Gaussian functional integral

A Gaussian functional integral has an exponent quadratic in a bosonic field. Completing the square expresses its source dependence through the inverse quadratic operator, the propagator.

For $S[\varphi ]=\frac{1}{2}\int (\partial \varphi {)}^2$ and a source appearing as $-\int J\varphi$ in the exponent, the positive quadratic operator is $A=-\Delta$. Completing the square gives $Z[J]=\mathcal{N}\exp [\frac{1}{2}J{A}^{-1}J]$. If one writes it with a negative quadratic source exponent, its kernel must be $G=-A^{-1}={\Delta }^{-1}$. In the plane $G(z,w)=(2\pi {)}^{-1}\log (|z-w|/L)$, up to harmonic additions and infrared conventions. The physical covariance is its negative. Zero modes are regulated or separated; oscillatory sources in a string path integral produce momentum conservation from the constant mode.

For a positive differentiable covariance $C_t$ and a sufficiently regular, integrable $W$, differentiating the normalized Gaussian integral and integrating by parts twice gives the displayed identity. An unnormalized integral has the additional field-independent term $\mathrm{Tr}(C_t^{-1}{\dot{C}}_t)/2$ times the integral. A finite-mode regulator makes the analogous Gaussian functional integral identity precise before taking a continuum limit.

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.

A Gaussian path integral over a bosonic field with quadratic operator $A$ produces a factor proportional to $(\det A{)}^{-1/2}$. When $A$ depends on background fields, this functional determinant contributes to their quantum effective action.

For a change of field variables $\varphi \mapsto F[\varphi ]$, the functional Jacobian is the functional determinant of the linearized change. At a finite regulator it is an ordinary Jacobian determinant on the finitely many field coordinates; the continuum expression denotes the regulated limit. It changes the functional integral measure and can depend on the background field. The Faddeev-Popov determinant is the example obtained by replacing the integration coordinate along a gauge orbit by a chosen gauge fixing condition.