Functional Integrals and Free Gaussian Fields

A functional integral becomes a defined calculation only after its variables, integration domain, regulator, and boundary data are specified. Free Gaussian fields provide an exact setting in which sources, state preparation, and the canonical description can be compared without an interaction expansion.

Enter this chapter

Begin with Regulated Bosonic Field Integrals to identify what is integrated. Then use Gaussian Fields and Sources for the central calculation of this chapter.

Topics

1. Regulated Bosonic Field Integrals — Defines finite-dimensional field integrals before taking continuum limits.
2. Time Slicing and Transition Amplitudes — Constructs transition kernels from short-time evolution.
3. Gaussian Fields and Sources — Computes covariances and source dependence by completing the square.
4. Boundaries and State Preparation — Turns endpoint kernels into amplitudes or traces with appropriate boundary states.
5. Saddles and the Semiclassical Expansion — Separates an exact Gaussian integral from a controlled saddle approximation.
6. Changes of Variables and Regulated Jacobians — Tracks Jacobians, integration domains, and branch conditions under field redefinitions.
7. Canonical–Functional Crosswalk for Regulated Systems — Compares canonical and integral formulas for the same regulated oscillators.

How the ideas fit together

For finitely many real Euclidean field coordinates, a positive-definite quadratic kernel gives a convergent Gaussian integral. Its inverse is the covariance, while its determinant controls the source-independent normalization. Dividing by the zero-source integral removes that normalization from normalized correlators, but it does not erase it from every physical question. See Schwartz 2014, §§ 14.1–14.3.

Boundary wave functions select a matrix element; identifying and integrating endpoints instead produces a trace. These are different uses of the same evolution kernel. A formal continuum measure or a saddle with zero modes requires additional analysis beyond the finite positive Gaussian.

Review the chapter

For positive $a$, let $Z(J)$ be the normalized integral of $\exp(-a x^2/2+Jx)$. What determines its mean and covariance?

Solution

Completing the square gives $Z(J)=\exp(J^2/2a)$. Thus $\partial_J\log Z=J/a$ is the mean and $\partial_J^2\log Z=1/a$ is the connected covariance. At zero source the mean vanishes, but fluctuations remain.

Where to go next

Continue to Correlators, Sources, and Effective Actions to organize these derivatives into full, connected, and irreducible quantities.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.