. Classically one could define the vacuum as a box containing no matter or
particles whatsoever or, more rigorously, a subspace $V$ of the $\mathbb{R}^3$ such
that $N(V)=0$, where $N$ denotes the number of particles detected by an observer in the
exterior of $V$. Intuitively this function $N:\mathbb{R}^3\rightarrow\mathbb{N}$ is an
invariant under coordinate transformations, since it shouldn't matter how one looks at
$V$: when there are no particles, there's nothing to detect by anyone! Even though the
geometry of $V$ is not an invariant (due to e.g. Lorentz contraction), I couldn't, a priori,
imagine how this would account for the creation of particles.\\
One of the most exhilarating results of Quantum Field Theory is the observation due to
Unruh\cite{unruh1} that $N$ is \emph{not} an invariant, i.e. there are coordinate
transformations $\tau$ such that $N(\tau V)\neq0$: the notion of the vacuum becomes
ambiguous. In fact, a uniformly accelerating observer will detect a thermal spectrum
radiating out of $V$.\\
So where does our intuitive definition of the vacuum break down?\\
When taking into account the study of quantum mechanics, first of all the notion of a
particle is blurred out, and we are to think about a wave-particle dualism. States are
represented by wave vectors $|\psi\rangle$ so our volume $V$ previously defined should
be replaced by such a vector, living in Hilbert space. Simultaneously, observables are
represented by operators. The function $N$ is turned into an \emph{number operator}
$\hat{N}$ which acts on the state of interest, and one is to calculate the expectation value
$\langle\hat{N}\rangle=\langle\psi|\hat{N}|\psi\rangle$ of this number operator to obtain
the expectation value of the number of particles contained in the state $|\psi\rangle$. Now
let us define two new operators, $a(x^\mu)$ and $a^\dag(x^\mu)$, the annihilation and
creation operator respectively, in terms of a set of local coordinates $x^\mu$. They are
defined so that if $n=\langle\hat{N}\rangle$ is the number of particles in state
$|n\rangle$, then $a|\psi\rangle\propto |n-1\rangle$ and $a^{\dag}|\psi\rangle\propto
|n+1\rangle$, where the constants of proportionality are obtained upon normalization.
Note that $N(x^\mu)=a^\dag(x^\mu)a(x^\mu)$. With the aid of these new operators, the
vacuum state $|0\rangle$ is defined such that $a|0\rangle=0$, where the state $|0\rangle$
is expressed in terms of the set $x^\mu$. Vacuum states with respect to different
coordinates need not equal each other, which is to say that $|0\rangle_a$ need not equal
$|0\rangle_b$ where $a$ and $b$ are two different sets of coordinates.\\
With this new definition of the vacuum, the question of wether or not the previously
defined \emph{function} $N(V)$ is invariant under coordinate transformations is turned
into the question wether the outcome of $\langle0|N|0\rangle$ is altered upon
transformation of $x^\mu$, or: \\Does $\langle0|N(x^\mu)|0\rangle$ equal
$\langle0|N(x'^\mu)|0\rangle$ for every coordinate transformation?\\
Ever since Unruh answered this question with a surprisingly \emph{no} we have to once
more change our intuition on the concepts of a particle and empty space.\\
Making a leap from the microscopic to the macroscopic, we encounter a likewise
counterintuitive result in the study of black hole mechanics. Where one would expect a
black hole to indeed be entirely black, i.e. not to radiate any form of particles,
Zel'dovich\cite{zel1}\cite{zel2} proposed on heuristic grounds that there might be such
radiation, a statement made rigorous by Hakwing\cite{haw1}\cite{haw2}. This
remarkably discovery, now referred to as Hawking radiation, turns out to depend, to some
extend, on the Unruh effect.\\
In this thesis I will examine these canonical field theoretic results and compare the Unruh
effect with Hawking radiation. After having introduced the technical tools of quantum
field theory required to derive the results stated above I will follow a subtle argument due
to Unruh to end up with an expression for the number of particles in a vacuum as
observed by a uniformly accelerating observer. Finally I will discuss whether the Unruh
effect has a classical counterpart and exactly how quantummechanical this result is.\\
Throughout this thesis I employ a system of units wherein $c=\hbar=k_b=G=1$.\newline
\newline \emph{Paul de Lange, Amsterdam}