Quantum field theory 1, lecture 01
Introduction
What is quantum field theory?
Quantum field theory is the modern theoretical framework to describe almost all phenomena in fundamental physics. This includes the standard model of elementary particle physics with the electromagnetic, the weak and the strong force and most likely, in one way or another, also dark matter and gravitation.
There are close connections to quantum mechanics and historically quantum field theory was developed as quantum theory for infinetly many degrees of freedom when it became clear that a relativistic version of quantum mechanics is not consistent. In the modern understanding quantum field theory in actually underlying non-relativistic quantum mechanics and the latter follows from the former in a limit. There is also a non-relativistic version of quantum field theory which can describe few-body physics of non-relativistic particles, but can also be used favorably to describe many-body physics and condensed matter.
Another very interesting connection is between quantum field theory and statistical field theory. Many of the concepts needed for relativistic quantum field theory can only be properly understood from the point of view of statistical physics and moreover, the same concepts can be used to describe stochastic theories where fluctuations are not of a quantum origin but have different reasons. This goes even beyond physics and the natural sciences.
Relativistic quantum field theories have also an interesting intersection with group theory, the theory of symmetries. Specifically Lie groups of various kinds play an important role to understand the phenomena of the standard model of elementary particle physics. Also consequences of space-time symmetries like conservation laws or the basic notion of what a particle actually is can be mentioned here.
There is also a very interesting relation to (quantum) information theory that is currently being explored in more detail. It is well possible that further insights into quantum field dynamics arise here in the coming years.
What concepts are needed to understand it?
Quantum theory
Symmetries and Lie group theory
Concepts from statistical physics
Basics of Lie groups
Symmetries and groups
Symmetry transformations
Studying symmetries and their consequences is one of the most fruitful ideas in physics. This holds especially in high energy and particle physics but by far not only there. To get started, we first define the notion of a symmetry transformation and relate it to the mathematical concept of a group.
It is natural to characterize a symmetry transformation by the following properties
One symmetry transformation followed by another should be a symmetry transformation itself.
There should be a unique (trivial) symmetry transformation doing nothing.
For each symmetry transformation there needs to be a unique symmetry transformation reversing it.
With these properties, the set of all symmetry transformations \(G\) forms a group in the mathematical sense.
Properties of groups
More formally, a group \(G\) has the following properties.
Closure: For all elements \(f, g \in G\) the composition \(g \cdot f\) is in \(G\). (We use here transformations acting to the right so that \(g \cdot f\) should be read as a transformation where we apply first \(f\) and then \(g\).)
Associativity: \(h \cdot (g \cdot f) = (h \cdot g) \cdot f\).
Identity element: There exists a unique unit element \(\mathbb{1}\) in the group, \(\mathbb{1} \in G\), such that \(\mathbb{1} \cdot f = f \cdot \mathbb{1} =f\) for all \(f \in G\).
Inverse element: For all elements \(f \in G\) there is a unique inverse \(f^{-1} \in G\) such that \(f \cdot f^{-1} = f^{-1} \cdot f = \mathbb{1}\).
These basic properties define groups of many kinds, both finite and infinite, discrete and continuous.
Representations
One distinguishes between groups as abstract entities and concrete representations. The abstract abstract group is defined through the set of its elements and composition law. A representation is a concrete realization of the group elements and their composition law, for example as matrices acting on a vector space or transformations of some type.
For example, a very simple group is \(\mathbb{Z}_2\). It has two elements, the unit element \(\mathbb{1}\) and an element \(R\) with \(R^2=\mathbb{1}\). A representation of \(R\) on the space of functions \(f(x)\) of a single variable \(x\) could be given by the parity transform \(f(x) = f(-x)\). The unit element is represented by the identity transform \(f(x)=f(x)\), and we thus have a representation of the group \(\mathbb{Z}_2\).
Abelian and non-abelian groups
A group is called abelian if the group product is commutative, \(f \cdot g = g \cdot f\) for all \(f, g \in G\). Otherwise the group is called non-abelian.
Examples for Lie groups
Lie groups can be defined as differentiable manifolds with a group structure. They have an infinite number of elements. Let us start with a few examples.
\(G=\mathbb{R}\), the additive group of real numbers. The group “product” is here the addition, the inverse of an element is its negative and the neutral or unit element is zero. This is clearly an abelian group.
\(G=\mathbb{R}^*_+\), the multiplicative group of positive real numbers. Also an abelian group.
\(G = \text{GL}(n,\mathbb{R})\), the general linear group of real \(n\times n\) matrices \(g\) with \(\det(g) \neq 0\) (such that they are invertible). Similarly, \(G = \text{GL}(n,\mathbb{C})\), the general linear group of complex \(n\times n\) matrices. These groups are non-abelian for \(n>1\).
\(G=\text{SL}(n,\mathbb{R})\) the special linear group is a subgroup of \(\text{GL}(n,\mathbb{R})\) with \(\det(g)=1\). This is a more general notion, the S for special usually means \(\det(g)=1\).
\(G=\text{O}(n)\), the orthogonal group of real \(n\times n\) matrices \(R\) with \(R^T R = \mathbb{1}\). This immediately implies \(\det(R)=\pm 1\). Again this is a subgroup of \(\text{GL}(n,\mathbb{R})\). As a manifold, \(\text{O}(n)\) is not connected. One component is the subgroup \(\text{SO}(n)\) with \(\det(R)=1\), the other is a separate submanifold where \(\det(R)=-1\). One can understand \(\text{O}(n)\) as the group of rotations and reflections in the \(n\)-dimensional Euclidean space. The simplest non-trivial case is for \(n=2\) where \(\text{SO}(2)\) consists of elements of the form \[R(\theta) = \begin{pmatrix} \cos(\theta) && -\sin(\theta) \\ \sin(\theta) && \cos(\theta) \end{pmatrix}.\] This is clearly isomorphic to the group \(\text{U}(1)\) of complex phases \(e^{i\theta}\). \(\text{SO}(n)\) is non-abelian for \(n>2\).
\(G=\text{U}(n)\), the unitary group of complex \(n\times n\) matrices \(U\) with \(U^\dagger U = \mathbb{1}\). Now we immediately infer that \(\det(U)\) is a complex number with absolute value \(1\). \(\text{U}(n)\) is non-abelian for \(n>1\).
\(G=\text{SU}(n)\), the special unitary group with unit determinant. Plays an important role in physics, most importantly \(\text{SU}(2)\) and \(\text{SU}(3)\).
\(G=\text{O}(r,n-r)\) the indefinite orthogonal group of \(n \times n\) matrices \(R\) that leaves the metric \(\eta=\text{diag}(-1,\ldots, -1, +1, \ldots, +1)\) with \(r\) entries \(-1\) and \(n-r\) entries \(+1\) invariant, in the sense that \(R^T \eta R = \eta\). An example is \(\text{O}(1,3)\), the group of Lorentz transformations in \(d=1+3\) dimensions.
\(G=\text{Sp}(2n,\mathbb{R})\) is the symplectic group of \(2n \times 2n\) matrices \(M\) that leaves a symplectic bilinear form \[\Omega = \begin{pmatrix} 0 && +\mathbb{1}_n \\ - \mathbb{1}_n && 0 \end{pmatrix} \label{eq:OmegaSpn}\] invariant in the sense that \(M^T \Omega M = \Omega\). Here \(\mathbb{1}_n\) is the \(n\) dimensional unit matrix and similarly \(0\). This is obviously a subgroup of \(\text{GL}(2n, \mathbb{R})\). There is also a complex version \(\text{Sp}(2n,\mathbb{C})\).
Lie groups have very nice features and a rich mathematical structure because they are both, groups and manifolds. We will now first introduce Lie groups and Lie algebras from an algebraic point of view, and subsequently also briefly introduce a differential-geometric characterization.