Classical field theory

B.1 Classical field theory

B.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),

(B.1.1)

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 .

(B.1.2)

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 .

(B.1.3)

Example B.1.1. We can confirm that this is equivalent to Newtonian mechanics by considering the simple Lagrangian:

L = \frac{1}{2} m ẋ^{2} - V (x) .

(B.1.4)

Plugging this into Eq. B.1.3 gives us:

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

(B.1.5)

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},

(B.1.6)

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 $\frac{ℏ}{S} \to 0$ ,¹ 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 [81] Chapter 9).

B.1.2 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},

(B.1.7)

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},

(B.1.8)

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}),

(B.1.9)

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.

B.1.3 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,

(B.1.10)

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

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

(B.1.11)

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}

(B.1.12)

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},

(B.1.13)

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}

(B.1.14)

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}

(B.1.15)

Example B.1.2. Revisiting the simple (non-field-theoretic) Lagrangian from Example B.1.1, we can derive the conjugate momentum to $x$ to be:

p = \frac{∂L}{∂ẋ} = mẋ,

(B.1.16)

and hence,

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

(B.1.17)

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}

(B.1.18)

The former is simply the definition of velocity, while the latter again reproduces Newton’s second law. Finally, we can explitly 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}

(B.1.19)

Free scalar field Hamiltonian

For the free scalar Lagrangian in Eq. 3.1.4, 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}),

(B.1.20)

where we plugged in the plane-wave solutions for $ϕ$ from Eq. B.1.9, and

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

(B.1.21)

This is, in fact, the same as the expression for energy we derived via Noether’s theorem in Eq. 3.1.14. 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 B.1.1.), is the reason the Lagrangian viewpoint is preferred in QFT.

¹If 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.