Introduction
Course details
This course provides an introduction to quantum field theory.
Acknowledgements. This version of Advanced Quantum Theory IV has been inherited from Silvia Nagy, Nabil Iqbal and Charlotte Sleight in reverse chronological order and largely follows the structure of their course. They also owe a lot to previous versions of the course by Marija Zamaklar and Kasper Peeters, the lecture notes by David Tong and the books by Peskin and Schroeder and Srednicki.
Outline.
Overview of the course. Review of the Lorentz group.
More on the Lorentz and Poincaré groups. Lagrangian methods for field theory.
Noether’s theorem, proof and examples.
Quantizing the free complex scalar field.
Hamiltonians and the infinite energy of the vacuum. Normal ordering.
Single- and multi-particle states. Charge and momentum operators.
Propagators.
Feynman propagator. Interacting theories.
More on interacting theories. Wick’s theorem and Feynman diagrams.
Feynman rules, scattering (non examinable).
More details. More details about the course, including information about lectures, problem classes and homework assignments, can be found on Blackboard at
https://blackboard.durham.ac.uk/ultra/courses/_68483_1/outline
Additional resources. There are many great books on Quantum Field Theory. Some of these include:
Peskin & Schroeder, “An Introduction to Quantum Field Theory” – the standard reference in the field.
Anthony Zee, “QFT in a Nutshell” – a delightful book and great for a physical understanding.
Mark Srednicki, “Quantum Field Theory” – very clear and a nice complement to Peskin & Schroeder. A version of the book is available for free from the author’s website:
http://web.physics.ucsb.edu/~mark/qft.html.
Steven Weinberg, “The Quantum Theory of Fields” – takes a unique route through the topic with many deep insights.
You can also find previous versions of the lecture notes for this course on Blackboard.
Why quantum field theory?
Quantum field theory is the mathematical framework behind some of our most well-tested and precise theories of the natural world. It is the language of nearly all modern research in quantum physics. The goal of this course is to introduce to the principles and methods of quantum field theory and use these to calculate observable quantities. Quantum field theory can be challenging. This stems from the fact that we still do not have a rigorous understanding of how it works. As a result there are many different approaches to the same questions. While challenging, this is also exciting. Quantum field theory lies at the cutting edge of modern theoretical and mathematical physics.
Let us discuss why we need quantum field theory. Consider the following map of physical theories
The goal of quantum field theory is to describe the physics of the very small and very fast. Some examples include:
Light. We are familiar with the idea of light as a wave but it can also behave like a particle. “Light particles” are called photons. It is natural to expect that these particles are “small” and move at the speed of light. Therefore, a quantum mechanical treatment of light needs quantum field theory.
Particle colliders. These are machines, such as the LHC at CERN, that accelerate individual partners to very high speeds and then collides them. They are at the cutting edge of modern experimental physics and are key for discovering new particles and understanding different states of matter. To work out what happens in the collision processes requires quantum field theory.
High temperature. Heating something up causes the particles in it to move around more quickly. If the speed of the particles comes close to the speed of light then quantum field theory is needed to describe the physics. This does not happen very often, but it does describe the very early universe, shortly after the “Big Bang”.
Quantum field theory is a theory that successfully combines quantum mechanics and special relativity. It is instructive to ask why it is not possible to simply make quantum mechanics relativistically invariant and why we need to introduce a new formalism. Consider a free particle with mass \(m\). In non-relativistic classical mechanics the energy of this particle is \[E_{\mathrm{non-relativistic}} = \frac{\vec{p}^2}{2m} ~,\] where \(\vec{p}\) is the spatial momentum of the particle. For a relativistic particle this is corrected to \[E_{\mathrm{relativistic}} = \sqrt{\vec{p}^2 c^2 + m^2 c^4} ~,\] where \(c\) is the speed of light. Expanding this equation for small momentum we find \[E_{\mathrm{relativistic}} = mc^2 + \frac{\vec{p}^2}{2m} + \mathcal{O}((\vec{p}^2)^2)) ~.\] Setting \(\vec{p} = 0\), we see that \(E_{\mathrm{relativistic}} = mc^2\) for a relativistic particle at rest. This says that a single particle has an intrinsic amount of energy \(mc^2\), which is called the rest mass. The non-relativistic kinetic energy appears as a correction to this rest mass. This tells us that particles always have energy, even if they are at rest. However, this suggests that we may be able to create particles by supplying enough energy. In particular, if we supply energy \(E = 2mc^2\) we might expect to be able to create a particle/anti-particle pair.
On the other hand, in quantum mechanics we know that quantities such as position, momentum, time and energy are uncertain and can fluctuate. More precisely, if we consider a state that is not an energy eigenstate then it will have a spread of energies \(\Delta E\). If \(\Delta E \approx 2mc^2\) then this suggests that the fluctuations of energy will be enough to create a particle/anti-particle pair. Therefore, if we are interested in combining quantum mechanics and special relativity into a single theory we will need to account for states where the particle number can change. This is not allowed in traditional quantum mechanics where we consider the Schrödinger equation for a fixed number of particles \(N\). Therefore, we need a new formalism in which the space of states allows the particle number to change. This is the formalism of quantum field theory.
Let us now work out an approximation for when we expect quantum field theory to be experimentally relevant. Consider a relativistic particle of mass \(m\) in a box of size \(L\). Since the position of the particle is known to an accuracy of at least \(L\), it follows from the Heisenberg uncertainty relation \[\begin{equation} \label{eq:heisenberguncertainty}
\Delta q \Delta p \geq \frac{\hbar}{2} ~, \end{equation}\] that the there is a lower bound on the uncertainty in the momentum \[\begin{equation} \label{eq:puncertainty}
\Delta p \geq \frac{\hbar}{2L} ~. \end{equation}\] Assuming the particle is in a highly relativistic regime, i.e. \(|\vec{p}| \gg m c\), we have \[\begin{equation} \label{eq:relenergy}
E_{\mathrm{relativistic}} = \sqrt{\vec{p}^2 c^2 + m^2 c^4} \approx |\vec{p}| c ~. \end{equation}\] From eqs. \(\eqref{eq:puncertainty}\) and \(\eqref{eq:relenergy}\) we find the following uncertainty in the energy \[\Delta E \approx c \Delta p \geq \frac{c \hbar}{2L} ~.\] As we have argued, we expect the effects of changing particle number to become important when \(\Delta E\) exceeds \(2mc^2\). Comparing these two expressions, we see that if the particle is localised within a distance of order \[L_{\mathrm{Compton}} \equiv \frac{\hbar}{mc} ~,\] then both quantum and relativistic effects become important. Note that \(L_{\mathrm{Compton}}\) combines both \(\hbar\) (Planck’s constant) and \(c\) (the speed of light) and it indicates that if we try to confine a particle of mass \(m\) to a box of size smaller than \(L_{\mathrm{Compton}}\) then we expect quantum fluctuations of the energy to create particle/anti-particle pairs from the vacuum. \(L_{\mathrm{Compton}}\) is called the Compton wavelength and for an electron is approximately \(10^{-12}m\).
Note that we have not shown that relativistic quantum mechanics does not work. It is an instructive exercise to try to build such a theory and see that it leads to inconsistencies such as negative probabilities and negative energies.
Before we start with a review of special relativity and Lorentz invariance, let us briefly describe some of the key features of quantum field theory. In classical mechanics, the degrees of freedom are real- or complex-valued functions of time \(q_a(t)\), \(p_a(t)\). Here \(a\) runs over the different degrees of freedom, e.g., \(a=1,2,3\) for a particle moving in \(\mathbb{R}^3\). In quantum mechanics, we canonically quantize these degrees of freedom following a well-established algorithm leading to operators \(\hat q_a\), \(\hat p_a\), which obey a Heisenberg uncertainty relation such as \(\eqref{eq:heisenberguncertainty}\).
In classical field theory, the basic degree of freedom is a field such as \(\phi(\vec{x},t)\). Here \(\phi(\vec{x},t)\) is a real scalar field and is a function from space-time to the real numbers \[\phi: \mathbb{R}^{1,3} \to \mathbb{R}~.\] We can have other types of field, e.g., a complex scalar field, or a vector field such as \(\vec{E}(\vec{x},t)\), the electric field from electrodynamics. The field \(\phi(\vec{x},t)\) has a conjugate momentum \(\pi(\vec{x},t)\). Canonically quantizing the classical fields \(\phi(\vec{x},t)\) and \(\pi(\vec{x},t)\) leads us to a quantum system with operators \(\hat \phi(\vec{x})\) and \(\hat \pi(\vec{x})\). This quantum system is a quantum field theory.
Since the classical fields \(\phi(\vec{x},t)\) and \(\pi(\vec{x},t)\) can take a different values at each point in space, they contain infinitely many degrees of freedom. This leads to infinitely many quantum operators, one at each point in space for each field, in the quantum field theory. Understanding how to organise and work with these infinities is one of the main successes of quantum field theory.
The development of quantum field theory leads to a number of profound physical insights that we will understand in this course:
There are different yet indistinguishable copies of elementary particles such as the electron.
There is a relationship between the statistics of particles (their behaviour under exchange) and their spin (their behaviour under rotation).
Particles can be created and destroyed, and anti-particles exist.
Forces can be interpreted as the exchange of particles.
Conventions
In this course we use natural units, that is \(\hbar = c = 1\). Factors of \(\hbar\) and \(c\) can be reinstated by dimensional analysis. We use signature \((-,+,+,+)\) for \(\mathbb{R}^{1,3}\).
Special Relativity and Lorentz Invariance
In this course we will study quantum field theory on the flat space-time \(\mathbb{R}^{1,3}\). The symmetry, or isometry, group of flat Minkowski space is the Poincaré group, which includes both Lorentz transformations and translations. Therefore, the laws of physics that we derive from quantum field theory should be invariant under these symmetries.
Rotational invariance in 2 dimensions
Let us start with a simpler example and consider flat 2-dimensional Euclidean space \(\mathbb{R}^2\). We let \((x,y)\) denote coordinates on \(\mathbb{R}^2\). Consider a second coordinate system \((x',y')\) related to the first by \[\begin{equation} \label{eq:2x2rotations}
\vec{x}' = \begin{pmatrix} x' \\ y' \end{pmatrix}
= \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} = R(\theta) \vec x ~. \end{equation}\] The matrix \(R(\theta)\) is an element of the group of rotations \(\mathrm{SO}(2)\) of \(\mathbb{R}^2\).
While the individual components \(x'\) and \(y'\) change under the rotation, the length of the vector is invariant. This is geometrically clear, but we can also check it explicitly by computing \[\vec x'{}^2 = x'{}^2 + y'{}^2 = (x \cos \theta + y \sin \theta)^2 + (y \cos \theta - x \sin \theta)^2 = x^2 + y^2 = \vec x^2 ~.\] More generally, the dot product of two vectors \(\vec{v} = (v^x,v^y)\) and \(\vec{w} = (w^x,w^y)\) is invariant, that is \(\vec{v}'\cdot \vec{w}' = \vec{v}\cdot \vec{w}\). The dot product can be written as \(\vec{v}\cdot \vec{w} = \vec{v}^T \vec{w}\), hence the condition that the dot product is invariant can then be written as \[\vec{v}'{}^T \vec{w}' = \vec{v}^T R^T R \vec{w} = \vec{v}^T \vec{w} ~.\] Since this should hold for any two vectors \(\vec{v}\) and \(\vec{w}\) it follows that \(R^T R = I\) where \(I\) is the identity matrix. This is the definition of the orthogonal group \(\mathrm{O}(2)\) in 2 dimensions. Elements of the special orthogonal group \(\mathrm{SO}(2)\) also satisfy \(\det R = 1\). It is easy to check that the rotation matrix \(R(\theta)\) \(\eqref{eq:2x2rotations}\) satisfies both \(R^T R = I\) and \(\det R = 1\).
It will also be useful to recall the triangle inequality. If we have three vectors \(\vec{v}\), \(\vec{w}\) and \(\vec{x} = \vec{v} + \vec{w}\), then the length of \(\vec{x}\) is less than or equal to the length of \(\vec{v}\) plus the length of \(\vec{w}\) \[\vec{x}^2 = (\vec{v} + \vec{w})^2 \leq \vec{v}^2 + \vec{w}^2 ~.\]
Basic kinematics of Lorentz invariance
Einstein defined special relativity from the following two postulates
Informally, if person A is at rest in an inertial frame and person B moves past at a constant speed, then they are also in an inertial frame.
We group space and time into space-time, which we denote \(\mathbb{R}^{1,3}\). Here, \(1\) denotes that we have one time direction, while \(3\) represents the three space directions. We label a point in space-time by \(x^\mu = (t,x^1,x^2,x^3) = (x^0,x^1,x^2,x^3)\). Depending on the context, we will use both \(t\) and \(x^0\) to denote time. We have also written \(x^\mu\) with a raised index. This is important since \(x_\mu\) with a lowered index will differ from \(x^\mu\) by a minus sign.
Now consider the following transformation to a new coordinate system \(x'{}^\mu\) \[\begin{equation} \label{eq:lorentzboost}
x'{}^\mu = \begin{pmatrix} t' \\ x'{}^1 \\ x'{}^2 \\ x'{}^3 \end{pmatrix}
=\begin{pmatrix} \gamma (t - v x^1) \\ \gamma (x^1 - vt) \\ x^2 \\ x^3 \end{pmatrix} ~,
\qquad
\gamma = \frac{1}{\sqrt{1-v^2}} ~, \end{equation}\] where \(v\in (-1,1)\). This is a Lorentz boost and is a transformation that mixes space and time. To understand what it means, consider a particle sitting at rest in the origin of the first coordinate system \[\begin{pmatrix} t \\ x^1 \\ x^2 \\ x^3 \end{pmatrix}_{\mathrm{particle}} = \begin{pmatrix} \tau \\ 0 \\ 0 \\ 0 \end{pmatrix} ~,\] where \(\tau\in\mathbb{R}\) parametrises the worldline of the particle. In the second coordinate system we find \[\begin{pmatrix} t' \\ x'{}^1 \\ x'{}^2 \\ x'{}^3 \end{pmatrix}_{\mathrm{particle}} = \begin{pmatrix} \gamma\tau \\ -v\gamma\tau \\ 0 \\ 0 \end{pmatrix} ~.\] This means that in this coordinate system the particle is moving since the position \(x'{}^1\) is not constant. The second set of coordinates correspond to an inertial frame that is moving with speed \(v\) with respect to the original frame. If we consider two subsequent Lorentz boosts parametrised by speeds \(v_1\) and \(v_2\), we find that this is equivalent to a single Lorentz boost parametrised by speed \(\frac{v_1+v_2}{1+v_1v_2}\). We can also consider a particle moving at the speed of light, which is equal to \(1\) \[\begin{pmatrix} t \\ x^1 \\ x^2 \\ x^3 \end{pmatrix}_{\mathrm{particle}} = \begin{pmatrix} \tau \\ \tau \\ 0 \\ 0 \end{pmatrix} ~.\] In the second coordinate system we find \[\begin{pmatrix} t' \\ x'{}^1 \\ x'{}^2 \\ x'{}^3 \end{pmatrix}_{\mathrm{particle}} = \begin{pmatrix} \gamma(1-v)\tau \\ \gamma(1-v)\tau \\ 0 \\ 0 \end{pmatrix} ~.\] This means that the speed has not changed, hence the speed of light is the same in all inertial frames.
Note that if we reinstate \(c\) by rescaling \(t\to c t\), \(t' \to c t'\) and \(v \to \frac{v}{c}\), the non-relativistic limit is given by taking \(c \to \infty\). Taking this limit the transformation \(\eqref{eq:lorentzboost}\) becomes \[x'{}^\mu = \begin{pmatrix} t' \\ x'{}^1 \\ x'{}^2 \\ x'{}^3 \end{pmatrix}
=\begin{pmatrix} t \\ (x^1 - vt) \\ x^2 \\ x^3 \end{pmatrix} ~,\] which is a Galilean boost familiar from Newtonian mechanics. In this limit we now have \(v \in (-\infty,\infty)\) and speeds are additive, i.e., two Galilean boosts by speeds \(v_1\) and \(v_2\) is equivalent to a Galiean boost by speed \(v_1 + v_2\).
In flat 2-dimensional Euclidean space we saw that certain quantities (lengths and dot products) are invariant under rotations. We now would like to construct similar invariants for Lorentz boosts. To do so we introduce the Minkowski metric on our flat 1+3-dimensional Lorentzian space-time \[\eta_{\mu\nu} = \begin{pmatrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}{\vphantom{\begin{pmatrix}0\\0\\0\end{pmatrix}}}_{\!\mu\nu} ~.\] The space-time metric allows us to lower indices \[x_\mu = (x_0,x_1,x_2,x_3) = \eta_{\mu\nu} x^\nu = (-x^0,x^1,x^2,x^3) ~,\] where we use the Einstein summation convention for repeated indices, i.e., \[\eta_{\mu\nu} x^\nu \equiv \sum_\nu \eta_{\mu\nu} x^\nu ~.\] The vector with a raised index is known as the contravariant vector, while the vector with a lowered index is the covariant vector. Due to the difference in the sign of the first component they transform differently under Lorentz transformations.
We also introduce the inverse metric \[\eta^{\mu\nu} = \begin{pmatrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}{\vphantom{\begin{pmatrix}0\\0\\0\end{pmatrix}}}^{\!\mu\nu} ~,\] which we can use to lower indices \[x^\mu = \eta^{\mu\nu} x_\nu ~.\] In the case of flat 1+3-dimensional Lorentzian space-time the matrix form of the metric and its inverse are the same as we have written them. This is not always the case and is a feature of the simplicity of flat space-time and our choice of coordinates. The statement that \(\eta^{\mu\nu}\) is the inverse of \(\eta_{\mu\nu}\) can be written as \[\eta^{\mu\nu}\eta_{\nu\rho} = \delta^\mu_\rho ~,\] where \(\delta^\mu_\rho\) is the identity matrix \[\delta^\mu_\rho = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}{\vphantom{\begin{pmatrix}0\\0\\0\end{pmatrix}}}^{\!\mu}_{\!\rho} ~.\]
We can now construct a scalar product between two vectors \(x^\mu\) and \(y^\mu\) \[x \cdot y = x^\mu y^\nu \eta_{\mu\nu} = x^T \eta y = \begin{pmatrix}x^0&x^1&x^2&x^3\end{pmatrix} \begin{pmatrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}y^0\\y^1\\y^2\\y^3\end{pmatrix} = -x^0y^0 + x^1y^2 + x^2y^2 + x^3y^3 ~.\] This is similar to the familiar dot product on Euclidean space, except that there is an extra minus sign appearing in front of the product of the time components of \(x^\mu\) and \(y^\mu\). This sign is important and can be understood as distinguishing the time direction from the space directions and ultimately is the origin of the different nature of time and space. Note that there are many different ways of writing the scalar product by raising and lowering indices and renaming dummy indices, i.e., indices that are summed over. For example, \[x\cdot y = x^\mu y^\nu \eta_{\mu\nu} = x^\mu y_\mu = x_\mu y^\mu ~.\]
Now that we have a scalar product between two vectors, we can ask what is the most general transformation of the coordinates that leaves this invariant. In other words, what is the most general \(4\times 4\) matrix \(\Lambda\) such that \[x' \cdot y' = (x')^\mu (y')_\mu = x^\mu y_\mu = x \cdot y ~, \qquad (x')^\mu = \Lambda^\mu{}_\nu x^\nu~, \qquad (y')^\mu = \Lambda^\mu{}_\nu y^\nu ~.\] Substituting in, we see that we require that \[x^T \Lambda^T \eta \Lambda y = x^T \eta y ~,\] which implies \[\begin{equation} \label{eq:lorentzrelation}
\Lambda^T \eta \Lambda = \eta ~, \qquad
\Lambda^\mu{}_\nu \eta_{\mu\rho}\Lambda^\rho{}_\sigma = \eta_{\nu\sigma} ~, \end{equation}\] where we have written the relation both as a matrix equation and explicitly with indices. This relation can be understood as saying that the transformations we are interested in leave the Minkowski metric invariant. Matrices \(\Lambda\) that satisfy this property are elements of a Lie group denoted \(\mathrm{O}(1,3)\), which is commonly known as the Lorentz group.
Finally, we check that the Lorentz boost in eq. \(\eqref{eq:lorentzboost}\) satisfies the property \(\eqref{eq:lorentzrelation}\). From eq. \(\eqref{eq:lorentzboost}\) we read off \[\begin{equation} \label{eq:lorentzboostmatrix}
\Lambda^\mu{}_\nu = \begin{pmatrix} \gamma & -v\gamma & 0 & 0 \\ -v\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}{\vphantom{\begin{pmatrix}0\\0\\0\end{pmatrix}}}^{\!\mu}{\vphantom{\begin{pmatrix}0\\0\\0\end{pmatrix}}}_{\!\nu} ~. \end{equation}\] Substituting into the left-hand side of \(\eqref{eq:lorentzrelation}\), we find \[\begin{split}
\begin{pmatrix} \gamma & -v\gamma & 0 & 0 \\ -v\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
&
\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} \gamma & -v\gamma & 0 & 0 \\ -v\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\\ & =
\begin{pmatrix} -\gamma^2(1-v^2) & 0 & 0 & 0 \\ 0 & \gamma^2(1-v^2) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
=
\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~,
\end{split}\] as required.
Group theory of the Lorentz group.
The Lorentz boost in eq. \(\eqref{eq:lorentzboostmatrix}\) is an example of a Lorentz transformation. Let us now determine the full set of Lorentz transformations that satisfy the property \(\eqref{eq:lorentzrelation}\). If we take the determinant of eq. \(\eqref{eq:lorentzrelation}\) we find \[(\det \Lambda)^2 \det\eta = \det\eta \qquad \Rightarrow \qquad \det \Lambda = \pm 1 ~.\] We can also set \(\nu = \sigma = 0\) in \(\eqref{eq:lorentzrelation}\) to give \[-(\Lambda^0{}_0)^2 + (\Lambda^1{}_0)^2 + (\Lambda^2{}_0)^2 + (\Lambda^3{}_0)^2 = -1 ~,\] which, noting that we are considering real matrices \(\Lambda\) to ensure the transformed coordinates are real, implies that \[(\Lambda^0{}_0)^2 \geq 1 \qquad \Rightarrow \qquad \Lambda^0{}_0 \geq 1 \mathrm{~or~} \Lambda^0{}_0 \leq -1 ~.\] Therefore, the set of all \(\Lambda\) can be split into four disjoint sets labelled by the sign of their determinant and the sign of \(\Lambda^0{}_0\).
We start by considering the set with \(\det \Lambda = 1\) and \(\Lambda^0{}_0 \geq 0\), which forms a subgroup of the Lorentz group. This subgroup is denoted \(\mathrm{SO}^+(1,3)\). This is the subgroup that is continuously connected to the identity, hence it is useful to consider the corresponding Lie algebra that describes the infinitesimal behaviour of the Lie group. Elements of the Lie algebra are often referred to as generators of the Lie group. To construct the Lie algebra, we consider a Lorentz transformation that takes the following form \[\begin{equation} \label{eq:identitycomponent}
\Lambda = \exp (\mathcal{M}) ~, \end{equation}\] where \(\mathcal{M}\) is a \(4\times 4\) matrix. To determine the set of all possible elements \(\mathcal{M}\) of the Lie algebra, we assume \(\mathcal{M}\) is small and expand the exponential in eq. \(\eqref{eq:identitycomponent}\) \[\Lambda = I + \mathcal{M} + \mathcal{O}(\mathcal{M}^2) ~, \qquad
\Lambda^\mu{}_\nu = \delta^\mu_\nu + \mathcal{M}^\mu{}_\nu + \mathcal{O}(\mathcal{M}^2) ~.\] Now substituting this into the group property \(\eqref{eq:lorentzrelation}\) we find \[\delta^\mu_\nu \eta_{\mu\rho}\delta^\rho_\sigma + \mathcal{M}^\mu{}_\nu \eta_{\mu\rho}\delta^\rho_\sigma + \delta^\mu_\nu \eta_{\mu\rho} \mathcal{M}^\rho{}_\sigma = \eta_{\nu\sigma} ~,\] where we have dropped terms of \(\mathcal{O}(\mathcal{M}^2)\). Simplifying the first term on the left-hand side, we find that it cancels with the single term on the right-hand side. Therefore, we are left with \[\mathcal{M}_{\sigma\nu} + \mathcal{M}_{\nu\sigma} = 0 ~,\] i.e., the matrix \(\mathcal{M}\) with lowered indices is antisymmetric. There are six linearly independent \(4 \times 4\) antisymmetric matrices. We consider the following basis of such matrices \[\begin{equation} \label{eq:so13down}
(M^{\rho\sigma})_{\mu\nu} = \delta^\sigma_\mu\delta^\rho_\nu - \delta^\rho_\mu\delta^\sigma _\nu ~. \end{equation}\] In this expression the indices \(\rho\) and \(\sigma\) label the elements of the basis. Note that the right-hand side is antisymmetric in these indices so there are only six linearly independent elements of the basis as expected. The indices \(\mu\) and \(\nu\) label the components of the \(4\times 4\) matrices. Since the right-hand side is also antisymmetric in these indices, we see that these matrices are antisymmetric as required. That is, for each choice of \(\rho = 0,1,2,3\) and \(\sigma = 0,1,2,3\), eq. \(\eqref{eq:so13down}\) defines a \(4\times4\) matrix that is a generator of the Lorentz group. Moreover, out of the 16 possible \(4\times4\) matrices only six are linearly independent since \(M^{\rho\sigma} = -M^{\sigma\rho}\). Finally, we can raise the index \(\mu\) on the basis \(\eqref{eq:so13down}\) to find a basis for the generators of the Lorentz group \[\begin{equation} \label{eq:so13generators}
(M^{\rho\sigma})^\mu{}_\nu = \eta^{\sigma\mu}\delta^\rho_\nu - \eta^{\rho\mu}\delta^\sigma _\nu ~. \end{equation}\]
We are now in a position to write down the most general Lorentz transformation connected to the identity. A general element of the Lie algebra is given by an arbitrary linear combination of all the elements of the basis of generators \[\mathcal{M}(\omega_{\rho\sigma}) = \omega_{\rho\sigma}M^{\rho \sigma} ~.\] A general Lorentz transformation is then given by the exponential of this expression, i.e., \[\Lambda(\omega_{\rho\sigma}) = \exp(\omega_{\rho\sigma}M^{\rho \sigma}) ~.\] The antisymmetry of \(M^{\rho \sigma}\) means that we can also take \(\omega_{\rho\sigma}\) to be antisymmetric, i.e. \(\omega_{\rho\sigma} = -\omega_{\sigma\rho}\), without loss of generality. Note that in these expressions we have suppressed the matrix indices \(\mu\) and \(\nu\).
Since Lorentz transformations form a group this means that the product of two Lorentz transformations is also a Lorentz transformation. How Lorentz transformations are composed with each other is encoded in the commutator of Lie algebra generators, e.g., through the Baker–Campbell–Hausdorff formula. Direct computation the definition \(\eqref{eq:so13generators}\) shows that \[\phantom{}[M^{\mu\nu}, M^{\rho\sigma}] = -\eta^{\nu\rho}M^{\mu\sigma} + \eta^{\mu\rho} M^{\nu\sigma} + \eta^{\nu\sigma} M^{\mu\rho} - \eta^{\mu\sigma}M^{\nu\rho} ~,\] where again we have suppressed the matrix indices. This commutator defines the Lorentz algebra.
Let us now consider some specific examples of Lorentz transformations. First we look at the generators \(M^{0i}\). From eq. \(\eqref{eq:so13generators}\) we find that \[M^{01} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~,
\qquad
M^{02} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~,
\qquad
M^{03} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} ~.\] We see that the labelling of the basis denotes which two dimensions are being transformed. The Lorentz transformations \(M^{0i}\) correspond to boosts in the \(x^i\) direction. Focusing on \(M^{01}\) (\(M^{02}\) and \(M^{03}\) behave similarly) we have \[\Lambda(\omega_{01}) = \exp(\omega_{01} M^{01}) = \begin{pmatrix} \cosh \omega_{01} & \sinh \omega_{01} & 0 & 0 \\ \sinh \omega_{01} & \cosh \omega_{01} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~.\] This agrees with eq. \(\eqref{eq:lorentzboostmatrix}\) if we relate the boost velocity \(v\) to the Lie algebra parameter \(\omega_{01}\) as \[\sinh\omega_{01} = \frac{-v}{\sqrt{1-v^2}} ~.\] The Lie algebra parameter \(\omega_{01}\) is often called the rapidity.
Now let us look at the generators \(M^{ij}\). From eq. \(\eqref{eq:so13generators}\) we find that \[M^{12} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~,
\qquad
M^{13} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} ~,
\qquad
M^{23} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix} ~.\] Focusing on \(M^{12}\) (\(M^{13}\) and \(M^{23}\) behave similarly) we have \[\Lambda(\omega_{12}) = \exp(\omega_{12}M^{12}) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\omega_{12} & -\sin\omega_{12} & 0 \\ 0 & \sin\omega_{12} & \cos\omega_{12} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~.\] This takes the form of an \(\mathrm{SO}(2)\) rotation of the \(x^1\) and \(x^2\) directions. Together with the rotations generated by \(M^{13}\) and \(M^{23}\) this gives the group of spatial rotations \(\mathrm{SO}(3)\), which is a subgroup of \(\mathrm{SO}^+(1,3)\).
Finally, let us return to those transformations that do not have \(\det\Lambda = 1\) and \(\Lambda^0{}_0 \geq 1\). Two important examples are time reversal, which acts as \(T: (x^0,x^1,x^2,x^3) \to (-x^0,x^1,x^2,x^3)\), and parity, which acts as \(P: (x^0,x^1,x^2,x^3) \to (x^0,-x^1,-x^2,-x^3)\) These change the direction of time and of all space directions respectively. In matrix form they are given by \[\Lambda_T = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~,
\qquad
\Lambda_P = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} ~.\] We can then recover the full Lorentz group \(\mathrm{O}(1,3)\) by composing time reversal, parity and their composition with elements of the identity component \(\mathrm{SO}^+(1,3)\) discussed above.
Translations and the Poincaré group
Having understood Lorentz transformations, which are the generalisation of rotations in flat Euclidean space to a relativistic flat Lorentzian space-time, we now turn to space-time translations, i.e., transformations of the form \[x^\mu \to x'{}^\mu = x^\mu - a^\mu ~,\] where \(a^\mu\) is a constant vector. These are symmetries of (sufficiently small) regions of empty space-time. The time component of \(a^\mu\) corresponds to a translation of the origin of time, while the space components correspond to translations of the origin of space. This tells us that the laws of physics are the same now and in the future and are the same no matter where we are located.
The combination of space-time translations with Lorentz transformations results in a Lie group called the Poincaré group. The Poincaré group acts on the coordinates of space-time as \[\begin{equation} \label{eq:Poincaretransformations}
x^\mu \to x'{}^\mu = \Lambda^\mu{}_\nu x^\nu - a^\mu ~. \end{equation}\] It has ten generators, six Lorentz generators \(M^{\rho\sigma}\) and four translations \(P^\mu\).
The translations do not commute with the Lorentz transformations and the full structure of the commutation relations of the corresponding Lie algebra is given by \[\begin{split}
&\phantom{}[M^{\mu\nu}, M^{\rho\sigma}] = -\eta^{\nu\rho}M^{\mu\sigma} + \eta^{\mu\rho} M^{\nu\sigma} + \eta^{\nu\sigma} M^{\mu\rho} - \eta^{\mu\sigma}M^{\nu\rho} ~,
\\
&\phantom{}[P^\mu,M^{\nu\rho}] = -\eta^{\mu\nu}P^\rho + \eta^{\mu\rho}P^\nu ~, \qquad
[P^\mu,P^\nu] = 0 ~.
\end{split}\] These are the commutation relations of the Poincaré algebra.
The twin paradox
We have seen that the symmetries of flat \(1+3\)-dimensional Lorentzian space-time \(\mathbb{R}^{1,3}\) with the Minkowski metric consist of Lorentz transformations and translation. Together these form the Poincaré group. Working with these transformations is not that different to working with ordinary rotations of flat Euclidean space, except that we need to keep track of signs. The signs turn out to be important, they have important consequences and capture the fact that time is different from space.
As an example, let us consider the twin paradox.
Here we have one twin that stays at home and follows the space-time trajectory OAB. The second twin first travels along OC and then CB. When the two twins meet again, the travelling twin is younger. To see this we note that the invariant notion of time elapsed along each trajectory is measured with the Minkowski metric. Therefore, if the stationary twin OAB measures time \(\Delta t\), the travelling twin OCB will measure time \(2\sqrt{(\frac{\Delta t}{2})^2 - (\Delta x)^2} < \Delta t\). The twin paradox is not really a paradox but is a consequence of the fact that we use the Minkowski metric to measure the invariant, or proper, time and that the travelling twin accelerates between two inertial frames.
Lagrangian Methods and Classical Field Theory
Lagrangian methods for classical mechanics
Before we introduce our first field theories, let us recall the Lagrangian formalism for classical mechanics. Let us take a system with \(N\) coordinates \(q_a\), where \(a = 1,\dots, N\). Then we can define a Lagrangian \(L(q_1,\dots,q_N,\dot{q}_1,\dots,\dot{q}_N)\) where \(\dot{q}_a = \frac{dq_a}{dt}\) that encodes the dynamics of the system. A typical Lagrangian might be \[L(q_1,\dots,q_N,\dot{q}_1,\dots,\dot{q}_N) = \sum_{a=1}^N \frac{1}{2}m\dot{q}_a^2 - V(q_1,\dots,q_N) ~,\] which takes the form of kinetic minus potential energy.
From the Lagrangian we can then determine the action \[S[q_a] = \int_{t_i}^{t_f} dt\, L(q_a,\dot{q}_a) ~.\] The action is a functional and is a map from the space of particle trajectories to \(\mathbb{R}\). The principle of least action tells us that the solution to the classical equations of motion is the one that extremises the action, i.e., if we consider a variation of the path \(q_a(t) \to q_a(t) + \delta q_a(t)\) then the variation of the action should be zero \[0 = \delta S[q_a] = \int_{t_i}^{t_f} dt \, \delta L(q_a,\dot q_a)
= \int_{t_i}^{t_f} dt \, \sum_{a=1}^N \Big(\frac{\partial L}{\partial q_a} \delta q_a + \frac{\partial L}{\partial \dot{q}_a} \delta \dot{q}_a \Big)
= \int_{t_i}^{t_f} dt \, \sum_{a=1}^N \Big(\frac{\partial L}{\partial q_a} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_a}\Big) \delta q_a ~,\] where we have integrated by parts and neglected a boundary term in the final equality. Formally, we demand that \(\delta q_a(t_i) = \delta q_a(t_f) = 0\). Demanding that this holds for all variations \(\delta q_a(t)\) we find that the coefficient of \(\delta q_a(t)\) in the integrand must vanish for each \(a\). This gives the Euler-Lagrange equations of motion \[\frac{\partial L}{\partial q_a} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_a} = 0 ~.\] We have obtained \(N\) equations of motion, which for classical mechanics are ODEs, from a single scalar \(L\) highlighting the benefit of the Lagrangian formalism.
Lagrangian methods for classical field theory
Our goal is to write down a Lagrangian for a classical scalar field \(\phi(x) = \phi(t,\vec{x})\). To gain some insight into the structure of the Lagrangian, consider a classical mechanical system with degrees of freedom \(q_a(t)\) and the index \(a\) running over the sites of a cubic lattice. As we take the size of the lattice to infinity and the lattice spacing to zero, we see that that the lattice approximates a continuum \(\mathbb{R}^3\). We then have that \(q_a \to \phi(t,\vec{x})\) and \(\sum_{a=1}^N \to \int d^3 x\). We can think of the spatial coordinate \(\vec{x}\) as replacing the index \(a\) and labelling the degrees of freedom.
We are therefore led to consider a Lagrangian of the form \[L = \int d^3x \, \mathcal{L}(\phi(x),\partial_\mu \phi(x)) ~,\] where \(\mathcal{L}\) is called the Lagrangian density. Substituting this into the general form of the action we find \[S[\phi] = \int dt d^3x \, \mathcal{L}(\phi(x),\partial_\mu \phi(x)) ~,\] where we recall that \(x^0 = t\). While it is possible to consider more complicated actions, this is the general form that we will consider. Let us note a few of its properties:
The measure in the integral is \(d^4x = dt dx^1 dx^2 dx^3\). In other words time and space are treated on a similar footing, which is important for relativistic invariance.
Locality is built into the action since fields only couple to each other at the same space-time point. We do not consider terms in the action of the form \(\phi(t,\vec{x})\phi(t,\vec{y})\) where we integrate over both \(\vec{x}\) and \(\vec{y}\).
The analogue of the Euler-Lagrange equations are again found by demanding that the variation of the action under \(\phi(x) \to \phi(x) + \delta \phi(x)\) vanishes, i.e., \[\begin{split}
0 = \delta S[\phi] & = \int d^4x \, \delta \mathcal{L}(\phi(x),\partial_\mu \phi(x)) = \int d^4x \, \Big( \frac{\partial\mathcal{L}}{\partial\phi}\delta\phi + \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu\delta\phi\Big)
\\ & = \int d^4x \, \Big( \frac{\partial\mathcal{L}}{\partial\phi} - \partial_\mu \Big(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big)\Big)\delta\phi ~.
\end{split}\] From this we can read off the Euler-Lagrange equation of motion for a classical field theory \[\begin{equation} \label{eq:el}
\frac{\partial\mathcal{L}}{\partial\phi} - \partial_\mu \Big(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big) = 0 ~. \end{equation}\] Since the field \(\phi\) is a function of space-time, the Lagrangian formalism yields a PDE from a single scalar density \(\mathcal{L}\). When a field configuration satisfies the Euler-Lagrange equation, we say that it is on-shell. Whenever we consider an on-shell solution, the action is stationary under small variations about this field configuration.
Action of a real scalar field
Let us now construct an action for a real scalar field by imposing some physical requirements on the Lagrangian density \(\mathcal{L}\). First, the action should be invariant under Poincaré transformations. Under translations \(x' = x - a\) and Lorentz transformations \(x' = \Lambda x\), the measure \(d^4x\) is invariant since \(a\) is constant and the Jacobian of the Lorentz transformation is \(|\det\Lambda|\), which is equal to 1. Therefore, requiring the action is invariant amounts to requiring that the Lagrangian density is invariant under Poincaré transformations. Second, to yield non-trivial dynamics the Euler-Lagrange equation \(\eqref{eq:el}\) should be a non-trivial PDE. In practice, this means that \(\mathcal{L}\) must depend on \(\partial_\mu \phi\).
The simplest Lagrangian density for a real scalar field that satisfies these two properties is \[\begin{equation} \label{eq:actionscalar}
\mathcal{L}(\phi,\partial_\mu\phi) = -\frac12 \partial_\mu\phi\partial^\mu\phi - V(\phi) ~. \end{equation}\] The first term is called the kinetic term, while \(V(\phi)\) is a function of one variable called the potential of the scalar field.
Expanding out the Lagrangian density \(\eqref{eq:actionscalar}\) we find \[\mathcal{L}= -\frac12\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - V(\phi)
= \frac12 (\partial_t \phi)^2 - \frac12 \vec{\nabla}\phi\cdot\vec{\nabla}\phi - V(\phi) ~.\] This is invariant under Lorentz transformations because we have contracted all the indices using the Minkowski metric and its inverse. Note that this has introduced a relative minus sign between the time and space derivatives. In analogy with classical mechanics, we can think of the first term \(\frac12 (\partial_t \phi)^2\) as a kinetic energy and the second two terms as a potential energy.
The simplest choice for \(V(\phi)\) is \[V(\phi) = \frac12 m^2\phi^2 ~.\] With this choice of potential the classical field theory with Lagrangian density \(\eqref{eq:actionscalar}\) describes a system of non-interacting particles with mass \(m\). To see this let us work out the Euler-Lagrange equation for this system. Computing the first term in eq. \(\eqref{eq:el}\) we have \[\frac{\partial\mathcal{L}}{\partial\phi} = - \frac{\partial V}{\partial\phi} = -m^2 \phi ~,\] since the kinetic term does not depend explicitly on \(\phi\). The second term in eq. \(\eqref{eq:el}\) is \[\partial_\mu \Big(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big) = \partial_\mu \Big(\frac{\partial}{\partial(\partial_\mu\phi)}\Big(-\frac12\eta^{\rho\sigma}\partial_\rho\phi\partial_\sigma\phi\Big)\Big) ~,\] where we have renamed dummy indices to ensure we are not using the same index twice. To compute this expression we use the identity \[\frac{\partial}{\partial(\partial_\sigma\phi)}(\partial_\rho\phi) = \delta_\rho^\sigma ~.\] This follows since the left-hand side vanishes if \(\sigma \neq \rho\) and equals 1 if \(\sigma = \rho\), which is the definition of the Kronecker delta. At first sight, it may be slightly surprising that the index \(\sigma\) is raised on the right-hand side. However, this can be shown to be consistent with Lorentz transformations. Using this identity gives \[\begin{split}
\partial_\mu \Big(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big) & = \partial_\mu \Big(\frac{\partial}{\partial(\partial_\mu\phi)}\Big(-\frac12\eta^{\rho\sigma}\partial_\rho\phi\partial_\sigma\phi\Big)\Big)
\\
& = -\frac12\eta^{\rho\sigma} (\delta_\rho^\mu \partial_\sigma \phi + \partial_\rho\phi \delta_\sigma^\mu) = - \eta^{\mu\sigma} \partial_\sigma \phi = - \partial^\mu \phi ~.
\end{split}\] Therefore, we find \[\frac{\partial\mathcal{L}}{\partial\phi} - \partial_\mu \Big(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big) = -m^2\phi - \partial_\mu (-\partial^\mu\phi) ~,\] hence, the Euler-Lagrange equation is \[\partial^\mu\partial_\mu \phi - m^2\phi = 0 ~.\] This is the Klein-Gordon equation. It is a linear PDE that describes a relativistically-invariant free wave equation. The differential operator \(\partial^\mu\partial_\mu = \eta^{\mu\nu}\partial_\mu\partial_\nu = -\partial_t^2 + \vec{\nabla}\cdot\vec{\nabla}\) takes the expected form for a wave equation, with the speed of wave propagation equal to 1 since we have set the speed of light \(c=1\).
Noether’s theorem
Noether’s theorem states that a continuous symmetry leads to a conserved current and is one of the deepest and most important results in mathematical and theoretical physics.