Chapter 1 Introduction

1.1 What is quantum field theory about?

As suggested by its name, quantum field theory (in short, QFT) is about studying the quantum theory of fields. Recall that a field is a concept from classical physics that is modeled by a function

\begin{flalign} \phi : M \longrightarrow \mathcal {Q}~,~~x\longmapsto \phi (x) \end{flalign} from a space or spacetime \(M\) to a set of values \(\mathcal {Q}\). It is important to note that a field is a local concept, in the sense that the assigned value \(\phi (x)\in \mathcal {Q}\) depends on the point \(x\in M\) in space or spacetime, i.e. it might vary if we vary \(x\). Fields come in different types, depending on what one chooses for the set of values \(\mathcal {Q}\). Physical examples that you may have come across in your studies are the mass density \(\rho (x)\) from continuum mechanics (this is a scalar field, i.e. \(\mathcal {Q}=\bbR \)), the electric \(\mathbf {E}(x)\) and magnetic \(\mathbf {B}(x)\) fields from electromagnetism (these are \(3\)-dimensional vector fields, i.e. \(\mathcal {Q}=\bbR ^3\)), and the metric field \(g_{\mu \nu }(x)\) from general relativity (this is a rank \(2\) tensor field, i.e. \(\mathcal {Q}=\mathrm {Mat}_{d\times d}(\bbR )\) with \(d\) the dimension of spacetime).

Informally speaking, a quantum field is what one gets when applying the principles of quantum mechanics to fields. The most crucial (in my personal opinion) feature of quantum mechanics is that the values of physical quantities, such as the position and momentum of a particle, are promoted to operators. Different operators do in general not commute with each other, for instance the position \(\hat {q}\) and momentum \(\hat {p}\) operators satisfy the famous commutation relations

\begin{flalign} \label {eqn:CCRQMintro} [\hat {q},\hat {q}] = 0 = [\hat {p},\hat {p}]\quad ,\qquad [\hat {q},\hat {p}] = \ii \,\hbar = -[\hat {p},\hat {q}]\quad , \end{flalign} where \([\hat {A},\hat {B}] := \hat {A}\,\hat {B} - \hat {B}\,\hat {A}\) denotes the commutator of two operators \(\hat {A},\hat {B}\). In the Heisenberg picture, operators evolve in time \(t\in \bbR \). The time evolution of an operator \(\hat {A}\) is described by Heisenberg’s equation

\begin{flalign} \frac {\dd \hat {A}(t)}{\dd t} = \frac {\ii }{\hbar }\,\big [\hat {H},\hat {A}(t)\big ]\quad , \end{flalign} subject to the initial condition \(\hat {A}(0)=\hat {A}\), where \(\hat {H}\) denotes the Hamiltonian operator of the system. Solving Heisenberg’s equation for the position operator \(\hat {q}\) yields an operator valued function

\begin{flalign} \hat {q}\,:\, \bbR \longrightarrow \mathrm {Operators}~,~~t\longmapsto \hat {q}(t) = e^{\frac {\ii t}{\hbar } \hat {H}}\, \hat {q}\, e^{-\frac {\ii t}{\hbar } \hat {H}} \quad , \end{flalign} which one can think of as a quantum version of the classical trajectory \(q:\bbR \to \bbR \, ,~t\mapsto q(t)\) of a particle. Note that, for the typical examples of Hamiltonians such as \(\hat {H} = \frac {\hat {p}^2}{2m} + V(\hat {q})\), the momentum operator \(\hat {p}(t)\) can be derived by taking a \(t\)-derivative of \(\hat {q}(t)\). Explicitly, one computes

\begin{flalign} \nn m\,\frac {\dd \hat {q}(t)}{\dd t} &= \frac {\ii \,m}{\hbar }\,\big [\hat {H},\hat {q}(t)\big ] = \frac {\ii \,m}{\hbar } \,e^{\frac {\ii t}{\hbar } \hat {H}}\, \big [\hat {H},\hat {q}\big ]\,e^{-\frac {\ii t}{\hbar } \hat {H}}\\ \nn &= \frac {\ii }{2\hbar } \,e^{\frac {\ii t}{\hbar } \hat {H}}\, \big [\hat {p}^2,\hat {q}\big ] \,e^{-\frac {\ii t}{\hbar } \hat {H}}\\ &= e^{\frac {\ii t}{\hbar } \hat {H}}\,\hat {p}\,e^{-\frac {\ii t}{\hbar } \hat {H}}= \hat {p}(t)\quad , \end{flalign} where steps three and four follow from the commutation relations (1.2). Hence, all information about this quantum system is encoded in \(\hat {q}(t)\). Since the field theoretic analog of the trajectory \(q(t)\) of a particle is the field \(\phi (x)\), it is natural to expect that operator valued functions

\begin{flalign} \hat {\phi } : M\longrightarrow \mathrm {Operators}~,~~x\longmapsto \hat {\phi }(x) \end{flalign} on a space or spacetime \(M\) will play a fundamental role in QFT. We will see later in this module that this is indeed the case.

What remains unclear from what was written so far is the scope of applicability of QFT. Since the concept of a quantum field is based on that of a classical field, it is not hard to believe that one may apply these techniques to, say, the electric and magnetic fields from electromagnetism in order to obtain their quantum analogs \(\hat {\mathbf {E}}(x)\) and \(\hat {\mathbf {B}}(x)\). (This is indeed the case, as we shall see later in this module.) But what about particles, such as electrons? These are typically described by their trajectories \(q(t)\) and not by a field, so there is no obvious classical field theory one could start from to derive a QFT for, say, electrons. This seeming incompatibility has caused a lot of confusion in the early days of QFT until it was realized that each quantum field has an associated theory of particles. These particles arise by studying representations of the operator valued functions \(\hat {\phi }(x)\) on a Hilbert space \(\HH \) and can be interpreted as excitations of the lowest energy (i.e. vacuum) state \(\ket {0}\in \HH \) of the QFT. But that is not all: Quantum fields naturally lead to a multiparticle theory in which the particle number is not necessarily conserved. This allows for physical processes in which particles get annihilated and/or new particles get created. Recall that many important processes in nature are of this type. For example, a single neutron can transform into a triple of particles, given by a proton, an electron and an antineutrino, under a process called beta decay. This involves annihilating the neutron and creating the three other particles. Such processes do not admit a description in terms of ordinary quantum mechanics, which from the start assumes that there is a fixed number of particles that is not allowed to change over time, but they make perfectly sense in the framework of QFT. The intrinsic properties of the particles associated with a QFT, such as their masses and spins, are encoded by the type of field one considers, i.e. its underlying set of values \(\mathcal {Q}\), and by its dynamics. The latter does also dictate the interactions between the particles. For example, we will see that the simplest case of a scalar field \(\hat {\Phi }(x)\) will give rise to spin \(0\) particles and that for electrons one needs a more sophisticated field \(\hat {\Psi }(x)\) whose classical set of values is the set of Dirac spinors. Furthermore, the particle associated with the electromagnetic fields \(\hat {\mathbf {E}}(x)\) and \(\hat {\mathbf {B}}(x)\) is the photon. These are only some of the many particles comprising the standard model of particle physics. It turns out that each fundamental particle (electron, quarks, neutrinos, …) and also each fundamental force (electromagnetic, weak and strong) arises from a quantum field of a suitable type and that their interplay/interaction is governed by the dynamics resulting from an action functional that is defined on the whole collection of fields. Unfortunately, such quantum field theoretical methods do not seem to be suitable for the gravitational field, and it is widely believed among physicists that a consistent theory of quantum gravity will go beyond the scope of QFT.

Historically, QFT was developed with the aim of providing the theoretical foundations for high-energy elementary particle physics. This endeavor has been very successful because the standard model of particle physics, which is build on the principles of QFT, describes the behavior of the elementary particles and their interactions with a remarkable accuracy. Over the past decades, the standard model got tested by more and more precise and higher energetic experiments, such as those performed at the Large Hadron Collider (LHC) at CERN, which so far have always agreed impressively well with the theoretical predictions. Because of this success, QFT has become an indispensable tool in the repertoire of every theoretical high-energy physicist. It is worthwhile to mention that the QFTs appearing in high-energy physics are defined on the Minkowski spacetime and that they are invariant under Poincaré transformations. This is because high-energy physics involves extremely fast particles, moving at almost the speed of light, which is the natural habitat of special relativity. In addition to high-energy physics, QFT has also contributed substantially to the development of other physical disciplines, such as condensed matter physics, cosmology and statistical physics. In condensed matter physics, quantum fields are used to model quasi-particles, such as phonons, magnons and others, that arise as excitations in a condensed matter system. These QFTs are usually defined on a nonrelativistic Newtonian spacetime because the velocities in a condensed matter system are small in comparison to the speed of light, however there exist materials (e.g. graphene) that require relativistic QFTs. In cosmology, the current understanding is that quantum fields in the early universe have generated quantum fluctuations that over time have grown to the galaxy structures we observe today. These QFTs are defined on curved spacetimes, in the sense of general relativity, which is an interesting topic that we however shall not touch in this module. The QFTs that arise in statistical physics are defined on Euclidean spaces, in contrast to spacetimes, and they are used to describe the probabilistic behavior of random systems.

I would like to conclude this section with some remarks about the complicated but fruitful relationship between QFT and mathematics, as this topic lies at the heart of my personal interests. Let me start with a warning: QFT is not (yet) a theory according to the high standards of pure mathematics. By this I mean that there does not yet exist a mathematical framework which provides an axiomatic definition of what a QFT should be and, importantly, accommodates the key examples from physics, such as the standard model of particle physics. This is also known as the Yang-Mills Millennium Prize Problem and whoever can solve it will be awarded \(\$ 1,000,000\) by the Clay Mathematics Institute. Mathematicians and mathematical physicists have tried to develop such axiomatic frameworks, which started with the Wightman axioms and culminated in modern approaches such as algebraic QFT, factorization algebras and functorial QFT. This has initiated an impressive cross-fertilization between physics and many branches of mathematics, most notably operator algebras, geometry, topology and category theory. While oversimplified toy-models of QFTs that share some (but not many) similarities with physically relevant models could be constructed (in a mathematical sense) in these frameworks, it still remains an open problem to construct Yang-Mills theory and the standard model of particle physics. The way how most of the theoretical/mathematical physics literature, and also our module, deals with these issues is as follows. First of all, instead of treating QFTs in a fully general fashion, we consider only examples of QFTs in which the interactions are described perturbatively, i.e. we expand nonlinearities order by order in the coupling constants and typically stop after the first few terms. While there is no mathematical reason why that should give something sensible, this approach is justified a posteriori by the fact that it matches the relevant physics with very high accuracy. (Unfortunately, nobody knows why …) Secondly, when analyzing perturbative QFTs, one finds infinities popping up all over the place. The origin of these infinities lies in the fact that QFT deals with infinitely many degrees of freedom (at least one for each point \(x\in M\), namely \(\hat {\phi }(x)\)) which sometimes add up to produce a divergent result in calculations. These infinities can be dealt with by redefining the parameters of the theory, which is called renormalization, and it turns out that this is not a bug, but rather a feature of QFT. At the end of this module you will understand what this means.