Zheng-Cheng Gu and Xiao-Gang Wen- A lattice bosonic model as a quantum theory of gravity

8/3/2019 Zheng-Cheng Gu and Xiao-Gang Wen- A lattice bosonic model as a quantum theory of gravity

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A lattice bosonic model as a quantum theory of gravity

Zheng-Cheng Gu, and Xiao-Gang Wen

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Center for Advanced Study, Tsinghua University, Beijing, China, 100084

(Dated: June 22, 2006)

A local quantum bosonic model on a lattice is constructed whose low energy excitations aregravitons described by linearized Einstein action. Thus the bosonic model is a quantum theory

of gravity, at least at the linear level. We find that the compactification and the discretization ofmetric tenor are crucial in obtaining a quantum theory of gravity.

Seven wonders of our universe: Our world hasmany mysteries and wonders. At the most fundamentallevel, there are may be seven deep mysteries and/or won-ders in our universe: (1) Identical particles. (2) Gaugeinteractions.13 (3) Fermi statistics.4,5 (4) Tiny masses offermions ( 1020 of the Planck mass).6,7,12 (5) Chiralfermions.8,9 (6) Lorentz invariance.10 (7) Gravity.11 Herewe would like to question where those wonderful and mys-terious properties come from. We wish to have a singleunified understanding of all of the above mysteries. Or

more precisely, we wish that we can start from a singlemodel to obtain all of the above wonderful properties.

Recent progresses showed that, starting from a sin-gle origin a local bosonic model (which is also calledspin model), the first four properties in the above listemerge at low energies.1215 Thus we may say that thelocal bosonic model provides a unified origin for iden-tical particles, gauge interactions, Fermi statistics andnear masslessness of the fermions. Gauge interaction andFermi statistics are unified under this point of view ofemergence. However, three more mysteries remain to beunderstood. In this paper, we will show that it is possibleto construct a local bosonic model from which gravitonsemerge. On one hand, such a model can be viewed asa theory of quantum gravity. It solves a long standingproblem of putting quantum mechanics and gravity to-gether. On the other hand, the model provides a designof a condensed matter system which has an emergentquantum gravity at low energies. This allows us to studysome quantum effects of gravity (a simulated one) in alaboratory.

The belief that all the wonderful phenomena of ouruniverse (such as gauge interaction, Fermi statistics, andgravitons) emerge from a lowly local bosonic model iscalled locality principle.12 Using local bosonic model asa underlying structure to understand the deep myster-ies of our universe represent a departure from the tra-ditional approach of gaining a better understanding bydividing things in to smaller parts. In the traditionalapproach, we assume that the division cannot continueforever and view everything as made of some simple indi-visible building blocks the elementary particles. How-ever, the traditional approach may represent a wrong di-rection. For example, phonons behave just like any otherelementary particles at low energies. But if we look atphonons closely, we do not see smaller parts that form a

phonon. We see the atoms that form the entire lattice.The phonons are not formed by those atoms, the phononsare simply collective motions of those atoms. This makesus to wonder that photons, electrons, gravitons, etc , mayalso be emergent phenomena just like phonons. Theymay not be the building blocks of everything. They maybe collective motions of a deeper underlying structure.

This paper uses a particular emergence approach: wetry to obtain everything from a local bosonic model. Thedetail form of the bosonic model is not important. Theimportant issue is how the bosons (or the spins) are or-ganized in the ground state. It is shown that if bosonsorganize into a string-net condensed state, then photons,electrons and quarks can emerge naturally as collectivemotions of the bosons.1215 In this paper, we will find anorganization of bosons such that the collective motionsof bosons lead to gravitons.

Other emergence approaches were developed in super-string theory16 in last 10 years, as demonstrated by theduality relations among various superstring models andmatrix models.17 The anti-de Sitter-space/conformal-field-theory duality even shows how space-time and grav-ity emerge from a gauge theory.18

The rule of game: There are many different ap-proaches to quantum gravity2022 based on different prin-ciples. Some approaches, such as loop quantum gravity,23

stress the gauge structure from the diffeomorphism ofthe space-time. Other approaches, such as superstringtheory,16,17 stress the renormalizability of the theory. Inthis paper, we follow a different rule of game by stressingfiniteness and locality.

Our rule of game is encoded in the following workingdefinition of quantum gravity. Quantum gravity is(a) a quantum theory.(b) Its Hilbert space has a finite dimension.(c) Its Hamiltonian is a sum of local operators.(d) The gapless helicity 2 excitations are the only lowenergy excitations.(e) The helicity 2 excitations have a linear dispersion.(f) The gravitons35 can interact in the way consistentwith experimental observations.

The condition (b) implies that the quantum gravityconsidered here has a finite cut-off. So the renormaliz-ability is not an issue here. The condition (c) is a localitycondition. It implies two additional things: (1) the totalHilbert space is a direct product of local Hilbert spaces


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Htot = nHn, (2) the local operators are defined as op-erators that act within each local Hilbert space Hn orfinite products of local operators. The conditions (a c) actually define a local bosonic model. Certainly, anyquantum spin models satisfy (a c). It is the conditions(df) that make the theory to look like gravity.

The condition (d) is very important. It is very easyto construct a quantum model that contains helicity 2gapless excitations, such as the theory described by thefollowing Lagrangian L = 1

20h

ij0hij 1

2kh

ijkhij .

Such a theory is not a theory of gravity since it alsocontain helicity 0, 1 gapless excitations.

Superstring theory16,17 satisfies the conditions (a), (e)and (f), but in general not (d) due to the presence ofdilatons (massless scaler particles). The superstring the-ory (or more precisely, the superstring field theory) alsodoes not satisfy the condition (b) since the cut-off is notexplicitly implemented. The spin network24 or the quan-tum computing25 approach to quantum gravity satisfiesthe condition (a,b) or (ac). But the properties (df) areremain to be shown. The induced gravity from superfluid3

He discussed in Ref. 26 does not satisfy the condition (d)due to the presence of gapless density mode. In Ref. 27,it is proposed that gravitons may emerge as edge exci-tations of a quantum Hall state in 4 spatial dimensions.Again the condition (d) is not satisfied due to the pres-ence of infinite massless helicity 1 ,2, 3, modes.

In Ref. 28, a very interesting bosonic model is con-structed which contains gapless helicity 0 and 2 exci-tations with quadratic dispersions. The model satisfies(ac), but not (d,e). In this paper, we will fix the twoproblems and construct a local bosonic model that satis-fies the conditions (a e) and possibly (f). To quadraticorder, the low energy effective theory of our model is thelinearized Einstein gravity. The key step in our approachis to discretize and compactify the metric tensor.

Review of emergence of U(1) gauge theory: Ourmodel for emergent quantum gravity is closely relatedto the rotor model that produces emergent U(1) gaugetheory.19,30 So we will first discuss the emergence ofU(1)gauge theory to explain the key steps in our argument ina simpler setting.

To describe the rotor model, We introduce an angularvariable aij aij + 2 and the corresponding angularmomentum Eij for each link of a cubic lattice. Here ilabels the sites of the cubic lattice and aij and Eij satisfyaij = aji and Eij = Eji. The phase space Lagrangianfor physical degrees of freedom aij and

Eij is given by

19

L =ij

Eij aij J2

ij

E2ij + gijkl

cosBijkl U2

i

Q2i

Bijkl = aij + ajk + akl + ali, Qi =

j next to i

Eij (1)

where

i sums over all sites,

ij over all links, andijkl over all square faces of the cubic lattice. We note

that after quantization, Eij are quantized as integers.

To obtain the low energy dynamics of the above ro-tor model, let us assume that the fluctuation of Eij arelarge and treat Eij as a continuous quantity. We alsoassume that the fluctuations of aij are small and expand(1) to the quadratic order of aij . Then we take the con-tinuum limit by introducing the continuous fields (Ei, ai)and identifying Eij =

ji

dxiEi and aij =ji

dxiai. Herewe have assumed that the lattice constant a = 1. The

resulting continuum effective theory is given by

L = Ei0ai 12

J(Ei)2 12

g(Bi)2 12

U(iEi)2, (2)

where Bi = ijkjak. We find that the rotor model hasthree low lying modes. Two of them are helicity 1modes with a linear dispersion k

gJ|k| and the

third mode is the helicity 0 mode with zero frequencyk = 0. We know that a U(1) gauge theory only havetwo helicity 1 modes at low energies. Thus the key tounderstand the emergence U(1) gauge theory is to un-derstand how the helicity 0 mode obtain an energy gap.

To understand why helicity 0 mode is gapped, let usconsider the quantum fluctuations ofEi and ai. We notethat the longitudinal mode and the transverse modes sep-arate. Introduce Ei = E|| +E and ai = a|| + a, we findthat the dynamics of the transverse mode is describedby L = a0E J2 E2 g2 iaia. At the latticescale x 1, the quantum fluctuations ofE and a aregiven by E

gJ and a

Jg . We see that the as-

sumptions that we used to derive the continuum limit arevalid when J g. In this limit we can trust the resultfrom the continuum effective theory and conclude thatthe transverse modes (or the helicity 1 modes) have alinear gapless dispersion.

The longitudinal mode is described by (f(x), (x))with ai = if and = iEi. Its dynamics is determinedby L|| = 0f J2 (2) 12 U 2. At the lattice scale,the quantum fluctuations of and f are given by = 0and f = . We see that a positive U and J will makethe fluctuations of f much bigger than the compactifi-cation size 2 and the fluctuations of much less thenthe discreteness ofEi which is 1. In this limit, the resultfrom the classical equation of motion cannot be trusted.

In fact the weak quantum fluctuations in the discretevariable and the strong quantum fluctuations in thecompact variable f indicate that the corresponding modeis gapped after the quantization. Since has weak fluctu-ations which is less than the discreteness of , the groundstate is basically given by = 0. A low lying excitationis then given by = 0 everywhere except in a unit cellwhere = 1. Such an excitation have an energy of or-der U. The gapping of helicity 0 mode is confirmed bymore careful calculations.19 From those calculations, wefind that the weak fluctuations of lead to a constraint = iEi = 0 and the strong fluctuations of f lead toa gauge transformation ai ai + if. The Lagrangian(2) equipped with the above constraint and the gaugetransformation becomes the Lagrangian of a U(1) gauge


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theory. We will use this kind of argument to argue theemergence of gravitons and the linearized Einstein action.

The emergence of quantum gravity: First, let usdescribe a model that will have emergent gravitons atlow energies. The model has six variables xx(i), yy(i),zz(i), L

xx(i), Lyy(i), and Lzz(i) on each vertex of acubic lattice. The model also has two variables on eachsquare face of the cubic lattice. For example, on the

square centered at i+x2 +

y

2 , the two variables are xy(i+x2

+ y2

) = yx(i +x2

+ y2

) and Lxy(i + x2

+ y2

) = Lyx(i +x2

+ y2

). The bosonic model is described by the followingphase space Lagrangian

L =

i,ab=xy,yz,zx

Lab(i +a

2+b

2)0ab(i +

a

2+b

2)

+

i,a=x,y,z

Laa(i)0aa(i) HU HJHg (3)

where

HU =nGU1i

a=x,y,z

{1 cos[2Q(i, i + a)/nG]}

+nGU2i

{1 cos[(i)]}

HJ =nGJ

i,a=x,y,z

{1 cos[2Laa(i)/nG]}

+2nGJi

ab=xy,yz,zx

{1 cos[2Lab(i + a2

+b

2)/nG]}

12

nGJi

{1 cos[2

a=x,y,z

Laa(i)/nG]}

Hg =nGg

4

i,a=x,y,z

{1 cos[aa(i)]}

+nGg

4

i,ab=xy,yz,zxsin[

ab (i)] sin[

ba(i)]} (4)

Here ij(i), (i) and Q(i, i + x) are defined as xx(i) =

zx(i + y +z2

+ x2

) + zx(i +z2

+ x2

) xy(i + z +x2

+ y2

) xy(i + x2 + y2 ), xy(i) = yz(i + z2 +y

2) yz(i + z2 y2 ) + 2yy(i + z) + 2yy(i), xz (i) =2zz(i+y) 2zz(i) + yz(i+ y2 + z2 ) + yz(i+ y2 z2 ),

(i) =

a=x,y,z

b=x,y,z[bb(i + a) + bb(i a) +

2bb(i)]

a=x,y,z[aa(i + a) + aa(i a) + 2aa(i)] ab=xy,yz,zx[ab(i +

a2

+ b2

) + ab(i a2 + b2 ) + ab(i +a2 b

2) + ab(i a2 b2 )], and Q(i, i+x) = Lxx(i+x) +

Lxx(i) + Lyx(i+ x2

+ y2

) + Lyx(i+ x2 y

2) + Lzx(i+ x

2+

z

2 ) + Lzx

(i+x

2 z

2 ). Other components are obtained bycycling xyz to yzx and zxy.Note that both ab and its canonical conjugate Lab are

compactified: ab ab + 2, Lab Lab + nG. Henceafter quantization they are both discretized and nG isan integer. Due to the compactification, only WabL =

e2 iLab/nG , Wab = e

iab and their products are physicaloperators. For a fix ab and i, WabL (i) and W

ab (i) satisfy

the algebra

WabL (i)Wab (i) = e

2 i/nGWab (i)WabL (i) (5)

Such an algebra has only one nG dimensional represen-tation. This nG dimensional representation becomes ourlocal Hilbert space Hi,ab. The total Hilbert space of ourmodel (3) is given by H = i,abHi,ab after quantization.In other words, there are n3G states on each vertex andnG states on each square face of the cubic lattice. Notethat the Hamiltonian

H = HU + HJ + Hg (6)

is a function of the physical operators WabL , Wab and

their hermitian conjugates. So the quantum model de-fined through the Hamiltonian (6) and the algebra (5)is a bosonic model whose local Hilbert spaces have finitedimensions.

Next, we would like to understand low energy excita-tions of the quantum bosonic model (6) in large nG limit.We first assume that the fluctuations of ij 2Lij/nGand ij are much bigger then 1/nG (the discreteness ofij and ij) so that we can treat

ij and ij as contin-uous variables. We also assume that the fluctuations of

ij

and ij are much smaller then 1 so that we can treatij and ij as small variables. Under those assumptions,we can use semiclassical approach to understand the lowenergy dynamics of the quantum bosonic model (6).

Expanding the Lagrangian (3) to quadratic order in ij

and ij , we can find the dispersions of collective modesof the bosonic model. There are total of six collectivemodes. We find four of them have zero frequency for allk, and two modes have a linear dispersion relation neark = (, , ). Near k = (, , ), the dynamics of thesix modes are described by the following continuum fieldtheory:

L = nGij

ij J

2

(ij

)2

(ii)2

2

g

2 ijRij

U12

(iij)2 U2

2(Rii)2

(7)

where Rij = imkjlnmlnk and we define the contin-uum field ab(x) as (1)i 1

2ab(i+ a

2+ b

2) for a = b and

as (1)iab(i) for a = b. The helicity 2 modes havea linear dispersion relation k

gJ|k|. We find that

for large nG, the quantum fluctuations of the helicity 2modes is of order ij, ij

1/nG (assuming U1,2, J

and g are of the same order). So the fluctuations of ij

and ij satisfy 1/nG ij , ij 1 and the semiclassi-cal approximation is valid for the helicity

2 modes. In

this case, the result k gJ|k| can be trusted.The helicity 1 modes and one of the helicity 0 mode

are described by ij = ij + ji and i = jji. Theirfrequency k = 0. For such modes, the Hamiltonianonly contains i. Thus the quantum fluctuations satis-fies i 1/nG and i 1. So the semiclassical ap-proximation is not valid and the result k = 0 cannot betrusted. Using the similar argument used in emergenceof U(1) gauge bosons, we conclude that those modes aregapped. The strong fluctuations ij = ij + ji 1


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and the weak fluctuations i 1/nG lead to gauge trans-formations and the constraints

ij ij + ij + ji, jji = 0 (8)The second helicity 0 mode is described by ij =

(ij2ij) and = (ij2ij)ij. Its frequency is

again k = 0. The Hamiltonian for such a mode contains

only . So the quantum fluctuations satisfies 1 and 1/nG. The second helicity 0 mode is also gapped.The strong fluctuations ij = (ij

2 ij) 1 andthe weak fluctuations = (ij

2 ij)ij 1/nG leadto a gauge transformation and a constraint

ij ij + (ij2ij), (ij2ij)ij = 0 (9)The Lagrangian (7) equipped with the gauge trans-

formations and the constraints (8,9) is nothing but thelinearized Einstein Lagrangian of gravity, where ij gij ij represents the fluctuations of the metric tenorgij around the flat space. So the linearized Einstein grav-ity emerge from the quantum model (6) in the large nG

limit. The local bosonic model (6) can be viewed as aquantum theory of gravity.

We have seen that the gapping of the helicity 0 mode inthe rotor model (1) leads to an emergence of U(1) gaugestructure at low energies. The emergence of a gaugestructure also represents a new kind of order quantumorder12,31 in the ground state. In Ref. 32, it was shownthat the emergent U(1) gauge invariance, and hence thequantum order, is robust against any local perturbationsof the rotor model. Thus the gaplessness of the emer-gent photon is protected by the quantum order.33 Sim-ilarly, the gapping of the two helicity 0 modes and the

helicity 1 modes in the bosonic model (6) leads to anemergent gauge invariance of the linearized coordinatetransformation. This indicates that the ground state ofthe bosonic model contains a new kind of quantum orderthat is different from those associated with emergent or-dinary gauge invariances of internal degrees of freedom.We expect such an emergent linearized diffeomorphisminvariance to be robust against any local perturbation of

the bosonic model. Thus the gaplessness of the emergentgravitons is protected by the quantum order.

The emergent gravitons in the model (3) naturally in-teract but the interaction may be different from that de-scribed by the higher order non-linear terms in Einsteingravity. However, those higher order terms are irrelevantat low energies. Thus it may be possible to generatethose higher order terms by fine tuning the lattice model(3), such as modifying the Hamiltonian (HJ and Hg),the constraints (HU), as well as the Berrys phase termin (3). So it may be possible that local bosonic modelscan generate proper non-linear terms to satisfy (f).29

Our result appears to contradict with the Weinberg-

Witten theorem34

which states that in all theories witha Lorentz-covariant energy-momentum tensor, compos-ite as well as elementary massless particles with helicityh > 1 are forbidden. However, the energy-momentumtensor in our model is not invariant under the linearizeddiffeomorphism (although the action is invariant). Thismay be the reason why emergent gravitons are possiblein our model.

We would like to thank J. Polchinski and E. Witten fortheir very helpful comments. This research is supportedby NSF grant No. DMR-0433632 and ARO grant No.W911NF-05-1-0474.

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(1973).7 H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).8 T. Lee and C. Yang, Phys. Rev. 104, 254 (1956).9 C. Wu et al., Phys. Rev. 105, 1413 (1957).10 A. Einstein, Annalen der Physik 17, 891 (1905).11 A. Einstein, Annalen der Physik 49, 769 (1916).12 X.-G. Wen, Phys. Rev. D 68, 065003 (2003).13 M. A. Levin and X.-G. Wen, Rev. Mod. Phys. 77, 871

(2005).14 M. Levin and X.-G. Wen, hep-th/0507118.15 X.-G. Wen, cond-mat/0508020.16 M. B. Green, J. H. Schwarz, and E. Witten, Superstring

Theory (Cambridge University Press, 1988).17 J. Polchinski, String Theory (Cambridge University Press,

1998).18 O. Aharony etal, Phys. Rept. 323, 183 (2000).19 X.-G. Wen, Quantum Field Theory of Many-Body Systems

From the Origin of Sound to an Origin of Light andElectrons (Oxford Univ. Press, Oxford, 2004).

20 T. Thiemann, gr-qc/0210094.21 S. Carlip, Rept. Prog. Phys. 64, 885 (2001).22 P. Peldan, Class. Quant. Grav. 11, 1087 (1994).23 L. Smolin, hep-th/0408048.24 C. Rovelli and L. Smolin, Phys.Rev. D52, 5743 (1995).25 L. Smolin, hep-th/0507235.26 G. Volovik, J. Low. Temp. Phys. 113, 667 (1998).27 S.-C. Zhang and J. Hu, Science 294, 823 (2001).28 C. Xu, cond-mat/0602443.29 Z.-C. Gu and X.-G. Wen, to appear.30 O. I. Motrunich and T. Senthil, Phys. Rev. Lett. 89,

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gapless helicity 2 excitations.

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Zheng-Cheng Gu and Xiao-Gang Wen- A lattice bosonic model as a quantum theory of gravity