How does the functional measure transform under a field redefinition?

Question

My question is: how does the path integral functional measure transform under the following field redefinitions (where $c$ is an arbitrary constant and $\phi$ is a scalar field): \begin{align} \phi(x)&=\theta(x)+c \,\theta^3(x) \tag{1}\\ \phi(x)&=c\,\theta^3(x) \tag{2}\\ \phi(x)&=\sinh\big(\theta(x)\big)\tag{3} \end{align} My naive guess for the transformation in Eq.(3) is \begin{align} \mathcal{D}\phi&=\mathcal{D}\theta\,\,\text{Det}\Bigg[\frac{\delta \phi(x)}{\delta\theta(x')}\Bigg]=\mathcal{D}\theta \,\,\text{Det}\bigg[\cosh(\theta(x))\delta(x-x')\bigg]\\ &=\mathcal{D}\theta\exp\bigg[\text{Tr}\,\Big(\log\big(\cosh(\theta(x))\delta(x-x')\big)\Big)\bigg]\\ &=\mathcal{D}\theta\exp\bigg[\int dx\,\,\log\Big(\cosh(\theta(x))\delta(x-x')\Big)\bigg] \end{align} But that seems very wrong.

Answer 1

1. It seems natural to generalize OP's setting to several fields $\phi^{\alpha}$ in $d$ spacetime dimensions. Consider for simplicity ultra-local field redefinitions$^1$ $$\begin{align} \phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),x)\cr ~=~&\phi^{\alpha}(x)-f^{\alpha}(\phi(x),x).\end{align}\tag{1} $$

2. The Jacobian functional determinant in the path/functional integral is formally given as as a functional Berezin integral $$ \begin{align} J~=~&{\rm Det} (\mathbb{M})\cr ~=~&\int \!{\cal D}c~{\cal D}\bar{c} \exp\left(\int\! d^dx^{\prime} \!\int\! d^dx ~\bar{c}_{\beta}(x^{\prime})\mathbb{M}^{\beta}{}_{\alpha}(x^{\prime},x)c^{\alpha}(x) \right) ,\end{align} \tag{2}$$ where $c^{\alpha}(x)$ and $\bar{c}_{\beta}(x^{\prime})$ are Grassmann-odd ghost fields; or as $$\begin{align} J~=~&{\rm Det} (\mathbb{M})\cr ~=~&\exp {\rm Tr}\ln (\mathbb{M})\cr ~=~& \exp\left(-\sum_{j=1}^{\infty} \frac{1}{j}{\rm Tr} (\mathbb{m}^j)\right) \cr ~=~& \exp\left(\delta^d(0) \int\! d^dx ~{\rm tr} (\ln M(x))\right),\end{align} \tag{3} $$ where we have defined $$\begin{align} \mathbb{M}~\equiv~&\mathbb{1}-\mathbb{m},\cr \mathbb{M}^{\beta}{}_{\alpha}(x^{\prime},x) ~:=~&\frac{\delta F^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr ~=~& M^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr M^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial F^{\beta}(x)}{\partial\phi^{\alpha}(x)}~=~\delta^{\beta}_{\alpha}-m^{\beta}{}_{\alpha}(x),\cr \mathbb{m}^{\beta}{}_{\alpha}(x^{\prime},x) ~:=~&\frac{\delta f^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr ~=~& m^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr m^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial f^{\beta}(x)}{\partial\phi^{\alpha}(x)}.\end{align} \tag{4}$$

3. A local field redefinition corresponds to insertion of UV-relevant/IR-irrelevant terms in the action, i.e. it doesn't change low-energy physics.

4. If we discretize spacetime, then the Jacobian (3) becomes a product of ordinary determinants $$ J~=~\prod_i \det (M(x_i)), \tag{5}$$ where the index $i$ labels lattice points $x_i$ of spacetime. The Dirac delta at zero $\delta^d(0)$ is here replaced by a reciprocal volume of a unit cell of the spacetime lattice, which can viewed as a UV regulator, cf. e.g. my Phys.SE answer here.

5. In dimensional regularization (DR), the Jacobian $J=1$ becomes one under local field redefinitions (if there are no anomalies present).

   • A ghost-loop diagram in momentum space is UV-divergent (because the ghost propagator is $1$ rather than $\frac{1}{k^2}$, so there are no negative powers of momenta in the diagram), cf. Refs. 1-7. The scaleless integral is then proportional to a positive power of an IR-mass regulator, and hence zero when the IR-mass regulator is removed.

   • The Dirac delta at zero $$\delta^d(0)~=~\int \!\frac{d^dk}{(2\pi)^d} (k^2)^0~=~0 \tag{6}$$ vanishes, cf. Refs. 1-7. Heuristically, DR only picks up residues of various finite parameters of the physical system, while contributions from infinite parameters are regularized to zero.

References:

1. G 't Hooft & M.J.G. Veltman, Diagrammar, CERN report, 1973; p. 46-51.

2. G. Leibbrandt, Introduction to the technique of dimensional regularization, Rev. Mod. Phys. 47 (1975) 849; Subsection IV.B.3 p. 864.

3. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; Subsection 18.2.4.

4. I.V. Tyutin, Once again on the equivalence theorem, arXiv:hep-th/0001050 .

5. R. Ferrari, M. Picariello & A. Quadri, An Approach to the Equivalence Theorem by the Slavnov-Taylor Identities, arXiv:hep-th/0203200.

6. A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863; p. 33-34 & p. 51.

7. J.C. Criado & M. Perez-Victoria, Field redefinitions in effective theories at higher orders, arXiv:1811.09413.

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$^1$ Much of this can be generalized to local field redefinitions $$ \begin{align}\phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x)\cr ~=~&\phi^{\alpha}(x)\cr ~-~&f^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x),\end{align}\tag{7}$$ and to a certain degree even beyond; and to derivatives $\partial^j\delta^d(0)$ of the Dirac delta at zero, cf. Refs. 1-7.

Answer 2

All three cases, (1)-(3), are local redefinitions, meaning that the value of $\phi(x)$ for any given $x$ is determined only by the value of $\theta(x)$ at that same value of $x$ (and conversely, assuming it's invertible).

Conceptually, the parameter $x$ is just a continuous index labeling different integration variables. In fact, the most generally-applicable way we have for defining a functional integral (at least in QFT) is to replace this continuous parameter with a discrete index. Then you have an ordinary multi-variable integral, and the rule for changing integration variables is the usual one. So the cases (1)-(3) just describe changes-of-variable in a bunch of single-variable integrals.

Thinking about things this way (with $x$ discretized) should help track down what's really going on with the $\delta(x-x')$ factor.

Answer 3

Consider the more general case of a (not necessarily local) field redefintion of the form: \begin{equation} \phi(x)=\int d y \,\,f(x,y)\,g\big(\theta(y)\big)\tag{1} \end{equation} For example, in the question above we have \begin{equation} \phi(x)=\int d y \,\,\delta(x-y)\,\sinh\big(\theta(y)\big) \end{equation} which is an example of a local field redefinition. In the discrete case where we think of the path integral as taking place on a lattice, Eq.(1) takes the form, where $\phi_i$ is shorthand for the discrete variable $\phi(x_i)$: \begin{equation} \phi_i=\sum_j F_{ij}\, g(\theta_j) \tag{2} \end{equation} where $F_{ij}=F(x_i,x_j)$ can be thought of as a matrix and $F_{ij}=\delta_{ij}$ corresponds to the case of a local transformation. Discretizing the path integral we can write the change of variables in Eq.(2) as (here $\wedge$ denotes the wedge product, which is always present for the tensor density $d^dx$ but I make it explicit here only to make the presence of the Jacobian apparent) \begin{align} \int \mathcal{D}\phi&=\int d\phi_1\wedge d\phi_2\wedge...\wedge d\phi_n\\ &= \int \frac{1}{n!}\epsilon_{i_1...i_n}d\phi^{i_1}\wedge...\wedge d\phi^{i_n}\\ &=\int \frac{1}{n!}\epsilon_{i_1...i_n}\,\,\frac{\partial\phi^{i_1}}{\partial\theta^{i_1'}}...\frac{\partial\phi^{i_1}}{\partial\theta^{i_n'}}d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\ &=\int \frac{1}{n!}\epsilon_{i_1...i_n}\,\,\bigg(F^{i_1i_1'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i_1'}}\bigg)...\bigg(F^{i_ni_n'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i_n'}}\bigg)d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\ &=\int\frac{1}{n!}\epsilon_{i_1'...i_n'}\text{Det}\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\ &=\int\text{Det}\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\ &=\int d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\,\,\exp\bigg\lbrace\text{Tr}\bigg(\log\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]\bigg)\bigg\rbrace\tag{3} \end{align} So for example if we have a local field redefintion $F_{ij}=\delta_{ij}$ then we encounter the term $Tr(\log(\delta_{ij}))=\log(n)$, where $n$ is the number of lattice sites. In the continuous case where $F_{ij}\rightarrow f(x,y)=\delta^{d}(x-y)$ we encounter the highly singular term \begin{equation} Tr(\log(\delta^d(x-y)))=\int d^dx\,\,\log\bigg(\delta^{d}(x-x)\bigg)=\int d^dx\,\,\log\bigg(\delta^{d}(0)\bigg) \end{equation} So to answer the original question, I think the measure transforms as: \begin{align} \mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(1+3c\theta^2)\big)\bigg)\tag{1}\\ \mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(3c\theta^2\big)\bigg)\tag{2}\\ \mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(\cosh(\theta(x))\big)\bigg)\tag{3} \end{align} Apparently one can ignore the $\delta^d(0)$ when working in dimensional regularization as we can interpret $\delta^d(0)$ as the volume of spacetime, which in $d-\epsilon$ dimensions is \begin{equation} \delta^{d}(0)=\frac{2\pi^{d/2}}{\Gamma(d/2)}\frac{\Gamma(-d)}{\Gamma(1-d)} \end{equation} which is $\frac{1}{\epsilon}$ dependent and hence we can use $\frac{1}{\epsilon}$ dependent counterterms to get rid of this term. However $\exp(\log(g'))$ terms still remain and it is unclear to me how to show that correlation functions will be unaffected by this term.