Lagrangian Field Theory — formula sheet
Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Euler-Lagrange Equations for Fields
The rate of change (in spacetime) of the field's "conjugate momentum density" \(\pi^\mu\equiv\partial\mathcal{L}/\partial(\partial_\mu\phi)\) is balanced against the "force density" \(\partial\mathcal{L}/\partial\phi\). It is the field analogue of \(\tfrac{d}{dt}(\partial L/\partial\dot q)=\partial L/\partial q\): the particle time-derivative \(d/dt\) is promoted to the spacetime divergence \(\partial_\mu\), and \(q(t)\) to \(\phi(x)\).
Noether's First Theorem
Each independent continuous symmetry of the action supplies one four-current whose four-divergence vanishes on physical field configurations. The current is built from the canonical momentum density \( \pi^\mu_a \equiv \partial\mathcal{L}/\partial(\partial_\mu\phi_a) \) contracted with the shape of the field variation \( \delta\phi_a \), minus the boundary term \( K^\mu \) that quantifies how much the Lagrangian fails to be strictly invariant. The vanishing divergence is a genuine continuity equation, so the spatial integral of \( j^0 \) is a time-independent charge. The theorem is constructive: give it the symmetry, it hands you the conserved quantity.
Canonical Stress-Energy Tensor
The first term is the "flux-of-field-momentum" piece — the canonical momentum density \(\pi^\mu_a=\partial\mathcal{L}/\partial(\partial_\mu\phi_a)\) contracted with the field's own gradient; the second subtracts the Lagrangian along the diagonal so that the \(\nu=0\) charge is the Hamiltonian. Explicitly \(T^{00}=\pi_a\dot\phi_a-\mathcal{L}=\mathcal{H}\) is the energy density, \(T^{0i}\) the momentum density (\(P^i=\int d^3x\,T^{0i}\)), and \(T^{ij}\) the stress: the \(i\)-momentum flowing in the \(j\)-direction. The single equation \(\partial_\mu T^{\mu\nu}=0\) is four continuity equations, one per conserved component of four-momentum.
Belinfante-Rosenfeld Improvement
The intrinsic spin content \(S^{\lambda\mu\nu}\) is folded back into the stress tensor as a total divergence. The result is a single symmetric, conserved, gauge-invariant tensor whose orbital moment \(x^{\mu}T_B^{\nu\lambda}-x^{\nu}T_B^{\mu\lambda}\) already carries the full angular momentum — spin and orbital are unified. Because it is symmetric and gauge invariant, \(T_B^{\mu\nu}\) coincides with the Hilbert tensor obtained by varying the action with respect to the metric, and it is this tensor, not \(T_c\), that couples to gravity.
Klein-Gordon Field from Its Lagrangian
The scalar Lagrangian forces every Fourier mode of the field to oscillate on a fixed "mass shell": its frequency and wavevector are locked by \(\omega^2=\mathbf{k}^2+m^2\). The kinetic term \(\tfrac12(\partial\varphi)^2\) supplies the \(\Box\) (the \(\omega^2-\mathbf{k}^2\)); the mass term supplies the constant \(m^2\) that gives even a mode at rest (\(\mathbf{k}=0\)) a nonzero frequency \(\omega=m\). In quantum language, restoring \(\hbar,c\), this is exactly \(E=\hbar\omega\), \(\mathbf{p}=\hbar\mathbf{k}\) plugged into Einstein's relation: the field's normal modes are relativistic particles of rest mass \(m\).
Representations of the Lorentz Group
Dirac Equation from the Spinor Lagrangian
One real Lagrangian produces three linked facts: the field's equation of motion (Dirac), a locally conserved four-current, and the guarantee that its density is a genuine probability — never negative. The first-order form and the anticommuting \(\gamma^\mu\) are not decorative; they are precisely what makes \(j^0\) a sum of squares while still reproducing the relativistic energy–momentum relation of Klein–Gordon.
Maxwell's Equations from −¼F²
Varying the potential in \(-\tfrac14F^2-j\!\cdot\!A\) says the divergence of the field strength equals the current: charges and currents source \(F\). This single covariant equation contains Gauss's law (\(\nu=0\)) and the Ampère–Maxwell law (\(\nu=i\)). The second equation is not a law but the statement that \(F\) is the curl (exterior derivative) of a potential; it delivers "no magnetic charge" and Faraday induction automatically. Conservation of charge, \(\partial_\mu j^\mu=0\), is not an extra assumption but a forced consequence of the antisymmetry of \(F\).
Local U(1) Gauge Invariance and Minimal Coupling
Local U(1) invariance is not free: it demands a compensating gauge field \( A_\mu \), packaged with \( \partial_\mu \) into a covariant derivative that transforms like the field it acts on. Expanding \( D_\mu \) manufactures the coupling \( e\,j^\mu A_\mu \) — the photon talks to the electron only through the conserved current \( j^\mu=\bar\psi\gamma^\mu\psi \). Because \( F^{\mu\nu} \) is antisymmetric, the field equation \( \partial_\mu F^{\mu\nu}=e\,j^\nu \) can only be consistent if \( \partial_\mu j^\mu=0 \): gauge symmetry and charge conservation are the same statement.
Proca Field and Its Degrees of Freedom
A vector field with a mass term is no longer gauge-redundant. Its equation of motion automatically enforces the Lorenz condition, which acts as a single scalar constraint that eliminates the unphysical time-like component. What remains is a massive spin-1 particle: two transverse polarizations (as for light) plus one longitudinal polarization that only a massive vector can carry. The field is short-ranged, with Yukawa-type behaviour \(\sim e^{-mr}/r\) rather than the Coulomb \(1/r\) of the massless case.
Hamiltonian Formulation for Fields
The momentum density \(\pi\) is what the Lagrangian pays per unit field velocity. The Hamiltonian density is the Legendre transform that trades \(\dot\phi\) for \(\pi\), and equals the field energy density. The two first-order Hamilton equations are fully equivalent to the single second-order Euler–Lagrange equation, and the canonical bracket \(\{\phi,\pi\}=\delta^3\) generates all of the dynamics through \(\dot F=\{F,H\}\).
Canonical Quantization of a Field
At one instant of time, the field amplitude at a point and its conjugate momentum at the same point are incompatible observables obeying an uncertainty relation, while the amplitude at one point commutes perfectly with the amplitude and the momentum at every other point. Space is treated as a continuum of independent quantum mechanical degrees of freedom, one canonical pair \((\hat\phi,\hat\pi)\) per point, glued together only by the Hamiltonian's gradient terms.
Mode Expansion and Fock Space
The quantized field is a superposition, over all momenta, of operators that destroy (\(\hat{a}\)) or create (\(\hat{a}^{\dagger}\)) a relativistic quantum of energy \(E_{\mathbf{p}}=\sqrt{|\mathbf{p}|^2+m^2}\). Their oscillator-like algebra means the Hilbert space is a Fock space: a vacuum, one-particle states \(\hat{a}^{\dagger}_{\mathbf{p}}|0\rangle\), and symmetric \(n\)-particle towers. The commutator's \(2E_{\mathbf{p}}\) is exactly the factor that makes single-particle states Lorentz-covariantly normalized, \(\langle\mathbf{p}|\mathbf{q}\rangle=(2\pi)^3 2E_{\mathbf{p}}\delta^{(3)}(\mathbf{p}-\mathbf{q})\).
The Feynman Propagator
The time-ordered vacuum expectation value \(\langle 0|T\phi(x)\phi(y)|0\rangle\) inverts the Klein–Gordon operator with a unit imaginary source at coincident points. Among all Green's functions of \((\Box+m^2)\), the \(+i\epsilon\) prescription selects the one that propagates positive-frequency modes forward in time and negative-frequency modes backward — the amplitude for the vacuum to create a quantum at the earlier point and absorb it at the later point (or the antiparticle in reverse). It is the internal line of every Feynman diagram.
Conserved Charge as Symmetry Generator
The conserved charge does not merely commute with the Hamiltonian and sit inert; commuting it against a field returns (up to \( -i \)) exactly the infinitesimal symmetry variation of that field. In operator language \( Q \) is the Lie-algebra generator whose exponential is the unitary symmetry transformation on the Hilbert space. Conservation (\( \dot Q=0 \)) guarantees the same operator generates the symmetry at every instant.
The Spin-Statistics Theorem
Whether the causal spacelike-vanishing bilinear is the commutator or the anticommutator is fixed entirely by the spin. The \((-1)^{2s}\) sign a field acquires under a \(2\pi\) rotation (and equivalently under the \(x\to -x\) exchange of the two mode terms) is exactly cancelled only when the graded bracket matches the spin: symmetric bracket for integer spin, antisymmetric for half-integer. Energy positivity gives the same verdict, so the two independent requirements agree.
Goldstone's Theorem
The curvature of the potential vanishes in every direction obtained by acting on the vacuum with a broken symmetry generator. Each such direction is a valley floor of the "Mexican-hat" landscape: rolling along it costs no energy, so the corresponding quantum has zero rest mass. The number of these massless Nambu–Goldstone bosons equals the number of symmetry generators that fail to leave the vacuum invariant — the dimension of the coset \(G/H\).
The Abelian Higgs Mechanism
Gauging a spontaneously broken \(U(1)\) gives the gauge boson a mass equal to the coupling times the vacuum expectation value. The would-be Goldstone boson is not a physical particle: it has been "eaten" to supply the longitudinal polarisation that a massive vector must carry. The surviving physical scalar is the radial Higgs excitation, with mass set by the potential's curvature. Degrees of freedom are exactly conserved.