Functional Integrals and Free Gaussian Fields

A functional integral is not first defined by writing a product over all spacetime points. In this chapter it begins as an ordinary integral over finitely many variables: lattice values, retained modes, or the variables introduced by a finite time slicing. The regulator, reference measure, normalization, boundary data, signature, and convergence prescription are part of the definition. Only after exact finite-system checks are in place does continuum notation become useful.

The free massive scalar is the recurring example. Its quadratic action makes the regulated integral Gaussian, so the inverse kernel, source dependence, covariance, and determinant can all be computed exactly. Time slicing then connects that calculation to a canonical transition kernel when the Hamiltonian, state, endpoints, ordering, regulator, and limiting procedure are matched. This is a controlled comparison, not a universal equivalence theorem for QFT formulations.

Choose an entry route · See every page · Review the chapter

A finite Gaussian is the starting object

Let $x\in\mathbb R^N$ collect every retained real degree of freedom and let $K_E=K_E^{\mathsf T}>0$ be the Euclidean quadratic kernel after the domain and boundary conditions have been fixed. With the reference measure displayed explicitly,

$$\begin{aligned} \mathcal Z_E[J;K_E] &= \int_{\mathbb R^N} \frac{\mathrm d^N x}{(2\pi)^{N/2}} \exp\!\left( -\frac12 x^{\mathsf T}K_E x+J^{\mathsf T}x \right) \\[2pt] &= (\det K_E)^{-1/2} \exp\!\left( \frac12 J^{\mathsf T}K_E^{-1}J \right). \end{aligned}$$

This equality is exact. Completing the square translates $x$ by $K_E^{-1}J$; diagonalizing the positive matrix reduces the remaining integral to $N$ one-dimensional Gaussians. The same calculation defines the normalized source functional

$$Z_E[J] \equiv \frac{\mathcal Z_E[J;K_E]}{\mathcal Z_E[0;K_E]} = \exp\!\left( \frac12 J^{\mathsf T}C_EJ \right), \qquad C_E=K_E^{-1},$$

and therefore

$$\frac{\partial\log Z_E}{\partial J_i} =(C_EJ)_i, \qquad \frac{\partial^2\log Z_E} {\partial J_i\,\partial J_j} =(C_E)_{ij}.$$

The source-independent determinant cancels in $Z_E[J]$, but it has not ceased to exist. It remains relevant for unnormalized amplitudes, comparisons that change the kernel or boundary data, gluing, vacuum energies, and free energies. If the coordinates carry units, a corresponding reference scale also belongs to the measure. If $K_E$ has a zero eigenvalue, neither the displayed inverse nor the determinant formula is available until the zero direction is treated explicitly. If the quadratic form is indefinite or the exponent is oscillatory, a contour or boundary-value prescription is additional data; finite $N$ alone does not guarantee convergence.

For a finite lattice or finite set of modes, $K_E$ may discretize $-\partial_E^2+m^2$ with $m>0$. The notation

$$\int \mathcal D\phi\, \exp\!\left[-S_E[\phi]+\int J\phi\right]$$

then summarizes a regulated family and a proposed limiting procedure. The symbol $\mathcal D\phi$ does not by itself define a flat infinite-dimensional product measure. Constructed Gaussian measures and continuum existence questions require additional hypotheses and belong to more specialized treatments. The progression from ordinary integrals to regularized field integrals is developed in Zinn-Justin 2021, §§ 1.1–1.4, pp. 1–5; §§ 7.1–7.5, pp. 126–138.

What must be specified before calculating

Two formulas that look identical can represent different calculations because the data surrounding the integral differ. Before calling an expression a regulated functional integral, record the following.

Datum	What must be named	Failure check
Variables and domain	Lattice sites, modes, time-slice variables, constraints, and the real domain or complex cycle	Can one state exactly what a point of the integration domain is?
Reference measure	Every finite-dimensional measure factor and any dimensionful normalization scale	Does a change of coordinates transform both the measure and the domain?
Action or kernel	The regulated action, including boundary terms and the operator domain encoded in $K$	Is the claimed inverse actually an inverse with those boundary conditions?
State and boundary data	Fixed endpoints, boundary wave functions, vacuum projection, or another preparation prescription	Is a transition kernel being mistaken for an expectation value in a state?
Signature and convergence	Euclidean damping, Lorentzian contour, $i0$, or another convergence prescription	Is an oscillatory formula being manipulated as though it were a positive measure?
Normalization	Whether $\mathcal Z[0]$ is retained, divided out, or compared between systems	Did a determinant cancel only because the same kernel and boundary data occur in numerator and denominator?
Observable	Kernel, amplitude, normalized expectation, ordered correlator, response, or determinant ratio	Are source derivatives being interpreted with the correct factors of $i$ and ordering?
Limits	Time-slice spacing, lattice spacing, mode cutoff, volume, time extent, mass, $i0$, or semiclassical parameter	Has the order of limits been stated rather than assumed to commute?

This information is also the comparison contract with the canonical formulation. Agreement of a denominator or a covariance after some choices have been suppressed is not enough.

Choose a route

The arrows below are suggested reading orders. They are not interchangeable: each route closes a different set of dependencies and answers a different question.

Reader goal	Suggested route	Observable exit
Learn what is actually integrated	The action principle → regulated bosonic field integrals	Identify the finite variables, domain, reference measure, and normalization, then state what the continuum symbol has not defined
Derive a transition amplitude	Regulated integrals plus Schrödinger wave functionals → time slicing	Exhibit the intermediate configurations, per-slice normalization, endpoints, ordering convention, and controlled slicing limit
Prepare or glue states	Time slicing → boundaries and state preparation	Distinguish a fixed-boundary kernel from a state amplitude and integrate a shared gluing boundary exactly once
Use source derivatives for the free field	Regulated integrals plus Klein–Gordon modes → Gaussian fields and sources	Recover the inverse kernel and pass a fully declared source convention to the correlator chapter
Organize a semiclassical approximation	Regulated integrals plus the action principle → saddles and the semiclassical expansion	State the expansion parameter and diagnose zero modes, negative modes, contour choices, determinant phases, and remainders
Change field variables	Regulated integrals → regulated Jacobians	Transform the finite domain, source, action, and Jacobian without inferring an anomaly or equivalence theorem
Compare canonical and functional answers	Time slicing plus Gaussian sources and the quantized real scalar → canonical–functional crosswalk	Match a free oscillator or field covariance after every regulator, state, boundary, normalization, and prescription choice has been aligned

Check your entry point

This overview has no prerequisite. Individual pages do. Use these prompts to locate the shortest useful repair before choosing a route through the chapter.

Try this	Ready when	Repair if unsure
Complete the square in a two-variable positive Gaussian	You can identify the shifted mean, inverse kernel, determinant, and condition for convergence	Review Measures and Measurable Functions and Gaussian Vectors, Processes, Random Distributions, and Wick Structure
Vary a quadratic scalar action without dropping its surface term	You obtain the bulk field equation and can name boundary data that make the variation well posed	Review The Action Principle and Field Equations
Interpret $\langle q_f	e^{-iHT}	q_i\rangle$
Distinguish a Euclidean covariance from a Feynman two-point function	You name the signature, boundary prescription, source convention, and contact equation rather than comparing denominators alone	Review Scalar Propagators, Ordered Correlators, and Sources
Justify moving a limit through an integral	You can state a convergence theorem and check its hypotheses, or say that no such justification has yet been supplied	Review Lebesgue Integration and Convergence Theorems

For a guided curriculum route, Functional integrals and correlators connects this chapter to its downstream source and correlator pages.

From time slicing to field variables

For a system with finitely many coordinates $q$, the transition kernel is

$$K(q_f,t_f;q_i,t_i) = \langle q_f|e^{-iH(t_f-t_i)}|q_i\rangle.$$

Writing $t_f-t_i=M\,\Delta t$ and inserting $M-1$ resolutions of the identity gives

$$K(q_f,t_f;q_i,t_i) = \lim_{M\to\infty} \int \prod_{r=1}^{M-1}\mathrm d q_r \prod_{r=0}^{M-1} \langle q_{r+1}|e^{-iH\Delta t}|q_r\rangle,$$

with $q_0=q_i$ and $q_M=q_f$. Position–momentum insertions or a controlled short-time approximation turn the right-hand side into a finite multiple integral before the limit is taken. Its per-slice factors are essential: they enforce kernel composition and the delta-function limit as $t_f\to t_i$. The derivation also exposes an ordering convention; replacing a noncommuting Hamiltonian by an apparently obvious discretized action can change terms at finite $\Delta t$. Standard derivations are given in Schwartz 2014, §§ 14.1–14.2, pp. 251–260 and Weinberg 1995, § 9.1, pp. 378–384.

For a spatially regulated field, each $q_r$ is itself a finite vector of site values or mode coefficients. The same insertion therefore produces an ordinary integral over all intermediate regulated field configurations. Boundary wave functions turn the fixed-endpoint kernel into a state amplitude,

$$\langle\psi_f|e^{-iHT}|\psi_i\rangle = \int \mathrm d q_f\,\mathrm d q_i\, \psi_f(q_f)^*\, K(q_f,T;q_i,0)\, \psi_i(q_i).$$

Likewise, composing two kernels requires one integration over their common boundary configuration. Fixed endpoints, integrated endpoints, Euclidean long-time projection, and closed-time contours prepare different objects; no one of them is implied by the bare symbol $\int\mathcal D\phi$.

The time-slice spacing $\Delta t$ is not the Feynman convergence parameter $i0$. Sending $\Delta t\to0$ controls a product approximation to time evolution. Sending $i0\to0^+$ selects a boundary value after a convergence or state prescription has been fixed. Keeping the symbols separate prevents two logically different limits from being merged.

The free scalar cross-check

The canonical chapter represents each retained massive free-field mode as a harmonic oscillator. Collecting finitely many real modes into vectors gives

$$H = \frac12 p^{\mathsf T}p +\frac12 q^{\mathsf T}\Omega^2q, \qquad \Omega>0.$$

For the selected ground state, the Euclidean ordered covariance is

$$C_E(\tau) = \frac12\Omega^{-1}e^{-\Omega|\tau|}, \qquad \left(-\partial_\tau^2+\Omega^2\right)C_E(\tau) = \delta(\tau)\,I,$$

where matrix functions of $\Omega$ are defined by its spectral decomposition. The Lorentzian Feynman two-point function is instead

$$D_F(t) = \frac12\Omega^{-1}e^{-i\Omega|t|}, \qquad \left(\partial_t^2+\Omega^2\right)D_F(t) = -i\,\delta(t)\,I.$$

The derivative jump at the origin supplies the contact term in each equation. A regulated Euclidean Gaussian reproduces $C_E$ when its kernel, temporal boundary behavior, and state preparation are those of the same oscillator system. A Lorentzian functional calculation reproduces $D_F$ only after its oscillatory contour or $i0$ prescription and source factors are matched. The different contact terms are a compact check that Euclidean source derivatives have not been mixed with Lorentzian $1/i$ derivatives.

For the full free scalar, the same check is performed mode by mode. It establishes agreement for the declared finite system and selected observable. It does not prove that regulator removal exists, that analytic continuation is valid in a broader interacting theory, or that every canonical and functional construction is equivalent. The complete canonical derivation of functional representations, including endpoints and ordering, is discussed in Weinberg 1995, §§ 9.1–9.4, pp. 378–398.

The seven pages in order

The list below is the sidebar order. Dependencies branch, so the numbering is not a claim that every page requires all earlier pages.

1. Regulated Bosonic Field Integrals. Begin here to identify the actual finite variables, domain, reference measure, action, normalization, boundary data, and limiting question. It requires the action principle. Continue to Gaussian sources for an exact calculation or to time slicing for the canonical origin of the measure.

2. Time Slicing and Transition Amplitudes. Take this route to derive a finite multiple integral from canonical time evolution while keeping endpoints, ordering, per-slice normalization, and approximation error visible. It requires the regulated-integral page and Schrödinger wave functionals. Continue to boundary state preparation or to the final crosswalk.

3. Gaussian Fields and Sources. Take the computational route to complete the square, invert the regulated kernel, retain or cancel its determinant consistently, and generate mean and covariance by source differentiation. It requires the regulated-integral page and Klein–Gordon modes. Continue to the correlator chapter once the signature and source convention are fixed.

4. Boundaries and State Preparation. Take the state route to distinguish fixed-boundary kernels, amplitudes between boundary wave functions, vacuum projection, and gluing. It requires time slicing; the Schrödinger wave-functional page is useful preparation. Continue to Lorentzian, Euclidean, or in–in formulations when the physical contour rather than the elementary boundary construction is central.

5. Saddles and the Semiclassical Expansion. Take the asymptotic route to expand a regulated integral near a stationary configuration and state the control parameter, Hessian, contour, phase, and remainder status. It requires the regulated-integral page and the action principle. Symmetry-generated zero modes require collective coordinates and the associated Jacobian; other null directions require a higher-order or degenerate-saddle analysis. Negative modes require a contour decision. Continue to Nonperturbative QFT for developed physical saddle sectors.

6. Changes of Variables and Regulated Jacobians. Take the transformation route to apply the finite-dimensional change-of-variables theorem to a regulated field integral, carrying the domain or cycle, action, source, and Jacobian together. It requires the regulated-integral page. Continue to Symmetry and Gauge Structure for anomalies or to Renormalization and EFT for field-redefinition equivalence questions.

7. Canonical–Functional Crosswalk for Regulated Systems. Finish here to compare the same free regulated system in operator and functional language. It requires time slicing, Gaussian sources, and the quantized real scalar; the boundary-state page is recommended. The conclusion is a matched calculation with explicit hypotheses, not a framework-level equivalence theorem.

One calculation, seven uses

The finite Gaussian appears throughout the chapter, but each page asks a different question of it.

Gaussian datum	Chapter use	Question that remains local to that page
$x$, its domain, and its reference measure	Regulated bosonic integral	What is integrated, and which continuum symbol is still only formal?
The assembly of $K$ from short time steps and boundary terms	Time slicing	Which Hamiltonian, ordering, endpoint prescription, and $\Delta t$ error produced it?
$K^{-1}$, $\det K$, and $J$	Gaussian fields and sources	Which mean, covariance, normalization, and source derivatives follow?
Fixed or integrated components of $x$	Boundaries and state preparation	Which kernel, state amplitude, projection, or gluing operation is computed?
$K$ as the transverse Hessian near a stationary point	Saddles	Which zero or negative directions, phases, collective coordinates, and remainders qualify the approximation?
$x=f(y)$ and its transformed domain	Regulated Jacobians	Which real or complex Jacobian is correct, and is the map globally one-to-one?
$K^{-1}$ compared with an oscillator correlator	Canonical–functional crosswalk	Have the system, state, observable, prescription, and order of limits actually been matched?

An exact quadratic integral is not merely a semiclassical approximation: for a genuinely quadratic action it is the whole regulated calculation. In an interacting problem the Gaussian supplied by a Hessian is instead the leading local term of an asymptotic expansion, and its validity must be justified in a stated parameter and integration cycle.

Chapter-wide conventions and checks

The chapter inherits the site’s $(+---)$ metric and natural units. Additional local conventions keep the finite and continuum claims distinguishable.

Convention	Use in this chapter	Check
Euclidean source weight	$\exp[-S_E+\int J\phi]$	For a positive kernel, ordinary source derivatives give $C_E=K_E^{-1}$
Lorentzian source weight	$\exp[iS_L+i\int J\phi]$	Derive every factor of $i$ and state the contour or $i0$ prescription
Unnormalized and normalized functionals	$\mathcal Z[J]$ retains overall factors; $Z[J]=\mathcal Z[J]/\mathcal Z[0]$ has $Z[0]=1$	Do not discard determinants in a comparison whose kernel, boundary, or domain changes
Kernel inverse	$K_E^{-1}$ is a covariance only after the domain and boundary conditions are fixed	Apply $K_E$ and recover the identity on the declared regulated space
Determinant square root	The positive root is used only for real positive-definite $K_E$	Indefinite or complex cases need a phase, branch, and contour; $\det'K$ needs an explicit projected subspace
Slice and boundary parameters	$\Delta t$ or $a_t$ denotes slicing; $i0$ denotes a boundary value	Never use one as an explanation for the other
Limit labels	Keep $N\to\infty$, $a\to0$, $\Delta t\to0$, $T\to\infty$, $L\to\infty$, $i0\to0^+$, and $\hbar\to0$ distinct	State the order and supply the appropriate convergence or asymptotic argument

For a real invertible finite-dimensional transformation $x=f(y)$, the ordinary Lebesgue measure uses $|\det Df|$. For an oriented complex contour deformation, the complex Jacobian and cycle orientation replace that absolute-value rule. Neither statement alone determines a continuum functional determinant or an anomaly.

What the chapter establishes

The chapter supports the following conditional chain.

• A finite regulator turns the chosen bosonic variables into a genuine finite-dimensional integration problem, provided its domain and convergence data are sufficient.
• A positive quadratic Euclidean problem is exactly soluble; its inverse kernel is the covariance and its determinant records the source-independent Gaussian normalization.
• Time slicing derives a regulated multiple integral from canonical evolution while exposing endpoints, ordering, short-time normalization, and the limiting step.
• Boundary wave functions and gluing operations turn fixed-boundary kernels into state amplitudes and composites without hiding the shared-boundary measure.
• Saddles and changes of variables can be analyzed first at finite regulator, where Hessians, cycles, Jacobians, and approximation errors are ordinary finite-dimensional data.
• The canonical and functional free-scalar answers agree when the complete comparison contract is matched.

The chain stops short of a general continuum product measure, regulator-independent interacting construction, anomaly theorem, field-redefinition equivalence theorem, unrestricted Wick rotation, or universal equivalence among formulations. Those questions require the neighboring chapters or specialist volumes named below.

Review the chapter

Each prompt includes a success criterion and a direct repair route; use them to diagnose the corresponding calculation or definition.

1. Gaussian normalization. Let $K=\begin{pmatrix}a&c\\c&b\end{pmatrix}$ with $a>0$ and $ab-c^2>0$. Compute $K^{-1}$ and identify what cancels between $\mathcal Z_E[J]$ and $\mathcal Z_E[0]$. A satisfactory response obtains $(ab-c^2)^{-1}\begin{pmatrix}b&-c\\-c&a\end{pmatrix}$, gives $Z_E[0]=1$, and identifies the covariance as $K^{-1}$. Repair: Gaussian Fields and Sources.

2. Signature translation. For one oscillator, apply $-\partial_\tau^2+\omega^2$ to $e^{-\omega|\tau|}/(2\omega)$ and $\partial_t^2+\omega^2$ to $e^{-i\omega|t|}/(2\omega)$. A satisfactory response obtains $\delta(\tau)$ in the Euclidean equation and $-i\delta(t)$ in the Feynman equation, and does not identify the Feynman function with a retarded kernel. Repair: Gaussian Fields and Sources and Scalar Propagators.

3. Boundary diagnosis. Compare a fixed-endpoint kernel, its integral against $\psi_i$ and $\psi_f^*$, and a Euclidean long-time construction. A satisfactory response says which endpoints are fixed or integrated, integrates a shared gluing boundary once, and states the spectral and overlap assumptions needed for vacuum projection. Repair: Boundaries and State Preparation.

4. Jacobian invariant. Under $x=By$ with real invertible $B$, transform $\mathrm d^N x$ and $K$. A satisfactory response obtains $\mathrm d^N x=|\det B|\mathrm d^N y$, $K_y=B^{\mathsf T}KB$, and $|\det B|(\det K_y)^{-1/2}=(\det K)^{-1/2}$, while also carrying the domain and source. It draws no anomaly conclusion. Repair: Changes of Variables and Regulated Jacobians and Product Measures, Fubini–Tonelli, and Change of Variables.

5. Saddle failure check. State the leading Euclidean Laplace term near a nondegenerate minimum, then replace one Hessian eigenvalue by zero or by a negative value. A satisfactory response distinguishes a symmetry-generated zero mode, which calls for a collective coordinate and its Jacobian, from another null direction, which may require higher-order terms or a degenerate-saddle treatment. It sends a negative direction to a contour and phase decision and does not write a naive determinant. Repair: Saddles and the Semiclassical Expansion, Second Variation, Hessians, and Jacobi Operators, and Laplace Method and Steepest Descent.

6. Crosswalk synthesis. Diagnose the statement “the same denominator proves canonical–functional equivalence.” A satisfactory response names the regulator, ordering, operator domain, state, endpoints, source normalization, contour or $i0$, observable, and order of limits, then uses an oscillator contact equation as an invariant check. Repair: Canonical–Functional Crosswalk for Regulated Systems.

Where to continue

• Build connected and 1PI objects: The Generating Functional is the direct handoff from Gaussian sources; Correlators, Sources, and Effective Actions then continues through connected $W$, Wick factorization, and the 1PI effective action.
• Choose a physical contour: Lorentzian, Euclidean, and In-In Formulations develops $i\epsilon$, Wick rotation, Euclidean Schwinger functions, and closed-time-path questions.
• Change field species: Free Fermion Fields replaces ordinary Gaussian integration by regulated Berezin integration for free fermions.
• Develop physical saddle sectors: Nonperturbative QFT treats instantons, bounces, thimbles, transseries, and the interpretation of negative modes.
• Interpret anomalous Jacobians: Symmetry and Gauge Structure develops regulated measure anomalies; Renormalization and Effective Field Theory develops field-redefinition and equivalence questions in EFT.
• Control regulator removal: Lattice Field Theory and Hamiltonian Methods develops lattice-regulated numerical continuum limits; Renormalization and Effective Field Theory develops renormalized continuum limits; Mathematical QFT develops constructed Gaussian measures on distribution spaces and theorem-level framework comparisons.
• Return to the shared model: the free-scalar thread follows the same finite system through canonical, functional, spectral, Euclidean, and structural checks.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.