Part I, Chapter 1 | Page 1 of 8

Lagrangian Field Theory

From discrete particle mechanics to continuous field theory

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Lecture 1: Classical Field Theories and Principle of Locality - MIT 8.323

Introduction to classical field theory and Lagrangian formalism (MIT QFT Course)

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1.1 Review: Lagrangian Mechanics for Particles

In classical mechanics, a system with n generalized coordinates $q^i(t)$ (i = 1, ..., n) is described by a Lagrangian:

$$L(q^i, \dot{q}^i, t) = T - V$$

where T is kinetic energy and V is potential energy. The action is:

$$S[q] = \int_{t_1}^{t_2} L(q^i, \dot{q}^i, t) \, dt$$

Hamilton's Principle

The physical trajectory extremizes the action: $\delta S = 0$. This yields the Euler-Lagrange equations:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^i} - \frac{\partial L}{\partial q^i} = 0$$

Canonical Momentum

The canonical momentum conjugate to $q^i$ is:

$$p_i = \frac{\partial L}{\partial \dot{q}^i}$$

Hamiltonian

Via Legendre transform, we obtain the Hamiltonian:

$$H(q^i, p_i, t) = \sum_i p_i \dot{q}^i - L$$

typically equal to the total energy H = T + V. Hamilton's equations:

$$\dot{q}^i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q^i}$$

1.2 From Discrete to Continuous Systems

Consider a 1D chain of N coupled oscillators with positions $q_n(t)$, separation a, and coupling constant k:

$$L = \sum_{n=1}^N \left[\frac{1}{2}m\dot{q}_n^2 - \frac{1}{2}k(q_{n+1} - q_n)^2\right]$$

Continuum Limit

Take the limit N → ∞, a → 0, with Na = L fixed. Define a continuous field:

$$\phi(x, t) = \lim_{a \to 0} q_n(t), \quad x = na$$

The discrete sum becomes an integral:

$$\sum_{n=1}^N (\cdots) a \to \int_0^L (\cdots) dx$$

The discrete derivative becomes a spatial derivative:

$$\frac{q_{n+1} - q_n}{a} \to \frac{\partial \phi}{\partial x}$$

Field Lagrangian

Define mass density ρ = m/a and spring constant μ = ka. The Lagrangian becomes:

$$L = \int_0^L \left[\frac{1}{2}\rho\left(\frac{\partial \phi}{\partial t}\right)^2 - \frac{1}{2}\mu\left(\frac{\partial \phi}{\partial x}\right)^2\right] dx$$

The integrand is the Lagrangian density $\mathcal{L}$:

$$L = \int_0^L \mathcal{L}(\phi, \partial_t \phi, \partial_x \phi) dx$$

This yields the wave equation:

$$\rho \frac{\partial^2 \phi}{\partial t^2} = \mu \frac{\partial^2 \phi}{\partial x^2}$$

with wave speed $v = \sqrt{\mu/\rho}$.

1.3 Relativistic Field Theory

Lorentz Covariance

In special relativity, spacetime events are labeled by 4-vectors:

$$x^\mu = (x^0, x^1, x^2, x^3) = (ct, x, y, z) = (ct, \vec{x})$$

Greek indices μ, ν, ... run from 0 to 3. The Minkowski metric (signature -,+,+,+):

$$\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$$

Raise and lower indices: $x_\mu = \eta_{\mu\nu} x^\nu$, so $x_0 = -ct$, $x_i = x^i$.

Invariant Interval

The spacetime interval is Lorentz invariant:

$$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

4-Gradient

The 4-gradient (natural units c = 1):

$$\partial_\mu = \frac{\partial}{\partial x^\mu} = \left(\frac{\partial}{\partial t}, \nabla\right), \quad \partial^\mu = \left(\frac{\partial}{\partial t}, -\nabla\right)$$

The d'Alembertian operator:

$$\Box = \partial_\mu \partial^\mu = \partial_t^2 - \nabla^2$$

Relativistic Action

For Lorentz covariance, the action must be a Lorentz scalar:

$$S[\phi] = \int d^4x \, \mathcal{L}(\phi, \partial_\mu \phi)$$

where $d^4x = dt \, d^3x$ is the invariant volume element and $\mathcal{L}$ is the Lagrangian density (a scalar).

Convention: We use natural units ℏ = c = 1 throughout. Restore them via dimensional analysis when needed.

Key Concepts (Page 1)

• Fields φ(x^μ) are functions of spacetime with infinitely many degrees of freedom
• The Lagrangian density $\mathcal{L}$ replaces the discrete Lagrangian L
• Action is a spacetime integral: S = ∫d⁴x $\mathcal{L}$
• Lorentz covariance requires $\mathcal{L}$ to be a scalar
• We use natural units ℏ = c = 1

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Chapter 1: Lagrangian Field Theory

Quantum Field Theory Course