Classical field theory

3.1 Classical field theory

Historically, field theory was in part an attempt to develop local theories rather than those, such as Newtonian gravity, implying action-at-a-distance.¹ The idea is to associate each point in space and time with a value or set of values $ϕ_{a} (x, t)$ , called fields. As long as these fields interact only at the same point in spacetime or, at most, with their immediate neighbours (via their derivatives), the theory is guaranteed to be local. Classic examples include the vector-valued electric and magnetic fields $E (x, t)$ and $B (x, t)$ . The behavior of the fields is encapsulated by the Lagrangian of the system, as we see next.

3.1.1 Lagrangian mechanics

Lagrangian mechanics is a formulation of classical mechanics based on the energies of a system, as opposed to the force-based Newtonian approach. We define the Lagrangian of a particle as the difference between its kinetic ( $T$ ) and potential energies ( $V$ ):

L (ẋ, x) = T (ẋ) - V (x),

(3.1)

where $x$ and $ẋ$ are the particle’s position and velocity, respectively. To determine the dynamics of the system, we assign a value based on $L$ to each possible path the particle can take between two points $t_{i}$ and $t_{f}$ , called the action $S$ :

S [x (t)] = \int_{t_{i}}^{t_{f}} L (ẋ (t), x (t)) dt .

(3.2)

The equations of motion (EOMs) are then derived from the principle of stationary action, which states that the true path is an extremum of $S$ . This condition yields the Euler-Lagrange (E-L) equations:

\frac{d}{dt} (\frac{∂L}{∂ẋ}) - \frac{∂L}{∂x} = 0 .

(3.3)

Example 3.1. We can confirm that this is equivalent to Newtonian mechanics by considering the simple Lagrangian:

L = \frac{1}{2} m ẋ^{2} - V (x) .

(3.4)

Plugging this into Eq. 3.3 gives us:

mẍ + \frac{dV}{dx} = 0 \Rightarrow mẍ = - \frac{dV}{dx} = F,

(3.5)

which is exactly Newton’s second law. Classically, Lagrangian mechanics has certain benefits over Newtonian mechanics, such as being based on scalars (energies) instead of vectors (forces), and ease of coordinate transformations. For us, as we will see, its main advantage is its natural generalization to fields rather than particles.

Path integral formulation of QM

Note that the principle of stationary action is based on the classical behavior of particles, in that they follow a single true path. However, in QM, (unobserved) particles are thought to traverse a superposition of all possible paths between two observed positions. This can be expressed with Feynman’s path integral formula, where the probability of observing a particle at position $q_{f}$ and time $T$ given it was at $q_{i}$ at $t = 0$ is based on its wavefunction

ψ (q_{f}, T) = \int_{q_{i}}^{q_{f}} D q (t) e^{iS [q (t)] ∕ ℏ},

(3.6)

where $\int_{q_{i}}^{q_{f}} D q (t)$ is an integral over all possible paths $q (t)$ between $q_{i}$ and $q_{f}$ , interfering through their complex phases $e^{iS [q (t)] ∕ ℏ}$ that are based on the action $S [q (t)]$ of the path divided by the reduced Planck constant $ℏ$ . In the classical limit $\frac{ℏ}{S} \to 0$ ,² by the stationary phase approximation, only the path that extremizes the action contributes, as we expect.

The path integral formulation was a critical development in QFT. The fact that the Lagrangian shows up naturally in this formulation is the reason why we “consider it the most fundamental specification of a QFT” (Peskin and Schroeder [10] Chapter 9).

3.1.2 Free scalar field theory

When dealing with fields $ϕ (x, t)$ instead of particles, the Lagrangian is written as a function of the field and its derivatives:

L (t) = \int d^{3} x L (\partial_{μ} ϕ, ϕ),

(3.7)

where $L$ is the Lagrangian density (but usually referred to as the Lagrangian as well). The action is the usual integral of $L$ over time, or $L$ over spacetime:

S = \int L dt = \int d^{4} x L,

(3.8)

with the principle of stationary action leading to the analogous E-L equations for fields:

\partial_{μ} (\frac{\partial L}{\partial (\partial_{μ} ϕ)}) - \frac{\partial L}{∂ϕ} = 0 .

(3.9)

The Klein-Gordon equation

A useful example is the Lagrangian for a free, scalar relativistic field $ϕ (x, t)$ , where “free” indicates no interactions, and “scalar” means the value of the field at each point is a single number:

L = \frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ - \frac{1}{2} m^{2} ϕ^{2},

(3.10)

The E-L equation for this Lagrangian is called the Klein-Gordon equation:

\partial_{μ} \partial^{μ} ϕ + m^{2} ϕ \equiv (□ + m^{2}) ϕ = 0,

(3.11)

where we’ve defined the d’Alembertian operator $□ = \partial_{μ} \partial^{μ} = \partial_{t}^{2} - \nabla^{2}$ .

The Klein-Gordon equation is essentially the relativistic generalization of the Schrödinger equation. Just as the Schrödinger equation quantizes the non-relativistic EOM $E = \frac{p^{2}}{2 m}$ , the Klein-Gordon equation converts the relativistic EOM for a free particle

E^{2} = p^{2} c^{2} + m^{2} c^{4}

(3.12)

into quantum operator form, with $E \to Ê = i ℏ \partial_{t}$ and $p \to \hat{p} = - i ℏ \nabla$ :

\begin{matrix} \begin{aligned} Ê^{2} & = {\hat{p}}^{2} c^{2} + m^{2} c^{4} \\ - ℏ^{2} \partial_{t}^{2} ϕ & = - ℏ^{2} c^{2} \nabla^{2} ϕ + m^{2} c^{4} ϕ \\ \Rightarrow (\partial_{t}^{2} & - c^{2} \nabla^{2} + \frac{m^{2} c^{4}}{ℏ^{2}}) ψ = 0 . \end{aligned} \end{matrix}

(3.13)

Natural units

From this point onwards, we will use natural units, as is conventional in high energy physics:

ℏ = c = 1 .

(3.14)

Besides being notationally convenient, this means all dimensionful physical quantities can be put on the same scale — conventionally, in terms of energy and, specifically, in units of electronvolts (eV). For example:

Mass: $E = m c^{2} \to m = E$ ,
Compton wavelength: $λ = \frac{ℏ}{mc} \to λ = \frac{1}{E}$ ,
Momentum: $p = \frac{ℏ}{λ} \to p = E$ .

We define each quantity to have a dimension in terms of energy, i.e. energy, mass, and momentum all have dimension $[E] = [m] = [p] = 1$ , while length has dimension $[λ] = - 1$ . Thus, in natural units Eqs. 3.11 and 3.13 are identical.

Solutions to the Klein-Gordon equation

Using the Fourier transform, we see the solutions to the Klein-Gordon equation are plane waves:

ϕ (x, t) = \int \frac{d^{3} p}{{(2 π)}^{3}} \tilde{ϕ} (p, t) e^{i p \cdot x},

(3.15)

with $\tilde{ϕ} (p, t)$ satisfying the simple-harmonic oscillator (SHO) equation

(\partial_{t}^{2} - | p |^{2} - m^{2}) \tilde{ϕ} (p, t) = 0 \Rightarrow \tilde{ϕ} (p, t) \propto e^{- i ω_{p} t},

(3.16)

with frequency $ω_{p} = | \sqrt{| p |^{2} + m^{2}} |$ . Thus,

ϕ (x, t) = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{\sqrt{2 ω_{p}}} (a (p) e^{ip \cdot x} + a^{*} (p) e^{- ip \cdot x}),

(3.17)

where $p \cdot x$ is the 4D spacetime inner product with $p_{μ} = (ω_{p}, p)$ , and the $\frac{1}{\sqrt{2 ω_{p}}}$ factor is conventional. The coefficients $a$ and $a^{*}$ are complex conjugates to ensure a real sum. As we will see, in quantum field theory, the form of the fields is quite similar but with $a$ and $a^{*}$ quantum operators.

3.1.3 Symmetries and Noether’s theorem

Noether’s theorem states an important consequence of continuous symmetries of a system: they are associated with a physical conserved currents. For example, translational and rotational invariance of the potential energy imply conservation of momentum and angular momentum, respectively.

More precisely, if a continuous transformation on the field

ϕ (x) \to ϕ^{'} (x) = ϕ (x) + 𝜖 Δ ϕ (x)

(3.18)

is a symmetry, it must leave the EOMs invariant. For this, it can change the Lagrangian by at most a total derivative:

L \to L^{'} = L + 𝜖 \partial_{μ} J^{μ} .

(3.19)

A total derivative in the Lagrangian contributes only a surface term $\int dσ n_{μ} J^{μ}$ to the action, which vanishes if we assume the fields are fixed at the boundaries for any variation $δϕ \to δ J$ (as we do in the derivation of the E-L equations). This can then be shown to imply the existence of a conserved current $j^{μ}$ :

\partial_{μ} j^{μ} = 0, j^{μ} = \frac{\partial L}{\partial (\partial_{μ} ϕ)} Δ ϕ - J^{μ},

(3.20)

which in turn implies the existence of a conserved charge $Q$

Q = \int_{allspace} d^{3} x j^{0} .

(3.21)

Example: translation symmetry

Consider a translation-invariant theory, such as for the free scalar field (Eq. 3.10). A spacetime translation $x^{μ} \to x^{μ} - a^{μ}$ leads to the transformation $ϕ (x) \to ϕ (x + a) ≃ ϕ (x) + a^{μ} \partial_{μ} ϕ (x)$ and $L (x) \to L + a^{μ} \partial_{μ} L$ . Comparing this to Eq. 3.19, we see that $(J^{μ})_{ν} = δ_{ν}^{μ} L$ for each of the four translations $a^{ν}$ . Thus, the conserved current is:

{(j^{μ})}_{ν} = \frac{\partial L}{\partial (\partial_{μ} ϕ)} \partial_{ν} ϕ - δ_{ν}^{μ} L,

(3.22)

which we call the energy-momentum tensor $T_{ν}^{μ}$ . The associated conserved quantities (or “charges”) are the total energy and momentum of the field configuration:

E = \int d^{3} x T^{00}, P^{i} = \int d^{3} x T^{0 i} .

(3.23)

For our free scalar field, this turns out to be:

\begin{matrix} \begin{aligned} E & = \int d^{3} x \frac{1}{2} {\dot{ϕ}}^{2} + \frac{1}{2} {(\nabla ϕ)}^{2} + \frac{1}{2} m^{2} ϕ^{2}, \\ P^{i} & = \int d^{3} x \dot{ϕ} \partial^{i} ϕ . \end{aligned} \end{matrix}

(3.24)

Perhaps surprisingly, the interpretation of these as energy and momenta will be more intuitive once we quantize the free scalar field (Section 3.2).

Example: a U(1) internal symmetry

So far the symmetries we have discussed, such as translational and rotational invariance, are spacetime symmetries. An internal symmetry is a transformation that acts only on the fields, at each point of spacetime. A simple example is for the complex scalar field $ψ (x)$ , composed of two real scalar fields $ϕ_{1} (x)$ and $ϕ_{2} (x)$ :

\begin{matrix} \begin{aligned} ψ (x) & = \frac{1}{\sqrt{2}} (ϕ_{1} (x) + i ϕ_{2} (x)), \\ ψ^{*} (x) & = \frac{1}{\sqrt{2}} (ϕ_{1} (x) - i ϕ_{2} (x)) . \end{aligned} \end{matrix}

(3.25)

for which we can write down the free Lagrangian:

L = \partial_{μ} ψ^{*} \partial^{μ} ψ - m^{2} ψ^{*} ψ .

(3.26)

This Lagrangian possesses an internal $U (1)$ symmetry: it is invariant under $ψ (x) \to e^{iα} ψ (x)$ for any constant $α$ . This might seem simply a mathematical quirk, rather than a “physical” symmetry like translation invariance; however, Noether’s theorem tells us this also has the important physical consequences of a conserved current and charge, which turn out to be:

j^{μ} = i (ψ^{*} \partial^{μ} ψ - ψ \partial^{μ} ψ^{*}), Q = \int d^{3} x i (ψ^{*} \partial^{0} ψ - ψ \partial^{0} ψ^{*}) .

(3.27)

Again, their interpretation will be more clear once quantized; we will see this exactly corresponds to the conservation of electric charge! In fact, we say that a field that transforms as so under a global $U (1)$ rotation

\begin{matrix} \begin{aligned} ψ (x) & \to e^{iqα} ψ (x), \\ ψ^{*} (x) & \to e^{- iqα} ψ^{*} (x), \end{aligned} \end{matrix}

(3.28)

is charged under the $U (1)$ symmetry, with charge $q$ (and its complex conjugate with charge $- q$ ). Of course, for a single field, the two constants $α$ and $q$ are redundant so the magnitude of the charge can be defined arbitrarily; however, generally, as in QED, we have multiple fields transforming under the same $U (1)$ symmetry with different constants $q$ , representing different charges.

3.1.4 Hamiltonian mechanics

In QM, the Hamiltonian formalism is most natural. In QFT, as well, it will prove useful for the canonical quantization of the fields in the next section. The Hamiltonian density is the Legendre transform of the Lagrangian:

H = π_{a} {\dot{ϕ}}_{a} - L,

(3.29)

where $\dot{ϕ}$ is the time derivative and

π_{a} = \frac{\partial L}{\partial {\dot{ϕ}}_{a}}

(3.30)

are the conjugate momenta to the fields $ϕ_{a}$ . The Hamiltonian generally has the interpretation of the energy of a system, or the energy operator in QM. The EOMs are Hamilton’s equations:

\begin{matrix} \begin{aligned} {\dot{ϕ}}_{a} & = \frac{\partial H}{\partial π_{a}}, \\ {\dot{π}}_{a} & = - \frac{\partial H}{\partial ϕ_{a}} . \end{aligned} \end{matrix}

(3.31)

Poisson brackets

The time evolution of a general quantity $f (ϕ, π)$ can be expressed as:

\frac{df (ϕ, π)}{dt} = \frac{∂f}{∂ϕ} \dot{ϕ} + \frac{∂f}{∂π} \dot{π} = \frac{∂f}{∂ϕ} \frac{\partial H}{∂π} - \frac{∂f}{∂π} \frac{\partial H}{∂ϕ} \equiv {f, H},

(3.32)

where the last step defines the Poisson bracket ${⋅,⋅}$ . In terms of Poisson brackets, Hamilton’s equations can be written as:

\begin{matrix} \begin{aligned} {\dot{ϕ}}_{a} & = {ϕ_{a}, H} = \frac{\partial H}{\partial π_{a}}, \\ {\dot{π}}_{a} & = {π_{a}, H} = - \frac{\partial H}{\partial ϕ_{a}} . \end{aligned} \end{matrix}

(3.33)

Importantly, the canonical fields of the Hamiltonian, $ϕ$ and $π$ , obey the canonical Poisson bracket relations:

\begin{matrix} \begin{aligned} {ϕ (x), ϕ (y)} & = 0, \\ {π (x), π (y)} & = 0, \\ {ϕ (x), π (y)} & = δ^{3} (x - y) . \end{aligned} \end{matrix}

(3.34)

Example 3.2. Revisiting the simple (non-field-theoretic) Lagrangian from Example 3.1, we can derive the conjugate momentum to $x$ to be:

p = \frac{∂L}{∂ẋ} = mẋ,

(3.35)

and hence,

H = pẋ - L = \frac{1}{2} m ẋ^{2} + V (x) = \frac{p^{2}}{2 m} + V (x),

(3.36)

which is the classical energy of a free particle. Note, as in the last step, we express the Hamiltonian as a function of the conjugate momenta $p$ rather than the time derivative of the coordinate $ẋ$ . Finally, the EOMs are:

\begin{matrix} \begin{aligned} ẋ & = \frac{∂H}{∂p} = \frac{p}{m}, \\ ṗ & = - \frac{∂H}{∂x} = - \frac{dV}{dx} . \end{aligned} \end{matrix}

(3.37)

The former is simply the definition of velocity, while the latter again reproduces Newton’s second law. Finally, we can explicitly confirm the canonical Poisson bracket relations for the canonical coordinates $x$ and $p$ :

\begin{matrix} \begin{aligned} {x, x} & = {p, p} = 0, \\ {x, p} & = \frac{∂x}{∂x} \frac{∂p}{∂p} - \frac{∂x}{∂p} \frac{∂p}{∂x} = 1 . \end{aligned} \end{matrix}

(3.38)

Free scalar field Hamiltonian

For the free scalar Lagrangian in Eq. 3.10, we find

π = \dot{ϕ} = - i \int \frac{d^{3} p}{{(2 π)}^{3}} \sqrt{\frac{ω_{p}}{2}} (a (p) e^{ip \cdot x} - a^{*} (p) e^{- ip \cdot x}),

(3.39)

where we plugged in the plane-wave solutions for $ϕ$ from Eq. 3.17, and

H = π \dot{ϕ} - L = \frac{1}{2} π^{2} + \frac{1}{2} {(\nabla ϕ)}^{2} + \frac{1}{2} m^{2} ϕ^{2} .

(3.40)

This is, in fact, the same as the expression for energy we derived via Noether’s theorem in Eq. 3.24. Note that, unlike the Lagrangian, the Hamiltonian is not Lorentz-invariant. This makes sense under the interpretation of the Hamiltonian as the energy, which is not a Lorentz scalar. Its Lorentz-invariance, as well as its natural connection to the path integral formulation (Section 3.1.1), is the reason the Lagrangian viewpoint is preferred in QFT.

¹See Weinberg’s notes on a history of QFT [25] for a nice summary of its historical development.

²If we take $S \sim energy \cdot T = \frac{ℏ c}{λ} \cdot T$ , then the classical limit $\frac{ℏ}{S} = \frac{λ}{cT} \to 0$ physically is the case where the de Broglie wavelength of the particle $λ$ is negligible compared to the relevant length scales.