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A background gauge field is a nondynamical source for a conserved current, rather than a field integrated over in the path integral. For a Dirac current with source term $+B_{\mu }{j}^{\mu }$, the combined transformations $\psi \mapsto e^{i\alpha }\psi$ and $B_{\mu }\mapsto B_{\mu }+{\partial }_{\mu }\alpha$ preserve the action. Whether this also preserves the generating functional depends on the absence of the corresponding gauge anomaly.

A perturbative correlator factors into an exponential of connected Vacuum Feynman diagrams times a sum of Feynman diagrams attached to external insertions. Dividing its path integral by the vacuum generating functional at zero source cancels that exponential, including all disconnected products of vacuum components. The corresponding operator formalism uses the same denominator in the vacuum expectation of the Dyson series. This leaves externally disconnected correlators intact; the connected generating functional removes their products.

A classical path joining prescribed endpoints is a stationary point of the action, satisfying the Euler-Lagrange equation and the endpoint boundary conditions. Expanding the path integral about each such path separates its classical phase from its quadratic fluctuation determinant. There may be multiple classical paths or none within a chosen real saddle class; the semiclassical propagator must respect that saddle structure rather than assuming a unique trajectory for every potential.

The achievement depended on complementary ways to organize the same physics. In his Nobel lecture, delivered in 1966 for the 1965 prize, Sin-Itiro Tomonaga described covariant contact transformations and replacing the unobservable bare mass and electric charge by experimental values. He also credited Julian Seymour Schwinger's magnetic-moment work and Richard Feynman's methods for extending the approximation.^[ref]

Richard Feynman's 1965 lecture traced his spacetime and path integral viewpoint and the Feynman diagrams that made perturbation theory easier to organize. He emphasized that the path integral viewpoint helped him find expressions without being essential to every calculation. These accounts support shared credit for workable renormalized QED, not attribution of the whole theory to a single person.^[ref]

An effective Lagrangian encodes the interactions of retained degrees of freedom after others have been removed, often by a path integral. For external energies below a massive field's excitation threshold, its effects admit a local expansion in external momenta divided by its mass. Matching scattering amplitudes fixes the coefficients of the resulting local operators; truncation has a specified order in that small ratio.

Wick rotation of the free-line propagator gives the displayed coincident-endpoint kernel for positive Euclidean duration $T$. Equivalently it is the path integral over Dirichlet fluctuations about the straight classical path, with its time-sliced Gaussian measure included. It is a propagator density, not a full thermal partition function: that trace additionally integrates position and, on a multiply connected configuration space, sums the winding sectors. For an angular coordinate replace $M$ by moment of inertia.

A path integral with weight $e^{-S_E}$ uses the Euclidean action obtained by Wick rotation. Bosonic convergence requires appropriate positivity and boundary conditions.

A thermal path integral represents the canonical partition function by paths periodic in Euclidean time of circumference $\beta$. For $H=p^2/2+V(x)$, its formal action is $S_E[x]={\int }_0^{\beta }({\dot{x}}^2/2+V(x))d\tau$. The periodic boundary condition results from taking the trace of the evolution kernel. The continuum measure requires a regularization and normalization; finite-dimensional measures make both choices explicit.

The functional measure is the formal integration measure over field configurations in a path integral. Its transformation under a symmetry can preserve the corresponding Ward identity or produce a quantum anomaly.

Completing the square in the free scalar path integral gives a quadratic exponential in the source. If ${\Delta }_F$ includes the factor $i$ in the momentum-space Feynman propagator, the normalized Minkowski functional is $\exp [-\int J{\Delta }_{F}J/2]$. Two source derivatives with their $1/i^2$ factor recover ${\Delta }_F$, while higher derivatives give Wick contractions.

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.

A quadratic path integral becomes a product of Gaussian integrals after diagonalizing its fluctuation operator. One real coordinate per mode gives an inverse square root of the functional determinant; one complex coordinate, or two real coordinates, gives its inverse. A regulator and normalization are essential: determinant ratios are often meaningful before an absolute determinant is defined.

A generating functional is a path integral with a source $J$ coupled to a field. Its functional derivatives with respect to $J$ generate time-ordered correlation functions.

The harmonic oscillator transition kernel is the position-space unitary time evolution amplitudeFor $0<T<\pi /\omega$,Here $\hslash$ is the reduced Planck constant. The interval fixes the initial square-root branch; extension beyond focusing times requires its continued phase.

In the path integral, write a path as the classical oscillator path plus a fluctuation vanishing at both endpoints. The quadratic action splits exactly into the classical action and the endpoint-independent fluctuation action. The Dirichlet oscillator determinant ratio gives the prefactor, while the classical boundary action gives the exponent. ^[ref]

As $\omega \to 0$ with $T$ fixed, the expression reduces to the free-particle propagatorAt $T=n\pi /\omega$, the coordinate kernel is distributional because classical paths refocus. The Feynman i-epsilon prescription fixes the continuation through these caustics. Integrating the kernel against an initial wavefunction gives its later wavefunction.

John McGreevy develops quantum field theory through connections with many-particle physics. His authored lectures construct coherent-state path integrals for both bosons and fermions, relating the choice of integration variables to their different quantum statistics. The same treatment derives low-energy phase dynamics for a superfluid. This gives a concrete route from microscopic quantum degrees of freedom to a collective field description, explaining why similar field-theory methods recur in particle and condensed-matter physics.^[ref]

Author of Physics 215C: Quantum Field Theory, Part 3 by John McGreevy. Course record: mcgreevy.physics.ucsd.edu/s22/lecture-notes.html

For positive kinetic coefficient $\zeta$ in ${\mathcal{L}}_0=\zeta ({\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi )$, the canonical momentum is ${\pi }_{\varphi }=\zeta {\dot{\varphi }}^{\ast }$. Positive-energy oscillator modes with unit oscillator commutators have amplitude $1/\sqrt{2\zeta E_{\mathbf{p}}}$. The mixed time-ordered propagator is $C={\zeta }^{-1}{D}_F$, with $D_F$ the ordinary scalar propagator. This also follows by inverting the quadratic form in the path integral. Rescaling $\varphi$ changes interaction coefficients as well as external correlation-function factors.

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.

For $M>0$, integrate the vector action by parts, dropping the boundary term, and write it as a quadratic functional:The explicit mass makes this Proca kernel invertible: the displayed action alone has no gauge redundancy left to fix. It is interpreted as unitary gauge when obtained from a Higgs or Stückelberg completion with its would-be Goldstone field removed.

Introduce a source $J_{\mu }$ and complete the square, $A=A^{\prime }-K^{-1}J$. The normalized Gaussian path integral isThe source-independent functional determinant cancels in $Z[J]/Z[0]$. For Fourier momentum $k$, the mixed-index kernel is ${\tilde{K}}^{\mu }{}_{\nu }=-(k^2+M^2){\delta }^{\mu }{}_{\nu }+k^{\mu }{k}_{\nu }$. For $k^2\ne 0$, use the projectorsTheir orthogonality givesCombining the projectors removes their apparent massless denominator. Adding the Feynman prescription gives the unitary-gauge propagatorMultiplication by $\tilde{K}$ verifies that it inverts the quadratic operator. On the physical pole $k^2=-M^2$, the numerator is transverse to $k$, as required for the three physical massive-vector polarizations.

The drawback is its off-shell ultraviolet behaviour: at large spacelike momentum the transverse term falls as $1/k^2$, but the longitudinal tensor term remains of order $1/M^2$. Naive loop power counting therefore looks worse and renormalizability is not manifest. In a finite $R_{\xi }$ gauge the corresponding expression iswhich falls as $1/k^2$ at fixed finite $\xi$. Its $\xi \to \infty$ limit gives the unitary-gauge longitudinal term, so taking that limit before the ultraviolet limit changes power counting.

This is not a claim that a renormalizable Higgs theory ceases to be physically renormalizable in this gauge. Nor does it mean that the free Proca Gaussian integral is inconsistent. For conserved external sources $k\cdot J=0$, the longitudinal numerator drops out of $J^{\mu }{D}_{\mu \nu }{J}^{\nu }$ and the tree exchange again falls as $1/k^2$. The difficulty concerns off-shell Green functions and manifest control of loop divergences, not the number of physical polarizations.

Use the source term $\int d^{4}x[J\varphi +\bar{\varsigma }\psi +\bar{\psi }\varsigma ]$ and left Grassmann functional derivatives. Differentiating its exponential inserts fields according toThe sign for $\bar{\psi }$ follows because a left derivative with respect to $\varsigma$ must cross $\bar{\psi }$. Keep the order $\bar{\psi }{\gamma }_5\psi \varphi$ when replacing these fields. Expand $e^{i{S}_{\mathrm{int}}}$ in the original path integral; each field monomial can now be generated by these ordered derivatives acting on the free functional. Resumming givesThe arguments refer respectively to $\varphi ,\bar{\psi },\psi$ at $x$. The free functional arises by performing the free Gaussian field integrals, and $\mathcal{N}$ may be chosen so that $Z[0]=1$.

No additional field-integration measure remains in this differential-operator formula. Such a measure is printed outside the operator in the source even though $Z_0$ has already integrated out those fields. At $\lambda =0$, that literal expression would contain $\int \mathcal{D}\varphi \mathcal{D}\bar{\psi }\mathcal{D}\psi 1$ rather than simply $Z_0$, which is not the free generating functional. Also, when ${\mathcal{L}}_{\mathrm{int}}$ denotes a density, the spacetime integral in the exponent is required.

The gauge-invariant functional integral counts every gauge orbit repeatedly. Insert a gauge-orbit identityDividing by the gauge-group volume and averaging $f$ with a Gaussian weight produces the term $-G^2/(2\xi )$. The determinant is represented by a pair of independent anticommuting scalar fields $c,\bar{c}$, giving the prescribed ghost density $-\bar{c}\mathcal{M}c$, where $\mathcal{M}=\delta G/\delta \theta$. The Grassmann integral gives a determinant rather than the inverse determinant of a commuting Gaussian integral; field-independent constants and overall determinant signs are absorbed in normalization.

Although the gauge group is Abelian, the determinant here is field dependent because the gauge condition contains the scalar fluctuation $b$. Thus Abelian ghosts need not decouple in a broken-phase $R_{\xi }$ gauge. The fields implement gauge-orbit counting and are not physical external particles.