lecture_iii_notes.pdf

8/8/2019 Lecture_III_notes.pdf

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Semi-Classical and Functional Methods: Lecture III notes

Subodh P. Patil∗ and Jaspreet Sandhu(Dated: November 27 2014)

We begin this lecture by consolidating on a few concepts discussed in the previous

lecture (and lecture notes). We begin with a discussion of the imaginary part of

the effective action (which we computed via the heat kernel in the last set of notes

to calculate particle production). After this, we close the circle on the abstract

definition of the effective action and the concrete implementation of the mean field

approximation (at the level of the equations of motion), the Coleman-Weinberg

potential and corrections to it from processes such as particle production. We then

move on to a discussion of dimensional regularization (and why only logarithmic

divergences are physical) as well as a discussion of the physical interpretation of UV

and IR divergences.

I. THE IMAGINARY PART OF THE EFFECTIVE ACTION

Reconsider the action of a scalar field coupled to an external source

S = S [φ] +

d4xjφ, −H I

(1)

where we consider the unperturbed system to evolve with the free Hamiltonian H 0 con-

structed from S 0, and treat the external source as an interaction. Imagine that we turn onthe source at some finite time, and subsequently switch it off at some later time adiabati-cally1. In this case, the vacuum in the asymptotic past and future relate to the vacuum atsome intermediate time, say t = 0 as

|0−∞J = T e− i 0

−∞ H I |0

|0+∞J = T e+ i ∞

0 H I |0

(2)

So that the overlap between these two states is given by

0+|0−J = 0|T e− i

∞−∞ H I |0 (3)∗Electronic address: [email protected] Adiabatic evolution means that any state initially corresponding to a particular energy eigenstate of H 0

remains in the corresponding eigenstate of H 0 + H I for all times, i.e. it does not make any transitions

over the course of the source turning on and off.


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Upon taking repeated functional derivatives of the above quantity with respect to the source,we find that

δ n0+|0−J δj (x1)...δj(xn)

=

i

n0|T φ(x1)...φ(xn)|0

= i

nGn(x1,...,xn)

(4)

so that the overlap is nothing other than the generating functional Z [ j] = ei

W [ j], which wefurther decompose as

0+|0−J = e iW [ j] ≡ ei

Re(W )+iIm(W )

(5)

Therefore the vacuum transition amplitude from the asymptotic past to the future is givenby

|0+|0−

J

|2 = e−2Im(W ) (6)

which differs from unity only if there is an imaginary component to the effective action.Given that the state began in the infinite past in the vacuum at that time, this could onlymean that time evolution evolves this state into states that are outside of the Hilbert spaceof our theory2, normally indicating that the φ quanta are being converted (or decaying) intoquanta of other degrees of freedom, possibly even degrees of freedom we had integrated outto obtain the effective action in the first place. Assuming that Im(W ) is small, we have

e−2Im(W ) ≈ 1 − 2Im(W ) = 1− P (7)where P/V is the particle production probability per unit space-time. In the appendix of lecture notes II, we compute the imaginary part of the effective action in the context of

inflationary cosmology, where the inflaton η couples to a heavy field χ which we nominallyintegrate out even though it becomes transiently light at a fixed value of η = η∗. The factthat χ quanta are produced when η = η∗ manifests as an imaginary contribution to theeffective action, computed using heat kernel techniques introduced in the previous lectures.It is also instructive to consider this same example in a more general form, and show howthis relates to the so-called mean field approximation at the level of the equations of motionand how one is essentially calculating corrections to the Coleman-Weinberg potential fromparticle production.

II. THE EFFECTIVE ACTION REDONE

A prototypical problem of interest, is when one has a light field η that couples to somemassive field χ which we integrate out. For illustrative purposes, we consider an action thatis at most quadratic in χ:

S =

√ −g

1

2ηη − V (η)

+

√ −g

1

2χχ− M

2χ(η)

2 χ2

(8)

2 Any imaginary component to the action cannot result in unitary evolution, i.e. probability is not being

conserved, so states must be evolving into states beyond the Hilbert space of our theory.


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where in general we allow for the mass of the χ field to depend on η (as in the examplediscussed in the appendix of lecture II). Defining the first contribution to the action aboveas S [η], we see that the effective action W [η] is obtained by functionally integrating over χis given by

e

i

W [η]

= e

i

S [η] Dχ e− i2 χ[−+M 2χ(η)]χ :=Z χ[η]

≡ ei

S [η]

det(− + M 2χ[η])

−1/2

(9)

So that formally, the effective action is

W [η] = S [η]− i ln Z χ[η] (10)The effective equations of motion for η are obtained by the functional variation

δW

δη = 0 (11)

implying the equations of motion

η − V ,η = M 2

[η]2 χ2η (12)

where

χ2η := Dχ2e− i2 χ[−+M 2χ(η)]χ De− i2 χ[−+M 2χ(η)]χ . (13)

Our goal will be to calculate this by more direct means and compare the outcome to theresults of our previous lectures. We begin by noting that if |0− is the vacuum of χ field inthe asymptotic past and uk, u

∗k are the associated mode functions, then at most, the effect

of time evolution would be to effect a Bogoliubov transformation to the mode functions:

vk

= αk

uk

+ β k

u∗k

(14)

v∗k = α∗kuk + β

∗ku

∗k.

Were the original mode functions uk, u∗k associated with the ladder operators ak, a

†k respec-

tively, then the ladder operators associated with vk, v∗k, denoted bk, b

†k, would be given by

the corresponding transformation

bk = αkak − β ∗ka†k (15)b†k = −β ∗kak + αka†k,

which the requirement of canonical normalization of the transformed mode functions imposesthe Wronksian condition

α∗kαk − β

∗kβ k = 1 (16)

so that we can parametrize αk = coshθk, β k = e−iδksinhθk. For transformations which only

rotates modes up to some fixed momentum scale, the unitary transformation 3 that formally

3 If the transformation angles were non-zero for arbitrarily large momenta (i.e. say αk and β k were constant

∀k), then famously, the Hilbert spaces spanned by ak, a†k and bk, b†k would be unitarily inequivalent. Thisis one of the most important ways that quantum mechanically, a finite number of degrees of freedom

is qualitatively different from having an infinite number of degrees of freedom, and is the mathematical

origin of spontaneous symmetry breaking– a phenomenon that does not occur for a system with a finite

number of degrees of freedom.


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effects this is given by

U (t,−∞) = e− 12

[Θha2

k−Θ∗

ka2

k ]d3k

Θk = θkeiδk

(17)

withαk = coshθk

β k = e−iδksinhθk

(18)

so that one can straightforwardly compute

χ2η = 0−|U (θ)χ2U (θ)|0−

= 1

(2π)3

d3k

2ωk[1 + 2|αkβ k|cos(2δ k) + 2|β k|2].

(19)

We see that we have compartmentalized the entire dependence of M 2

χ(η) on the dynamicsof η into the Bogoliubov coefficients αk, β k, which implicitly depends on the field profile forη. In the limit where one can neglect the backreaction of the produced χ particles on the dynamics of η, which we can take as some classical external background field (the mean field approximation), then one can avail of various techniques to compute αk, β k precisely as we implicitly did with the heat kernel in the appendix of the notes to lecture II. In the limit where this approximation is not valid, one must perturbatively solve for the dynamics of ηitself, and consequently the mode functions (and by extension, the Green’s functions) of ηand χ iteratively. This is essentially the content of the so called Schwinger-Dyson equations.

We note that in the absence of any χ particle production (i.e. the evolution stays adiabaticthroughout the dynamics of η), then αk, β k

≡0, so that we are only left with the first term in

the above so that the equation of motion (12) becomes (after discarding power law divergentterms cf. section III):

η − V ,η = M 2(η)

4(2π)3

Λ2 d3k√ k2 + M 2

= M 2

(η)

32π2 M 2(η)log

M 2

Λ2

≡ δV CW [η]δη

.

(20)

Therefore we see how (19) contains (through αk, β k) the non-adiabatic corrections to theusual Coleman-Weinberg contribution to the effective equations of motion which can bethought of as due to particle production. As we have shown in the appendix of lecture II, incertain cases (where we can neglect the backreaction onto the dynamics of the backgroundη solution) this can be calculated using heat kernel techniques.

III. DIMENSIONAL REGULARIZATION

We now finally address a point that we have been skirting so far in our lectures– thatof regularizing divergent integrals properly. So far in the interests of making connection


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with Wilson’s initial idea of integrating out modes above a particular scale, we have stuckto regulating our integrals by introducing a hard cutoff in momentum space. Althoughthis allows us to arrive at the answers we’ve been dealing in rather quickly, there are severalreasons one ought to be suspicious of this method in both a general context and in the contextof gravity (or more generally, in any theories with a particular set of low energy symmetries,

e.g. gauge symmetries). Specifically, the low energy effective expansion that is our EFTmust be consistent with symmetries we demand of the theory. Introducing regularizationschemes that violate these symmetries will make the resulting amplitudes (and consequently,the resulting counter terms needed to subtract off the relevant divergences) not respect thesesymmetries, in conflict with the symmetry requirements of the low energy expansion. Forexample, using cutoff renormalization in Q.E.D. naively requires counter terms that givesthe photon a mass.

Furthermore we have seen that in regulating certain integrals with a hard momentumcutoff, we obtain power law divergences. These divergences need to be subtracted off toobtain a finite answer. Once this is done, there is no residual reference to the scale at whichthis has been performed– power law divergences in some sense are a legitimate artefact

of the fact that our short distance description of the theory is incomplete. This is notso for any logarithmically divergent contributions since the logarithm of any dimensionfulquantity needs a corresponding dimensionful quantity to render the argument dimensionless.This quantity (call it µ) can be thought of as the renormalization scale, and tracks thescale dependence of the finite part of the couplings which is the only physically relevantconsequence of renormalization group flow.

Therefore as a regularization scheme, we find dimensional regularization4 to be par-ticularly useful in that it eliminates all power law divergences, preserves diffeomorphisminvariance, and moreover is particularly suited to the so-called minimal subtraction schemethat is widespread in the wider literature. For further evidence that dimensional regular-

ization captures the physically relevant divergences of the theory5

, reference [1] has a nicediscussion of the meaning of hierarchies and naturalness issues in terms of sensitivity toinitial conditions in renormalization group flow– where sensitivity to the UV completion of the theory manifests as a quadratic dependence on the heavy mass scales defining this UVtheory– something which dimensional regularization captures accurately.

We begin our introduction to dimensional regularization by first considering the proto-typical loop integral in λφ4 theory

I = −λ

d4 p

(2π)41

p2 + m2 (21)

Exploiting the so called Schwinger ‘proper time’ parametrisation [2]:

1

p2 + m2 =

∞0

dτ e−τ [ p2+m2], (22)

4 Where integrals are regulated by considering them in arbitrary spacetime dimensions where they converge,

and then analytically continuing the answers to D = 4.5 i.e. one may be suspicious of this method as it appears to rephrase the hierarchy problem and the

cosmological constant problem in terms to sensitivity to the highest masses one has integrate out rather

than the naive cutoff that defines where new physics is expected to enter.


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or more generally1

( p2 + m2)a =

1

Γ(a)

∞0

dτ τ a−1 e−τ [ p2+m2] (23)

we can consider the loop integral

dD p(2π)D

1( p2 + m2)a

= 1Γ(a)

∞0

dτ e−τ m2

τ 1−a

dD p(2π)D

e−τ p2

dp2π

e−τp2

D

= 1

Γ(a)

∞0

dτ e−τ m2

τ 1−a

1

4πτ

D/2

= 1

Γ(a)(4π)D/2

∞0

dτ e−τ m2

τ 1−a+D/2

(24)

Presuming that a and D are such that this integral converges we have

1

Γ(a)(4π)D/2

∞0

dτ e−τ m2

τ 1−a+D/2 =

Γ

a− D2

Γ(a)(4π)D/2

m2[D2 −a]. (25)

The integral converges for a − 1 − D2

> −1, or if 2a > D. By defining this integral overall integer values where it converges and analytically continuing elsewhere on the complexplane, we can obtain a finite expression for arbitrary D. This is the crux of dimensionalregularisation.

We recall the expression for the leading terms in the curvature expansion for the effectiveaction obtained by integrating out a massive scalar field (cf. eq. (37) in lecture II notes):

W = 1

32π2

∞0

e−m2s

s3 a = 0, D = 4

+ R

16π21

12

∞0

e−m2s

s2 a = 1, D = 4

(26)

where we realize that the integrals left to perform are of the form (25) with the correspondingvalues for a in D = 4 indicated above. Therefore by defining

ω := D

2

(27)

we find

W = Γ(−ω)2(4π)ω

m2[ω] + R

(4π)ω1

12Γ(1 − ω)m2[ω−1] (28)

Since we are interested in going to the limit D → 4 ≡ ω → 2, we find that the terms abovecan be expanded as

limω→2

Γ(−ω)(4π)2

m2[ω] = −m4

32π2(ω − 2) m2[ω] +

m4

32π2(ω − 2)

3− 2γ − 2log m

2

µ2

(29)


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limω→2

Γ(1 − ω)(4π)2

m2[ω−1] = −m2

32π2(ω − 2)m2[ω] +

m2

16π2

− 1 + γ + log m

2

µ2

(30)

Where the quantity µ is required to be introduced to keep the argument of the logarithmdimensionless. Therefore, were we to begin with the ‘bare’ action

S = 1

16π2GB

√ −gR − √ −gΛB + 12

√ −gφ[ + m2]φ (31)and integrating out φ, we would have obtained the effective action

W = 1

16π2GR

√ −gR − √ −gΛR (32)where through (29) and (30) we have the relations between the bare and the renormalizedquantities

1

GR(µ)

= 1

GB

+ m2

12

1

ω − 2 +

m2

12 − 1 + γ + log m2

µ2 (33)

ΛR(µ) = ΛB − m4

64π2(ω − 2) + m4

64π2

3

2 − γ − log m

2

µ2

(34)

By absorbing the ω−2 poles into the bare quantities (equivalent to adding counterterms withcoefficients to cancel off these divergences), and performing a gravitational experiment withcharacteristic momentum transfer µ to determine the couplings GR(µ), ΛR(µ), we will havefixed the finite part of the bare quantities to have renormalized the gravitational effectiveaction to one loop. Some further remarks about dimensionally regularized integrals are inorder. Consider the following integral and the subsequent manipulations:

dD p(2π)D

m2

p2( p2 + m2) =

dD p2πD

1 p2 − 1

( p2 + m2)

= m2

dD p

(2π)D

∞0

dτ 1dτ 2e−(τ 1+τ 2) p2−τ 2m2

= m2

(4π)D/2

∞0

dτ 1dτ 2(τ 1 + τ 2)−D/2e−τ 2m

2

= m2

(4π)D/2

∞0

dτ 2

∞τ 2

dτ −D/212 e

−τ 2m2

= m2

(4π)D/2 ∞

0

dτ 2−τ −D

2 +1

2 e−τ 2m2

1− D2

= −(m2

)

D2 −1

(4π)D/2 Γ(1 − D

2 )

(35)

Compare this to (24) for the case a = 1

−

dD p

(2π)D1

p2 + m2 = −(m

2)D2 −1

(4π)D2

Γ(1 − D2

) (36)

from which it appears to follow that dimensional regularization implies

dD p

(2π)D1

p2 ≡ 0 (37)


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converting the integrand above into polar coordinates, this implies

S D(2π)D

∞0

dkkD−3 = 0 (38)

which must vanish for all D. Therefore as a corollary, it must also be true that dD p

(2π)D1

( p2)a = 0; a, D ∈ C (39)

Recalling that1

( p2)a =

1

Γ(a)

∞0

dτ τ a−1e−τ p2

(40)

this implies that

dD p

(2π)D1

( p2)a =

1

Γ(a)(4π)D/2 ∞

0

dτ 1

τ 1−a+D2

≡ 0 (41)

where the rhs of the above is typical of the integral one would have to perform in thecourse of computing the heat kernel (cf. (26)). This has as a corollary, that integrating out massless fields does not contribute to renormalizations of the cosmological constant and Newton’s constant.

IV. HIGHER ORDER CURVATURE CORRECTIONS

Recalling that so far, we have only computed the first order in curvature corrections to theeffective action. We did this by expanding the diffusion kernel in powers of the curvature by

working in Riemann normal coordinates based at xµ

. If x

µ

− xµ

:= yµ

, then we can expandthe metric tensor at an arbitrary point xµ as [3]

gµν (x) = ηµν − 1

3Rµανβy

αyβ + 1

6Rµανβ;γ y

αyβyγ

+ 2

45RαµβλR

λγν δ −

1

20Rµανβ;γδ

yαyβyγ yδ

(42)

where we have now included corrections up to quadratic order in curvature. Expanding thediffusion operator

∂ s − + m2 = ∂ s −0 + m2 A − [aµν ∂ µ∂ ν + b

µ∂ µ] B (43)we have previously computed up to first order in curvature that

O(R) =

aµν = 13

Rµ ν ρ λbµ = −2

3Rµλy

λ (44)

Evidently, the way to proceed would be to calculate aµν and bµ to the next order and toevaluate

x, τ | 1∂ − + m2

A−B|x, 0 (45)


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using the expansion

⇒ 1A−B =

1

A +

1

A B

1

A +

1

AB

1

AB

1

A + ... (46)

to quadratic order in curvature. This would be a rather involved (though still relativelytractable) calculation we will not concern ourselves with here. Instead we quote the resultof the calculation done elsewhere [4]

W 2 = 12880π2

√ −g

aRµνλρRµνλβ + (b− 2a)Rµν Rµν

+

d − a − b3

R2 + cR

(47)

where the coefficients a, b, c, d depend on the spin of the field integrated out:

spin a b c d

0 1 1 30ξ − 6 90(ξ − 1/6)21/2 −7/2 −11 6 01 −13 62 18 02 212 0 0 717/4

where ξ is the so-called non-minimal coupling parameter that results from a term in theLagrangian of the form ∆L = ξφ2R.

V. IR DIVERGENCES

So far, we have learnt how to deal with UV divergences and understood them for themathematical artefact that they are– corresponding merely to a breakdown in the descriptionof the physics of the system at short distances due to extrapolating a long wavelengthdescription into regimes where they are not applicable6. We learnt how to deal with them,and how to extract sensible physical answers from physical systems even though we lackknowledge of the true UV physics through the renormalization procedure. However wemight also encounter IR divergences in our calculations, and these require more care in theirinterpretation.

We first deal with the most trivial type– that associated with zero modes which only

exist for systems which propagate long range forces– i.e. those with corresponding massless quanta . This divergence actually has a physical basis. Coulomb like forces have an infinitecross section, which is simply the statement that given enough time, two particles thatexchange massless quanta between each other are guaranteed to interact. This divergenceis the result of an interaction that does not die off fast at long distances to overcome thephase space volume available to it. One easy way to regulate such divergences is simply to

6 A situation that we encounter just as readily in classical mechanics– think the divergent forces and

potential energies in electrostatics at small distances.


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place the system in a finite volume. Alternatively, one can introduce a small fictitious massto the interaction and take the limit that this mass vanishes at the end of the calculation.

A much more serious IR divergence arises in the case where the extremum of the classicalaction we expand around does not correspond to a stable solution (i.e. the vacuum isunstable). Consider for example, φ4 theory with m2


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where y(n)R indicates that this is the complete solution incorporating all terms in up to that

order:

y(0)R + y

(1)R = 0 ⇒ y(0)R =

eit

2 + c.c.

ÿ(2)R + y(2)R = −ẏ(2)R ⇒ y(2)R = 12eit

1− 2

4 − t2 + c.c.,

(54)

and so on. Notice what happens if we expand y(2)R up to second order:

y(2)R (t) =

1

2eit

1 −

2t +

2

8 (t2 − it)

+ c.c. + O(3) (55)

which is to be compared to (52). Clearly the solution remains bounded for all time

|y

(2)

R

(t)| ≤

1 ∀

t

reiterating that the large time divergence we had found previously was merely an artefactof our perturbation scheme.

In summary, although the physical interpretation of UV divergences is relatively straigh- forward (being a mathematical artefect of our long wavelength description breaking down at short wavelengths), IR divergences could either have an underlying physical significance, or they could merely be artefacts of our particular perturbation scheme that can be cured by an appropriate resummation– one has to think about the problem at hand carefully.

[1] L. Alvarez-Gaume and M. A. Vazquez-Mozo, “An invitation to quantum field theory,” Lect.

Notes Phys. 839, 1 (2012), Chapter 12.

[2] H. Kleinert and V. Schulte-Frohlinde, “Critical properties of phi**4-theories,” River Edge,

USA: World Scientific (2001) 489 p, Chapter 8.

[3] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,”

[4] D. V. Vassilevich, “Heat kernel expansion: User’s manual,” Phys. Rept. 388, 279 (2003) [hep-

th/0306138].

[5] J. Berges, “Introduction to nonequilibrium quantum field theory,” AIP Conf. Proc. 739, 3

(2005) [hep-ph/0409233].

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