Contents

# Contents

## Idea

In perturbative quantum field theory via the method of effective quantum field theories what is called Wilsonian RG (remark below) and specifically Polchinski’s flow equation (prop. below) is a characterization of the (infinitesimal) dependence of relative effective actions $S_{eff,\Lambda}$ (“effective potentials”) on the choice of UV cutoff-scale $\Lambda$.

Solving Polchinki’s flow equation with a choice of initial conditions may be used to choose a ("re"-)normalization of an interacting perturbative QFT.

This is related to, but conceptually different from, the renormalization group flow via beta functions in the sense of Gell-Mann-Low renormalization cocycles.

## Statement

###### Remark

(Wilsonian groupoid of effective quantum field theories)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to this def.) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (this def.).

Then the relative effective actions $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ (this def.) satisfy

$S_{eff, \Lambda', \Lambda_0} \;=\; \left( \mathcal{S}_{\Lambda'}^{-1} \circ \mathcal{S}_\Lambda \right) \left( S_{eff, \Lambda, \Lambda_0} \right) \phantom{AAA} \text{for} \, \Lambda,\Lambda' \in [0,\infty) \,,\, \Lambda_0 \in [0,\infty) \sqcup \{\infty\} \,.$

This is similar to a group of UV-cutoff scale-transformations. But since the composition operations are only sensible when the UV-cutoff labels match, as shown, it is really a groupoid action.

This is often called the Wilsonian RG, following (Wilson 71).

We now consider the infinitesimal version of this “flow”:

###### Proposition

(Polchinski's flow equation)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to this def.), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ), such that $\Lambda \mapsto \Delta_{F,\Lambda}$ is differentiable.

Then for every choice of UV regularization $\mathcal{S}_\infty$ (this prop.) the corresponding relative effective actions $S_{eff,\Lambda}$ (this def.) satisfy the following differential equation:

$\frac{d}{d \Lambda} S_{eff,\Lambda} \;=\; - \frac{1}{2} \frac{1}{i \hbar} \frac{d}{d \Lambda'} \left( S_{eff,\Lambda} \star_{F,\Lambda'} S_{eff,\Lambda} \right)\vert_{\Lambda' = \Lambda} \,,$

where on the right we have the star product induced by $\Delta_{F,\Lambda'}$ (this def.).

This goes back to (Polchinski 84, (27)). The rigorous formulation and proof is due to (Brunetti-Dütsch-Fredenhagen 09, prop. 5.2, Dütsch 10, theorem 2).

###### Proof

First observe that for any polynomial observable $O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

\begin{aligned} & \frac{1}{(k+2)!} \frac{d}{d \Lambda} ( \underset{ k+2 \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } ) \\ & = \frac{1}{(k+2)!} \frac{d}{d \Lambda} \left( prod \circ \exp\left( \hbar \underset{1 \leq i \lt j \leq k}{\sum} \left\langle \Delta_{F,\Lambda} , \frac{\delta}{\delta \mathbf{\Phi}_i} \frac{\delta}{\delta \mathbf{\Phi}_j} \right\rangle \right) ( \underset{ k + 2 \, \text{factors} }{ \underbrace{ O \otimes \cdots \otimes O } } ) \right) \\ & = \underset{ = \frac{1}{2} \frac{1}{k!} }{ \underbrace{ \frac{1}{(k+2)!} \left( k + 2 \atop 2 \right) }} \left( \frac{d}{d \Lambda} O \star_{F,\Lambda} O \right) \star_{F,\Lambda} \underset{ k \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } \end{aligned}

Here $\frac{\delta}{\delta \mathbf{\Phi}_i}$ denotes the functional derivative of the $i$th tensor factor of $O$, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of $k+2$ tensor factors, where we use that the star product $\star_{F,\Lambda}$ is commutative (by symmetry of $\Delta_{F,\Lambda}$) and associative (by this prop.).

With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) = \mathcal{S}(g S_{int} + j A)$ (this equation) we compute as follows:

\begin{aligned} 0 & = \frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A) \\ & = \frac{d}{d \Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) + \left( \frac{d}{d \Lambda} \mathcal{S}_{\Lambda} \right) \left( S_{eff, \Lambda} \right) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \;+\; \frac{1}{2} \frac{d}{d \Lambda'} \left( \frac{1}{i \hbar} S_{eff,\Lambda} \star_{F,\Lambda'} \frac{1}{i \hbar} S_{eff, \Lambda} \right) \vert_{\Lambda' = \Lambda} \star_{F,\Lambda} \mathcal{S}_\Lambda \left( S_{eff, \Lambda} \right) \end{aligned}

Acting on this equation with the multiplicative inverse $(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )$ (using that $\star_{F,\Lambda}$ is a commutative product, so that exponentials behave as usual) this yields the claimed equation.

## References

The idea of effective quantum field theory was promoted in

• Kenneth Wilson, Renormalization group and critical phenomena , I., Physical review B 4(9) (1971).

The flow equation in its original form is due to

• Joseph Polchinski, equation (27) in Renormalization and effective Lagrangians , Nuclear Phys. B B231, 1984 (pdf)

The rigorous formulation and proof in causal perturbation theory/perturbative AQFT is due to

reviewed in

Last revised on August 29, 2018 at 05:27:48. See the history of this page for a list of all contributions to it.