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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.

In a temporal winding sector of a quantum rotor, decompose the lifted angle into a straight winding path and a fluctuation vanishing at both endpoints. The cross term in the kinetic action vanishes, so all sectors have the same Gaussian path integral. With inverse-energy imaginary time its value is the displayed diagonal free-particle kernel on the line. Integrating the initial angle separately supplies $2\pi$. Thus this fixed-endpoint factor is an amplitude density, not an additional rotor partition function or an uncounted angular zero mode.

After the shift $x(\tau )=x_0(\tau )+y(\tau )$, the endpoint values of $y$ are both zero. Expanding the Euclidean action givesBy integration by parts, the cross term isThe first term vanishes by the homogeneous Dirichlet boundary conditions, and the second by the Euler-Lagrange equation. The translation has unit Jacobian determinant in the time-sliced configuration-space path integral, so $\mathcal{D}x=\mathcal{D}y$. Consequently classical-path factorization of a quadratic path integral is exact here:All dependence on the common endpoint $x$ lies inThe endpoint domain and the integrand defining $Z$ are the same for every $x$. This $Z$ is a zero-endpoint Gaussian path integral; the canonical partition function instead integrates the entire diagonal kernel over $x$. The final sentence of the printed part omits $/\hslash$ from its exponential: the physical Boltzmann factor has the dimensionless exponent $-S/\hslash$.

An optional normalization check uses the Dirichlet oscillator fluctuation determinant. Its eigenvalues are $m[(n\pi /\mathcal{T}{)}^2+{\omega }^2]$, $n\ge 1$. Relative to the free-particle Gaussian path integral, the hyperbolic-sine infinite product givesThus $Z=\sqrt{m\omega /[2\pi \hslash \sinh (\beta \hslash \omega )]}$. Integrating the resulting diagonal thermal kernel of a quantum harmonic oscillator by a Gaussian integral yields $\mathrm{Tr}{e}^{-\beta H}=[2\sinh (\beta \hslash \omega /2){]}^{-1}$, including the zero-point energy.

Use Minkowski spacetime with signature $(+,-,\dots ,-)$ and $\hslash =1$, with a vacuum time-ordering prescription. At a regulator, define the normalized generating functionalThe normalized vacuum correlation functions of a time-ordered product areThe connected generating functional isThe logarithm selects connected source diagrams, or equivalently cumulants. With the assumed vanishing one-point function, derivatives of $Z=e^{iW}$ give the centered four-point cumulant decomposition:In detail, four differentiations either act on one connected block or form one of three pair partitions. Contributions with blocks of size one vanish. If $W_{ij}$ and $W_{1234}$ denote source derivatives at zero, the same relation is $G_{1234}=i{W}_{1234}-W_{12}{W}_{34}-W_{13}{W}_{24}-W_{14}{W}_{23}$, since $C_2=-i{W}_2$ and $C_4=i{W}_4$.

Let $\phi (x)=\delta W/\delta J(x)$ be the source-dependent mean field. On a locally invertible source-to-field branch, define the quantum effective action by the Minkowski Legendre transformIts variation is $\delta \Gamma =-\int J\delta \phi$, so $\delta \Gamma /\delta \phi =-J$. Differentiating the two inverse source-field maps yields the Minkowski inverse-Hessian relation for an effective action:Thus ${\Gamma }^{(2)}=-(W^{(2)}{)}^{-1}$. The inverse is understood at the regulator and on a nonsingular fluctuation sector; it is not an assertion that every source-field map is globally invertible.

For a free real scalar field, integrate by parts to writewhere repeated spacetime variables are integrated. Specify the propagator convention explicitly:Completing the Gaussian path integral around $\varphi =-A_{ϵ}^{-1}J$ cancels the source-independent determinant between numerator and denominator. ThereforeThe positive imaginary term in $A_{ϵ}$ damps the oscillatory Gaussian. Together with vacuum projection at the time boundaries, it selects the Feynman poles and the time-ordered, in-out correlation function rather than a retarded inverse. It can also be obtained by continuation from the Euclidean vacuum integral. A denominator convention without the numerator $i$ redistributes the displayed factors of $i$.

Finally $\phi =-A_{ϵ}^{-1}J$, hence $J=-A_{ϵ}\phi$. Substitution into the Legendre transform gives the vacuum-normalized effective action of a free scalar field:All connected functions beyond order two vanish and all proper vertices beyond the classical quadratic kernel vanish. Vacuum normalization fixes the additive constant; without it, the quantum effective action equals the classical action up to a field-independent determinant contribution.

Set $\Delta =T/M$, $q_0=q$, and $q_M=q^{\prime }$. The Trotter product formula separates the kinetic and potential energy evolution to first order in $\Delta$. Inserting $M-1$ resolutions of the identity in the position operator basis and using the free short-time kernel givesThe exponent converges formally to the action along the path. The prefactors and intermediate integrations define the time-sliced configuration-space path integral measure, soA real-time path integral is an oscillatory limit with the Feynman i-epsilon prescription, rather than an ordinary probability integral. The operator derivation assumes the usual self-adjointness and product-formula hypotheses for the Hamiltonian.

For a quadratic potential energy write $q=q_c+\eta$, with $\eta (0)=\eta (T)=0$ and ${\ddot{q}}_c+V^{\prime }(q_c)=0$. The first variation vanishes by the Euler-Lagrange equation and the fixed endpoints. Because $V$ is quadratic, the remaining expansion is exact:Here integration by parts has no boundary contribution. Translation of the integration variables leaves the time-sliced measure unchanged. Thus the Gaussian path integral over $\eta$ is independent of the endpoint values, and the entire endpoint dependence lies in $e^{iS[q_c]}$.

Expanding $\eta$ in eigenfunctions satisfying Dirichlet boundary conditions, each real mode contributes an inverse square root of its quadratic eigenvalue, with the oscillatory phase fixed by continuation. Normalizing against $A_0={\partial }_t^2$ gives a precise determinant-ratio form:The determinant ratio is meaningful after a common regulator; the phase is part of the prescription. It is equivalent to writing $N=C(T)({\det }_{D}A{)}^{-1/2}$ with an endpoint-independent normalization.

For example, if $V^{\prime \prime }={\omega }^2$, the Dirichlet oscillator determinant ratio isA linear or constant term in $V$ changes the action but not this prefactor. The formula holds away from conjugate times, when the boundary problem and determinant are nonsingular. At $\sin \omega T=0$, it must be continued as a distribution with its Maslov index phase; the assertion of an ordinary finite prefactor cannot be used literally there. For the unshifted oscillator, at $T=r\pi /\omega$ the kernel is $e^{-ir\pi /2}\delta (q^{\prime }-(-1{)}^{r}q)$.

For the full-line source integral, impose vacuum boundary conditions by adiabatic damping. Take the source initially as a test function, so its pairings with the Green function are defined. After integration by parts, writeWith the Fourier transform convention $G(t)=\int d\omega e^{i\omega t}\tilde{G}(\omega )/(2\pi )$, the inverse isSet $q=\eta -GJ$. Completing the square givesand consequentlyThe Gaussian functional integral $Z[0]$ is source independent and contains the regulated determinant. This is most naturally read as a normalized vacuum generating functional. For $m>0$, closing the frequency contour gives $G(t)=-i{e}^{-im|t|}/(2m)$; the time-ordered oscillator two-point function is $iG(t)$. This checks the sign of the source exponent and identifies the required vacuum prescription.

Write $z=z_c+\eta$, with $\eta (0)=\eta (T)=0$. Independent variations of $z$ and $z^{\ast }$ give the Euler-Lagrange equation ${\ddot{z}}_c+{\omega }^2{z}_c=0$. Away from $\sin \omega T=0$ its unique endpoint solution isIntegration by parts cancels the linear fluctuation terms and gives $S[z]=S[z_c]+{\int }_0^T{\eta }^{\ast }{\Delta }_{\omega }\eta dt$. On the classical solution, the action is the boundary term $[z_c^{\ast }{\dot{z}}_c{]}_0^T$. Substituting the endpoint derivatives therefore givesThe remaining Gaussian path integral contains two real fluctuation coordinates per mode, so it contributes an inverse functional determinant, rather than its inverse square root. Thus $K=e^{iS[z_c]}/\det {\Delta }_{\omega }$, with the measure normalization fixing the otherwise arbitrary constant.

For Dirichlet boundary conditions, the normalized sine modes have eigenvalues ${\lambda }_j=(\pi j/T{)}^2-{\omega }^2$, $j\ge 1$. Their determinant ratio is the convergent Dirichlet oscillator determinant ratioThe last equality is the sine infinite product. With the prescribed free determinant this givesThe kernel uses the Feynman i-epsilon prescription; at a caustic it is a distributional limit, not an ordinary finite function. Its $\omega \to 0$ limit is $(\pi iT{)}^{-1}\exp (i|z_f-z_i{|}^2/T)$.

There is a sign error in the printed complex-integral hint. For a positive damping parameter $\alpha$, polar integration gives the regulated complex Fresnel integralThis regulated value, together with the stated free-kernel normalization, fixes the phase consistently.

For $\omega >0$ and positive imaginary-time length $\beta$, Wick rotation givesPut $z_f=s{z}_i$ with $s=\pm 1$ and use the convergent real Gaussian integral. Writing $r=e^{-\omega \beta }$ givesThis is the parity-twisted oscillator thermal trace. The complex coordinate describes two independent real quantum harmonic oscillators, each with mass two in these units. Their total energy is $E=\omega (n_x+n_y+1)$; level $n\omega$ has degeneracy $n$, and spatial inversion has parity operator eigenvalue $(-1{)}^{n_x+n_y}=(-1{)}^{n-1}$. The plus sign is $\mathrm{Tr}{e}^{-\beta H}$; the minus sign is $\mathrm{Tr}(P{e}^{-\beta H})$. The alternating trace inserts parity into a bosonic system; it does not change the oscillators into fermions.

Insert position resolutions between short-time evolution factors for $H=p^2/(2M)+V(q)$. Integrating the momenta gives the path integral, with real-time slice factor $[M/(2\pi i\hslash \Delta t){]}^{1/2}$ and weight $e^{iS[q]/\hslash }$. A real classical trajectory cannot pass through a barrier with insufficient energy, but the path integral includes nonclassical paths.

Continue $t=-is$, with s physical imaginary time. The damped kernel isThe Euclidean action saddle obeys $M\ddot{q}=V^{\prime }(q)$: motion in the inverted potential. For degenerate minima at $\pm a$, subtract their common potential energy. A trajectory joining them with zero Euclidean energy satisfiesIt is an instanton; reversing its direction gives an anti-instanton. Thus a crossing amplitude is nonperturbative, proportional to $e^{-S_I/\hslash }$. For the representative quartic well $V=M{\omega }_0^2(q^2-a^2{)}^2/(8a^2)$, direct integration givesThe arbitrary center $s_0$ is an instanton translation zero mode. Expanding $q=q_I+\eta$ gives a Gaussian path integral involving $L_I=-{\partial }_s^2+V^{\prime \prime }(q_I)/M$. Treating the zero mode as a collective coordinate rather than a zero Gaussian denominator produces the instanton fluctuation prefactorCommon endpoint regularization defines the functional determinant ratio; K and kappa have inverse-time units. The prime removes only the zero eigenvalue, while integrating the center supplies the total available time. Large $S_I/\hslash$ and $\kappa \ll {\omega }_0$ justify a dilute instanton gas.

Multiple crossings alternate direction. Integrating n ordered centers gives ${\mathcal{T}}^n/n!$. Return paths contain even n, and transfer paths contain odd n, so their common local-well factor multiplies $\cosh (\kappa \mathcal{T})$ and $\sinh (\kappa \mathcal{T})$ respectively. The two lowest levels are consequentlyThe lower state is the even combination of the two localized states. Local anharmonic corrections modify $E_{\mathrm{well}}\simeq \hslash {\omega }_0/2$ and higher semiclassical corrections modify K. The thermal trace as a closed Euclidean path integral gives $Z\simeq 2e^{-\beta E_{\mathrm{well}}}\cosh (\beta \hslash \kappa )$ when higher local levels are negligible. This describes coherent level splitting in a stable well, not irreversible decay.

For a scattering barrier at energy E, the leading WKB tunnelling exponent gives an amplitude $\exp [-{\int }_{q_1}^{q_2}\sqrt{2M(V-E)}dq/\hslash ]$ and a transmission probability with twice that exponent. Endpoint matching fixes prefactors. Metastable escape instead uses a Euclidean bounce, with a negative fluctuation mode and a decay rate. Neither its negative mode nor the probability/amplitude factor should be transplanted into the stable double-well splitting. ^[ref]

In winding sector m write the lifted path asThe cross term vanishes since ${\int }_0^{\beta }\dot{\eta }d\tau =0$. The action separates into $I(2\pi m{)}^2/(2{\hslash }^2\beta )$ and the same fixed-endpoint Gaussian path integral for every m. Neither term depends on the initial angle, so its integral supplies $2\pi$:The fixed-endpoint rotor fluctuation normalization isIt is the diagonal free-particle thermal kernel on the lifted line, or the amplitude density for a zero-displacement fluctuation bridge. It is not by itself the circle partition function or a separately countable angular zero mode. The integration over ${\varphi }_0$ has already counted that mode. Quadraticity makes the saddle-plus-fluctuation evaluation exact.

Keep $\lambda$ for the cubic coupling and $j$ for the external source. Expanding the interaction part of the scalar field path integral yieldsA functional derivative of the source exponential inserts $i\varphi (y)$. In particular, ${\delta }_j(y{)}^3{e}^{i\int j\varphi }=-i\varphi (y{)}^3{e}^{i\int j\varphi }$, so $-i\lambda \varphi (y{)}^3$ can be replaced by $\lambda {\delta }_j(y{)}^3$. The functional derivatives commute and can be taken outside the free path integral. Resumming the exponential series givesThis is a formal perturbation series; a real cubic scalar potential is unbounded and the expression does not assert a stable nonperturbative vacuum state.

To evaluate the free Gaussian path integral, use integration by parts to writeThe inverse kernel isCompleting the square with $\varphi =\chi -K^{-1}j$ changes the exponent to $i\int \chi K\chi /2-i\int j{K}^{-1}j/2$. The translation-invariant functional measure makes the remaining Gaussian path integral precisely $Z[0;0]$, henceFor $ϵ>0$, the exponent includes $-ϵ\int {\varphi }^2/2$, which damps the regulated finite-dimensional integrals. The Feynman i-epsilon prescription displaces the positive-energy pole below the real $p^0$ axis and the negative-energy pole above it, selecting the time-ordered vacuum expectation value. Indeed the Feynman propagator is ${\Delta }_F=iD$. Thus $iϵ$ specifies both convergence and the vacuum state boundary conditions, with $ϵ\to 0^{+}$ taken after integration.

Write $S=\frac{1}{2}\varphi A\varphi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the Gaussian path integral gives $Z[J]/Z[0]=\exp [-iJ{A}^{-1}J/2]$, hence $W=-J{A}^{-1}J/2$, $\phi =-A^{-1}J$ and $\Gamma =\phi A\phi /2=S[\phi ]$. The source-independent determinant cancels in vacuum normalization. Without that normalization, the equality holds up to a field-independent vacuum constant.

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