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For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

For the convention $s\bar{c}=-b$, the action $S_{\mathrm{YM}}-s\Psi$ contains $-b\partial \cdot A+\xi b^2/2-\bar{c}\partial \cdot Dc$ after integration by parts. Integrating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$. The Grassmann Gaussian integral represents the Faddeev-Popov determinant.

The generators in the Spinor representation of the Lorentz group satisfy $M^{†}{\gamma }^0+{\gamma }^{0}M=0$, so the finite transformation preserves the indefinite form $S^{†}{\gamma }^{0}S={\gamma }^0$. This makes $\bar{\psi }\psi$ a Lorentz scalar through the Dirac adjoint. Boost generators are Hermitian rather than anti-Hermitian: a boost has eigenvalues $e^{\pm \eta /2}$ and is not unitary in the ordinary positive-definite component norm. Rotations are unitary. This component representation is distinct from the unitary action on the physical state space.

For an autonomous Lagrangian $L(x,\dot{x})$, define its action $A=\int Ldt$ and Hamiltonian $H=\dot{x}{L}_{\dot{x}}-L$. The chain rule givesThus the identity holds along forced trajectories as well as solutions of the homogeneous Euler-Lagrange equation. Explicit time dependence adds $-{\partial }_{t}L$; stochastic work requires a consistent regularization or the Stratonovich chain rule.

Expanding an action about a classical stationary field and retaining its second variation gives a Gaussian path integral. Its nonzero fluctuation modes determine a functional determinant and a one-loop soliton mass correction. Symmetry-generated zero modes in field theory must be removed from that determinant and integrated as collective coordinates.

For action $\frac{m}{2}\int ({\dot{q}}^2-{\omega }^2{q}^2)dt$, the kernel is $\sqrt{m\omega /(2\pi i\sin \omega T)}\exp \{im\omega [(q_f^2+q_i^2)\cos \omega T-2q_f{q}_i]/(2\sin \omega T)\}$. The Dirichlet oscillator determinant ratio supplies its prefactor, while the classical boundary action supplies its exponent. The Feynman i-epsilon prescription specifies its continuation through caustics.

An internal symmetry changes field values while leaving spacetime coordinates fixed, preserving the action. A constant phase rotation of a complex scalar field is an example. This distinguishes internal charges from the spacetime translation charges in the canonical stress-energy tensor. Localizing the parameter generally requires gauge fields; a global internal symmetry is not automatically a gauge symmetry.

A Lagrangian density is integrated over spacetime to give the action $S=\int d^{d}x\mathcal{L}$.

For single-field adiabatic inflation, the Mukhanov-Sasaki variable rescales the comoving curvature perturbation by $z=a\sqrt{2ϵ}{M}_{\mathrm{Pl}}$, where $M_{\mathrm{Pl}}$ is the reduced Planck mass. Its quadratic bulk action is $\frac{1}{2}\int [(v^{\prime }{)}^2-(\nabla v{)}^2+(z^{\prime \prime }/z)v^2]$, after a boundary term is removed. A Fourier transform mode obeys $v_k^{\prime \prime }+(k^2-z^{\prime \prime }/z)v_k=0$ and has canonical Wronskian normalization. The mass scale and sound-speed factors depend on the underlying scalar theory; this definition assumes the canonical single-field action and positive slow-roll kinetic coefficient.

A functional derivative of the quantum effective action gives an amputated one-particle-irreducible correlation function. Its perturbative graphs stay connected after cutting any single internal propagator. A free quadratic effective action has no vertices of order greater than two; interactions generate higher vertices.

With a time-slice regulator, inserting field-coordinate resolutions of identity in evolution kernels produces a bosonic phase-space path integral; integrating the canonical momentum gives the configuration-space action. For fermions, resolutions in fermionic coherent states instead produce a Berezin integral and the first-order kinetic term. Vacuum endpoint projection supplies the Feynman i-epsilon prescription. Both constructions reproduce the same free oscillator correlators and their Wick theorem expansions, and thus agree order by order in a regulated perturbation series.