nLab Polchinski's flow equation

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.

Related concepts

• renormalization group flow

• gauge coupling unification

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

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, prop. 5.2 of Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

• Michael Dütsch, theorem 2 in Connection between the renormalization groups of Stückelberg-Petermann and Wilson, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 (arXiv:1012.5604)

reviewed in

• Michael Dütsch, section 3.8 of From classical field theory to perturbative quantum field theory, 2018