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8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Quantum geometry and quantum dynamics

at the Planck scale

Martin Bojowald

The Pennsylvania State UniversityInstitute for Gravitation and the Cosmos

University Park, PA

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Space-time in canonical quantum gravity

Canonical formulations provide insights in underlyingsymmetries, corresponding to general covariance for gravity.

Quantization implies correction terms which can change theunderlying symmetries or even provide new quantum degrees offreedom.

In loop quantum gravity, corrections arise from quantumgeometry (spatial structure) as well as quantum dynamics.

Main recent developments (model systems or perturbations):

Consistent deformations of classical gravity. Effective description to derive interacting quantum states

and quantum corrections in equations of motion.

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Canonical gravity

Infinite dimensional phase space of fields qab (spatial metric) and

pab (momenta, related to extrinsic curvature).Other components N and Na of the space-time metric

ds2 = gdxdx = N2dt2 + qab(dxa + Nadt)(dxb + Nbdt)

are not dynamical since N and Na do not occur in the action.

Thus, momenta pN = S/N and pNa vanish identically, and somust pN and pNa . Implies constraints

C = det q16G (3)R 16Gdet q (pabpab12(paa)2) = 0 , Ca = 2Dbpba = 0

Diffeomorphism constraint D[Na] = d3xNaCa generates

spatial diffeomorphisms, Hamiltonian constraint C completesspace-time transformations.

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Constrained dynamics

Dynamics of general relativity in canonical formulation

determined by constraints: H[N, Na] =

d3x(N C+ NaCa) = 0

for all multiplier functions N, Na.

Generates equations of motion f = {f, H[N, Na]} for any phasespace function f(q, p), dot refers to time gauge as given by lapseN and shift Na.

Consistency requirement:

H[M, Ma] = {H[M, Ma], H[N, Na]} = 0 must vanish for all Nand Na for fields satisfying the constraints

first class constraint algebra, to be realized also afterincluding quantum corrections. Quite restrictive.

Loop Quantum Cosmology p.

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Constraint algebra

Specific constraint algebra for gravity: hypersurfacedeformations

{H[N1], H[N2]} = D qab(N1bN2 N2bN1)

N

N

NN

N

2

2

1

1

a

Basic information about space-time manifold. Not very sensitive

to dynamics, e.g. higher curvature actions.

Loop Quantum Cosmology p.

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Quantum corrections

After quantization, constraints will change and show quantumspace-time structure without direct reference to manifold orcoordinates.

Three types of corrections, in general equally important:

Entire states evolve which spread and deform. Quantumfluctuations, correlations and higher moments are independent

variables back-reactingon expectation values.

In loop quantum gravity, holonomieshe(A) = Pexp(e A

iaidt) as non-local, non-linear functions

imply higher order corrections.

In loop quantum gravity, fluxes quantizing metric havediscrete spectra containing zero. Inverse metric components

receive corrections for discrete (lattice-like) states with smallelementary areas.

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Quantum Friedmann equation

[PRL 100 (2008) 221301]a

a

2=

8G

3

1 Q

crit

1

2

1 Q

crit ( P) +(

P)2

( + P)2 2

where P is pressure and parameterizes quantum correlations,

Q := + 0crit + ( P)k=0

k+1( P)k/( + P)k

with fluctuation parameters k; crit = 3/8G2 with scale .Simple behavior if = P (free, massless scalar): bounSingh.[P. Singh, PRD 73 (2006) 063508] Also if = 0 when Q = crit.

But: so far no consistent inhomogeneous formulation withholonomy corrections.

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Inverse metric corrections

[with G. Hossain, M. Kagan, S. Shankaranarayanan: PRD 78 (2008) 063547]Corrected Hamiltonian, perturbative around FRW:

HQgrav :=1

16Gd3xNH

(0) + (2)

H(0) +

HQ(2)

+ NHQ(1)

with H(0) = 6H2a and

HQ(1) =

4(1 + f)

HacjK

jc

(1 + g)

H2

a

jcEcj +

2

a

cjEcj

HQ(2) = aKjc Kkd ckdj a(Kjc cj)2 2H

aEcjK

jc

H2

2a3 E

c

jE

d

k

k

c

j

d + H2

4a3 (E

c

j

j

c)

2

(1 + h) jk

2a3(cE

cj )(dE

dk)

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

(r)

r=1/2r=3/4

r=1r=3/2

r=2

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Anomaly cancellation

First class algebra to second order if 2f + g = 0 and

h

f +

a

a

= 0

f g 2a fa

a

a= 0

f + g

a

g

a+

a

a= 0

1

6

a

Ecja3

+(2)

(Eai )(aj

ci cjai ) = 0

as well as a condition for matter correction functions in terms of2 (if matter is present).

All coefficients fixed in terms of , which can be derived in

models.

Loop Quantum Cosmology p.

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Quantum constraint algebra

Anomaly-free constraints including quantum corrections exist:consistent deformation. Underlying discreteness (inverse metriccorrections) does not destroy general covariance.

Constraint algebra of hypersurface deformations quantumcorrected(contains inverse metric):

{HQ

[N1], HQ

[N2]} = D 2N a1/2a(N2 N1)

N

N

NN

N

2

2

1

1

a

Loop Quantum Cosmology p.1

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Quantum constraint algebra

Anomaly-free constraints including quantum corrections exist:consistent deformation. Underlying discreteness (inverse metriccorrections) does not destroy general covariance.

Constraint algebra of hypersurface deformations quantumcorrected(contains inverse metric):

{HQ

[N1], HQ

[N2]} = D 2N a1/2a(N2 N1)

Provides consistent cosmological perturbation equations:

Hamiltonian constraint, diffeomorphism constraint and evolutionequations.

Quantum corrections to constraints also change gauge invariant

variables. Consistent perturbation equations can be formulatedin terms of only gauge invariant quantities.

Loop Quantum Cosmology p.1

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Corrected perturbation equations

c

+ H(1 + f)

= G

c

GI

(2

) 3H(1 + f) + H(1 + f)= 4G

(1 + f3)

GI 2(1 + f1) + a2V,()GI

+ H

2

1 a

2

d

da

+ (1 + f)

+2 H + H21 + a

2

df

da a

2

d

da(1 + f)

= 4G

GI a2V,()GI

Now being used for cosmological applications (power spectrum),e.g. non-conservation of power on large scales.

Loop Quantum Cosmology p.1

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Number of atoms

Corrections depend on discrete building blocks. Spatialgeometry subdivided dynamically. (Avoids macroscopic lattice.)

Elementary size 0 (coordinate length), total volume V0. Number

of atoms of geometry: N= V0/30, densityN/V0 = 30 .Basic holonomies exp(i0a/N) and fluxes =

20a

2/2P ifgeometry nearly isotropic.

V0

0l0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

(r)

r=1/2r=3/4

r=1r=3/2

r=2

Loop Quantum Cosmology p.1

8/3/2019 Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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Number of atoms

Corrections depend on discrete building blocks. Spatialgeometry subdivided dynamically. (Avoids macroscopic lattice.)

Elementary size 0 (coordinate length), total volume V0. Number

of atoms of geometry: N= V0/30, densityN/V0 = 30 .Basic holonomies exp(i0a/N) and fluxes =

20a

2/2P ifgeometry nearly isotropic.

Holonomy corrections when curvature a/N k = (N/V0)1/3.Inverse metric corrections large when a a = (N/V0)1/3P.

(For a a, physical vertex densityN/a3

V0 = (a/a)3

/3

P nearone per Planck volume.)

Classical range: a/N k, a a, determined by vertexdensity

N/V0 of quantum geometry state.

Loop Quantum Cosmology p.1

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