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In quantum field theory and statistical mechanics, an exact renormalization group equation is an exact (meaning precisely that) renormalization group equation unlike approximate ones like the Callan-Symanzik equation which doesn't take irrelevant couplings into account. There is more than one ERGE.

Wilson ERGE

This is the simplest conceptually but is practically impossible to do real calculations with. Fourier transform into momentum space after Wick rotating into Euclidean space. Insist upon a hard momentum cutoff, p 2 ≤ Λ 2 {\displaystyle p^{2}\leq \Lambda ^{2}} so that the only degrees of freedom are those with momenta less than Λ. The partition function is

Z = ∫ D ϕ p 2 ≤ Λ 2 exp ⁡ [ − S Λ [ ϕ ] ] {\displaystyle Z=\int {\begin{matrix}{\mathcal {D}}\phi \\p^{2}\leq \Lambda ^{2}\end{matrix}}\exp \left\right]}

For any positive Λ' less than Λ, define SΛ (a functional over field configurations φ whose Fourier transform has momentum support within p 2 ≤ Λ ′ 2 {\displaystyle p^{2}\leq \Lambda '^{2}} ) as

exp ⁡ ( − S Λ ′ [ ϕ ] ) ≡ ∫ D ϕ Λ ′ ≤ p ≤ Λ exp ⁡ [ − S Λ [ ϕ ] ] {\displaystyle \exp \left(-S_{\Lambda }'\right)\equiv \int {\begin{matrix}{\mathcal {D}}\phi \\\Lambda '\leq p\leq \Lambda \end{matrix}}\exp \left\right]}

Obviously,

Z = ∫ D ϕ p 2 ≤ Λ ′ 2 exp ⁡ [ − S Λ ′ [ ϕ ] ] {\displaystyle Z=\int {\begin{matrix}{\mathcal {D}}\phi \\p^{2}\leq \Lambda '^{2}\end{matrix}}\exp \left\right]}

In fact, this transformation is transitive. If you compute SΛ' from SΛ and then compute SΛ' ' from SΛ', this gives you the same Wilsonian action as computing SΛ' ' directly from SΛ.

Polchinski ERGE

This involves a smooth UV regulator cutoff.

Basically, the idea is an improvement over the Wilson ERGE. Instead of a sharp momentum cutoff, we use a smooth cutoff. Essentially, we suppress contributions from momenta greater than Λ heavily. The smoothness of the cutoff, however, allows us to derive a functional differential equation in the cutoff scale Λ. As in Wilson's approach, we have a different action functional for each cutoff energy scale Λ. Each of these actions are supposed to describe exactly the same model which means that their partition functionals have to match exactly.

In other words, (for a real scalar field; generalizations to other fields are obvious)

Z Λ [ J ] = ∫ D ϕ exp ⁡ ( − S Λ [ ϕ ] + J ⋅ ϕ ) = ∫ D ϕ exp ⁡ ( − 1 2 ϕ ⋅ R Λ ⋅ ϕ − S i n t Λ [ ϕ ] + J ⋅ ϕ ) {\displaystyle Z_{\Lambda }=\int {\mathcal {D}}\phi \exp \left(-S_{\Lambda }+J\cdot \phi \right)=\int {\mathcal {D}}\phi \exp \left(-{\frac {1}{2}}\phi \cdot R_{\Lambda }\cdot \phi -S_{int\Lambda }+J\cdot \phi \right)}

and ZΛ is really independent of Λ! We have used the condensed deWitt notation here. We have also split the bare action SΛ into a quadratic kinetic part and an interacting part Sint Λ. This split most certainly isn't clean. The "interacting" part can very well also contain quadratic kinetic terms. In fact, if there is any wave function renormalization, it most certainly will. This can be somewhat reduced by introducing field rescalings. RΛ is a function of the momentum p and the second term in the exponent is

1 2 ∫ d d p ( 2 π ) d ϕ ~ ∗ ( p ) R Λ ( p ) ϕ ~ ( p ) {\displaystyle {\frac {1}{2}}\int {\frac {d^{d}p}{(2\pi )^{d}}}{\tilde {\phi }}^{*}(p)R_{\Lambda }(p){\tilde {\phi }}(p)}

when expanded. When p ≪ Λ {\displaystyle p\ll \Lambda } , RΛ(p)/p^2 is essentially 1. When p ≫ Λ {\displaystyle p\gg \Lambda } , RΛ(p)/p^2 becomes very very huge and approaches infinity. RΛ(p)/p^2 is always greater than or equal to 1 and is smooth. Basically, what this does is to leave the fluctuations with momenta less than the cutoff 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

d d Λ Z Λ = 0 {\displaystyle {\frac {d}{d\Lambda }}Z_{\Lambda }=0}

can be satisfied by (but not only by)

d d Λ S i n t Λ = 1 2 δ S i n t Λ δ ϕ ⋅ ( d d Λ R Λ − 1 ) ⋅ δ S i n t Λ δ ϕ − 1 2 T r [ δ 2 S i n t Λ δ ϕ δ ϕ ⋅ R Λ − 1 ] {\displaystyle {\frac {d}{d\Lambda }}S_{int\Lambda }={\frac {1}{2}}{\frac {\delta S_{int\Lambda }}{\delta \phi }}\cdot \left({\frac {d}{d\Lambda }}R_{\Lambda }^{-1}\right)\cdot {\frac {\delta S_{int\Lambda }}{\delta \phi }}-{\frac {1}{2}}Tr\left}

Jacques Distler claimed without proof that this ERGE isn't correct nonperturbatively.

Effective average action ERGE

This involves a smooth IR regulator cutoff.

Basically, the idea is to take all fluctuations right up to a IR scale k into account and then applying mean field theory to all other fluctuations below that scale. As is well known from the study of critical phenomena, mean field theory can be completely way off. So, we'd expect that the effective average action will only be accurate for fluctuations with momenta larger than k. But the smaller k is, the more accurate the effective average action will be. By the same reasoning, the large k is, the closer the effective action will be to the "bare action". So, the effective average action interpolates between the "bare action" and the effective action.

Details

For a real scalar field, we add an IR cutoff

1 2 ∫ d d p ( 2 π ) d ϕ ~ ∗ ( p ) R k ( p ) ϕ ~ ( p ) {\displaystyle {\frac {1}{2}}\int {\frac {d^{d}p}{(2\pi )^{d}}}{\tilde {\phi }}^{*}(p)R_{k}(p){\tilde {\phi }}(p)}

to the action S where Rk is a function of both k and p such that for p ≫ k {\displaystyle p\gg k} , Rk(p) is very tiny and approaches 0 and for p ≪ k {\displaystyle p\ll k} , Failed to parse (unknown function "\gsim"): {\displaystyle R_k(p)\gsim k^2} . Rk 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. We will use the condensed deWitt notation

1 2 ϕ ⋅ R k ⋅ ϕ {\displaystyle {\frac {1}{2}}\phi \cdot R_{k}\cdot \phi }

for this IR regulator.

So,

exp ⁡ ( W k [ J ] ) = Z k [ J ] = ∫ D ϕ exp ⁡ ( − S [ ϕ ] − 1 2 ϕ ⋅ R k ⋅ ϕ + J ⋅ ϕ ) {\displaystyle \exp \left(W_{k}\right)=Z_{k}=\int {\mathcal {D}}\phi \exp \left(-S-{\frac {1}{2}}\phi \cdot R_{k}\cdot \phi +J\cdot \phi \right)}

where J is the source field. The Legendre transform of Wk ordinarily gives the effective action. However, the action that we started off with is really S+1/2 φ⋅Rk⋅φ and so, to get the effective average action, we subtract off 1/2 φ⋅Rk⋅φ. In other words,

ϕ [ J ; k ] = δ W k δ J [ J ] {\displaystyle \phi ={\frac {\delta W_{k}}{\delta J}}}

can be inverted to give Jk and we define the effective average action Γk as

Γ k [ ϕ ] ≡ ( − W [ J k [ ϕ ] ] + J k [ ϕ ] ⋅ ϕ ) − 1 2 ϕ ⋅ R k ⋅ ϕ {\displaystyle \Gamma _{k}\equiv \left(-W\left\right]+J_{k}\cdot \phi \right)-{\frac {1}{2}}\phi \cdot R_{k}\cdot \phi }

So,

d d k Γ k [ ϕ ] = − d d k W k [ J k [ ϕ ] ] − δ W k δ J ⋅ d d k J k [ ϕ ] + d d k J k [ ϕ ] ⋅ ϕ − 1 2 ϕ ⋅ d d k R k ⋅ ϕ = − d d k W k [ J k [ ϕ ] ] − 1 2 ϕ ⋅ d d k R k ⋅ ϕ = 1 2 ⟨ ϕ ⋅ d d k R k ⋅ ϕ ⟩ J k [ ϕ ] ; k − 1 2 ϕ ⋅ d d k R k ⋅ ϕ = 1 2 T r [ ( δ J k δ ϕ ) − 1 ⋅ d d k R k ] = 1 2 T r [ ( δ 2 Γ k δ ϕ δ ϕ + R k ) − 1 ⋅ d d k R k ] {\displaystyle {\frac {d}{dk}}\Gamma _{k}=-{\frac {d}{dk}}W_{k}]-{\frac {\delta W_{k}}{\delta J}}\cdot {\frac {d}{dk}}J_{k}+{\frac {d}{dk}}J_{k}\cdot \phi -{\frac {1}{2}}\phi \cdot {\frac {d}{dk}}R_{k}\cdot \phi =-{\frac {d}{dk}}W_{k}]-{\frac {1}{2}}\phi \cdot {\frac {d}{dk}}R_{k}\cdot \phi ={\frac {1}{2}}\left\langle \phi \cdot {\frac {d}{dk}}R_{k}\cdot \phi \right\rangle _{J_{k};k}-{\frac {1}{2}}\phi \cdot {\frac {d}{dk}}R_{k}\cdot \phi ={\frac {1}{2}}Tr\left={\frac {1}{2}}Tr\left}
d d k Γ k = 1 2 T r [ ( δ 2 Γ k δ ϕ δ ϕ + R k ) − 1 ⋅ d d k R k ] {\displaystyle {\frac {d}{dk}}\Gamma _{k}={\frac {1}{2}}Tr\left}

is the ERGE.

As there are infinitely many choices of Rk, there are also infinitely many different interpolating ERGEs.

We only looked at the case of a scalar field but the generalization to other fields like spinorial fields is obvious.

Differences

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 we neglect the IR contributions to the effective action whereas in the Polchinski ERGE, we fix the QFT once and for all but vary the "bare action" 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.

References

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