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\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}