Background Independent Quantum Field Theory and the Cosmological Constant Problem

0.12 MB·English

by  Olaf Dreyer

Background Independent Quantum Field Theory
and the Cosmological Constant Problem
Olaf Dreyer∗
Perimeter Institute for Theoretical Physics, 35 King Street North, Waterloo, Ontario N2J 2W9, Canada
(Dated: February 1, 2008)
We introduce the notion of background independent quantum field theory. The distinguishing
feature of this theory is that the dynamics can be formulated without recourse to a background
metricstructure. Weshowinasimplemodelhowthemetricpropertiesofspacetimecanberecovered
from thedynamics. Background independenceis not only conceptually desirable but allows for the
resolutionofaproblemhauntingordinaryquantumfieldtheory: thecosmologicalconstantproblem.
PACSnumbers: 03.70.+k,98.80.-k,03.30.+p
5
0 I. INTRODUCTION ity is background-independent, namely, space and time
0 are dynamical degrees of freedom, it is difficult to see
2 howgravitycanbe includedinthissecondpointofview.
There are currently two competing views of the role
n In this paper, we want to address this objection by
quantumfieldtheoryplaysinourtheoreticalunderstand-
a revisitingthesolidstatemodelsfromabackgroundinde-
J ingofnature. Inoneview,quantumfieldtheorydescribes pendentperspective. Onecanformulatethedynamicsof
8 the dynamics of the elementary constituents of matter. a largeclassofmodels withoutrecourseto a background
2 Thejobofthe physicististofigureoutwhatthe elemen- structure. Notionsofdistancecanthenberecoveredfrom
taryparticlesareandtofindtheappropriateLagrangian
the dynamics of the theory.
2 that describes the interactions. The Standard Model of
Thelackofbackgroundindependenceisnotjustacon-
v
elementary particle physics is the epitome of this view
8 ceptual shortcoming. We argue that if the theory is for-
(see[1]foranauthoritativeexpositionofthisview). The
4 mulated in a background independent way it allows for
other point of view likens the use of quantum field the-
0 thesolutionofoneofthelong-standingproblemsofquan-
9 oryto its useinotherareasofphysics,mostimportantly tum field theory: the cosmologicalconstant problem.
0 in solid state physics. Here, quantum field theory is not
Thisproblemariseswhenoneviewsquantumfieldthe-
4 used to describe the dynamics of elementary particles.
oryasatheorydescribingfieldslivingonacurvedspace-
0 In solid state physics, this would be fruitless, since the
time. This view runs into a serious problem when one
/
h dynamics of a large number of atoms is usually beyond considers the effect the quantum fields should have on
t anyone’s ability to compute. It turns out, however, that
- spacetime. Sinceallthemodesofthequantumfieldhave
p these large numbers of constituents often have collective a zero energy of ±1/2~ω[12], one expects a contribution
e excitations that can be well-described by quantum field
to the vacuum energy on the order of
h
theory and that are responsible for the physical proper-
: Ξ
v tiesofthematerial. Theviewisthenthattheelementary dω ~ω3 ∼~Ξ4, (1)
i particlesoftheStandardModelarelikethecollectiveex- Z
X
citations of solid state physics. The world we live in is
whereΞissomehighenergycut-off. Ifonetakesthiscut-
r
just another materialwhose excitations are described by
a off to be the Planck energy the vacuum energy is some
the Standard Model. The point of view was introduced
123 orders of magnitude away from the observed value
by P. W. Anderson [2] and has since found a large fol-
of the cosmological constant [6], making this the worst
lowing (see e.g. [3, 4, 5] and references therein).
predictionin theoreticalphysics (for detailed discussions
The search for a quantum theory of gravity requires of this problem see [7, 8]).
a unification of quantum field theory and general rela- We will see that the cosmological constant problem
tivity. If the second point of view is correct, it should arisesonly whenoneregardsthequantumfieldsasliving
not be restricted to the Standard Model, but also in- on the background. This is where this problem connects
clude gravity. From this perspective, the second point with the conceptual problemof backgrounddependence.
of view possesses one objectionable feature: the mate- If, instead, it is the quantum fields that make the space-
rial whose excitations give rise to the elementary parti- timeappearinthefirstplace,andtheyarenottreatedas
cles around us is assumed to rest in a Newtonian world livingonthebackground,thenthecosmologicalconstant
whose notions of distance are taken over to formulate problem disappears.
the theory. That is, the condensed matter models are Theorganizationofthepaperisasfollows. Inthenext
manifestly background-dependent. Since general relativ- sectionwedemonstratewhatwemeanbybackgroundin-
dependent quantum field theory in a very simple model,
inwhich we showhow the dynamics of the model canbe
formulatedwithoutagivenbackgroundstructure. Wein-
∗Electronicaddress: [email protected] dicate how the spacetime picture can be recovered from
2
the dynamics of the theory. Section III contains the ar-
gumentwhythe cosmologicalconstantproblemdoesnot
appearinbackgroundindependentquantumfieldtheory.
Toshowthiswe lookatthe coarsegraineddescriptionof
the model in section II. We show that where there was
no cosmological constant problem in the original model,
itappearsifabackgroundis assumed. Inthe concluding
sectionIV, we remarkon the rolespecial relativityplays
in our approach and discuss what our results mean for
the search for a theory of quantum gravity.
FIG. 1: The dispersion relation for the one-dimensional XY-
model. In the state where all the negative energy levels are
II. A MODEL filled the dispersion relation becomes linear at the points A
and B.Close to thesepointsthespin chain looks relativistic.
We will illustrate background independent quantum
fieldtheory ona verysimple example: The XY-modelin
one spatial dimension. The Hamiltonian is given by After performing a Fourier transformation,the Hamilto-
nian now takes the simple form:
N
H =J (S+S− +S−S+ ). (2) N
⊥ i i+1 i i+1 H = ε(k)f†f , (10)
Xi=1 k k
Xk=1
TheoperatorsS± areformedfromthePaulimatricesσa,
where the energy is given by
a=1,2,3:
2π
S± = S1±iS2, (3) ε(k)=4πJ⊥cos k. (11)
N
1
Sa = σa. (4) The formof the spectrum is shownin Figure 1. We thus
2 find a system of free fermions. If we choose as a ground
The index i ranges over the N lattice sites of the one- stateofthesystemthestatewhereallthenegativemodes
dimensionallattice. Wehaveassumedonlynearestneigh- are filled, we find that the excitations over this ground
bor coupling and have not included the interaction be- state have a linear dispersion relation
tweenthe 3-componentsof the spins. We are thus in the 2π
quantumlimitoftheone-dimensionalspinchain. Wewill ∆ε=4πJ⊥ ∆k ≡vF∆k. (12)
N
impose acircletopology,sothatN+1isto be identified
Note that, to derive the above spectrum, we did not
with 1.
The S± satisfy mixed commutation relations. They assume any lattice spacing. It is enough to know which
spins are in relation to each other and how they inter-
commute for different sites and anti-commute on the
act. Inthis sensethe systemis backgroundindependent.
same site:
That is, we do not need to assume a background metric
{S±,S−} = 1, (5) structure to derive the dynamics of the system.
i i
How then can we define notions like distance in our
system?
[S+,S−] = 0, for i6=j. (6)
i j The answer lies in the excitation spectrum that we
have just derived. A point in the system can be defined
The S± can thus be thought of as creating or annihilat-
by the intersection of two traveling wave-packets made
ing hard-core bosons. To have more standard commuta-
up of the fermionic excitations of equation (7). The dis-
tion relations,and alsoto more readilysolve the system,
tancebetweenpointsisdeterminedoncethespeedofthe
one can perform a Jordan-Wigner transformation [9] to
traveling wave-packets is defined. This is analogous to
purely fermionic variables, f , i=1,...,N:
i how we practically define distance, by the value we give
f =U S−, f† =S+U† (7) to the speed of light c. Since the dispersion relation is
i i i i i i
linear, all wave-packetswill travel with the same speed.
In the limit where the width of the wave-packetscom-
prisesofa largenumber ofspins, the discretespinmodel
i−1
U =exp iπ S+S− . (8) willappearsmoothtoanobserverinthemodel. Because
i (cid:18) j j (cid:19) alltheexcitationshavethesamespeed,themodelisthen
Xj=1
perceivedas two-dimensionalMinkowskispace. Thus, in
These operators now behave like real fermions: the background-independent form of this model, the ge-
ometry of Minkowskispace is containedin the dynamics
{f†,f }=δ . (9) of the model and not in the kinematics.
i j ij
3
Theobserver’scosmologicalconstantproblemisreally
TABLE I:Fundamentaland emergent view.
aparadoxthatcanberesolvedifonecomparestheemer-
fundamental view emergent view gent view, described by the ψ’s, with the more funda-
fermionic mentalview,describedbythef’s(seetableI). TheFock
Hilbert H=(C2)⊗N Fock space vacuum|0iistobecomparedtothefilledFermisea|gndi
space
H− and the 1~ω in the ψ-description are like the energy of
2
the filled Fermi sea in the f-description. The ground
N
dimH 2 ∞ state |gndi does not have any a priori metric proper-
ties. It is only when the excitations are examined that
Minkowski the two-dimensional Minkowski space appears, for exci-
ground filled Fermi
state sea |gndi spvaacceuu+mF|o0cik tations over the ground state |gndi. Thus, the 21~ω are
not amounts of energy sitting on a spacetime and curv-
ingit. Theyarepartofapre-geometryandcannotcurve
excitations fi ψα the effective spacetime.
Weseethattherootofthecosmologicalconstantprob-
cocsomnosltoagnitcal – −21 k~ωk. lemliesinthefactthatwehavetreatedasdistinctobjects
P whichin factare one andthe same. If we treatquantum
fieldsaslivingonaspacetime,thenwewillencounterthe
cosmologicalconstantproblem. If,ontheother,handwe
III. THE COSMOLOGICAL CONSTANT
realizethatitisonlythroughtheexcitationsdescribedby
PROBLEM
the quantum fields that a spacetime appears in the first
place, the cosmologicalconstant problem disappears.
How then would an observer in the model, without
access to the microscopic spins, describe the physics of
the system? First, she will find that she lives in a
IV. CONCLUSIONS
two-dimensional Minkowski space. Second, she will find
that the particle content can be well described by two-
“Theworldisnotgiventwice”. WiththesewordsLeib-
component spinor fields ψ , α = 1,2, with Hamiltonian
α
niztriedtoconvinceClarke,intheircorrespondence,that
itiswrongtoviewtheworldasbeingembeddedinafur-
H = dx ψ†(x)βi∂ ψ(x), (13) thercontainer[11]. Whathemeantwasthatitisenough
x
Z to give the relationsbetween the materialthings making
up the world. To add the metric information was like
where
giving the world twice.
Inthispaperwearguedthatthecosmologicalconstant
0 1
β =σ1 = . (14) problem arises because we describe the worldas if it has
(cid:18)1 0(cid:19)
to be given twice. First, there are the quantum fields,
The relation of the spinor field ψ to the variables f of then there is the spacetime on which they live. Using a
α i
the previous section is given as follows: First we define simple model fromsolidstate physics we showedthatno
the two-component spinor ψ¯ , α=1,2: metric informationis neededinthe initialformulationof
α
the theory, but a two-dimensional Minkowski spacetime
ψ¯1(m) = i2m f(2m) (15) canbe recoveredfromthe dynamics. This is becausethe
ψ¯2(m) = i2m+1 f(2m+1), (16) relations between the excitations give the appearance of
spacetime in the appropriate limits.
for m = 1,...,N/2. The two-component spinor ψ is In summary, we constructed a background indepen-
α
then obtained from ψ¯ by appropriate rescaling so that dentquantum field theoryusing the simplest model pos-
α
theFermivelocityvF becomesunity(formoredetailssee sible. We want to stress though that the resolution of
[10, chapter 4]). the cosmological constant problem does not depend on
When she quantizes the system, she will do so by con- the model or its dimensionality. This simple model was
structing a fermionic Fock space H−. The Fock vacuum sufficienttopresenttheargument,butthegoalhastobe
|0i∈H− isinterpretedbythecoarse-grainedobserveras amodelthatgivesrisetogravityaswellasthe Standard
empty two-dimensional Minkowski space. Model. One expects that such a model will have more
At this point, she runs into the cosmological constant complicated couplings. Models that show how fermions
problem. She will notice that all the modes in H− con- and gauge theories arise exist [5] and fit into our ap-
tribute −1~ω to the vacuum energy. The contribution proach. Theinclusionofgravityiscurrentlyunderinves-
2
to the cosmological constant should thus be tigation.
Inourmodel,wewereabletoshowthatthe cosmolog-
1
− ~ω . (17) ical constant problem is a paradox that appears only if
k
2Xk spacetime is regarded as fundamental rather then emer-
4
gent. Eventhoughwedealtexclusivelywithaflatspace- A consequence of the setup as it is presented here is
time, this is appropriate for the cosmological constant that special relativity is no longer fundamental. It only
problem since it is usually presented in exactly this con- arises in the limit in which the spin model looks smooth
text. We argued that the solution of the problem can andit does soonly because the dispersionrelationis rel-
also be given without the inclusion of gravity. ativistic. This is to be contrasted with the usual view of
In quantum field theory one encounters a number of quantum field theory in which special relativity plays a
effects that are due to vacuum fluctuations. Examples fundamentalrole. Inourapproach,theemergenceofspe-
includetheCasimireffectandtheLambshift. Onemight cial relativity is a dynamical effect. One striking feature
wonder whether these observed effects disappear in our of special relativity is that all particles have the same
viewofquantumfieldtheory. Theeasiestwaytoseethat light cone. This raises an important question: What
this is not a concern is to realize that the description of dynamical effect ensures that all particles of the theory
thelowenergyphysicsintermsofaquantumfieldtheory share the same light cone?
isavaliddescriptioninitsdomain. Lowenergyphenom- It is important that, while the argument presented in
enalikethe Casimireffectorthe Lambshift thusremain this paper can only be conclusive in a full treatment in-
untouched. The point of the argument is to show that cluding gravity, it also indicates where one should look
the quantum field theory picture is just that: a picture. foraquantummechanicaltheorythatencompassesgrav-
It runs into trouble when applied outside its domain. ity. Itmakesclearthatanyapproachtoquantumgravity
Recent experimental data [6] indicates that we live in shouldtreatmatterandspacetimeastwomanifestations
universe with a small cosmological constant. The argu- of the same thing. An approach that puts matter on a
ment presented here says that there should be no cos- spacetimewillencounterthecosmologicalconstantprob-
mological constant that has its origin in the zero energy lem.
of quantum fields. The argumentdoes not exclude other
sources for a cosmological constant.
Inourconstruction,Fock-space-likestructuresareonly
approximately present. We can exactly identify one-
Acknowledgments
particle Hilbert spaces but, for interacting theories, we
canidentify manyparticle statesonly whenthe particles
are sufficiently far apart. That Fock spaces are only ap- The author would like to thank J. Barbour, T. Ja-
proximate is also clear from the fact that the dimension cobson,F. Markopoulou-Kalamara,D. Oriti, H. Pfeiffer,
of our Hilbert space is finite. It does not have enough L. Smolin, J. Stachel, A. Valentini, and H. Westman for
room for an infinite-dimensional Fock space. fruitful discussions and constructive criticism.
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[2] P.W. Anderson, Sience 177, 393 (1972). [9] P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
[3] G.E.Volovik, The universe in a helium droplet (Oxford [10] E. Fradkin, Field theories of condensed matter systems
UniversityPress, 2003). (Perseus Books, Cambridge, 1991).
[4] R.B. Laughlin, Int.J. Mod. Phys. A18, 831 (2003). [11] H. G. Alexander, The Leibniz-Clarke Correspondence
[5] X.-G. Wen, Quantum field theory of many-body systems (Manchester UniversityPress, 1956).
(Oxford University Press, 2004). [12] The sign depends on whether the particles are fermions
[6] D. N. Spergel et al, Astrophys. J., Suppl. Ser. 148, 175 or bosons.
(2003).