Renormalization of initial conditions and the trans-Planckian problem of inflation

0.39 MB·English

by  Hael Collins

Renormalization of initial conditions and the trans-Planckian problem of inflation
Hael Collins∗
Department of Physics, University of Massachusetts, Amherst MA 01003
R. Holman†
Department of Physics, Carnegie Mellon University, Pittsburgh PA 15213
(Dated: February 1, 2008)
Understanding how a field theory propagates the information contained in a given initial state
is essential for quantifying the sensitivity of the cosmic microwave background to physics above
the Hubble scale during inflation. Here we examine the renormalization of a scalar theory with
nontrivial initial conditions in the simpler setting of flat space. The renormalization of the bulk
theory proceeds exactly as for the standard vacuum state. However, the short distance features of
theinitial conditionscan introducenewdivergenceswhich areconfinedtothesurface onwhich the
5 initial conditions areimposed. Weshow howtheaddition of boundarycountertermsremovesthese
0 divergences and inducesa renormalization group flow in thespace of initial conditions.
0
2 PACSnumbers: 11.10.Gh,11.10.Hi,11.15.Bt,98.80.Cq

n
a
J I. INTRODUCTION have had their origin in fluctuations at sub-Planckian
scales during the early universe. Most models for infla-
9
1 tion typically produce significantly more expansion than
A fundamental principle of field theory is that we do
this minimum requirement. This apparent violation of
not need to know the details of a particular theory at
1 decoupling has been called the “trans-Planckian prob-
arbitrarily short distances to be able to use it to make
v lem” of inflation [3], but it can also be regarded as an
8 physical predictions. This ignorance is permissible not
opportunity,since it suggeststhat physics thatis impor-
5 because the corrections from high energies are small; in
tant at length scales much smaller than those accessible
1 fact they usually are infinitely large. Instead, our igno-
to accelerator experiments could have left its imprint on
1 ranceofthe shortdistancebehaviorofthe theorycanbe
0 the CMB.
absorbedinto the redefinition of the parametersdescrib-
5 ing the theory. The predictions can then be expressedin Because of this unique opportunity, much effort [4, 5,
0
terms of a finite number ofparametersassociatedwith a 6,7,8,9]hasbeendevotedrecentlytodeterminingunder
/
h setoflocaloperators. Inthisrenormalizedtheory,higher whatconditionssucheffectscouldbeseen,usuallywithin
-t order corrections,which are usually defined as power se- a particular framework for the near-Planck scale physics
p ries in a small rescaled coupling, remain small. What that determines the state of the inflaton. The expansion
e we have lost in this process is the idea of fixed, constant rateduringinflationdefines anaturalscale,H,the Hub-
h
parameters; coupling “constants” now depend upon the ble scale, and if the new physics above this scale has a
:
v scale at which they have been defined. mass M associated with it, then the “trans-Planckian”
i signal is generally found to be suppressed by H/M, or
X The behavior of a field theory in the early universe
more, relative to the vacuum result. Here the vacuum
r appears to violate this principle of decoupling—at least
a correspondstothe statethatis invariantunderthe sym-
at a first glance. A distinctive feature of inflation [1]
metries of the background space-time and that matches
is the superluminal stretching of length scales which al-
with the usual idea for the Minkowski space vacuum at
lows quantum field fluctuations during the inflationary
distances much shorter than the curvature of the back-
phase to induce the metric perturbations that seed the
ground, 1/H. While these approaches have been very
large scale structure of the universe. The observed spec-
illuminating since they have provided a clear estimate
trum of acoustic peaks in the cosmic microwave back-
for the expected size of the trans-Planckiansignalin the
ground (CMB) and, even more strikingly, the detection
CMB, they have generally been rather ad hoc in that
of anticorrelations in the polarization and temperature
they assumesome particularfeature for the physics near
anisotropies at superhorizon scales are both completely
the Planck scale or they chose a particular initial state.
consistentwiththepredictionsofinflation[2]. Howmuch
In this sense, they do not really address the underlying
expansion occurred during inflation depends on the ex-
trans-Planckian problem—why we should expect to ob-
pansionrateanditsduration,butwithmorethanthe60–
serve a CMB spectrum that essentially agrees with that
70 e-folds usually demanded of inflation, the scales asso-
produced by assuming that the universe is in its vac-
ciated with the large-scale structure we see today would
uumstateatallscales,withoutanyapriori assumptions
about physics near the Planck scale.
The setting for a field theory in the early universe is
∗Electronicaddress: [email protected] somewhatdifferentthanthatgenerallyassumedinanS-
†Electronicaddress: [email protected] matrix calculation [10]. The field begins in some state
2
specified at an initial time, t , which does not need to probablyalwayshavealargererrorthanwewouldlike—
0
correspondtoanasymptoticallywell-behavedstateinthe thenweneedonlythatfinitesetofparametersdescribing
infinitepastorwhichwasanenergyeigenstateofthefree the initial state suppressed by no more than n powers of
Hamiltonian. Depending upon the space-time geometry, ∆/M wherenisdefinedby(∆/M)n δ. Thereareaddi-
∼
a globally conserved energy may not even exist. This tionalparametersassociatedwiththestillfinerdetailsof
initial state can have short distance features imprinted theinitialstate,buttheseareincreasingirrelevant,being
upon it either froma preceding phase where heavyfields suppressedbyfurtherpowersof∆/M andarenotneeded
were excited or at length scales shorter than the Planck inpracticeuntilwecanmakemoreprecisemeasurements
length, where gravitationaleffects can become strong. (decreasing δ) or are able to probe the shorter distance
Whatisthereforeneededisanextensionoftheideasof features directly (increasing ∆). What has happened is
effectivefieldtheory[11]fordescribingthefeaturesofan that we have replaced an assumption about the details
arbitraryinitialstate. Inthisarticlewedescribehowthe ofthePlanckscalephysicssettingtheinitialstatewitha
renormalizationof this shortdistance information in the smallsetofparametersneeded to describe how the state
initialstateproceedsinMinkowskispace. Aninitialstate differs from the vacuum at short distances and which is
can be broadly classified as renormalizable or nonrenor- applicable for any short distance completion of the the-
malizable by how its behavior differs from the vacuum ory.
state atshortdistances. The new divergencesassociated Despite the power and the generality of the effec-
with the short distance features of the initial state are tive field theory perspective, this approachhas not been
cancelled by counterterms localized at the surface where widely applied to inflation. Ref. [5] derivedthe expected
the initial state is defined. Not surprisingly, the renor- size of the correctionsto the CMB power spectrum from
malizable initial states are associated with relevant or higher order operators, but still worked in the standard
marginal operators on the initial surface while the non- vacuum state. One of the earliest attempts to model
renormalizable initial states require irrelevant boundary initial state effects in the spirit of effective field theory
counterterms. integrated out the dynamics of a heavy field to learn of
To begin, we must understand what constitutes a its imprint on the CMB [7]; but the first attempt for
renormalizableinitialstate—wherethedivergencesinthe a general effective description of initial state effects has
bare theory occur and how they can be removed by the only appeared quite recently [12, 13].
appropriate counterterms which are consistent with the The essential feature of the renormalization of an ini-
symmetries left unbroken by the state. An important tial state to understand is how to control the infinities
aspect is the statement of a renormalization condition that arise from having short distance features in this
for this setting, since it is through this condition that state that differ from the standard vacuum. It is impor-
the renormalized theory acquires its dependence on the tant to note that for a Robertson-Walker universe the
renormalization scale. The Callan-Symanzik equation vacuum corresponds to the state that matches with the
then determines the running of the usual couplings and Minkowski space vacuum at distances sufficiently small
field rescalings as well as the scale dependence of the ef- that the background curvature is not noticeable. Since
fects associated with the initial condition. the curvature is unimportant where the renormalization
The secondimportantaspectofaneffective treatment is needed, the appropriate setting in which to begin is
of the initial state is characterizing and understanding Minkowski space. It is also simpler in flat space to ex-
the role of nonrenormalizable initial conditions. In the amine the structure of a state and its renormalization
modernviewoffieldtheory,nonrenormalizableoperators analytically. As we consider a completely general ini-
are notunphysicalbut are merely a sign thata theory is tial state, we shall use the Schwinger-Keldysh formal-
notintendedtobevalidtoarbitrarilyhighenergies. The ism [14, 15, 16] to define the time-evolution of a general
coefficient of a nonrenormalizable operator of dimension Green’s function in this setting.
n > 4 is of the order 1/Mn−4, where M is a mass scale. Oncewehaveunderstoodtheinitialstaterenormaliza-
If we study phenomena at an energy Λ, then the effects tioninflatspace,weshallhavethe necessaryfoundation
ofthenonrenormalizableoperatorswillbesuppressedby forproceedingtoageneralRobertson-Walkerspace-time.
powersΛ/M. As long as Λ M andwe do notdemand The expansion of the background affects how and when
≪
an arbitrary accuracy of our prediction, we only need weshouldchooseaninitialstate. TheexpansionrateH,
to consider a finite set of operators. As Λ M, the froman effective theory perspective, just sets the appro-
→
effective description breaks down. But once we are able priate scale with respect to which we define the infrared
to probe such energies, we should see the dynamics that (IR)andultraviolet(UV)detailsoftheinitialstate. But
gave rise to the higher dimension operators of our lower the continual blueshifting that occurs as we look further
energyeffectivetheoryandthatdescriptionisreplacedby backduring inflationmeans thatwe mustalso choosean
anothereffectivetheoryappropriateforthescaleΛ M. appropriate initial time at which to set the initial condi-
∼
Similarly, nonrenormalizable initial conditions natu- tionsdefining the state[13],whichisnaturallyandread-
rally introduce a mass scale M. To make a prediction at ilyaccomplishedintheSchwinger-Keldyshpicture. Both
alength1/∆sufficientlylargerthan1/M andtospecified of these details will be addressed in [17].
accuracyδ—andinthemeasurementoftheCMBweshall The next section begins with a description of how an
3
initial condition alters the structure of the propagator. One of the constants of integration is already fixed by
The need for the consistency of the propagator with the the equal time commutator,
stateisquitefamiliarfromstudiesofinteractingtheories
[18,19,20]fortheα-vacuaofdeSitterspace[21]. Oneof π(t,~x),ϕ(t,~y) = iδ3(~x ~y), π = ∂ϕ, (2.5)
thenewfeatureshereisapartofthepropagatorthatde- − − ∂t
(cid:2) (cid:3)
pends onthe initialstate,whichgivesrisetodivergences
which imposes
not present for the vacuum state. Section III describes
how to renormalize the theory in this setting, by es- U ∂ U∗ U∗∂ U =i. (2.6)
tablishing an appropriate renormalization condition and k t k − k t k
showing that we obtain the usual running for the state-
Applying this Wronskian condition to Eq. (2.4) yields
independent part of the theory. Section IV then shows
that all of the initial-state-dependent divergences are lo- 1
c 2 d 2 = . (2.7)
calized on the boundary where the initial condition was k k
| | −| | 2ω
k
imposed. Wethenremovethesedivergenceswithbound-
ary counterterms and derive their running dependence The secondofthe constantsin Eq.(2.4) is determined
on the renormalization scale. We describe nonrenormal- by an additional, physically motivated condition. The
izable initial conditions in Sec. V using the language of standard condition is to choose the modes to be those
effective field theory. Section VI concludes with com- associated with the vacuum; selecting only the positive
ments on the extension of this approach to Robertson- energy states imposes d = 0, which together with the
k
Walker universes and to the problem of backreaction. equal time commutation relation establishes the usual
Twomoredetailedpointsofourdiscussionarepresented form for the modes,
in the appendices. The first provides a brief introduc-
tion to the Schwinger-Keldysh formalism while the sec- UE(t)= 1 e−iωkt, (2.8)
ond shows how a resummation removes some spurious k √2ω
k
logarithmicdivergencesatthe initial boundarywhichdo
not affect the renormalization. uptoanarbitraryphase. The“E”signalsthatthemodes
are those of the vacuum.
Notice that the state annihilated by all of the a ,
~k
II. BOUNDARY CONDITIONS IN FLAT SPACE
a 0 =0, (2.9)
~k| i
Consider a free massive scalar field propagating in a isavacuumstatebothinthesenseofbeinginvariantun-
flat space-time, derthe generatorsofthe Poincar´egroupaswellasbeing
the lowest energy eigenstate with respect to the globally
conservedHamiltonianassociatedwiththeKillingvector
S =Z d4x 12∂µϕ∂µϕ− 21m2ϕ2 . (2.1) ∂∂t. In a de Sitter or a Robertson-Walker background,
(cid:2) (cid:3) a globally defined time-like Killing vector may not exist
The usual expansion of the field in creation and annihi- andsoforthesespace-timeswecanonlydefineavacuum
lation operators with respect to the vacuum is in terms of its symmetries.
For amore generalinitialstate, we fix the secondcon-
ϕ(t,~x)=Z (2π)d33√~k2ωk he−iωktei~k·~xa~k+eiωkte−i~k·~xa~†ki, sttihtoaennteoanorlfythineutnemigvoreadrtesiseo.nwThihneirsEeaqtp.hpe(r2or.a4ac)phiwdisitehexsppaaenncsiiainolilntyiaduloseecfsounlndoiint-
(2.2)
guarantee the existence of a noninteracting state in the
where the frequency is ω =√k2+m2.
k asymptoticpast. Forsimplicitywerestrictourselvestoa
We shall generalize this mode expansion to the situa-
first-orderconstraintthatislinearinthemodefunctions,
tion where the behavior of the field is specified along an
initial space-likesurface, t=t0, so it is instructive to re- ∂
callthe originofthe form ofthe modes. Since the initial ∂tUk(t)(cid:12) =−i̟kUk(t0). (2.10)
surface is spatially flat, the spatial modes are still plane (cid:12)t=t0
(cid:12)
wavesbut we leave the time-dependent part unspecified, (cid:12)
The particular boundary condition with ̟ = ω re-
k k
storesthe full Poincar´einvarianceof the state, but other
ϕ(t,~x)= d3~k U (t)ei~k·~xa +U (t)∗e−i~k·~xa† . values for ̟k break the symmetry. Applying this con-
Z (2π)3 h k ~k k ~ki dition to the general mode function in Eq. (2.4) with
(2.3) Eq. (2.7) yields
The time-dependent part of the mode functions, U (t),
k
is the solution to the Klein-Gordon equation, U (t)= |ωk+̟k| e−iωkt+ ωk−̟keiωk(t−2t0) .
k
2ω √2Re̟ (cid:20) ω +̟ (cid:21)
k k k k
U (t)=c e−iωkt+d eiωkt. (2.4) (2.11)
k k k
4
We can rewrite these mode functions in a form that A. Propagation
emphasizes the similarity of these modes to the mode
functions used for the α-states of de Sitter space [21]. The propagator must also be consistent with the
Define boundary conditions, which implies some additional
ω ̟ structure beyond that of the usual, Poincar´e invariant
eαk k− k (2.12)
Feynman propagator,
≡ ω +̟
k k
along with an image time, iGE(x,x′) = Θ(t t′) 0ϕ(x)ϕ(x′)0
− F − h | | i
+Θ(t′ t) 0ϕ(x′)ϕ(x)0
t 2t t, (2.13) − h | | i
I ≡ 0− d4k ie−ik·(x−x′)
= , (2.22)
so that Z (2π)4 k2 m2+iǫ
−
N
Uα(t)= k e−iωkt+eαke−iωktI (2.14) since the derivativesin the boundary conditions canalso
k √2ω
k (cid:2) (cid:3) act on the Θ-functions. As with the fields, it is simplest
to describe the constraint on the propagator in terms of
with
its spatial Fourier transform,
−1/2 ω +̟
Nk ≡h1−eαk+α∗ki = 2|√ωkkRe̟k|k. (2.15) Gαk(x,x′)= d3~k ei~k·(~x−~x′)Gαk(t,t′). (2.23)
F Z (2π)3 k
In terms of the Poincar´einvariant modes,
If we impose the conditions,
e−iωkt
UE(t)= , (2.16)
k √2ωk [∂t−i̟k]tt′=>tt00 Gαkk(t,t′) = 0
the modes satisfying the boundary condition are then [∂t′ −i̟k]tt′>=tt00 Gαkk(t,t′) = 0, (2.24)
Ukα(t)=Nk UkE(t)+eαkUkE(tI) . (2.17) then the propagator in the region t,t′ >t is
0
(cid:2) (cid:3)
The “α” indicates the mode functions, and later the Gαk(t,t′)=A GE(t,t′)+eαkGE(t ,t′) . (2.25)
Green’s functions, that are consistent with the bound- k k k k I
(cid:2) (cid:3)
ary condition of Eq. (2.10). where GE(t,t′) is the momentum representation of the
k
Since the modes can be written in terms of the flat standard vacuum propagator of Eq. (2.22). The same
space modes, the creationandannihilation operatorsas- propagator is used in [12].
sociated with the initial state can be correspondingly In this form, the propagator appears to contain two
written as a Bogoliubov transformation of the flat space sources, at x = x′ and at x = x′. The first term cor-
I
operators, responds to the effect of a point particle and the stan-
dardnormalizationof the residue at the physicalpole at
a~αkk =Nkha~k+eα∗ke2iωkt0a†−~ki. (2.18) k2 =m2 fixes Ak =1,
Let us write the state that is annihilated by the aαk op- Gαk(t,t′)=GE(t,t′)+eαkGE(t ,t′). (2.26)
~k k k k I
erators at t=t as
0
The second term represents a source in the unphysical
a~αkk|αk(t0)i=0. (2.19) regiont<t0; throughit we have exchangeda constraint
on the boundary with a bulk effect. The apparent non-
For simplicity, we shall often write the initial state with- locality in the correlated motion of the particle with its
out its argument, image encodes the fact that we have specified the value
of the field over an entire space-like hypersurface.
α α (t ) . (2.20)
k k 0 ThepropagatorinEq.(2.26)alsofollowsfromagener-
| i≡| i
alized construction for the time-ordering operator. Con-
In the interaction picture, which we follow here, the
sider a time-ordering which includes the image time de-
time-evolutionof operatorsin the theory is givenby free
fined by
Hamiltonian while the interacting part of the Hamilto-
nian generates the evolution of the states, T ϕ(x)ϕ(x′) (2.27)
αk
|αk(t)i=U(t,t0)|αk(t0)i, (2.21) (cid:0)= Θˆ1(x,x(cid:1)′)ϕ(x)ϕ(x′)+Θˆ2(x,x′)ϕ(x′)ϕ(x)
+Θˆ (x,x′)ϕ(x )ϕ(x′)+Θˆ (x,x′)ϕ(x′)ϕ(x )
where U(t,t0) is the operator solving Dyson’s equation 3 I 4 I
andiswritteninEq.(A3). Butbeforeanalyzinganinter-
where
actingtheorywemustfirstestablishadescriptionforthe
propagationof a field in the regiont>t0 which respects Θˆ (x,x′) = AˆΘ(t t′)+ eαk(eαkAˆ−Bˆ)
the initial constraint, Eq. (2.10). 1 − 1 e2αk
−
5
Θˆ2(x,x′) = AˆΘ(t′−t)+ eα∗k1(eα∗keAˆ2α−∗k Bˆ) ntheegarteigvieonimta≥gitn0apryropvaidrte.dTwheearsesfuorme,eathltahto̟ugkhhEasq.a(s2m.3a2l)l
eαk(−eαkBˆ Aˆ) more closely resembles the standard expression for the
Θˆ (x,x′) = BˆΘ(t t′)+ − flat space propagatorintegratedoverk as well, we shall
3 I− 1−e2αk continue to write the propagator in it0s explicitly time-
Θˆ (x,x′) = BˆΘ(t′ t )+ eα∗k(eα∗kBˆ−Aˆ). orderedform with an integralover only the spatial wave
4 − I 1 e2α∗k vector.
−
Letus take the Fourier transformofthe propagatorin
The propagatorin Eq. (2.26) is obtained when
Eq. (2.32),
Aˆ=1 Bˆ =eαk (2.28) Gαk(x,x′) = d3~k ei~k·(~x−~x′)Gαk(t,t′). (2.34)
F Z (2π)3 k
by evaluating this time-ordering indan α states,
k
| i Inserting Θ-functions appropriately, we obtain
−iGαFk(x,x′)=hαk|Tαk ϕ(x)ϕ(x′) |αki. (2.29) Gαk(t,t′) (2.35)
(cid:0) (cid:1) k
Here, the operators are defined in terms of their action = Θ(t t′) dk0 e−ik0(t−t′)
on the Fourier components of the field, as for example, − I 2π k2 (ω2 iǫ)
lower 0 − k−
eαkf(t,~x)=Z (2dπ3~k)3 ei~k·~xeαkf~k(t). (2.30) +Θ(t′−t)Iupper d2kπ0 k02−eik(·ω(tk2′−−t)iǫ)
Note thdat in the particular case where eαk is real, which +Θ(t t′) dk0 k0+̟k e−ik·(tI−t′)
I− I 2π k ̟ k2 (ω2 iǫ)
implies that ̟k is also real, the time ordering becomes lower 0− k 0− k−
especially simple, +Θ(t′ t ) dk0 k0+̟k eik·(t′−tI−t′) ,
T ϕ(x)ϕ(x′) − I Iupper 2π k0−̟k k02−(ωk2−iǫ)
αk
=(cid:0) Θ(t t′)ϕ(cid:1)(x)ϕ(x′)+Θ(t′ t)ϕ(x′)ϕ(x) which allows us to integrate eachof the terms by closing
− − the contour in the upper or the lower half plane as indi-
−eαk Θ(t′−tI)ϕ(xI)ϕ(x′) cated. Note that although we have inserted Θ-functions
d+(cid:2)Θ(tI−t′)ϕ(x′)ϕ(xI) . (2.31) fsoinrctehewiemaargeerteimstreicatsewdetlol,Θt,(tt′I>−tt′).=T0hearnedfoΘre(,t′t−hteI)th=ird1
(cid:3) 0
The time-ordering in the image Θ-functions is the oppo- term always vanishes in the physical region. Assuming
site that of the fields—forward propagation of the phys- that Im̟ <0, the extra factor in the fourth term does
k
ical particle corresponds to the backwardpropagationof not produce any new poles so that the contour integral
its image. gives only the usual result,
We would like to show the relation between propaga-
i
torderivedaboveandthatusedin[12];bothpropagators Gαk(t,t′) = Θ(t t′) e−iωk(t−t′)
k − 2ω
are essentially the same—and agree within the physical k
i
regiont,t′ >t0—butthestatementoftheboundarycon- +Θ(t′ t) e−iωk·(t′−t)
ditions differs slightly from that which we have used in − 2ωk
Eq.(2.10). In[12]thepropagator1iswrittenintheform, +Θ(t t′)ieαke−iωk·(tI−t′)
I − 2ω
k
Gαk(x,x′)= d4k e−ik·(x−x′)+eαk(k0)e−ik·(xI−x′) +Θ(t′ t )ieαke−iωk·(t′−tI), (2.36)
F Z (2π)4 k2 m2+iǫ − I 2ω
− k
(2.32)
It is important to note that the third term in this equa-
where xµ = (t ,~x). Unlike the single-source propagator
I I tion did not result from the third term in Eq. (2.35).
usedinthePoincar´e-invariantvacuumstateofEq.(2.22),
Sincebothvanishfort,t′ >t ,wehaveformallyincluded
this propagatorcontains anadditionaldependence onk 0
0 the appropriate counterpart of the final image term to
through the prefactor of the image term,
produceanimage propagator whenthefullpropagatoris
k +̟ written in its time-ordered form. This result establishes
eαk(k0) = 0 k (2.33) that the propagators in Eqs. (2.26) and (2.32) agree in
k ̟
0− k the physical region.
This coefficient leads to a spurious new pole term, at
k =̟ ,whichdoesnotactuallyaffectthepropagatorin
0 k
B. Generating functionals
To generate a propagator with two sources, the free
1 SeeforexampleEq.(2.17)of[12]. field generating functional should include a current J(x)
6
that couples to the field simultaneously at the point x III. RENORMALIZATION CONDITIONS
and its image x . This apparent nonlocality in the bulk
I
physics encodes the effect ofhaving imposeda boundary
In the standard formulation of field theory, the per-
conditionuponthe spatialhypersurface,t=t . The free
0 turbative corrections to a given process often diverge as
field generating functional is thus
we sum over more and more of the short distance be-
havior. A theory remains predictive since, in the case of
W0α[J]=Z DϕeiR d4x[L0+(aˆ(x)ϕ(x)+ˆb(xI)ϕ(xI))J(x)] renormalizable interactions, it is possible to absorb this
(2.37) divergent behavior into a redefinition of the parameters
with ofthetheory. Thephysicalparameters,eachgivenbythe
sumof the correspondinginfinite bare parameterandits
L0 = 21∂µϕ∂µϕ− 21m2ϕ2. (2.38) infinite radiative corrections, remain finite. In this pro-
cess, the parameters acquire a dependence on the scale
To obtainthe correctpropagatorwe require that the co-
at which they are defined.
efficients of the currents should satisfy
When we consider a nonstandardboundary condition,
1 1/2 we introduce an additional type of renormalization. In
aˆ = Aˆ+ Aˆ2 Bˆ2
√2h q − i this case, we encounter new divergences related to sum-
ming over the shortdistance features of the initial state.
1 Bˆ
ˆb = (2.39) For the theory to remain predictive, we need a compa-
√2 Aˆ+ Aˆ2 Bˆ2 1/2 rablemethodfor absorbingourignoranceofthe extreme
−
(cid:2) p (cid:3) shortdistancestructureofthestatewithacorresponding
whichyieldsthe propagatorEq.(2.29)whenwe differen-
infinitecounterterm. Sincethesenewdivergencesarefea-
tiate with respect to the currents,
tures of the initial conditions and not the bulk physics,
the new counterterms should be confined to the initial
δ δ
iGαk(x,x′)= i i Wα[J] . surface.
− F (cid:20)− δJ(x)(cid:21)(cid:20)− δJ(x′)(cid:21) 0 (cid:12)J=0
(cid:12)(2.40) The physical setting is one in which the field may not
(cid:12)
necessarily be in its vacuum state, so we shall adopt the
Whenthegeneratingfunctionalisgeneralizedtoafully
methods usually applied in nonequilibrium field theory
interacting theory, it is important to distinguish the lo-
[22] to establish the renormalization conditions that de-
calityofthe bulk theoryfromthe nonlocalityintroduced
termine the scale dependence of the various parameters
by the boundary conditions. The underlying locality of
of the theory. For the “bulk” 3+1 dimensional physics,
the theory implies that the free Lagrangian remains of
these conditions produce the same anomalous dimen-
the form in Eq. (2.38), but to obtain the correct prop-
sions and β-functions we would have anticipated from
agators for the internal lines of a general graph in an
the S-matrix. This agreement between the renormaliza-
interacting theory requires that the interactions should
tion group running of the bulk properties of the theory
have the form [18, 20],
obtained for a general initial state and the running ob-
1 tainedusingtheS-matrixisanecessaryandnaturalcon-
= 1∂ ϕ∂µϕ 1m2ϕ2 λ ϕ˜n (2.41)
L 2 µ − 2 − n! n sequence of the fact that we have not modified the short
nX≥3
distancepropertiesofthetheory. Atveryshortdistances
where and away from the boundary, the field is not sensitive
to the details of the initial state. The bulk divergences
ϕ˜(x) aˆ(x)ϕ(x)+ˆb(x )ϕ(x ). (2.42)
≡ I I should therefore be unaltered by the initial conditions.
The interacting theory generating function is then This behavior still leaves the possibility of new diver-
gences that are associated with the short distance de-
Wα[J]= ϕei d4x[L0(x)− n≥3λnn!ϕ˜n(x)+ϕ˜(x)J(x)] tails of the initial state. Since any state other than the
N Z D R P vacuum state necessarilybreaksthe underlying Poincar´e
(2.43) invariance of the background, the counterterms needed
with to cancel these boundary divergencesareconsistent only
1 =Z DϕeiR d4x[L0(x)−Pn≥3λnn!ϕ˜n(x)]. (2.44) wthiethnethwisdbivreorkgeenncseysmtmhaettray.risIenfrfoamct,tahsewimeasghealtlersmhowin,
N the propagator only appear at t = t . Therefore the
0
A more convenient form for the generating functional
renormalization of the initial state corresponds to the
when calculating perturbative corrections is
appearance boundary counterterms that depend on the
Wα[J]=e−i d4x n≥3λnn![−iδJδ(x)]nWα[J], (2.45) initial condition imposed and that run with the renor-
R P 0 malization scale.
with the free field generating functional of Eq. (2.37) Our ultimate goal [17] is eventually to establish a
rewritten in the form frameworkthatcanbe appliedto the earlyuniverse. We
thereforetreatthescalarfieldinmuchthesamewayasif
W0α[J]=e(i/2)R d4xd4x′J(x)GαFk(x,x′)J(x′). (2.46) itwereaninflatoninanexpandingbackground,dividing
7
titiminetoanadcalassmsicaalllϕflz(etu,rco~xt)uma=otidφoen(tφ)ψ(+(tt),ψ~xw()ht,,i~xch).only depend(s3.o1n) x.............................................................................................................................................y............................................................................................................................................................. + x................................................................................................................................................................................................................................y..............................................................................................................................................................................................................
x
Tofhtehesitmapdlpeostler[e2nα2o,kr(2mt3)]a—ψliz+isa(txio)nαcko(nt)dit=io0n.—the vanis(h3i.n2g) + ...................................................................................................................................................................y.....................................................................................................................................................z..........................................................................................................................................................
h | | i
FIG. 1: The leading contributions to therunningof therun-
Inthe interactionpicture,the timeevolutionofthe state ning of the mass m and the coupling λ in a ϕ4 theory. The
isdeterminedbytheinteractingpartoftheHamiltonian.
solid lines represent propagating ψ fields while the dashed
In the Schwinger-Keldysh approach, which determines lines correspond to thezero mode φ.
the time-evolution of the full matrix element starting
from a specified initial state, this condition becomes
scalar theory we can have
hαk|Tα ψ+(x)e−iRt∞0 dt[HI(φ,ψ+)−HI(φ,ψ−)] |αki =0,
hαk(cid:0)|Tαe−iRt∞0 dt[HI(φ,ψ+)−HI(φ,ψ−)]|αk(cid:1)i St=t0 =Z d3~x 12z0ϕ∂tϕ+ 21z1mϕ2+ 16z2ϕ3 .
(3.3) t=t0 (cid:8) (cid:9)
(3.7)
where the denominator removes the vacuum to vacuum
We have included a factor of m in the quadratic term so
graphs. The + and superscripts refer to the result
− thatallthez aredimensionless. Notethatwehaveused
of time-evolving both the in-state and the out-state of i
capital Z ’s for the bulk renormalization and lowercase
the matrix element, respectively. Appendix A briefly re- i
z ’s for the boundary renormalization. Since we have
views the Schwinger-Keldysh approach and further de- i
broken time-translation invariance, operators such as
fines some of the notation we have used.
We illustrate the appearance of new boundary diver-
ϕ∂ ϕ= 1∂ ϕ2, ϕ∂2ϕ, ∂ ϕ∂ ϕ, ... (3.8)
gences by studying a scalar field with a simple quartic t 2 t t t t
self-coupling,
are allowedon the initial surface, although only the first
= 1∂ ϕ∂µϕ 1m2ϕ2 1 λϕ4. (3.4) operatorismarginalsincethe fieldhasamassdimension
L 2 µ − 2 − 24 ofone. ThesurfaceisstillO(3)invariant,sothe firstop-
This Lagrangian describes the bare theory. The per- eratorwith a spatialderivative only appearsas the irrel-
turbative corrections to a Green’s function, such as the evant,dimensionfouroperator, ~ϕ ~ϕ. Inourexample
∇ ·∇
one-pointfunctionweshallexamine,containdivergences ofaϕ4 theory,theLagrangianhasanadditionalϕ ϕ
↔−
whichcanbeabsorbedbyrescalingtheparametersofthe invariance so we have in this case that z =0 automati-
2
theory, cally. Thus, a renormalizableboundary condition in this
example only requires two types of boundary renormal-
Z Z
ϕ=Z1/2ϕ , m2 = 0m2, λ= 1λ . (3.5) ization. Inthissectionweshallshowhowtocharacterize
3 R Z3 R Z32 R such renormalizable initial conditions and to determine
how they run under a renormalizationgroup flow.
In terms of the renormalized theory, ϕ ,m ,λ , the
R R R Because of its coupling to the zero mode, the vanish-
{ }
perturbative corrections are finite. Equivalently, we
ing of the one-point Green’s function for the fluctuation
could write the Lagrangianin terms of the renormalized containsmuch information. Fromthe perspective of ψ±,
parameters,
the interacting part of the Hamiltonian is
= 1∂ ϕ ∂µϕ 1m2ϕ2 1 λ ϕ4.
L 2 µ R R− 2 R R− 24 R R H (φ,ψ±) = d3~y ψ±(φ¨+m2φ+ 1λφ3)
+21(Z3−1)∂µϕR∂µϕR− 12(Z0−1)m2Rϕ2R I Z h 6
−214(Z1−1)λRϕ4R. (3.6) +41λφ2ψ±2+ 16λφψ±3+ 214λψ±4 .
i
The terms on the last two lines correspond to the coun- (3.9)
terterms needed to render the theory finite.
The image partsofthe propagatorproducefurther di- Forsimplicity,weshalltreattheφ2ψ±2 termasaninter-
vergences on the initial boundary so the theory also re- action rather than as an effective mass term; this treat-
quiresanadditionalrenormalization. Theseboundarydi- ment is consistent when λφ2 m2. More generally, we
≪
vergences are renormalizable in the sense that they can can resum the effects of this term as has been done in
be removed by a set of relevant or marginal operators Appendix B. For this interaction, the connected part of
localizedatt=t ,whichareconsistentwiththesymme- the expectation value of ψ, expanded to second order in
0
tries left unbroken by the boundary. For example, for a H , yields
I
8
0 = α (t )ψ+(x)α (t )
k f k f
h | | i
tf
= dt d3~y G>(x,y) G<(x,y)
−Z Z −
t0 (cid:2) (cid:3)
λ iλ iλ2 t
φ¨+m2φ+ φ3 φG>(y,y)+ φ dt′φ2(t′) d3~z G>(y,z)G>(y,z) G<(y,z)G<(y,z)
×n 6 − 2 α 4 Zt0 Z (cid:2) α α − α α (cid:3)
λ t λ iλ
φ2 dt′ d3~z G>(y,z) G<(y,z) φ¨(t′)+m2φ(t′)+ φ3(t′) φ(t′)G>(z,z)+
−2 Zt0 Z (cid:2) − (cid:3)n 6 − 2 α ···o
iλ t λ iλ
+ G>(y,y) dt′ d3~z G>(y,z) G<(y,z) φ¨(t′)+m2φ(t′)+ φ3(t′) φ(t′)G>(z,z)+
2 α Zt0 Z (cid:2) − (cid:3)n 6 − 2 α ···o
λ2 t
+ φ(t) dt′ d3~zG>(z,z) G>(y,z)G>(y,z) G<(y,z)G<(y,z)
4 Z Z α α α − α α
t0 (cid:2) (cid:3)
λ2 t
+ dt′ d3~zφ(t′) G>(y,z)G>(y,z)G>(y,z) G<(y,z)G<(y,z)G<(y,z)
6 Z Z α α α − α α α
t0 (cid:2) (cid:3)
+ (3.10)
···o
mwhaenrefuxnc=tio(tnfs,,~xG),>αy,<=(dx3(,~kty,)~y,)aarenddezfi=ned(t′b,y~z). The Wight- x....................................................y........................................................z.............................................................................................................................................................................................................................................. + x....................................................y...................................................................z................................................................................................................................................................................
G>(y,z) = i ei~k·(y~−~z)
α Z 2ω (2π)3
k
×he−iωk(t−t′)+eαkeiωk(2t0−t−t′)i + x.........................................................................................y.................................................................................................................................................................z................................................................................................................................. + x..........................................................................................y..........................................................................................................................................................z...........................................................................................
d3~k
G<(y,z) = i ei~k·(y~−~z)
α Z×2hωeikω(k2(πt−)3t′)+eαkeiωk(2t0−t−t′)i.(3.11) + x.................................z.............................................................................................................................y...................................................................................................................................................................... + x..................................y.............................................................................................................................................................................................................z........................
Those without the α subscript are the vacuum mode FIG. 2: Further graphs obtained by expanding the exponen-
Wightman functions (for eαk = 0) given in Eq. (A16) tial in Eq. (3.3) to second order. The last of these graphs
of the Appendix. The diagrams for the leading order containstheleadingnontrivialcorrectiontothewavefunction
corrections to the mass and the coupling are those as- renormalization.
sociated with the first line within the braces and are
shown in Fig. 1. The diagrams for the remaining terms 2+m2
inEq.(3.10)areshowninFig.2. Inthesefigures,asolid x............................................................................................................................................y.............................................................................................................................................................. = x....................................................................t....................y..........................................................
linerepresentsthefluctuatingpartofthefield,ψ,whilea
dashedline indicates the zeromode, φ. The shaded blob
represents an insertφi¨o+n omf2tφhe+o16pλerφa3t,or, (3.12) + x..................................................................................16......y.............l........................................................................................................................................................................
FIG. 3: The shaded blob corresponds to the following two
whichactsontheoutgoingzeromode,asshowninFig.3.
graphs. The time derivativesact on theclassical φ(t) field.
To calculate the mass renormalizationto first order in
λ and the coupling renormalizationto secondorder,it is to consider mass and coupling renormalization,
sufficient to set the first line in the braces to zero. The
secondlinecorrespondstoaself-energycorrectiontoone Z =1+ (λ), Z =1+ (λ), Z =1+ (λ2),
0 1 3
O O O
of the external legs, as can be seen in the figure, while (3.13)
the third, forth and fifth lines are order λ2 self-energy since the leading correction to wave function renormal-
corrections. Totheorderweshallcalculate,weonlyneed ization is order λ2 and is from the two-loopcontribution
9
given by the fifth line within the braces of Eq. (3.10). the divergent and finite parts. Note that since we are
Setting the integral for the first line of the integrand only integrating over the spatial momenta, the integral
in Eq. (3.10) to zero, we find the effect of the interac- is already Euclidean and no Wick rotation is needed,
tions onthe equationof motion for the zero mode to the
specifiedorder. Substitutingintheformoftheboundary d3−2ǫ~k µ2ǫ
I(ǫ,α) =
propagator,Fig. 1 then gives Z (2π)3−2ǫ ωα
k
0 = −Zt0tf dtsin[m(mtf −t)] = 8√ππ2Γ(ǫΓ−(α23)−2α)(cid:20)4mπµ22(cid:21)ǫm3−α. (3.16)
λ d3~k 1 λ
φ¨(t)+φ(t) m2+ + φ3(t) Herewehaveincludedamassscaleµtokeepthecoupling
×(cid:26) (cid:20) 2 Z (2π)3 2ωk(cid:21) 6 dimensionless, λ µ2ǫλ.
→
λ2 t ∞ d3~k sin(2ω (t t′)) The leadingtadpolecorrectionto the massisfromthe
φ(t) dt′φ2(t′) k − term,
− 8 Z Z (2π)3 ω2
t0 0 k
+λφ(t) d3~k 1 eαke−2iωk(t−t0) λ d3~k 1 = λm2 1 +1 γ+ln4πµ2 ,
2 Z (2π)3 2ωk 4 Z (2π)3 ωk −32π2 (cid:20)ǫ − m2 (cid:21)
iλ2 t ∞ d3~k eαk (3.17)
φ(t) dt′φ2(t′) while the leading correctionto the coupling is through
− 8 Z Z (2π)3 ω2
t0 0 k
e−2iωk(t−t0) e−2iωk(t′−t0) λ2 t d3~k sin(2ω (t t′))
×h − i φ(t) dt′φ2(t′) k − . (3.18)
− 8 Z Z (2π)3 ω2
t0 k
+ . (3.14)
···(cid:27)
To extract the divergent piece, integrate by parts with
respect to t′,
The first two lines of the integrand are purely bulk ef-
fectsandarepresentevenforthePoincar´e-invariantvac-
uum state. The last two terms depend explicitly on the λ2φ(t) tdt′φ2(t′) d3~k sin(2ωk(t−t′))
boundaryconditions,indicatedbythefactorsofeαk. We − 8 Z Z (2π)3 ω2
t0 k
treatthe renormalizationofeachsetofterms separately, λ2 λ2
discussing the more familiar bulk renormalization first. = φ3(t)K(0)+ φ(t)φ2(t )K(t t )
0 0
−16 16 −
λ2 t
+ φ(t) dt′φ(t′)φ˙(t′)K(t t′), (3.19)
A. Bulk renormalization 8 Z −
t0
where we have defined
Thebulkdivergencesoccuratanarbitrarytime inthe
evolution, so for the purpose of isolating and cancelling
d3~k cos(2ω (t t′))
these divergences, the modified equation of motion for K(t t′)= k − . (3.20)
the zero mode, the integrand of Eq. (3.14), is sufficient. − Z (2π)3 ω3
k
However,unlikethestandardnonequilibriumcalculation,
wemustbemorecarefulsincethetheorycontainsinitial- Thiskernelfunctiondivergeslogarithmicallywhenitsar-
time divergences. In particular, the zero mode equation gumentvanishes;sincethedivergenceisonlylogarithmic,
ofmotioncontainsspuriousdivergencesthatvanishupon the third term on the right side of Eq. (3.19) is finite
integrating its product with the externalpropagatorleg. upon integration but the first two terms are divergent—
Such divergences do not lead to µǫ/ǫ poles in the matrix the first for an arbitraryt and the second only at t=t0.
element—when dimensionally regulating the theory, for Since the first term is proportional to φ3(t), it corre-
example—andthusdonotaffecttherunningoftheinitial sponds to a divergent correction to the coupling and is
conditions. responsible for the familiar running of λ. Applying the
In the renormalization of the bulk and boundary ef- dimensionalregularizationresultforα=3 inEq.(3.16),
fects, we encounter many integrals of a similar general K(0) is
form,
d3~k 1 1 1 4πµ2
d3~k 1 d3~k 1 K(0)=Z (2π)3 ω3 = 4π2 (cid:20)ǫ −γ+ln m2 (cid:21). (3.21)
I(0,α)= = . (3.15) k
Z (2π)3 ωα Z (2π)3 [~k2+m2]α/2
k
Collecting the results of performing the loop integrals
When α 3, these loop integrals are potentially diver- for the standard bulk terms for the finite time-evolved
≤
gentandwecanusedimensionalregularizationtoextract matrix element, we obtain
10
0 = α (t )ψ+(x)α (t )
k f k f
h | | i
tf sin[m(t t)]
f
= dt −
−Z m
t0
λ µ λ 1 λ d3~k eαk
φ¨(t)+m2φ(t) 1 ln +1 γ+ln4π + φ(t) e−2iωk(t−t0)
×(cid:26) (cid:20) − 16π2 m − 32π2 (cid:18)ǫ − (cid:19)(cid:21) 4 Z (2π)3 ω
k
1 3λ2 µ 3λ2 1 λ2 t
+ φ3(t) λ ln γ+ln4π + φ(t) dt′φ(t′)φ˙(t′)K(t t′)
6 (cid:20) − 16π2 m − 32π2 (cid:18)ǫ − (cid:19)(cid:21) 8 Z −
t0
+λ2φ(t)φ2(t )K(t t ) iλ2φ(t) tdt′φ2(t′) d3~k eαk e−2iωk(t−t0) e−2iωk(t′−t0) + . (3.22)
8 0 − 0 − 8 Zt0 Z (2π)3 ωk2 h − i ···(cid:27)
We now rescale the bare parameters of the theory, as Theimportantfeaturetonotefromthebulkrenormal-
in Eq. (3.5), to absorb the ǫ 0 divergences. To order ization so far is that it is, in its short distance behavior,
→
λ, only Z contributes to the mass renormalization, entirely independent of the initial state used. This be-
0
havior is consistent with the principles of effective field
λ 1 theory[11]. Inchoosingthe initialstatetobe otherthan
Z =1+ +1 γ+ln4π , (3.23)
0 32π2 (cid:20)ǫ − (cid:21) the standard vacuum, we have not changed the bulk dy-
namics of the theory and the short distance features of
andonlyZ1 contributesto the couplingrenormalization, thetheoryshouldnotknowaboutthestatewehavecho-
sen once we are sufficiently far from the initial surface.
3λ 1
Defining γ (λ ) and β(λ ) to be
Z =1+ γ+ln4π , (3.24) m R R
1 32π2 (cid:20)ǫ − (cid:21)
µ dm dλ
R R
γ (λ )= , β(λ )=µ , (3.26)
m R R
where we have applied the MS renormalization scheme. m dµ dµ
R
In terms of the renormalized parameters and neglecting
weobtainthestandardrunningofthemassandcoupling
terms of higher order in λ , Eq. (3.22) becomes
R for a ϕ4 theory from Eq. (3.25),
0 = αR(t )ψ+(x)αR(t )
h k tff | sRin[m|R(ktf f it)] γm(λR)= 3λ2Rπ2, β(λR)= 136λπ2R2, (3.27)
= dt −
−Z m
t0 R to leading order in λ .
R
λ µ
φ¨(t)+m2φ(t) 1 R ln It might appear that we have neglected a possible di-
×(cid:26) R (cid:20) − 16π2 mR(cid:21) vergence from the K(t t0) term. Although this term
−
λ 3λ µ only diverges at the initial boundary, it is independent
+ Rφ3(t) 1 R ln of the state and apparently produces a new divergence
6 (cid:20) − 16π2 m (cid:21)
R
even for the vacuum state. Since the divergence is only
λ2 t
+ Rφ(t) dt′φ(t′)φ˙(t′)K(t t′) logarithmic, we actually obtain a finite result upon per-
8 Zt0 − forming the final dt integral. This term provides a first
λ exampleofthesortofspuriousdivergenceswemusttreat
+ Rφ(t)φ2(t )K(t t )
0 0 carefully when deriving the running of the initial state.
8 −
+λRφ(t) d3~k eαke−2iωk(t−t0) Tishtehraetastohneywdeomnuostt ninetgrloedctuctheeasenypoaldedsiitniotnhaeliµntedgerpaennd-
4 Z (2π)3 ω
k dence not already present in λ , m , etc. For example,
R R
iλ2 t d3~k eαk the µ-independence of the bare matrix element,
Rφ(t) dt′φ2(t′)
− 8 Z Z (2π)3 ω2
t0 k µ d α (t )ψ+(x)α (t ) =0, (3.28)
e−2iωk(t−t0) e−2iωk(t′−t0) dµh k f | | k f i
×h − i
implies that the renormalized matrix element satisfies,
+ . (3.25)
···(cid:27)
∂ ∂ ∂
µ +β(λ ) +γ (λ )m
R m R R
The αR(t ) indicates the time-evolved state using the (cid:20) ∂µ ∂λR ∂mR
| k f i
interaction Hamiltonian written in terms of the renor-
+γ(λ )+ αR(t )ψ+(x)αR(t ) =0, (3.29)
malized parameters. R ···(cid:21)h k f | R | k f i