Quantum Field Theory

Prerequisites: Canonical Quantum Field Theory. Full path (20 pages)

• Intuitive
• Concrete
• Abstract
• Formulations
• Roadmaps

Intuitive

Quantum field theory is the framework in which the Standard Model of particle physics and much of modern condensed matter physics are written. Its basic objects are fields: quantities that have a value at every point of space and time, like the electric field or the temperature in a room. Each kind of particle belongs to a field that fills all of space. There is an electron field, a photon field, a quark field for each type of quark, and so on.

A useful picture is a large mattress made of balls connected by springs. If you push one ball, a wave travels across the mattress. In classical physics a wave can have any amplitude. In quantum mechanics the energy of each vibration mode comes in discrete packets, and these packets are what we call particles. A photon is one quantum of vibration of the electromagnetic field; an electron is a quantum of the electron field. This explains why all electrons are exactly identical: they are excitations of the same field.

The theory was developed because quantum mechanics with a fixed number of particles does not fit together with special relativity. At high energies, energy can be converted into mass, so particles are created and destroyed, for example when an electron and a positron annihilate into two photons. A field can gain or lose quanta, so particle creation and annihilation are built into the description from the start. Interactions appear as couplings between fields: one field can lose a quantum while another gains one. Forces between particles, such as the electric repulsion of two electrons, come from the exchange of energy and momentum through such couplings.

In practice physicists use quantum field theory to compute rates of particle collisions and decays, the magnetic moment of the electron, and the behaviour of materials near phase transitions. Its predictions for the electron's magnetic moment agree with measurements to better than one part in a billion.

Concrete

We set $\hbar = c = 1$ and use the metric signature $(+,-,-,-)$. The simplest example is a real scalar field $\phi(x)$ with Lagrangian density

$$\mathcal{L} = \frac12\partial_\mu\phi\,\partial^\mu\phi - \frac12 m^2\phi^2 .$$

The Euler-Lagrange equation is the Klein-Gordon equation $(\partial_\mu\partial^\mu + m^2)\phi = 0$.

Canonical quantization. The conjugate momentum is $\pi = \partial\mathcal{L}/\partial\dot\phi = \dot\phi$. We promote $\phi$ and $\pi$ to operators with equal-time commutators

$$[\phi(t,\vec x), \pi(t,\vec y)] = i\,\delta^3(\vec x - \vec y), \qquad [\phi(t,\vec x),\phi(t,\vec y)] = [\pi(t,\vec x),\pi(t,\vec y)] = 0 .$$

The general solution of the field equation is a superposition of plane waves, and the commutators are satisfied by

$$\phi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}\left(a_{\vec p}\,e^{-ip\cdot x} + a_{\vec p}^\dagger\, e^{ip\cdot x}\right), \qquad [a_{\vec p}, a_{\vec q}^\dagger] = (2\pi)^3\delta^3(\vec p - \vec q),$$

with $p^0 = \omega_p = \sqrt{\vec p^{\,2} + m^2}$. Each Fourier mode is a harmonic oscillator. After normal ordering, which removes an infinite zero-point constant,

$$H = \int\frac{d^3p}{(2\pi)^3}\,\omega_p\, a_{\vec p}^\dagger a_{\vec p}, \qquad \vec P = \int\frac{d^3p}{(2\pi)^3}\,\vec p\, a_{\vec p}^\dagger a_{\vec p}.$$

The vacuum $|0\rangle$ is defined by $a_{\vec p}|0\rangle = 0$. The state $a_{\vec p}^\dagger|0\rangle$ has momentum $\vec p$ and energy $\sqrt{\vec p^{\,2}+m^2}$: it is one particle of mass $m$. Applying several creation operators gives multi-particle states, and since the $a^\dagger$ commute these states are symmetric under exchange, so the particles are identical bosons. The space spanned by all these states is the Fock space. For spin-$\frac12$ fields, such as the electron field described by the Dirac equation, one must use anticommutators instead, and the particles are fermions.

Interactions and perturbation theory. Adding a term such as $-\frac{\lambda}{4!}\phi^4$ to $\mathcal{L}$ makes the theory interacting. It can no longer be solved exactly, and one expands observables in powers of $\lambda$. The objects one computes are correlation functions $\langle\Omega|T\phi(x_1)\cdots\phi(x_n)|\Omega\rangle$, from which scattering amplitudes follow by the LSZ reduction formula. Each term of the expansion is represented by a Feynman diagram, built from the propagator

$$D_F(x-y) = \langle 0|T\phi(x)\phi(y)|0\rangle = \int\frac{d^4p}{(2\pi)^4}\frac{i\,e^{-ip\cdot(x-y)}}{p^2 - m^2 + i\epsilon}$$

and a vertex factor $-i\lambda$ for each interaction point. Loop diagrams contain divergent momentum integrals, which are handled by renormalization: the parameters in $\mathcal{L}$ are traded for measured quantities such as the physical mass and coupling.

Path integral. Equivalently, correlation functions are derivatives of the generating functional

$$Z[J] = \int\mathcal{D}\phi\;\exp\left(iS[\phi] + i\int d^4x\,J(x)\phi(x)\right),$$

which is the starting point for gauge theories, for the Faddeev-Popov treatment of gauge fixing and for lattice simulations after rotating to Euclidean time. The Standard Model is a quantum field theory of this kind, with gauge group $SU(3)\times SU(2)\times U(1)$, spin-$\frac12$ quark and lepton fields, gauge fields and a scalar Higgs doublet.

• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, chapter "Bird's-Eye View of Quantum Field Theory"

Abstract

There is no single accepted mathematical definition of an interacting quantum field theory in four dimensions. Several frameworks capture different parts of the structure.

Particles and symmetry. In a relativistic theory the Hilbert space carries a unitary representation of the universal cover of the Poincaré group, $\mathbb{R}^{1,3}\rtimes SL(2,\mathbb{C})$. Wigner's classification identifies one-particle states with irreducible unitary representations of positive energy, labelled by the mass $m \geq 0$ and the spin $s$ for $m > 0$ or the helicity for $m = 0$ (leaving aside the massless continuous-spin representations, which do not describe known particles). Given a one-particle space $\mathcal{H}_1$, the free-field Fock space is $\mathcal{F}(\mathcal{H}_1) = \bigoplus_{n\geq 0} S^n\mathcal{H}_1$ for bosons, with the antisymmetric powers $\Lambda^n\mathcal{H}_1$ for fermions. The free field is an operator-valued distribution on this space built from creation and annihilation operators. For interacting theories Haag's theorem shows that the interacting fields cannot be unitarily related to the free fields on a common Fock space, which is why the interaction picture is only a heuristic.

Axiomatic frameworks. The Wightman axioms characterize a theory by operator-valued tempered distributions on a Hilbert space with a Poincaré invariant vacuum, satisfying covariance, locality and the spectrum condition. The Haag-Kastler framework assigns to each region $\mathcal{O}$ of spacetime a C*-algebra of observables $\mathfrak{A}(\mathcal{O})$, with isotony, Einstein causality and covariance. The Osterwalder-Schrader axioms describe the Euclidean correlation functions; with reflection positivity they reconstruct a Wightman theory. In the Euclidean setting a bosonic theory is ideally a probability measure on the space of distributions $\mathcal{S}'(\mathbb{R}^d)$. See axiomatic quantum field theory.

Functorial view. Following Atiyah and Segal, a $d$-dimensional quantum field theory assigns a vector space $Z(\Sigma)$ (the space of states) to each closed $(d-1)$-manifold $\Sigma$ and a linear map $Z(M): Z(\Sigma_{\text{in}})\to Z(\Sigma_{\text{out}})$ to each $d$-dimensional bordism $M$, compatibly with gluing and disjoint union. In other words, it is a symmetric monoidal functor from a bordism category, with whatever geometric structure the theory needs (metric, spin structure, orientation), to vector spaces. For topological field theories no geometric structure beyond the topology is needed, and this definition is fully rigorous; for theories that depend on a metric it serves as a guide.

Perturbative and effective theory. Perturbation theory defines correlation functions as formal power series in the coupling. Gauge theories are quantized perturbatively with BRST or, more generally, Batalin-Vilkovisky methods, where gauge invariance becomes a cohomological condition. Renormalization organizes theories by their behaviour under changes of scale: the space of couplings carries a flow, fixed points of the flow are scale invariant theories, typically conformal field theories, and a quantum field theory with a cutoff is then an effective description valid below that scale. In this view, a continuum QFT is a renormalization group trajectory that starts at an ultraviolet fixed point.

Formulations

The most popular formulations of quantum field theory are

• Canonical Quantum Field Theory $\leftrightarrow$ Quantum Field Theory in Hilbert space,
• Path Integral Quantum Field Theory $\leftrightarrow$ Quantum Field Theory in configuration space

Canonical quantum field theory can be summarized as follows:

The main components of quantum field theory are:

1. Theory of free fields. Here we neglect any interaction terms and only calculate what different fields do when they are on their own.
2. Theory of interacting fields. Here we derive the correct terms in our equations that describe interactions using gauge symmetry. Unfortunately, it turns out that we can't solve these equations in closed form but only approximately. There is a powerful mathematical machinery that allows us to calculate results approximately. At the heart of this machinery are the famous Feynman diagrams. In addition, an important trick (the interaction picture) allows us to reuse most of the things we derived for free fields.
3. Renormalization. The main problem in quantum field theory is that calulcations are technically extremely challenging. One reason for this is that when we calculate things naively we end always end up with the result "infinity". This is physically nonsensical and we need to take greater care. The mathematical machinery that allows us to tame these infinities is known as renormalization.

Recommended Resources

• What is Quantum Field Theory, and What Did We Think It Is? by S. Weinberg
• Quantum Field Theory by Frank Wilczek

Roadmaps

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Special Relativity
14. Lagrangian Formalism
15. Phase Space
16. Classical Field Theory
17. Hamiltonian Formalism
18. Schrödinger Equation
19. Canonical Quantum Mechanics
20. Canonical Quantum Field Theory