Reformulation of the Photon Equations of Motion

We will now introduce a reformulation of the Eoms which is of little practical use if one is interested in solving the full Eoms numerically, but which is very instructive for learning about an interesting feature of theEoms which we will discuss in great detail later on.

We start by introducing an auxiliary field, the so-calledNakanishi–Lautrup (Nl) fieldB [Nak66,Lau66] which is commonly used in the path integral quantization of gauge theories in order to close theBrstalgebra. Only the gauge part of the classical action is modified, and the action involving the auxiliary field now reads:

SNL[A, B] =

Z

x −1

4FµνF^µν +B ∂^µAµ+ξ 2B²

!

=Sg[A] +SB[A, B] (3.52)

with

SB[A, B] =

Z

x B ∂^µAµ+ ξ 2B²

!

= 1 2ξ

Z

x

(∂^µAµ+ξ B)²−(∂^µAµ)²

=Sgf[A] + 1 2ξ

Z

x(∂^µAµ+ξ B)², (3.53)

where we have completed the square in the second line.³⁴ Note that there is no kinetic term forB, and since it appears only quadratically in the action, it can easily be integrated out in the path integral.

It is now convenient to introduce the composite field(A^e_m) = (A_µ, B)(withm= 0, . . . ,4, so that A^eµ=Aµ and A^e4 =B). Then the action (3.52) can be written equivalently as

SNL[A] =^e i 2

Z

x,y

Aem(x)D^f−1₀ ^mn(x, y)A^en(y) (3.54) with the inverse free propagator

iD^f−1₀ ^mn(x, y) = δ²SNL[A]^e

δA^em(x)δA^en(y), (3.55) so that

iD^f−1₀ ^µν(x, y) =g^µνx−∂_x^µ∂_x^νδ⁴(x−y), iD^f−1₀ ^µ4(x, y) =−∂_x^µδ⁴(x−y),

iD^f−1₀ ^4µ(x, y) =∂_x^µδ⁴(x−y), iD^f−1₀ ⁴⁴(x, y) =ξ δ⁴(x−y)

(3.56)

or

iD^f−1₀ ^mn(x, y) =

δ_µ^mδ_νⁿ(g^µνx−∂_x^µ∂_x^ν) + (δ₄^mδⁿ_µ−δ^m_µδ₄ⁿ)∂_x^µ+δ₄^mδ₄ⁿξ

δ⁴(x−y). (3.57) Since the free inverse propagators are translation invariant, we can Fourier transform them

34If we quantize the theory by establishing a path integral, the introduction of the Nl field can be interpreted as a Hubbard–Stratonovich transformation since

exp

−1

2ξ(∂^µAµ)²

=N Z

DB exp ξ

2B²+B ∂^µAµ

,

so that

N Z

DB e^iSB[A,B]= e^iSgf[A] .

to obtain

Df⁻¹₀ ^µν(p) = i(g^µνp²−p^µp^ν),

Df₀^−1µ4(p) = p^µ,

Df₀^−14µ(p) = −p^µ,

Df⁻¹₀ ⁴⁴(p) = −iξ .

(3.58)

Employing the formula³⁵

det A B C D

!

= det(D) det(A−BD⁻¹C) for the determinant of a block matrix, we obtain:

deti(D^f−1₀ )^mn(p)

=−det(D^f−1₀ )^µν(p)−(D^f₀⁻¹)^µ4(p)(D^f₀⁻¹)⁴⁴(p)⁻¹(D^f−1₀ )^4ν(p)det(D^f₀⁻¹)⁴⁴(p)

=−ξdet g^µνp²− 1−1 ξ

!

p^µp^ν

!

=p², (3.59)

where in the last step we have employed the matrix determinant lemma. Note that the determinant of iD^f−1₀ is nonvanishing, so that D^f₀⁻¹ is invertible (in contrast to D₀⁻¹), and independent of ξ. Therefore, the free propagator exists for each choice of gauge. We find:

Df0µν(p) =D0µν(p) = − i p²

"

gµν −(1−ξ)pµpν

p²

#

, Df0µ4(p) =D_µ^(AB)(p) =−pµ

p² , Df_04µ(p) =D₀^(BA)_µ (p) = p_µ

p² , Df₀₄₄(p) =D₀^(BB)(p) = 0.

(3.60)

Defining

Dfmn(x, y) = hTA^em(x)A^en(y)i, (3.61) we have

Dfµν(x, y) = hTA^eµ(x)A^eν(y)i=hTAµ(x)Aν(y)i=D^(AA)_µν (x, y) = Dµν(x, y), Dfµ4(x, y) = hTA^eµ(x)A^e4(y)i=hTAµ(x)B(y)i=D^(AB)_µ (x, y),

Df4µ(x, y) = hTA^e4(x)A^eµ(y)i=hTB(x)Aµ(y)i=D^(BA)_µ (x, y), Df₄₄(x, y) = hTA^e₄(x)A^e₄(y)i=hTB(x)B(y)i=D^(BB)(x, y),

(3.62)

35SinceDis just a number (i. e. a one-by-one matrix), the formula can further be simplified to det(AD− BC).

or

Dfmn(x, y) = δ_m^µδ_n^νDµν(x, y) +δ_m^µδ_n⁴D^(AB)_µ (x, y) +δ_m⁴δ_n^µD^(BA)_µ (x, y) +δ_m⁴δ⁴_nD^(BB)(x, y). (3.63) The first correlation function is just the usual photon propagator, while the others contain at least one Nlfield operator. The 2Pieffective action then reads:³⁶

Γ2PI[D, S] =^f i

2Tr lnD^f−1+ i

2TrD^f−1₀ D^f−i Tr lnS⁻¹ −i TrS₀⁻¹S+ Γ2[S,D]^f . (3.64) Here we have assumed that the expectation values of the photon and fermion fields vanish.

We further assume that the expectation value of the Nl field vanishes; in fact, in the operator formalism this is a consequence of a certain physicality condition (see App. A).

From the stationarity condition of the photon part, δΓ_2PI[D, S]^f

δD^fmn(x, y) = 0, (3.65)

it follows that theEom is given by:

Df^−1mn(x, y) =D^f−1₀ ^mn(x, y)−Π^emn(x, y) (3.66) with the self-energy

Πe^mn(x, y) = 2 i δΓ2PI[D, S]^f

δD^fmn(x, y). (3.67)

Now, the important observation is that the 2Pi part of the 2Pi effective action does not depend on any correlators involving the Nl field, i. e. instead of on the full D, it only^f depends on the pure photon correlator D. The reason is simple: The only way to include a correlator involving an Nl field into a given diagram is to replace a photon line by a photon line which is connected to a mixed correlator; schematically

D=D^(AA) → D^(AA)D^(AB)M(D^(AA), D^(AB), D^(BA), D^(BB), S)D^(BA)D^(AA)

=DD^(AB)M(D, S)D^{f (BA)}D ,

where M is a Lorentz scalar potentially depending on all possible propagators. Such a diagram, however, can never be 2Pi, since cutting the two photon lines leaves us with a diagram D^(AB)M(D, S)D^{f (BA)} which is disconnected from the rest. We will see a concrete example in Sec. 6.1 when we consider a finite truncation of the 2Pi effective action.

36Note that the matrices consisting of theAA-,AB-,BA- andBB-components are not block diagonal, so that the terms in the one-loop part of the 2Pieffective action do not factorize.

Therefore, Γ2PI may only depend on D, not on (all components of) D. Then, in terms^f of its components, we have

Πe^µν(x, y) = 2 i δΓ2PI[D, S]^f

δD^fµν(x, y) = 2 i δΓ2PI[D, S]^f

δDµν(x, y) = Π^µν(x, y), Πe^µ4(x, y) = 2 i δΓ2PI[D, S]^f

δD^fµ4(x, y) = 2 i δΓ2PI[D, S]^f

δD^(AB)µ (x, y) = Π^(AB)µ(x, y) = 0, Πe^4µ(x, y) = 2 i δΓ2PI[D, S]^f

δD^f_4µ(x, y) = 2 i δΓ2PI[D, S]^f

δD^(BA)µ (x, y) = Π^(BA)µ(x, y) = 0, Πe⁴⁴(x, y) = 2 i δΓ2PI[D, S]^f

δD^f₄₄(x, y) = 2 i δΓ2PI[D, S]^f

δD^(BB)(x, y) = Π^(BB)(x, y) = 0,

(3.68)

or

Πe^mn(x, y) = δ_µ^mδⁿ_νΠ^µν(x, y). (3.69) The Eom (3.66) can then be rewritten as

Z

z

Df₀^−1mk(x, z)D^f_kn(z, y) =δ_n^mδ⁴(x−y) +

Z

z

Πe^mk(x, z)D^f_kn(z, y), (3.70) so that

Z

z

Df⁻¹₀ ^µλ(x, z)D^fλν(z, y) +D^f0−1µ4

(x, z)D^f4ν(z, y)

=δ_ν^µδ⁴(x−y) +

Z

z

Πe^µλ(x, z)D^fλν(z, y), (3.71a)

Z

z

fD₀^−1µν(x, z)D^fν4(z, y) +D^f0−1µ4

(x, z)D^f44(z, y)

=

Z

z

Πe^µν(x, z)D^fν4(z, y), (3.71b)

Z

z

Df0−14µ

(x, z)D^fµν(z, y) +D^f0−144

(x, z)D^f4ν(z, y)

= 0, (3.71c)

Z

z

Df0−14µ

(x, z)D^fµ4(z, y) +D^f0−144

(x, z)D^f44(z, y)

=δ⁴(x−y). (3.71d) With the free inverse propagators (3.56), we then have:

g^µλx−∂_x^µ∂_x^λDλν(x, y)−∂_x^µD_ν^(BA)(x, y) = iδ^µ_νδ⁴(x−y) + i

Z

zΠ^µλ(x, z)Dλν(z, y), (3.72a)

g^µνx−∂_x^µ∂^ν_xD^(AB)_ν (x, y)−∂^µ_xD^(BB)(x, y) = i

Z

zΠ^µν(x, z)D_ν^(AB)(z, y), (3.72b)

∂_x^µDµν(x, y) +ξ D_ν^(BA)(x, y) = 0, (3.72c)

∂_x^µD_µ^(AB)(x, y) +ξ D^(BB)(x, y) = iδ⁴(x−y). (3.72d) Plugging (3.72c) into (3.72a), we obtain:

xDµν(x, y)−(1−ξ)∂xµD^(BA)_ν (x, y) = igµνδ⁴(x−y) + i

Z

zΠµλ(x, z)Dλν(z, y), (3.73)

Further, applying ∂xµ to (3.72a), we have:³⁷ xD^(BA)_ν (x, y) =−i∂xνδ⁴(x−y)−i

Z

z∂xµΠ^µλ(x, z)Dλν(z, y). (3.74) Decomposing these equations in their spectral and statistical components, we obtain:

xρ^(g)_µν(x, y)−(1−ξ)∂xµρ^(BA)_ν (x, y) =

Z _x⁰

y⁰ dzΠ(ρ)µλ(x, z)ρ^(g)_λν(z, y), (3.75a) xF_µν^(g)(x, y)−(1−ξ)∂xµF_ν^(BA)(x, y) =

Z _x⁰

0 dzΠ(ρ)µλ(x, z)F_λν^(g)(z, y)

−

Z _y⁰

0 dzΠ(F)µλ(x, z)ρ^(g)_λν(z, y), (3.75b) and

xρ^(BA)_ν (x, y) =−

Z _x⁰

y⁰ dz ∂xµΠ^µλ_(ρ)(x, z)ρ^(g)_λν(z, y), (3.76a) xF_ν^(BA)(x, y) =−

Z _x⁰

0 dz ∂xµΠ^µλ_(ρ)(x, z)F_λν^(g)(z, y) +

Z _y⁰

0 dz ∂xµΠ^µλ_(F)(x, z)ρ^(g)_λν(z, y). (3.76b) On first sight, it looks as if the Eoms for ρ^(BA)_µ and F_µ^(BA) are free, since only the longitu-dinal part ∂_xµΠ^µν(x, y) of the photon self-energy enters the memory integrals, and that is guaranteed to vanish by the Ward identities. If this were the case, one could easily solve

37Similarly, plugging (3.72d) into (3.72b), we obtain:

xD^(AB)_µ (x, y)−(1−ξ)∂xµD^(BB)(x, y) = i∂xµδ⁴(x−y) + i Z

z

Πµν(x, z)D^(AB)_ν (z, y), while applying∂xµ to (3.72b), we have:

xD^(BB)(x, y) =−i Z

z

∂xµΠ^µν(x, z)D^(AB)_ν (z, y).

Various other identities can be derived as well. For instance, combining Eqs. (3.72c) and (3.72d), we obtain

∂_x^µ∂_y^νDµν(x, y) +ξ

iδ⁴(x−y)−ξ D^(BB)(x, y)

= 0.

In the exact theory, whereD^(BB) satisfies a freeEom, which, together with its initial conditions, implies that it vanishes identically, this identity reduces to

∂_x^µ∂_y^νDµν(x, y) =−iξ δ⁴(x−y),

which is a well-known expression of the Ward identity stating that the longitudinal part of the photon propagator does not get modified by quantum fluctuations.

Finally note that from Eq. (3.72c), it follows that∂_x^µ∂_x^λDλν(x, y) +ξ ∂_x^µDν^(BA)(x, y) = 0, so that

∂_x^µD^(BA)_ν (x, y) =−1

ξ∂_x^µ∂_x^λDλν(x, y). Plugging this into Eq. (3.72a), we get back the originalEoms.

their Eoms analytically (given their initial conditions which will be provided in Sec. 3.4) and plug the results back into the photonEoms (3.75), thereby getting rid of any reference to the auxiliary field correlators altogether.

For a finitely truncated 2Pi effective action, however, it turns out that photon self-energies as defined previously are not purely transverse, see Chap. 5.³⁸ Therefore, the Eoms for the auxiliary correlators cannot easily be solved analytically, and instead of 32 Eoms, we now have 40Eoms. As already stated at the beginning of this section, however, the aim of this reformulation of the photon Eoms is not to make the equations more tractable from a practical point of view, but to understand their structure better. After we have provided the initial conditions needed to solve the Eoms in the next section, we will solve the free photon Eoms (for which the photon self-energy vanishes identically so that the Eoms of the auxiliary field correlators are in fact free) by employing the reformulation presented in this section.

Im Dokument Nonequilibrium Formulation of Abelian Gauge Theories (Seite 67-73)