Classical Field Theory and Variational Principles

Manual: General · Subject: Theoretical Physics

Develop the covariant action formalism for scalar, spinor, gauge, and gravitational fields, including constraints and canonical structure.

The Action Principle

From dynamics to variational calculus

The action functional $S[\phi]$ is the central object of classical field theory. Physical trajectories and field histories extremize $S$ under variations that vanish at the boundary. This unifies mechanics, electromagnetism, fluid dynamics, and general relativity within one formal language.

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Practical rule

If you can state a theory in terms of an action, then symmetry, quantization, and renormalization become much more tractable.

Canonical examples

Scalar field

S=\int d^dx\,\left(\frac12\partial_\mu\phi\partial^\mu\phi - V(\phi)\right)

Electromagnetism

S=-\frac14\int d^dx\,F_{\mu\nu}F^{\mu\nu}

Dirac field

S=\int d^dx\,\bar\psi(i\gamma^\mu\partial_\mu-m)\psi

Einstein-Hilbert action

S=\frac{1}{16\pi G}\int d^dx\,\sqrt{-g}\,R

Euler-Lagrange Equations and Constraints

Field equations from stationarity

For a field $\phi$ , the Euler-Lagrange equation reads $\frac{\partial \mathcal{L}}{\partial \phi}-\partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\right)=0.$ Gauge theories and gravity introduce constraints because not all components are independent physical degrees of freedom; the Lagrangian is often singular, requiring careful treatment of primary and secondary constraints.

Common field types

Field	Spin / structure	Main issue
Scalar $\phi$	Spin 0	Potential stability and renormalization
Vector $A_\mu$	Spin 1	Gauge redundancy
Spinor $\psi$	Spin $1/2$	Chirality and anomaly constraints
Metric $g_{\mu\nu}$	Spin 2	Diffeomorphism invariance and constraints

What does varying the action with respect to the metric produce in general relativity?

Maxwell's equations Einstein's field equations Schrödinger's equation The virial theorem

What is the field-strength tensor for an Abelian gauge field $A_\mu$ ?

Deriving equations of motion from an action

1

Write the Lagrangian density $\mathcal{L}$ as a scalar under the relevant symmetries.
2

Vary the action $S=\int d^dx\,\mathcal{L}$ with respect to the fields.
3

Integrate by parts and discard boundary terms consistent with the variational principle.
4

Set the coefficient of each independent variation to zero.
5

Interpret the resulting differential equations and constraints physically.

Why are gauge theories said to contain redundant variables?

Because the physical fields are nonlocal Because gauge transformations change the description but not observables Because they violate causality Because they are only valid in flat spacetime

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