Regulated Bosonic Field Integrals

A regulated bosonic field integral is an ordinary finite-dimensional integral only after the regulator has specified the independent variables, their domain or integration cycle, a reference measure with its units, the regulated action and boundary data, the observable, the normalization, and a convergence prescription. For a real Euclidean scalar with a stable finite quadratic action, these data produce a genuine normalized measure. Finite dimensionality alone does not guarantee convergence, and a Lorentzian weight $e^{iS}$ is not a probability density.

The formal continuum symbol $\mathcal D\phi$ supplies none of those missing data by itself. It can summarize a declared regulated family and a proposed limit, but it does not define a flat continuum product measure or establish that a continuum theory exists. This page makes that distinction precise and works the first normalization check for finitely many coupled scalar modes. Time slicing, source derivatives, and continuum construction are treated separately on the pages linked below.

Required background. The Action Principle and Field Equations supplies the scalar action, allowed field configurations, and boundary terms that must be retained when a regulator turns the action into a finite function.

Helpful background. Measures and Measurable Functions distinguishes an integration domain from a measure. Lebesgue Integration and Convergence Theorems and Lp Spaces, Inequalities, and Weak Convergence help with integrability and regulator limits. Probability Spaces, Random Variables, and Conditional Expectation supplies expectation-value language when the normalized Euclidean weight is genuinely nonnegative.

A complete regulator produces finitely many bosonic coordinates

For the unconstrained real scalar considered here, a complete finite regulator $R$ supplies a finite-dimensional configuration space

$$D_R\subseteq V_R\simeq\mathbb R^{N_R}, \qquad q=(q_1,\ldots,q_{N_R}).$$

Take $D_R\subseteq\mathbb R^{N_R}$ to be Borel measurable, with its relative Borel $\sigma$-algebra. A declared real coordinate system permits the restricted product-Lebesgue reference measure used below. The action principle supplies a function $S_{E,R}:D_R\to\mathbb R$ after the regulator, geometry, and boundary conditions have been specified. A measurable observable is another function $F_R:D_R\to\mathbb C$.

Three constructions that are often called “a cutoff” do not all have the same status.

Construction	Independent variables	Is the integration finite-dimensional?
Finite Euclidean spacetime lattice	One real field value at each of finitely many sites	Yes: $D_R\subseteq\mathbb R^{N_R}$
Finite spacetime expansion in a declared real basis	Finitely many real coefficients	Yes: $D_R\subseteq\mathbb R^{N_R}$
Spatial mode cutoff with continuous time	Finitely many functions $q_a(t)$	No: each retained coordinate is still a whole path; a time regulator or another construction is still required

For a real field expanded in complex Fourier modes, the coefficients obey $\widetilde\phi(-k)=\widetilde\phi(k)^*$. Integrating the $k$ and $-k$ coefficients as unrelated complex variables would double the real degrees of freedom. A real sine–cosine basis, or an explicit treatment of the reality constraint and self-conjugate modes, avoids that error.

Finite time slicing and finite field-eigenstate insertions are what turn canonical transition amplitudes into ordinary multiple integrals in Schwartz 2014, § 14.2, pp. 254–260 and Weinberg 1995, § 9.1, pp. 378–384. The derivation, ordering choice, endpoints, and slice normalization are developed on the next page; here they serve only as examples of data a complete regulator must expose.

Reference measure, weight, and expectation are different objects

Let $\mathrm d\lambda_R(q)$ be a declared finite-dimensional reference measure on $D_R$. In the Euclidean setting, define

$$\begin{aligned} \mathcal Z_R &= \int_{D_R} \mathrm d\lambda_R(q)\, e^{-S_{E,R}(q)}, \\ \langle F_R\rangle_R &= \frac{1}{\mathcal Z_R} \int_{D_R} \mathrm d\lambda_R(q)\, F_R(q)e^{-S_{E,R}(q)}. \end{aligned}$$

The second line is defined only when

$$0<\mathcal Z_R<\infty, \qquad \int_{D_R} \mathrm d\lambda_R(q)\, |F_R(q)|e^{-S_{E,R}(q)}<\infty.$$

Under those conditions,

$$\mathrm dP_R(q) = \mathcal Z_R^{-1} e^{-S_{E,R}(q)}\, \mathrm d\lambda_R(q)$$

is a probability measure and $\langle F_R\rangle_R=\int F_R\,\mathrm dP_R$. The roles are distinct: $D_R$ is the domain, $\mathrm d\lambda_R$ is the reference measure, $e^{-S_{E,R}}$ is the weight, $\mathcal Z_R$ normalizes that weight, and $F_R$ says which observable is being computed. Regularized Gaussian and field integrals with these distinctions kept explicit are treated in Zinn-Justin 2021, §§ 1.1–1.4, pp. 1–5; §§ 7.1–7.5, pp. 126–138.

Dimensionful fields require a reference scale

If $q_\alpha$ has physical units, the bare factor $\mathrm d q_\alpha$ has those units as well. Choose a reference scale $s_\alpha$ with the same units and define a dimensionless coordinate $y_\alpha=q_\alpha/s_\alpha$. One convenient convention is

$$\mathrm d\lambda_s(q) = \prod_{\alpha=1}^{N_R} \frac{\mathrm d q_\alpha} {\sqrt{2\pi}\,s_\alpha} = \prod_{\alpha=1}^{N_R} \frac{\mathrm d y_\alpha}{\sqrt{2\pi}}.$$

For a quadratic action $S_E=\tfrac12q^{\mathsf T}Aq$, let $R_s=\operatorname{diag}(s_1,\ldots,s_{N_R})$ and $K=R_s^{\mathsf T}AR_s$. Then $S_E=\tfrac12y^{\mathsf T}Ky$ and $K$ is dimensionless. When $K$ is real symmetric and positive definite,

$$\mathcal Z_s =(\det K)^{-1/2} = \left(\prod_{\alpha=1}^{N_R}s_\alpha^{-1}\right) (\det A)^{-1/2}.$$

Multiplying the reference measure by a constant $C_R>0$ multiplies $\mathcal Z_R$ by $C_R$ but cancels from normalized expectations at the same regulator. It does not cancel from an unnormalized amplitude, a free energy, a gluing law, or a comparison in which the number of variables, reference scales, domain, kernel, or boundary conditions change. A coordinate transformation is a different operation: its Jacobian, transformed domain, action, and observable must all be carried together.

Oscillatory weights require more data

With the site’s Lorentzian convention, a finite-regulator expression has weight $e^{iS_R}$. Its modulus is one for real $S_R$, so an integral over an unbounded real domain is generally not absolutely integrable. A damping boundary value, a declared complex cycle, or another oscillatory-integral prescription may define an amplitude, but the result is not a probability measure and its square-root phases and branches are part of the prescription. Time Slicing and Transition Amplitudes develops the canonical origin of that data.

Two coupled scalar modes give the first normalization check

Take two dimensionless real scalar coordinates with Euclidean action

$$\begin{aligned} S_E(q_1,q_2) &= \frac12\mu^2(q_1^2+q_2^2) +\frac12\kappa(q_1-q_2)^2 \\ &= \frac12 \begin{pmatrix}q_1&q_2\end{pmatrix} \begin{pmatrix} \mu^2+\kappa&-\kappa\\ -\kappa&\mu^2+\kappa \end{pmatrix} \begin{pmatrix}q_1\\q_2\end{pmatrix}. \end{aligned}$$

The orthogonal coordinates

$$q_+=\frac{q_1+q_2}{\sqrt2}, \qquad q_-=\frac{q_1-q_2}{\sqrt2}$$

preserve $\mathrm d q_1\,\mathrm d q_2$ and diagonalize the action:

$$S_E = \frac12\mu^2q_+^2 +\frac12(\mu^2+2\kappa)q_-^2.$$

Thus the integral with reference measure $\mathrm d q_1\,\mathrm d q_2/(2\pi)$ converges exactly when

$$\mu^2>0, \qquad \mu^2+2\kappa>0,$$

and in that domain

$$\mathcal Z_2 = \frac{1} {\sqrt{\mu^2(\mu^2+2\kappa)}}.$$

The explicitly normalized measure is therefore

$$\mathrm dP_2 = \frac{\sqrt{\mu^2(\mu^2+2\kappa)}}{2\pi} e^{-S_E(q_1,q_2)} \mathrm d q_1\,\mathrm d q_2, \qquad \int_{\mathbb R^2}\mathrm dP_2=1.$$

Three independent checks are immediate. When $\kappa=0$, the answer factorizes into two one-dimensional integrals. As $\mu^2\to0^+$, the $q_+$ direction becomes flat and the normalization diverges. If $\mu^2+2\kappa<0$, the exponential grows along $q_-$ rather than damping it. The checks established here are measure normalization, positivity, and convergence. Adding a source, completing the shifted square, and deriving the covariance $K^{-1}$ are developed in Gaussian Fields and Sources.

A finite lattice is a finite collection of coupled modes

Consider a real scalar on a periodic $d$-dimensional Euclidean hypercubic lattice. Let the spacing be $a$, let direction $\mu$ contain $N_\mu$ sites, and define

$$N_\Lambda=\prod_{\mu=1}^{d}N_\mu, \qquad L_\mu=aN_\mu.$$

The continuum-normalized scalar has mass dimension $(d-2)/2$. The lattice coordinate

$$\varphi_n = a^{(d-2)/2}\phi(x_n)$$

is dimensionless, so a complete finite reference measure is

$$\mathrm d\lambda_a(\varphi) = \prod_{n\in\Lambda} \frac{\mathrm d\varphi_n}{\sqrt{2\pi}} = \prod_{n\in\Lambda} \frac{a^{(d-2)/2}\mathrm d\phi_n}{\sqrt{2\pi}}.$$

For the free massive field, choose the nearest-neighbor action

$$S_{E,a}[\varphi] = \frac12 \sum_{n\in\Lambda} \left[ \sum_{\mu=1}^{d} (\varphi_{n+\hat\mu}-\varphi_n)^2 +(am)^2\varphi_n^2 \right].$$

This is $\tfrac12\varphi^{\mathsf T}K_a\varphi$, with

$$(K_a)_{nm} = \bigl[2d+(am)^2\bigr]\delta_{nm} - \sum_{\mu=1}^{d} \left( \delta_{n+\hat\mu,m} +\delta_{n-\hat\mu,m} \right).$$

Periodic plane waves diagonalize this matrix. For

$$k_\mu=\frac{2\pi\ell_\mu}{L_\mu}, \qquad \ell_\mu=0,\ldots,N_\mu-1,$$

the eigenvalue is

$$\lambda_a(k) =(am)^2 +4\sum_{\mu=1}^{d} \sin^2\!\left(\frac{k_\mu a}{2}\right).$$

The replacement of a continuum field by finitely many site variables, together with the resulting difference action and regulator-dependent symmetry reduction, is described in Zinn-Justin 2021, § 8.7, p. 175.

For $m>0$, every eigenvalue is positive and the finite integral is

$$\mathcal Z_{E,a} =(\det K_a)^{-1/2} = \prod_{\alpha=1}^{N_\Lambda} \lambda_\alpha^{-1/2}.$$

A real orthogonal eigenbasis $O$ makes the relation to coupled modes exact:

$$\varphi_n = \sum_{\alpha=1}^{N_\Lambda} O_{n\alpha}q_\alpha, \qquad O^{\mathsf T}O=I, \qquad \mathrm d^{N_\Lambda}\varphi = \mathrm d^{N_\Lambda}q.$$

Using every eigenvector is a change of coordinates on the same finite system. Discarding some eigenvectors is a new mode-cutoff regulator and requires a new statement of its domain, measure, symmetries, and limits.

For $m=0$ with periodic boundary conditions, $\lambda_a(0)=0$. The constant coordinate

$$q_0 = \frac{1}{\sqrt{N_\Lambda}} \sum_{n\in\Lambda}\varphi_n$$

drops out of the action, leaving a factor $\int_{\mathbb R}\mathrm d q_0=\infty$. A primed determinant cannot silently repair the problem: projecting the average field, compactifying the target, adding a mass, or otherwise treating the zero direction defines different data. The infrared meaning of those choices is developed on Massless Scalars, Zero Modes, and Infrared Limits.

Finite-dimensional does not mean convergent

The defining test is the integral, not the number of variables. Assume first that $e^{-S_{E,R}}$ is locally integrable, as it is when $S_{E,R}$ is continuous. A useful sufficient condition for control at infinity on $D_R=\mathbb R^{N_R}$ is

$$S_{E,R}(q) \geq c\lVert q\rVert^p-C$$

outside a compact set, with $c>0$ and $p>0$. Under the local-integrability assumption, this bound makes $e^{-S_{E,R}}$ integrable and makes every observable with polynomial growth absolutely integrable: outside a sufficiently large ball,

$$e^{-S_{E,R}(q)} \leq e^C e^{-c\lVert q\rVert^p},$$

whose polar-coordinate integral remains finite after multiplication by any polynomial. For a real quadratic action on all of $\mathbb R^{N_R}$, convergence is equivalent to its symmetric kernel being positive definite.

The basic failures are already visible in one or two variables:

• a zero eigenvalue leaves an infinite flat-volume factor;
• a negative eigenvalue makes the Euclidean exponential grow along one direction;
• an action such as $S_E(x)=-\lambda x^4$ with $\lambda>0$ gives $e^{+\lambda x^4}$ and diverges; and
• $e^{iS(x)}$ on an unbounded real cycle is oscillatory, not absolutely integrable, even when $S$ is a nondegenerate quadratic polynomial.

A small damping factor can be part of a Lorentzian prescription, but removing it requires a declared boundary value and branch. It does not turn the oscillatory weight into a positive Euclidean measure.

A regulator fixes symmetries and several different limits

A regulator is more than a number called “the cutoff.” It includes the map from continuum symbols to finite variables and therefore determines which relations are exact before any limit.

Regulator	Exact finite data	Typical structural cost
Periodic hypercubic lattice	Site variables, spacing, volume, boundary condition, and finite product measure	Preserves discrete translations and the subgroup of the hypercubic group compatible with the finite extents and boundary conditions, not continuous rotations or Lorentz symmetry
Finite mode set	A declared real basis, retained index set, coefficient measure, and projection rule	Can preserve translations or selected rotations, but generally makes position-space locality or other symmetries less transparent
Finite time slicing of a spatially finite system	Slice variables, endpoints, step size, short-time rule, and per-slice normalization	Introduces ordering and discretization choices; continuous time is recovered only through a controlled limit

Removing one regulator does not remove the others. On the periodic lattice:

• $a\to0$ with every $L_\mu$ fixed requires $N_\mu\to\infty$ and is a continuum limit at fixed volume;
• $L_\mu\to\infty$ with $a$ fixed is an infinite-volume limit at fixed lattice spacing; and
• $a\to0$ with every $N_\mu$ fixed shrinks the physical box rather than producing the intended fixed-volume continuum.

The mode cutoff, time-slice spacing, volume, mass, and boundary removal are additional limits and need not commute. Restoration of a broken symmetry is a claim about limiting observables or identities, often after parameter tuning; it is not guaranteed by writing the cutoff parameter as zero or infinity. Different regulators can preserve different symmetries and need not approach the same continuum theory without a matching argument.

What continuum notation has—and has not—defined

At finite $R$, the product in $\mathrm d\lambda_R$ contains exactly $N_R$ ordinary factors. In a fixed-volume lattice limit, $N_R\to\infty$. The expression

$$\frac{ \int\mathcal D\phi\, F[\phi]e^{-S_E[\phi]} }{ \int\mathcal D\phi\, e^{-S_E[\phi]} }$$

does not by itself specify:

• the continuum configuration space, topology, or measurable sets;
• a measure, its normalization, or the class of configurations on which it is supported;
• the meaning of the action and observable on those configurations;
• a family of finite regulators and the parameter choices made along it;
• the topology or mode of convergence of the desired observables; or
• the order of continuum, volume, mass, boundary, and prescription limits.

The notation is therefore a compact instruction only when those data have been supplied elsewhere. It is not an infinite-dimensional analogue of finite flat Lebesgue measure merely because it resembles $\prod_a\mathrm d q_a$. This qualification does not rule out continuum measure constructions. Their configuration spaces, covariances, support, and existence hypotheses belong to Gaussian Euclidean Fields as Measures; no such construction is asserted on this page.

For a continuum claim obtained as a limit of the regulated family treated here, a defensible statement has the form

$$\langle F\rangle = \lim_{R\to\infty} \langle F_R\rangle_R,$$

where “$R\to\infty$” has been unpacked into specific limits, $F_R$ has been matched to the intended observable, regulator-dependent parameters have been tuned when necessary, and convergence has been proved or supported at the level claimed. The formal symbol alone establishes none of those steps. Zinn-Justin explicitly assumes a regularization before using these field-integral identities and cautions that the formal identities make sense only when both sides exist Zinn-Justin 2021, §§ 7.1–7.5, pp. 126–138.

Checks for a reproducible regulated definition

Before using a bosonic functional-integral symbol, another reader should be able to reconstruct the same finite integral from the following information.

Check	Required statement	Quick verification
Independent variables	Number, reality conditions, constraints, and coordinate basis	Count real degrees of freedom without double-counting conjugate modes
Domain or cycle	$D_R\subseteq\mathbb R^{N_R}$ or an oriented finite-dimensional complex cycle	State endpoints, boundaries, and excluded configurations
Reference measure	Every factor, scale, and Jacobian convention	Verify that the measure has the declared dimensions or is dimensionless
Regulated action	Difference operator or mode kernel, parameters, and boundary terms	Check that $S_R$ is dimensionless under $\hbar=1$
Convergence	Positivity/growth for a Euclidean integral or a contour/boundary prescription for an oscillatory one	Establish $0<\mathcal Z_R<\infty$ before calling $\mathrm dP_R$ a probability measure
Observable and normalization	$F_R$, normalized expectation, amplitude, or determinant ratio	Confirm whether an overall measure factor cancels in the actual comparison
Finite symmetries	Transformations that preserve the domain, measure, and action	Test the finite integral, not only the continuum expression
Limits	Parameter varied, quantities held fixed, order, topology, and error criterion	Distinguish continuum, volume, mass, time-slice, and prescription limits

For the two-mode example, $\int\mathrm dP_2=1$, factorization at $\kappa=0$, and loss of normalization at a zero eigenvalue provide three independent checks. For the lattice example, the plane-wave eigenvalue and dimensionless action supply corresponding checks.

Common pitfalls

A spatial mode cutoff is already a finite-dimensional path integral. It leaves finitely many functions of continuous time, not finitely many real numbers. Time slicing or another complete temporal construction is still needed.

Finite $N_R$ guarantees a finite answer. A flat direction, an unbounded Euclidean action, or an oscillatory real-cycle weight can all fail even for one variable. Convergence or a prescription is part of the definition.

The formal product defines the continuum measure. The symbol $\prod_x\mathrm d\phi(x)$ does not declare a configuration space, measurable structure, measure, normalization, or limit. It is shorthand until a construction supplies those objects.

Normalization constants are always disposable. A common constant cancels from a matched normalized expectation. It can remain in unnormalized amplitudes, gluing, free energies, and comparisons whose kernels, domains, boundaries, or number of variables differ.

Every Euclidean-looking weight is a probability density. The weight must be real, nonnegative, and integrable, and $\mathcal Z_R$ must be positive and finite. A complex action or sign problem does not satisfy that criterion.

Fourier modes automatically give independent complex coordinates. A real field imposes a conjugation constraint. A complete real orthogonal basis is a coordinate change; discarding modes is a new regulator.

A primed determinant harmlessly deletes zero modes. The prime must name the projected subspace and the treatment of the omitted direction. Projection, compactification, gauge fixing, and collective-coordinate constructions are different definitions.

Check your understanding

Use these problems to test whether the regulated definition and its convergence conditions have been kept distinct.

1. List every datum needed to define $\langle F_R\rangle_R$.

   Answer

   Give the regulator and number of independent real variables, their domain or cycle and measurable structure, the reference measure with all scales, the regulated action and boundary conditions, the signature and convergence prescription, the observable, the normalization convention, the finite symmetries, and every proposed limit with its order. Omitting any of the first seven prevents reconstruction of the finite expectation itself.

2. Diagonalize the two-mode action and diagnose $\mu=0$.

   Answer

   The orthogonal modes $q_\pm=(q_1\pm q_2)/\sqrt2$ have eigenvalues $\mu^2$ and $\mu^2+2\kappa$. Therefore $\mathcal Z_2=[\mu^2(\mu^2+2\kappa)]^{-1/2}$ in the positive domain. At $\mu=0$, the action is independent of $q_+$, so the integral contains the divergent factor $\int_{\mathbb R}\mathrm d q_+$.

3. Derive the lattice eigenvalue by applying $K_a$ to $e^{ik\cdot x_n}$.

   Answer

   A forward or backward shift contributes $e^{\pm ik_\mu a}$. Hence

   $$\begin{aligned} \lambda_a(k) &=(am)^2 +\sum_\mu \left(2-e^{ik_\mu a}-e^{-ik_\mu a}\right) \\ &=(am)^2 +4\sum_\mu \sin^2\!\left(\frac{k_\mu a}{2}\right). \end{aligned}$$

   The constant periodic mode has $k=0$ and becomes a zero mode precisely at $m=0$.

4. Compare three lattice limits: $a\to0$ at fixed $N_\mu$, $a\to0$ at fixed $L_\mu$, and $L_\mu\to\infty$ at fixed $a$.

   Answer

   The first shrinks the physical box because $L_\mu=aN_\mu\to0$. The second is the fixed-volume continuum limit and requires $N_\mu\to\infty$. The third is the infinite-volume limit at fixed lattice spacing. None determines the others, and a massless limit can interact with each.

5. Classify the following: a positive Euclidean Gaussian, the same Gaussian with a zero eigenvalue, a damped Fresnel limit, and a bare $\mathcal D\phi$ expression.

   Answer

   After normalization, the first is a probability measure. The second diverges on the flat direction. The third can define a prescribed oscillatory amplitude, not a probability measure, and must retain its damping limit and branch. The fourth is formal continuum notation until its space, measure or regulated family, observable, convergence, and order of limits are supplied.

6. Choose the next page for each task: deriving the integral from a transition amplitude, adding sources and covariance, constructing a continuum Gaussian measure, stating a lattice continuum target, and removing a regulator from a renormalized prediction.

   Answer

   Use Time Slicing and Transition Amplitudes for the transition-kernel derivation; Gaussian Fields and Sources for sources and covariance; Gaussian Euclidean Fields as Measures for a constructed continuum measure; Lattice Regulators and Target Continuum Theories for the lattice target and extrapolation; and Regulator Removal and Renormalized Predictions for renormalized removal.

Where the finite definition leads

• Derive the finite multiple integral: Time Slicing and Transition Amplitudes supplies state insertions, ordering, endpoints, and slice normalization.
• Add sources and compute the inverse kernel: Gaussian Fields and Sources develops square completion, covariance, source derivatives, and determinant ratios.
• Change variables at fixed regulator: Changes of Variables and Regulated Jacobians carries the domain, action, source, and finite Jacobian together.
• Construct a continuum Gaussian measure: Gaussian Euclidean Fields as Measures supplies the measurable-space and distributional framework absent from flat $\mathcal D\phi$ notation.
• State a lattice continuum target: Lattice Regulators and Target Continuum Theories treats lattice regulator data, tuning, universality evidence, and extrapolation.
• Remove a regulator in a renormalized prediction: Regulator Removal and Renormalized Predictions supplies the renormalization conditions and regulator-independent observable.
• Track formal-proof provenance when the finite model is encoded: Machine-Checked Theorem Records and Formal-Proof Provenance explains how an encoded finite or free model is related to the narrower physical claim it licenses.

The direct answer is now explicit: a regulated bosonic field integral integrates finitely many declared real coordinates against an ordinary reference measure and a specified weight. Continuum notation becomes meaningful only through an additional construction or a controlled limit of such finite data.

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.