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The Renormalization Group Method An overview Florin Spineanu National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
in collaboration with Madalina Vlad
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The ubiquitous infinity When we want to calculate a physical quantity in a nonlinear model we often find an infinite value even if we know it should be finite. At a careful examination it results that this is due to the fact that we do not know to handle very small scales in the system. Or very high energies, as rare fluctuations. We would like to discard them but we know it is dangerous. The RNG just teaches us how to do it. When possible. • Latices (Ising) • particle mass in QM • self-interacting scalar fields
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The path-integral formalism for field theory
This is from van Baal. The scalar field 1 1 2 2 μ L = (∂μ φ) (∂ φ) − m φ − V [φ] − J (x) φ (x) 2 2 where V [φ] = =
g3 3 φ + ... 3! gl φl l! l≥3
We define the action functional and the partition function Z J, gl
1
1 ∂μ φ ∂ μ φ − m2 φ2 − V [φ] − Jφ i d4 x 2 2 1 1 2 2 μ
4 4 ∂μ φ ∂ φ − m φ − Jφ exp −i d xV [φ] D [φ] exp i d x 2 2 4 exp −i d xV [φ] Z2 [J]
= = =
D [φ] exp
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where Z2 [J]
1 1 (∂μ φ) (∂ μ φ) − m2 φ2 − Jφ D [φ] exp i d4 x 2 2 i d4 x d4 yJ (x) G (x − y) J (y) exp − 2
= =
We now replace in the expression of the potential exp −i d4 xV [φ] δ = exp −i d4 xV δ [−iJ (x)] ⎫ ⎧ l ⎬ ⎨ gl δ = exp −i d4 x ⎩ l! δ [−iJ (x)] ⎭ l≥3
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Introduce the Fourier transforms φ (x) J (x)
= =
1 (2π) 1 (2π)
4/2
4/2
d4 k exp (−ikx) φ (k) d4 k exp (−ikx) J (k)
Then the exponential of the potential V [φ] can be expressed in terms of derivatives to the Fourier components of the external current J (k) ⎫ ⎧ l⎬ ⎨ gl δ exp −i d4 x ⎩ l! δ [−iJ (x)] ⎭ l≥3 ⎫ ⎧ ⎞ ⎛ l l ⎬ ⎨ g 4 d k δ (j) l 4 (2π) δ ⎝ k(j) ⎠ = exp −i 4/2 ⎩ l! δ −iJ k(j) ⎭ j=1 j=1 (2π) l≥3 which shows that every vertex of order l will contribute with a δ
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Dirac function that ensures the conservation of the momenta k(1) + k(2) + ... + k(l) = 0 for the vertex of order l In the original expression of the exponential we have, for the vertex of order l, a power l of the functional derivative l δ δ [−iJ (x)] This is implicitely a product of l factors. This, after the Fourier transformation, is represented again by a product of l factors, this time the factors being functional derivatives to the Fourier components of the current
δ δ [−iJ (x)]
l
l d4 k(j)
δ → 4/2 δ −iJ k(j) j=1 (2π)
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In this process the factor 1 l! is absorbed in order to ensure the ennumeration of the permutations of the momenta k(j) . There arise integrations over the momenta k(j) . 4 But the integration over the space d x which is present in the first exponential leads to the Dirac δ function. On the other hand we have i exp − d4 x d4 yJ (x) G (x − y) J (y) 2 1 i 4 (−k) d k J (k) 2 J = exp − 2 k − m2 + iε The normalization is obtained taking Z [J = 0, gl = 0] = 1
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The definition GJ ≡ ln (Z [J, gl ]) This is a functional which is proved to be the sum over all connected diagrams. To prove that the functional GJ is a sum over all connected diagrams, van Baal uses a simplified formulation δ exp [Y (J)] GJ = ln exp X δ (−iJ (x)) and it is used the Campbell-Baker-Hausdorf formula 1
1
1
eA eB = eA+B+ 2 [A,B]+ 12 [A,[A,B]]+ 12 [B,[B,A]]+... here GJ = X + Y +
1 1 1 [X, Y ] + [X, [X, Y ]] + [Y, [Y, X]] + ... 2 12 12 F. Spineanu – Reading 2012 –
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This is considered as acting on the state 1. All components that commute will be eliminated in the calculation. All components that do not commute will be connected via the commutators. Calculation of the numerical coefficients of the terms (i.e. diagrams) representing multiplicities. For this a simplified model of the field theory is employed. It is a model in zero-dimensions. φ (x)
→ φ
D [φ]
→ dφ no
d4 x
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Z [J, g]
g 3 = dφ exp φM φ − φ − Jφ 3! 3 g i δ = exp −i exp − JM −1 J 3! δ (−iJ) 2
To calculate the average or the correlation of the product of functions φ’s, we will take the external current to zero, J →0 When we expand the two exponentials and apply each term of the first expansion on the terms of the second expansion, we will have contributions only when the number of derivations in the term of the first expansion is equal with the number of J’s in the term of the second expansion. Otherwise the terms where a J survives the derivation will be zero after taking J → 0. Or, if the number of J’s in a term of the second expansion is less than the number of derivations
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contained in a term of the first expansion, then the result will again be zero. Then
Z [J = 0] = 1 +
g 1 −i 2 3!
δ δ (−iJ)
3 2
i 1 − JM −1 J 3! 2
3 + ...
In the second term we have six operators of derivation acting on a product of six functions J’s. Other similar terms will result. They can be again written as an exponential 3 3 1 1 δ 2+3 2+3 2 −1 (i) Z [J = 0] = exp (−1) JM J 3·2 3 2+1 g δJ (−i) (2) (2) (3!)
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We have (−1)
5
(i)5 (−i)6
=
−1
=
(i)
and the coefficient is i and the result is Z [J = 0]
=
= i8 i3 = i3 = −i
g2 (2)4 (3!)3
=
11
2
3 +O g
5g exp i 24M 3 1 1 diag1 + diag2 + O g 3 exp 12 8
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Self-energy or renormalized mass due to self_interaction
Lectures, Pierre van Baal
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Eddy viscosity determined using the RNG approach The paper PHF00143 RNG Nagano. The Navier-Stokes equations 1 ∂p ∂Ui ∂ 2 Ui ∂Ui + Uj j = − + ν0 i ∂t ∂x ρ0 ∂x ∂x2j
(1)
∂Ui =0 ∂xi
(2)
ν0 ≡ molecular kinematic viscosity
(3)
and continuity
here The Fourier representation only in space of the averaged velocity and of the perturbation. U i (x, t) = U i (k, t) exp (ik · x) (4) kΛe
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ui (x, t) =
u (k, t) exp (ik · x)
(5)
kΛe
where the wavelength Λe is the limit of the spectrum (eddies) where the energy is conserved. The amount of energy in the small wavelength range is much smaller than the rest ∞ Λ0 Λe dkE (k) dkE (k) ≈ dkE (k) (6) Λe
0
Λe
The Kolmogorov spectrum E (k) = αε2/3 k −5/3 and the turbulent kinetic energy Λ0 3α 2/3 −2/3 K= ε Λe dkE (k) = 2 Λe
(7)
(8)
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wher α
≡
ε ≡
Kolmogorov constant
(9)
energy dissipation rate of K
The constant Λ0 represents a very high wavenumber (so very small space scale) Λ0 Λe (10) that marks the limit of the dissipation range of the turbulent kinetic energy K) 3/2 ε 3α (11) Λe = 3/2 2 K the turbulence Raynolds stress K2 Ret = ν0 ε
(12)
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The average 1 ∂p ∂ ∂U i ∂2U i ∂U i + Uj j = − + ν0 − j ui uj ∂t ∂x ρ0 ∂xi ∂x2j ∂x
(13)
1 ∂p ∂ ∂U i ∂ui ∂ 2 ui ∂ui + uj j + U j j = − + ν − (ui uj − ui uj ) 0 i j j j ∂t ∂x ∂x ρ0 ∂x ∂x ∂x ∂x (14) Now we go to the Fourier representation for the x variables. The equation for the perturbation ui is in Fourier space i ∂ 2 + ν0 k ui (k, t) = − Pimn (k) um (q, t) un (k − q, t) (15) ∂t 2 q −iPimn (k) U m (q, t) un (k − q,t) q
where Pimn (k) = km Pin (k) + kn Pim (k)
(16)
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Pij (k) = δij −
ki kj k2
(17)
The continuity equation ki ui (k, t) = 0
(18)
At this moment the allowed wavenumbers of the theory are between the limits Λe ≤ k < Λ0
(19)
the maximum wavenumber corresponds to the minimum spatial length admissible in the description, limited by the molecular processes. It can be estimated on the basis of the dissipation integral ∞ 2k 2 dk E (k) ν0 (20) ε= Λe
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where one inserts the Kolmogorov spectrum E (k) = αε2/3 k −5/3 and taking the upper limit Λ0 instead of ∞ we have 3/4 2 Λ0 ≈ kd 3α
(21)
(22)
where kd
≡ =
Kolmogorov dissipation scale 1/4 ε ν03
(23)
These approximations are valid when the energy-containing part of the spectrum and the region of the spectrum where the dissipation takes place are very far apart.
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A property of the projection operator Pij (k) results from taking the product of Eq.(15) by ki and summing, using also Eq. of continuity ki Pij (k) = 0 The Reynolds stress has the follwoing representation in Fourier space ui uj = Rij (k, t) exp (ik · x) kΛe
where Rij (k) =
ui (q, t) uj (k − q, t)
q
Application of the Renormalization Group Method. Define a new reference wavenumber Λ1 F. Spineanu – Reading 2012 –
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and separate the spectrum into two ranges: Λe
≤
k < Λ1
Λ1
(q, t) for Λ1 < q < Λ0 i
Since Λ0 is a cutoff required by the unavoidable existence of a minimum length in the system, the range Λ1 < q < Λ0 is a range F. Spineanu – Reading 2012 –
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close to the cutoff. The product of two fluctuating velocities summed over the whole spactrum appears fro example in the Reynolds stress ui (q, t) uj (k − q, t) q
=
< u< i (q, t) uj (k − q, t)
q
+
< u> i (q, t) uj (k − q, t)
q
+
> u< i (q, t) uj (k − q, t)
q
+
< u< i (q, t) uj (k − q, t)
q
Now we will average over the fluctuations belonging to the band close
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to the cutoff c It is a formal operator acting on a statistical ensemble of realizations of the fluctuating field. " ! # $ < < < < ui (q, t) uj (k − q, t) = ui (q, t) uj (k − q, t) c q
q
c
=
< u< (q, t) u i j (k − q, t)
q
since the velocities in this expression are not dependent on the part of the spectrum between Λ1 and the cutoff Λ0 . The average on the terms involving velocities belonging to both ranges are not immediate since a velocity of the spectral band still depends on the velocities from the large spectrum, due to the
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Navier-Stokes equations. Then one introduces another velocity > > u> (q, t) = v (q, t) + Δ i i i (q, t)
where vi> is taken independent on the velocities u< i and the average is zero. # # $ $ < > < > ui (q, t) uj (k − q, t) c = vi (q, t) c uj (k − q, t) (this term is zero) q
q
+
#
Δ> i
$ < (q, t) c uj (k − q, t)
q
=
# $ < > Δi (q, t) c uj (k − q, t) q
=
O (λm ) ≈ 0 , m > 1
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and # q
u< i
(q, t) u> j
$ (k − q, t) c
=
u< i
q
+
u< i
q
=
q
=
# > $ (q, t) vj (k − q, t) c (this term is zero)
u< i
# > $ (q, t) Δj (k − q, t) c
# > $ (q, t) Δj (k − q, t) c
O (λm ) ≈ 0 , m > 1
Now we have to take the average # $ > > ui (q) uj (k − q) c q
We will construct an equation for this correlation.
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For this we will write an equation for > u> i (q) uj (k − q)
by 1. writting the equation for u> i (q) and multiplying by the function u> j (k − q). > 2. writting the equation for u> j (k − q) and multiplying by ui (q).
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Then these equations are added ∂ > + ν0 q 2 + ν0 |k − q|2 u> i (q) uj (k − q) ∂t > U m (r, t) un (k − q − r, t) u> = −iPjmn (k − q) i (q, t) > (q) −iPimn
r
U m (r, t) un (q − r, t) u> j (k − q, t)
r
i > − Pimn (q) um (r, t) un (q − r, t) u> j (k − q, t) 2 rm i > − Pjmn (k − q) um (r, t) un (k − q − r, t) u> i (q, t) 2 rm Now, this equation can formally be solved. Our intention is to calculate # $ > > ui (q) uj (k − q) c q
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This means
=
> (k − q) ui (q) u> j c q −1
∂ > 2 2 −i (k − q) U m (r, t) un (k − q − r, t) u> (q, t) Pjmn + ν0 q + ν0 |k − q| i c ∂t q r > +Pimn (q) U m (r, t) un (q − r, t) u> (k − q, t) j c 1 > > + Pimn (q) um (r, t) un (q − r, t) uj (k − q, t) c 2 1 > > + Pjmn (k − q) um (r, t) un (k − q − r, t) ui (q, t) c 2
This formula is replaced by an approximative one where the correlations of the full velocitties ui are replaced by averages of the essential velocities vi plus the averages over the difference Δi .
> ui (q) u> (k − q) j c q
=
−i ×
q
r
∂ ∂t
+ ν0 q 2 + ν0 |k − q|2
−1
> > > Pjmn (k − q) U m (r, t) vn (k − q − r, t) vi (q, t)
> > > Pimn (q) U m (r, t) vn (q − r, t) vj (k − q, t) c
c
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From here we obtain # q
=
−i
u> i
(q, t) u> j
(k − q, t)
2π 3 q
L
$ c
1 ν0 q2 + ν0 |k − q|
2
% > > × Pjmn (k − q) Pin (q) Q> v (q) > > (q) Pjn +Pimn
(k −
q) Q> v
(|k − q|)
&
×U m (k, t) where the operator ∂/∂t has been neglected. This is the Markovian approximation. Now, since the band Λ1 ≤ k < Λ0 F. Spineanu – Reading 2012 –
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is very narrow, we can approximate |k − q| ∼ |q| = q > Q> v (|k − q|) ∼ Qv (q)
Then
=
> > ui (q, t) uj (k − q, t) c q
2π 3 1 > > > > > Pjmn (k − q) Pin (q) + Pimn (q) Pjn (k − q) Qv (q) U m (k, t) −i 2 L 2ν0 q q
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The projection operators can be transformed > > > > Pimn (q) Pjn (k − q) + Pjmn (k − q) Pin (q)
=
kj δim + ki δjm k j qi qm k i qj qm k n qj qn k n qi qn − − − δ − δjm im 2 2 2 2 q q q q k n qi qj qm qn +2 q4
The renormalized form of the Reynolds stress so that the effects of all the eliminated components u> in the spectral band Λ1 < k < Λ0
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are replaced by a viscosity coefficient ν1 calculated as ui (q) uj (k − q) c q
≈
< u< i (q) uj (k − q) −
q
−ν1
'
( iki U j (k, t) + ikj Ui (k, t)
where 7 ν1 ≡ Δν0 = 30ν0
Λ0 Λ1
E (q) dq q
This is the first change of the visocity coefficient due to the elimination of a band of spectral fluctuations close to the cutoff. We now continue this operation, introducing another limit wavelength, defining another spectral band that we want to suppress.
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Take
⎧ ⎨ u< (q, t) for Λ < q < Λ e 2 i ui (q, t) = ⎩ u> (q, t) for Λ2 < q < Λ1 i
Similar to the calculation presented above we get ui (q, t) uj (k − q, t) c q
