Lagrangian and Hamiltonian Field Theory

The preliminaries introduce classical fields, the Lagrangian density, and the Euler–Lagrange equations in compact form. This page develops that material in full: the action principle for fields, the passage to the canonical (Hamiltonian) formulation, functional derivatives, and the counting of degrees of freedom. It is the classical groundwork on which canonical quantization is built, and its symmetry content is developed on the companion page Symmetries, Noether's Theorem and Currents.

Conventions follow the rest of the QFT section: natural units $\hslash =c=1$ and the mostly-minus metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

From particles to fields

Classical mechanics describes a finite set of generalized coordinates $q_i(t)$ evolving in time. A classical field theory is the continuum limit: the discrete index $i$ is replaced by a continuous spatial label $\mathbf{x}$, and the dynamical variable becomes a field $\varphi (\mathbf{x},t)=\varphi (x)$ defined at every point of spacetime.

The crucial feature of the relativistic case is that space and time must enter on an equal footing. The Lagrangian $L$ of mechanics singles out the time derivative; a Lorentz-invariant field theory instead builds everything from the Lagrangian density $\mathcal{L}$, a local Lorentz scalar depending on the field and all its spacetime derivatives ${\partial }_{\mu }\varphi$ symmetrically.

The action principle for fields (Postulate — specializes QFT Postulate 9)

A field theory is specified by a Lagrangian density $\mathcal{L}$, a local function of the fields ${\varphi }_a(x)$ and their first derivatives ${\partial }_{\mu }{\varphi }_a(x)$ (the label $a$ runs over all fields and their internal/Lorentz components):

$$\mathcal{L}=\mathcal{L}({\varphi }_a,{\partial }_{\mu }{\varphi }_a).$$

The action is its spacetime integral over a region $\Omega$:

$$S[\varphi ]={\int }_{\Omega }{d}^{4}x\mathcal{L}({\varphi }_a(x),{\partial }_{\mu }{\varphi }_a(x)).$$

That the action is the integral of a local Lagrangian density built from fields and their first derivatives at a single spacetime point is exactly the content of QFT Postulate 9. Restricting to first derivatives keeps the field equations second-order (Ostrogradsky's theorem shows higher-derivative Lagrangians generically carry ghost instabilities); locality and Lorentz invariance of $\mathcal{L}$ are what deliver microcausality and Poincaré covariance in the quantum theory.

Hamilton's principle. The physical field configuration is a stationary point of $S$ under variations ${\varphi }_a\to {\varphi }_a+\delta {\varphi }_a$ that vanish on the boundary $\partial \Omega$:

$$\boxed{\delta S=0\text{for all }\delta {\varphi }_a\text{ with }\delta {\varphi }_a{|}_{\partial \Omega }=0.}$$

The Euler–Lagrange equations

Vary the action and expand to first order:

$$\delta S={\int }_{\Omega }{d}^{4}x\left[\frac{\partial \mathcal{L}}{\partial {\varphi }_a}\delta {\varphi }_a+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta ({\partial }_{\mu }{\varphi }_a)\right].$$

Since variation commutes with differentiation, $\delta ({\partial }_{\mu }{\varphi }_a)={\partial }_{\mu }(\delta {\varphi }_a)$. Integrate the second term by parts:

$$\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}{\partial }_{\mu }(\delta {\varphi }_a)={\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a\right)-{\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)\delta {\varphi }_a.$$

The total-derivative term integrates to a boundary contribution over $\partial \Omega$, which vanishes because $\delta {\varphi }_a=0$ there. What remains is

$$\delta S={\int }_{\Omega }{d}^{4}x\left[\frac{\partial \mathcal{L}}{\partial {\varphi }_a}-{\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)\right]\delta {\varphi }_a.$$

For $\delta S$ to vanish for arbitrary interior variations $\delta {\varphi }_a$, the bracket must vanish pointwise. This gives the Euler–Lagrange equations of motion, one for each field component:

$$\boxed{{\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)-\frac{\partial \mathcal{L}}{\partial {\varphi }_a}=0.}$$

These are the field-theory analogue of $\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$, with the single time derivative $\frac{d}{dt}$ promoted to the four-divergence ${\partial }_{\mu }$.

Worked example: the free scalar field

For the free real scalar Lagrangian (derived and motivated in preliminaries § Worked example),

$$\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2,$$

the two pieces of the Euler–Lagrange equation are

$$\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}={\partial }^{\mu }\varphi ,\frac{\partial \mathcal{L}}{\partial \varphi }=-m^2\varphi ,$$

so that ${\partial }_{\mu }{\partial }^{\mu }\varphi +m^2\varphi =0$, the Klein–Gordon equation $(\square +m^2)\varphi =0$. This is the classical field equation that the free scalar field will be quantized around.

Functional derivatives

It is often cleaner to phrase the variation in terms of the functional derivative $\delta S/\delta {\varphi }_a(x)$, defined by

$$\delta S=\int d^{4}x\frac{\delta S}{\delta {\varphi }_a(x)}\delta {\varphi }_a(x).$$

Comparing with the variation above, the Euler–Lagrange equation is simply the statement that the functional derivative of the action vanishes:

$$\frac{\delta S}{\delta {\varphi }_a(x)}=\frac{\partial \mathcal{L}}{\partial {\varphi }_a}-{\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)=0.$$

The basic functional-derivative identity, which recurs throughout the path-integral formulation, is

$$\frac{\delta {\varphi }_a(x)}{\delta {\varphi }_b(y)}={\delta }_{ab}{\delta }^4(x-y).$$

The Dirac delta here is the continuum analogue of the Kronecker ${\delta }_{ij}$ that would appear in $\partial q_i/\partial q_j$ in mechanics.

The Hamiltonian (canonical) formulation

Passing to the Hamiltonian picture is the essential preparatory step for canonical quantization, where equal-time (anti)commutation relations are imposed on a field and its conjugate momentum.

Conjugate momentum

The momentum density conjugate to ${\varphi }_a$ singles out the time derivative ${\partial }_0{\varphi }_a={\dot{\varphi }}_a$ (this is the step that breaks manifest Lorentz covariance — the price of the Hamiltonian formulation):

$$\boxed{{\pi }^a(x)=\frac{\partial \mathcal{L}}{\partial ({\partial }_0{\varphi }_a)}=\frac{\partial \mathcal{L}}{\partial {\dot{\varphi }}_a}.}$$

This matches the definition quoted in preliminaries § Canonical Structure.

Hamiltonian density

The Hamiltonian density is the Legendre transform of $\mathcal{L}$ with respect to ${\dot{\varphi }}_a$:

$$\boxed{\mathcal{H}={\pi }^a{\dot{\varphi }}_a-\mathcal{L},}$$

with ${\dot{\varphi }}_a$ expressed in terms of ${\pi }^a$. The total Hamiltonian is $H=\int d^{3}x\mathcal{H}$, and it is (for a theory with no explicit time dependence) the conserved energy — the $T^{00}$ component of the energy–momentum tensor derived on the Noether page.

Worked example: scalar field Hamiltonian

For $\mathcal{L}=\frac{1}{2}{\dot{\varphi }}^2-\frac{1}{2}(\nabla \varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2$ (expanding ${\partial }_{\mu }\varphi {\partial }^{\mu }\varphi ={\dot{\varphi }}^2-(\nabla \varphi {)}^2$ in the mostly-minus metric), the conjugate momentum is

$$\pi =\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=\dot{\varphi },$$

and the Hamiltonian density is

$$\mathcal{H}=\pi \dot{\varphi }-\mathcal{L}=\frac{1}{2}{\pi }^2+\frac{1}{2}(\nabla \varphi {)}^2+\frac{1}{2}{m}^2{\varphi }^2.$$

Every term is manifestly non-negative — the classical energy is bounded below, which is exactly the property that the quantum spectrum condition will promote to positivity of $P^0$.

Poisson brackets

The classical Poisson bracket of two functionals $F,G$ of the fields and momenta is

$$\{F,G\}=\int d^{3}x\left(\frac{\delta F}{\delta {\varphi }_a(\mathbf{x})}\frac{\delta G}{\delta {\pi }^a(\mathbf{x})}-\frac{\delta F}{\delta {\pi }^a(\mathbf{x})}\frac{\delta G}{\delta {\varphi }_a(\mathbf{x})}\right).$$

The fundamental equal-time brackets are

$$\{{\varphi }_a(\mathbf{x}),{\pi }^b(\mathbf{y})\}={\delta }_a^b{\delta }^3(\mathbf{x}-\mathbf{y}),\{{\varphi }_a,{\varphi }_b\}=\{{\pi }^a,{\pi }^b\}=0,$$

and Hamilton's equations take the form ${\dot{\varphi }}_a=\{{\varphi }_a,H\}$, ${\dot{\pi }}^a=\{{\pi }^a,H\}$. Canonical quantization is the rule $\{\cdot ,\cdot \}\to -i[\cdot ,\cdot ]$ (Dirac's prescription), turning the first bracket into the equal-time commutator $[{\varphi }_a(\mathbf{x}),{\pi }^b(\mathbf{y})]=i{\delta }_a^b{\delta }^3(\mathbf{x}-\mathbf{y})$ that opens the scalar-field construction. For half-integer spin the bracket is instead replaced by an anticommutator, as forced by spin–statistics; see the Dirac field.

Counting degrees of freedom

A recurring bookkeeping question is how many independent field degrees of freedom (dof) a Lagrangian carries — this controls the number of physical particle polarizations after quantization.

The mismatch between the number of Lagrangian components and the number of physical polarizations is what forces the careful treatment of constraints in the vector field — and, in the non-abelian case, the Faddeev–Popov machinery.

Where this leads

• Symmetries of the action → conserved currents and charges, and the energy–momentum tensor: Symmetries, Noether's Theorem and Currents.
• Canonical quantization of the fields built here: the free scalar field, the Dirac field, the vector field.
• The alternative, path-integral route that takes the same action $S[\varphi ]$ as input without passing through the Hamiltonian: the generating functional.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 2.1–2.2.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 7.
• Goldstein, Poole & Safko, Classical Mechanics, Ch. 13 (classical field theory).