Editing Renormalization group (section)

==Exact renormalization group equations==
An '''exact renormalization group equation''' ('''ERGE''') is one that takes [[irrelevant]] couplings into account. There are several formulations.

The '''Wilson ERGE''' is the simplest conceptually, but is practically impossible to implement. [[Fourier transform]] into [[momentum space]] after [[Wick rotation|Wick rotating]] into [[Euclidean space]]. Insist upon a hard momentum [[Cutoff (physics)|cutoff]], {{math|''p''<sup>2</sup> ≤ Λ<sup>2</sup>}} so that the only degrees of freedom are those with momenta less than {{mvar|Λ}}. The [[partition function (quantum field theory)|partition function]] is 
<math display="block">Z=\int_{p^2\leq \Lambda^2} \mathcal{D}\varphi \exp\left[-S_\Lambda[\varphi]\right].</math>

For any positive Λ&prime; less than Λ, define ''S''<sub>Λ&prime;</sub> (a functional over field configurations  {{mvar|φ}} whose Fourier transform has momentum support within {{math|''p''<sup>2</sup> ≤ Λ&prime;<sup>2</sup>}}) as
<math display="block">\exp\left(-S_{\Lambda'}[\varphi]\right)\ \stackrel{\mathrm{def}}{=}\  \int_{\Lambda'  \leq p \leq \Lambda} \mathcal{D}\varphi   \exp\left[-S_\Lambda[\varphi]\right].</math>

If {{math|''S''<sub>Λ</sub>}} depends only on {{Mvar|&varphi;}} and not on derivatives of {{Mvar|&varphi;}}, this may be rewritten as
<math display=block>\exp\left(-S_{\Lambda'}[\varphi]\right)\ \stackrel{\mathrm{def}}{=}\  \prod_{\Lambda'  \leq p \leq \Lambda}\int d\varphi(p) \exp\left[-S_\Lambda[\varphi(p)]\right],</math>
in which it becomes clear that, since only functions ''&phi;'' with support between  {{mvar|Λ'}}  and  {{mvar|Λ}} are integrated over, the left hand side may still depend on {{Math|''&varphi;''}} with support outside that range. Obviously,
<math display="block">Z=\int_{p^2\leq {\Lambda'}^2}\mathcal{D}\varphi \exp\left[-S_{\Lambda'}[\varphi]\right].</math>

In fact, this transformation is [[transitive relation|transitive]]. If you compute  {{math|''S''<sub>{{prime|Λ}}</sub>}} from  {{math|''S''<sub>Λ</sub>}} and then compute {{math|''S<sub>{{prime|Λ}}{{prime}}</sub>''}} from {{math|''S''<sub>{{prime|Λ}}{{prime}}</sub>}}, this gives you the same Wilsonian action as computing ''S''<sub>Λ&Prime;</sub> directly from ''S''<sub>Λ</sub>.

The '''Polchinski ERGE''' involves a [[smooth function|smooth]] UV [[regularization (physics)|regulator]] [[Cutoff (physics)|cutoff]].<ref>{{cite journal |last1=Polchinski |first1=Joseph |title=Renormalization and Effective Lagrangians |journal=Nucl. Phys. B |date=1984 |volume=231 |issue=2 |page=269 |doi=10.1016/0550-3213(84)90287-6 |bibcode=1984NuPhB.231..269P}}</ref> 
Basically, the idea is an improvement over the Wilson ERGE. Instead of a sharp momentum cutoff, it uses a smooth cutoff. Essentially, we suppress contributions from momenta greater than {{mvar|Λ}} heavily. The smoothness of the cutoff, however, allows us to derive a functional [[differential equation]] in the cutoff scale  {{mvar|Λ}}. As in Wilson's approach, we have a different action functional for each cutoff energy scale  {{mvar|Λ}}. Each of these actions are supposed to describe exactly the same model which means that their [[partition function (quantum field theory)|partition functional]]s have to match exactly.

In other words, (for a real scalar field; generalizations to other fields are obvious),
<math display="block">Z_\Lambda[J]=\int \mathcal{D}\varphi \exp\left(-S_\Lambda[\varphi]+J\cdot \varphi\right)=\int \mathcal{D}\varphi \exp\left(-\tfrac{1}{2}\varphi\cdot R_\Lambda \cdot \varphi-S_{\operatorname{int}\Lambda}[\varphi]+J\cdot\varphi\right)</math>
and ''Z''<sub>Λ</sub> is really independent of {{mvar|Λ}}! We have used the condensed [[deWitt notation]] here. We have also split the bare action ''S''<sub>Λ</sub> into a quadratic kinetic part and an interacting part ''S''<sub>int Λ</sub>. This split most certainly isn't clean. The "interacting" part can very well also contain quadratic [[kinetic term]]s. In fact, if there is any [[wave function renormalization]], it most certainly will. This can be somewhat reduced by introducing field rescalings. R<sub>Λ</sub> is a function of the momentum p and the second term in the exponent is
<math display="block">\frac{1}{2}\int \frac{d^dp}{(2\pi)^d}\tilde{\varphi}^*(p)R_\Lambda(p)\tilde{\varphi}(p)</math>
when expanded.

When <math>p \ll \Lambda</math>, {{math|''R''<sub>Λ</sub>(''p'')/''p''<sup>2</sup>}} is essentially 1. When <math>p \gg \Lambda</math>, {{math|''R''<sub>Λ</sub>(''p'')/''p''<sup>2</sup>}} becomes very very huge and approaches infinity. {{math|''R''<sub>Λ</sub>(''p'')/''p''<sup>2</sup>}} is always greater than or equal to 1 and is smooth. Basically, this leaves the fluctuations with momenta less than the cutoff {{mvar|Λ}} unaffected but heavily suppresses contributions from fluctuations with momenta greater than the cutoff. This is obviously a huge improvement over Wilson.

The condition that
<math display="block">\frac{d}{d\Lambda}Z_\Lambda=0</math>
can be satisfied by (but not only by)
<math display="block">\frac{d}{d\Lambda}S_{\operatorname{int}\Lambda}=\frac{1}{2}\frac{\delta S_{\operatorname{int}\Lambda}}{\delta \varphi}\cdot \left(\frac{d}{d\Lambda}R_\Lambda^{-1}\right)\cdot \frac{\delta S_{\operatorname{int}\Lambda}}{\delta \varphi}-\frac{1}{2}\operatorname{Tr}\left[\frac{\delta^2 S_{\operatorname{int}\Lambda}}{\delta \varphi\, \delta\varphi}\cdot R_\Lambda^{-1}\right].</math>

[[Jacques Distler]] claimed without proof that this ERGE is not correct [[nonperturbative]]ly.<ref>{{cite web |author-link=Jacques Distler |first=Jacques |last=Distler |url=http://golem.ph.utexas.edu/~distler/blog/archives/000648.html |title=000648.html  |website=golem.ph.utexas.edu}}</ref>

The '''effective average action ERGE''' involves a smooth IR regulator cutoff. 
The idea is to take all fluctuations right up to an IR scale {{mvar|k}} into account.  The '''effective average action(EAA)''' will be accurate for fluctuations with momenta larger than {{mvar|k}}. As the parameter {{mvar|k}} is lowered, the effective average action approaches the [[effective action]] which includes all quantum and classical fluctuations. In contrast, for large {{mvar|k}} the effective average action is close to the "bare action". So, the effective average action interpolates between the "bare action" and the [[effective action]].

For a real [[scalar field]], one adds an IR cutoff
<math display="block">\frac{1}{2}\int \frac{d^dp}{(2\pi)^d} \tilde{\varphi}^*(p)R_k(p)\tilde{\varphi}(p)</math>
to the [[action (physics)|action]]  {{mvar|S}}, where ''R''<sub>''k''</sub> is a function of both  {{mvar|k}} and  {{mvar|p}} such that for <math>p \gg k</math>, R<sub>k</sub>(p) is very tiny and approaches 0 and for <math>p \ll k</math>, <math>R_k(p)\gtrsim k^2</math>. ''R''<sub>''k''</sub> is both smooth and nonnegative. Its large value for small momenta leads to a suppression of their contribution to the partition function which is effectively the same thing as neglecting large-scale fluctuations.

One can use the condensed [[deWitt notation]]
<math display="block">\frac{1}{2} \varphi\cdot R_k \cdot \varphi</math>
for this IR regulator.

So,
<math display="block">\exp\left(W_k[J]\right)=Z_k[J]=\int \mathcal{D}\varphi \exp\left(-S[\varphi]-\frac{1}{2}\varphi \cdot R_k \cdot \varphi +J\cdot\varphi\right)</math>
where  {{mvar|J}} is the [[source field]]. The [[Legendre transform]] of ''W''<sub>''k''</sub> ordinarily gives the [[effective action]]. However, the action that we started off with is really ''S''[''φ'']&nbsp;+&nbsp;1/2 ''φ⋅R''<sub>''k''</sub>⋅''φ'' and so, to get the effective average action, we subtract off 1/2&nbsp;''φ''⋅''R''<sub>''k''</sub>⋅''φ''. In other words,
<math display="block">\varphi[J;k]=\frac{\delta W_k}{\delta J}[J]</math>
can be inverted to give ''J''<sub>''k''</sub>[''φ''] and we define the effective average action Γ<sub>''k''</sub> as
<math display="block">\Gamma_k[\varphi]\ \stackrel{\mathrm{def}}{=}\  \left(-W \left[J_k[\varphi]\right] + J_k[\varphi]\cdot\varphi\right)-\tfrac{1}{2}\varphi\cdot R_k\cdot \varphi.</math>

Hence,
<math display="block">\begin{align}
\frac{d}{dk}\Gamma_k[\varphi] &=-\frac{d}{dk}W_k[J_k[\varphi]]-\frac{\delta W_k}{\delta J}\cdot\frac{d}{dk}J_k[\varphi]+\frac{d}{dk}J_k[\varphi]\cdot \varphi-\tfrac{1}{2}\varphi\cdot \frac{d}{dk}R_k \cdot \varphi \\
&=-\frac{d}{dk}W_k[J_k[\varphi]]-\tfrac{1}{2}\varphi\cdot \frac{d}{dk}R_k \cdot \varphi \\
&=\tfrac{1}{2}\left\langle\varphi \cdot \frac{d}{dk}R_k \cdot \varphi\right\rangle_{J_k[\varphi];k}-\tfrac{1}{2} \varphi\cdot \frac{d}{dk}R_k \cdot \varphi \\
&=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta J_k}{\delta \varphi}\right)^{-1}\cdot\frac{d}{dk}R_k\right] \\
&=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^2 \Gamma_k}{\delta \varphi \delta \varphi} + R_k \right)^{-1}\cdot\frac{d}{dk}R_k\right]
\end{align}</math>
thus
<math display="block">\frac{d}{dk}\Gamma_k[\varphi] =\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^2 \Gamma_k}{\delta \varphi \delta \varphi}+R_k\right)^{-1}\cdot\frac{d}{dk}R_k\right]</math>
is the ERGE which is also known as the [[Christof Wetterich|Wetterich]] equation.<ref>{{cite journal |last1=Wetterich |first1=Christof |title=Exact evolution equations for the effective potential|journal=Phys. Lett. B |date=1993 |volume=301 |issue=1 |page=90 |doi=10.1016/0370-2693(93)90726-X|arxiv=1710.05815 |bibcode=1993PhLB..301...90W }}</ref>

As shown by Morris the effective action Γ<sub>k</sub> is in fact simply related to Polchinski's effective action S<sub>int</sub> via a Legendre transform relation.<ref>{{cite journal |last1=Morris |first1=Tim R. |title=The Exact renormalization group and approximate solutions|journal=Int. J. Mod. Phys. A |date=1994 |volume=9 |issue=14 |page=2411 |doi=10.1142/S0217751X94000972 |arxiv=hep-ph/9308265 |bibcode=1994IJMPA...9.2411M |s2cid=15749927 }}</ref>

As there are infinitely many choices of {{mvar|R}}<sub>''k''</sub>, there are also infinitely many different interpolating ERGEs.
Generalization to other fields like spinorial fields is straightforward.

Although the Polchinski ERGE and the effective average action ERGE look similar, they are based upon very different philosophies. In the effective average action ERGE, the bare action is left unchanged (and the UV cutoff scale—if there is one—is also left unchanged) but the IR contributions to the effective action are suppressed whereas in the Polchinski ERGE, the QFT is fixed once and for all but the "bare action" is varied at different energy scales to reproduce the prespecified model. Polchinski's version is certainly much closer to Wilson's idea in spirit. Note that one uses "bare actions" whereas the other uses effective (average) actions.