Changes of Variables and Regulated Jacobians

At a fixed finite regulator, a bosonic field redefinition is an ordinary change of variables only when the map, its image or branch structure, the integration domain or cycle, and the Jacobian are all defined. The resulting integral is exactly the same one only if the action, reference measure, source, insertion, constraints, and boundary data are transformed together. This finite-dimensional identity does not by itself define a continuum functional determinant, establish an anomaly, or prove an equivalence theorem.

Required background. Regulated Bosonic Field Integrals supplies the finite coordinates, domain or cycle, reference measure, normalization, and order of limits used below.

Helpful background. Product Measures, Fubini–Tonelli, and Change of Variables states the global change-of-variables theorem and explains why local invertibility is not enough.

A regulated field redefinition transforms the whole integral

Fix a regulator $R$ that leaves $N_R<\infty$ real commuting coordinates. In this section the integration domains are open subsets of $\mathbb R^{N_R}$, the weight is Euclidean, and $\rho_R\geq0$ is a density relative to coordinate Lebesgue measure. Boundary sets of measure zero may be restored by the usual limiting argument; constraints supported on a lower-dimensional set require their own induced measure. We write the old coordinate as a function of the new one,

$$q=f_R(\chi).$$

For an integrable observable $A_R$ and source $j_R$, consider

$$\mathcal I_R[A,j] = \int_{D_R} \mathrm d^{N_R}q\,\rho_R(q)A_R(q) e^{-S_{E,R}(q)+j_R^{\mathsf T}q}.$$

Suppose $f_R:D'_R\to D_R$ is a $C^1$ diffeomorphism. Then

$$\begin{aligned} \mathcal I_R[A,j] =\int_{D'_R}\mathrm d^{N_R}\chi\;& \rho_R(f_R(\chi)) \left|\det Df_R(\chi)\right| \\ &\times A_R(f_R(\chi)) e^{-S_{E,R}(f_R(\chi))+j_R^{\mathsf T}f_R(\chi)}. \end{aligned}$$

This is the finite-dimensional theorem, with its absolute Jacobian and global bijectivity hypotheses, applied to the complete integrand (Dyatlov 2022, § 10.1.3, Theorem 10.5, p. 108, PDF). Notice three points. The new domain is $D'_R=f_R^{-1}(D_R)$; a nonlinear source becomes $j_R^{\mathsf T}f_R(\chi)$ rather than $j_R^{\mathsf T}\chi$; and an old observable becomes $A_R(f_R(\chi))$. Boundary conditions, delta constraints, and any normalization denominator must be pulled back in the same way.

A reproducible fixed-regulator procedure

1. Declare the inputs. Record $R$, the retained variables, domain or oriented cycle, reference measure, action, sources, insertions, boundary data, map direction, and intended order of limits.
2. Control the map globally. Determine its image, inverse or branches, differentiability, and every point where the determinant vanishes or changes sign or phase.
3. Pull back every ingredient. Transform the domain or cycle, density, action, source, observable, constraints, and boundary data before simplifying the Jacobian.
4. Validate independently. Check a known normalization, a solvable Gaussian, an inverse round trip, or a parameter derivative that reduces to a vanishing boundary term.
5. Only then study limits. Ask separately whether the map, inverse, determinant, observables, and error bounds have a common $R\to\infty$ limit.

The output is an exact alternative representation of the same regulated integral, not an assertion that the old and new coordinate polynomials have the same expectation values. Computing a dense $N_R\times N_R$ determinant by standard factorization generally costs $O(N_R^3)$ operations; a diagonal or triangular Jacobian costs only $O(N_R)$. Stop if global invertibility, multiplicity, the transformed domain or cycle, determinant sign or phase, boundary behavior, regulator compatibility, or limit control is unknown.

The Jacobian may stay in the measure or enter the action

Keeping the Jacobian explicit makes the theorem easiest to check. It can instead be included in a transformed action, but the logarithm must be a dimensionless density ratio. If the original weight is $e^{-S_{E,R}/\hbar}$ and the new reference density is $\rho'_R(\chi)>0$, set

$$\mathscr R_R(\chi) = \frac{ \rho_R(f_R(\chi))\left|\det Df_R(\chi)\right| }{ \rho'_R(\chi) }.$$

Where $\mathscr R_R>0$,

$$S'_{E,R}(\chi) = S_{E,R}(f_R(\chi)) -\hbar\log \mathscr R_R(\chi).$$

The minus sign follows directly from $\mathscr R_R e^{-S_{E,R}/\hbar} =e^{-[S_{E,R}-\hbar\log\mathscr R_R]/\hbar}$. For flat reference densities and dimensionless coordinates, $\mathscr R_R$ reduces to $|\det Df_R|$. A bare logarithm of a dimensionful determinant is not meaningful, and a complex determinant needs an explicit phase and logarithm branch rather than this positive-density formula.

For a finite infinitesimal map

$$f_{R,\epsilon}(\chi)=\chi+\epsilon F_R(\chi), \qquad (M_R)_{ab}=\frac{\partial F_{R,a}}{\partial\chi_b},$$

the derivative matrix is $I+\epsilon M_R$. As long as the determinant stays positive and the matrix logarithm is followed continuously,

$$\begin{aligned} \log\det(I+\epsilon M_R) &= \operatorname{Tr}\log(I+\epsilon M_R) \\ &= \epsilon\operatorname{Tr}M_R -\frac{\epsilon^2}{2}\operatorname{Tr}(M_R^2) +O(\epsilon^3). \end{aligned}$$

In particular, $\det(I+\epsilon M_R)=1+\epsilon\operatorname{Tr}M_R+O(\epsilon^2)$ (Zinn-Justin 2021, § 1.6, p. 12). For an $\hbar$-independent map, the term $-\hbar\log\mathscr R_R$ is of order $\hbar$ when placed in the action. That is exact measure bookkeeping; calling it a physical one-loop effect or an anomaly would require additional arguments.

A linear Gaussian gives the determinant round trip

Let $K$ be a real symmetric positive-definite $N\times N$ matrix. With the declared coordinate measure,

$$\mathcal Z_K(j) = \int_{\mathbb R^N} \frac{\mathrm d^Nq}{(2\pi)^{N/2}} e^{-q^{\mathsf T}Kq/2+j^{\mathsf T}q} = \frac{e^{j^{\mathsf T}K^{-1}j/2}}{\sqrt{\det K}}.$$

Take $q=B\chi$ with real invertible $B$. The transformed data are

$$K_\chi=B^{\mathsf T}KB, \qquad j_\chi=B^{\mathsf T}j, \qquad \mathrm d^Nq=|\det B|\,\mathrm d^N\chi.$$

Since $\det K_\chi=(\det B)^2\det K$, the positive square root gives

$$|\det B|\,(\det K_\chi)^{-1/2} = (\det K)^{-1/2}.$$

The source dependence returns as well:

$$j_\chi^{\mathsf T}K_\chi^{-1}j_\chi = j^{\mathsf T}K^{-1}j.$$

Thus the Jacobian cancels the determinant change while the pulled-back source preserves the completed-square exponent. These are independent checks of the same substitution. They rely on a positive real Gaussian; complex contours also require determinant phases and square-root branches.

A nonlinear retained scalar mode gives an exact test

For one dimensionless real mode, $m>0$, and real $j$,

$$\mathcal Z(j) = \int_{\mathbb R}\frac{\mathrm dq}{\sqrt{2\pi}} e^{-m^2q^2/2+jq} = \frac{1}{m}e^{j^2/(2m^2)}.$$

Now set

$$q=f_\alpha(\chi)=\chi+\alpha\chi^3, \qquad \alpha\geq0.$$

Its derivative is $f'_\alpha=1+3\alpha\chi^2>0$, and $f_\alpha(\chi)\to\pm\infty$ as $\chi\to\pm\infty$. The map is therefore a smooth global diffeomorphism of $\mathbb R$, including the identity at $\alpha=0$. The exact transformed integral is

$$\begin{aligned} \mathcal Z(j) = \int_{\mathbb R}\frac{\mathrm d\chi}{\sqrt{2\pi}} &(1+3\alpha\chi^2) \\ &\times \exp\!\left[ -\frac{m^2}{2}(\chi+\alpha\chi^3)^2 +j(\chi+\alpha\chi^3) \right]. \end{aligned}$$

The right side looks interacting, but it is exactly the same Gaussian. An independent check follows from

$$\frac{\partial}{\partial\alpha} \left[ f'_\alpha e^{-m^2f_\alpha^2/2+jf_\alpha} \right] = \frac{\partial}{\partial\chi} \left[ \chi^3e^{-m^2f_\alpha^2/2+jf_\alpha} \right].$$

The boundary term vanishes, so $\partial_\alpha Z=0$. Source derivatives also show what equality means for observables:

$$\langle q^n\rangle = \left\langle f_\alpha(\chi)^n\right\rangle_{\mathrm{transformed}},$$

not $\langle\chi^n\rangle$. In particular, $\langle f_\alpha(\chi)^2\rangle=1/m^2$ at $j=0$.

The componentwise finite-mode extension is

$$q_a=\chi_a+\alpha\chi_a^3, \qquad \mathcal J_R(\chi) = \prod_{a=1}^{N_R}(1+3\alpha\chi_a^2).$$

A coupled action and source become $S_{E,R}(f_\alpha(\chi))$ and $j_R^{\mathsf T}f_\alpha(\chi)$. This map is componentwise in the retained coordinates. It is spatially local only if those coordinates are site variables; a pointwise nonlinear field map need not preserve a spectral cutoff subspace.

Local invertibility is not global equivalence

The map $q=\chi^2$ exposes three failures at once: its image is $[0,\infty)$, it is two-to-one away from the origin, and its derivative vanishes at the origin. For integrable $F$ on $[0,\infty)$,

$$\int_{\mathbb R}\mathrm d\chi\,|2\chi|F(\chi^2) = 2\int_0^\infty\mathrm dq\,F(q).$$

Integrating both branches therefore double-counts the target. One may restrict to a single branch or use a correctly weighted multiplicity formula, but neither repairs the image mismatch if the old domain was all of $\mathbb R$.

A nonzero derivative on one patch guarantees only a local inverse. Likewise, a field redefinition with an inverse known as a truncated power series gives an order-by-order identity only after every action, source, observable, and Jacobian term is retained to the same order. It is not an exact global substitution.

Real measures and complex cycles use different Jacobians

The word “measure” can hide different geometric objects:

Integration object	Transformation factor
Positive density on $\mathbb R^N$	$\lvert\det_{\mathbb R}Df\rvert$
Oriented real top form	$\det_{\mathbb R}Df$, with source orientation
Holomorphic $N$-form on an oriented complex contour	$\det_{\mathbb C}(\partial f/\partial w)$, with transformed contour and orientation
Real volume on $\mathbb C^N\simeq\mathbb R^{2N}$	$\lvert\det_{\mathbb C}(\partial f/\partial w)\rvert^2$ for holomorphic $f$
Grassmann variables	Berezinian rule; not the bosonic formula

Positive Lebesgue measure uses an absolute determinant, whereas pullback of an oriented top form retains the determinant sign (Eliashberg 2018, § 10.7, Theorem 10.32 and Corollary 10.33, pp. 137–138, PDF). For a holomorphic contour integral, replacing the complex determinant by its absolute value destroys phase and orientation information. Conversely, treating $\mathbb C^N$ simply as $\mathbb R^{2N}$ invokes the real-volume rule and produces the squared modulus. Constrained and gauge-redundant variables need an induced or quotient measure before any determinant formula is used.

Fixed-regulator equality does not define a continuum Jacobian

At fixed $R$, the positive-measure Jacobian has the finite logarithm

$$\log\mathcal J_R = \log\left|\det Df_R\right|.$$

For a near-identity map on a continuously chosen matrix-logarithm branch, this can be written as $\operatorname{Tr}_R\log Df_R$. Both expressions concern a finite matrix. For a diagonal map on a lattice the logarithm is a finite sum over sites. Replacing that sum by a continuum integral introduces powers of the cutoff and can produce local regulator-dependent terms. For a spectral cutoff, even defining $f_R$ may require a projection back into the retained subspace; that projection changes both the map and its Jacobian.

Continuum notation such as

$$\det\!\left[ \delta^{(d)}(x-y) +\epsilon\frac{\delta F(x;\chi)}{\delta\chi(y)} \right]$$

is therefore a prompt to specify a regulator, not already a number. The standard formal infinitesimal manipulation is displayed in Zinn-Justin 2021, § 7.5.3, pp. 136–137; its trace contains a coincident kernel that the fixed-mode matrix avoided. In a standard anomaly calculation, the corresponding singular trace is regulated before any finite conclusion is drawn (Schwartz 2014, § 30.3, pp. 628–630).

Whether a regulated local contribution vanishes, is absorbed into counterterms, or leaves a finite effect depends on the regulator, renormalization conditions, symmetry requirements, and limit. An anomaly is not a failure of finite-dimensional substitution. Nor does equality of the regulated integrals prove that untransformed off-shell correlators or S-matrix elements agree: source insertions have changed, and an equivalence theorem has further locality, invertibility, boundary, LSZ, truncation, and renormalization hypotheses.

Perform the exact substitution at fixed $R$ first. Then ask whether $f_R$, its inverse, the transformed observables, $\log\mathcal J_R$, and the needed convergence bounds have a common regulator limit. Do not assume that this limit commutes with infinite volume, a massless limit, contour removal, or a semiclassical limit.

Common pitfalls

Using the inverse determinant in the substitution. Writing $q=f_R(\chi)$ uses $|\det Df_R(\chi)|$. A pushed-forward density evaluated at $q$ instead uses the determinant of the inverse map; the two formulas answer different questions.

Leaving the source or insertion unchanged. A nonlinear substitution turns $j^{\mathsf T}q$ into $j^{\mathsf T}f(\chi)$ and $A(q)$ into $A(f(\chi))$. Equality does not identify moments of $q$ with moments of $\chi$.

Treating a local inverse as a global one. A nonsingular derivative at one point says nothing about distant branches, the image, or boundary behavior. Prove global control or decompose the map into branches.

Using an absolute value on a complex contour. A holomorphic form carries a complex determinant and the oriented cycle. Taking an absolute value changes the integral rather than reparametrizing it.

Calling every nonunit Jacobian an anomaly. At finite $R$, a field-dependent Jacobian is ordinary coordinate bookkeeping. The anomaly question begins only after a symmetry transformation, regulator, measure prescription, and continuum limit are specified.

Promoting a finite trace to a continuum result. Symbols such as $\delta^{(d)}(0)$ are not regulator-independent numbers. Keep the finite matrix until the regulator dependence and counterterms are controlled.

Check your understanding

1. Starting from $K_\chi=B^{\mathsf T}KB$, prove both the determinant and source-exponent identities in the linear Gaussian example.

   Solution

   Multiplicativity gives $\det K_\chi=(\det B)^2\det K$. Positivity of $K$ and $K_\chi$ fixes the positive square roots, so $|\det B|/\sqrt{\det K_\chi}=1/\sqrt{\det K}$. Also $K_\chi^{-1}=B^{-1}K^{-1}B^{-\mathsf T}$; inserting $j_\chi=B^{\mathsf T}j$ cancels the four $B$ factors and gives $j^{\mathsf T}K^{-1}j$.

2. Differentiate the transformed one-mode integral with respect to $\alpha$. Why does the result also show that $\langle f_\alpha(\chi)^2\rangle=1/m^2$ at zero source?

   Solution

   The displayed derivative is a total $\chi$ derivative. Its boundary term vanishes because the Gaussian, or the stronger large-$|\chi|$ decay for $\alpha>0$, dominates the polynomial. Hence $\mathcal Z(j)$ is independent of $\alpha$ and equals $m^{-1}e^{j^2/(2m^2)}$. Two source derivatives at $j=0$, divided by $\mathcal Z(0)$, insert $f_\alpha(\chi)^2$ and give $1/m^2$.

3. Explain the factor of two obtained from $q=\chi^2$ when both real branches are integrated.

   Solution

   Split the $\chi$ integral at zero. On each half-line, the absolute Jacobian $|2\chi|$ maps that branch once onto $q\in[0,\infty)$. The two equal branch integrals add, so the target integral is counted twice.

4. Restore $\hbar$ and absorb a positive Jacobian ratio $\mathscr R_R$ into the action. Determine its sign and explain why the result is not by itself an anomaly.

   Solution

   Since $\mathscr R_R e^{-S/\hbar} =e^{-[S-\hbar\log\mathscr R_R]/\hbar}$, the transformed action contains $-\hbar\log\mathscr R_R$. This identity holds for an ordinary finite change of coordinates. An anomaly claim additionally concerns a desired symmetry and whether its regulated measure and continuum renormalization can preserve that symmetry.

Where to continue

• Gaussian Fields and Sources supplies further normalization and source checks for the linear example.
• Saddles and the Semiclassical Expansion uses Jacobians when collective coordinates replace zero-mode amplitudes.
• Schwinger–Dyson Identities develops the hierarchy seeded by regulated infinitesimal substitutions.
• Grassmann Functional Integrals for Free Fermions gives the odd-variable determinant rule.
• Regulated Jacobians and Measure Variation treats anomaly-facing regulator and local-density questions.
• Local Field Redefinitions and the Equivalence Theorem states the extra hypotheses behind on-shell equivalence and redundant operators.

References

• Dyatlov, Semyon. Lecture Notes for 18.155: Distributions, Elliptic Regularity, and Applications to PDEs. Massachusetts Institute of Technology, 2022. Official course notes, PDF.
• Eliashberg, Yakov. Multilinear Algebra, Differential Forms and Stokes’ Theorem. Stanford University, 2018. Official course notes, PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford University Press, 2021. doi:10.1093/oso/9780198834625.001.0001.