A discrete space and time before the big bang Martin Bojowald The - - PowerPoint PPT Presentation

A discrete space and time before the big bang

Martin Bojowald The Pennsylvania State University Institute for Gravitation and the Cosmos University Park, PA

Time before the big bang – p.1

Gravity

The gravitational field is the only known fundamental force not yet quantized completely, despite of more than six decades of

• research. Difficulties arise due to two key properties:

Time before the big bang – p.2

Gravity

The gravitational field is the only known fundamental force not yet quantized completely, despite of more than six decades of

• research. Difficulties arise due to two key properties:

→ Although it is weak in usual regimes of particle physics, it becomes the dominant player on cosmic scales. Strong quantum gravity effects must appear in large gravitational fields: very early universe and black holes. Then, the classical field grows without bound, implying space-time singularities.

Time before the big bang – p.2

Gravity

The gravitational field is the only known fundamental force not yet quantized completely, despite of more than six decades of

• research. Difficulties arise due to two key properties:

→ Although it is weak in usual regimes of particle physics, it becomes the dominant player on cosmic scales. Strong quantum gravity effects must appear in large gravitational fields: very early universe and black holes. Then, the classical field grows without bound, implying space-time singularities. → Equivalence principle: gravity is a manifestation of space-time ge-

• metry.

The full space-time met- ric gµν is thus the physical object to be quantized non-perturbatively, rather than using perturbations hµν

• n a background space-time ηµν.

Time before the big bang – p.2

Canonical quantum gravity

Combine general relativity and quantum mechanics: “Substitutions” (q, p) − → (qab, pab) where qab is the spatial metric appearing in the line element ds2 = −N2dt2 + qab(dxa + Nadt)(dxb + Nbdt) for a dynamical, curved space-time, and pab its momentum related geometrically to extrinsic curvature Kab =

1 2N (Ltqab − 2D(aNb)).

Time before the big bang – p.3

Canonical quantum gravity

Combine general relativity and quantum mechanics: “Substitutions” (q, p) − → (qab, pab) where qab is the spatial metric appearing in the line element ds2 = −N2dt2 + qab(dxa + Nadt)(dxb + Nbdt) for a dynamical, curved space-time, and pab its momentum related geometrically to extrinsic curvature Kab =

1 2N (Ltqab − 2D(aNb)).

Wave function: ψ(q) − → ψ[qab], subject to i ∂ ∂tψ(q) = ˆ Hψ(q) − → ˆ Hψ[qab] = 0 , ˆ Dψ[qab] = 0 Wheeler–DeWitt equation

Time before the big bang – p.3

Space-time structure

Difficulty: tensor fields such as qab subject to transformation laws, but theory must be coordinate invariant. In generally covariant systems, non-linear change of coordinates qab → ∂x′a′ ∂xa ∂x′b′ ∂xb qa′b′ would lead to coordinate dependent factors not represented on Hilbert space. Moreover, manifold itself is part of the solution in general relativity, not known before quantum operators are defined.

Time before the big bang – p.4

Space-time structure

Difficulty: tensor fields such as qab subject to transformation laws, but theory must be coordinate invariant. In generally covariant systems, non-linear change of coordinates qab → ∂x′a′ ∂xa ∂x′b′ ∂xb qa′b′ would lead to coordinate dependent factors not represented on Hilbert space. Moreover, manifold itself is part of the solution in general relativity, not known before quantum operators are defined. Guides search for suitable building blocks of quantum gravity. One solution: Use index-free objects, holonomies/fluxes. Consequence: Configuration space given by holonomies is compact, geometrical fluxes become derivative operators on compact space with discrete spectra: discrete space.

Time before the big bang – p.4

Loop quantum gravity

Scalar objects based on new variables: canonical transformation (qab, pab) → (−ǫijkeb

j(∂[aek b] + 1 2ec kel a∂[cel b]) + eb iKab, | det(ej b)|ea i )

using co-triad ei

a (three co-vector fields) such that ei aei b = qab.

and its inverse ea

i .

More compactly: (qab, pcd) → (Ai

a, Eb j) with Ashtekar connection

Ai

a and densitized triad Eb

• j. Gauge group SO(3) for rotations.

Time before the big bang – p.5

Loop quantum gravity

Scalar objects based on new variables: canonical transformation (qab, pab) → (−ǫijkeb

j(∂[aek b] + 1 2ec kel a∂[cel b]) + eb iKab, | det(ej b)|ea i )

using co-triad ei

a (three co-vector fields) such that ei aei b = qab.

and its inverse ea

i .

More compactly: (qab, pcd) → (Ai

a, Eb j) with Ashtekar connection

Ai

a and densitized triad Eb

• j. Gauge group SO(3) for rotations.

Variables as in non-Abelian gauge theories: use “lattice” formulation. For any curve e and surface S in space, define holonomies and fluxes he(A) = P exp

• e

Ai

aτi ˙

eadt , FS(E) =

• S

d2ynaEa

i τi

with tangent vector ˙ ea, co-normal na and Pauli matrices τi.

Time before the big bang – p.5

Representation

Holonomies give connection, can be used as non-tensorial configuration variables: wave functions ψ[he(Ai

a)].

Take values in SU(2): compact.

[Quantization on compact space, e.g. circle: wave functions ψn(φ) = exp(inφ) with integer n. Momentum eigenvalues discrete: ˆ pψn(φ) = −idψn(φ)/dφ = nψn(φ).]

Fluxes canonically conjugate, become derivative operators on SU(2), analogous to angular momentum: discrete spectra.

Time before the big bang – p.6

Representation

Holonomies give connection, can be used as non-tensorial configuration variables: wave functions ψ[he(Ai

a)].

Take values in SU(2): compact.

[Quantization on compact space, e.g. circle: wave functions ψn(φ) = exp(inφ) with integer n. Momentum eigenvalues discrete: ˆ pψn(φ) = −idψn(φ)/dφ = nψn(φ).]

Fluxes canonically conjugate, become derivative operators on SU(2), analogous to angular momentum: discrete spectra. Construction: spatial metric qab − → densitized triad Ea

i −

→ flux

• perator. Thus, spatial geometry is discrete.

− → Volumes of point sets can only increase in discrete steps when they are enlarged. − → Dynamical growth such as universe expansion appears in discrete steps.

Time before the big bang – p.6

Scale of discreteness

Dimensional argument: Planck length ℓP = √ G ≈ 10−35m.

[Analogy in quantum mechanics: Bohr radius ∝ 2/mee2]

Precise role of Planck length to be determined from calculations. Loop quantum gravity: √γℓP with γ ≈ 0.238 (from black hole entropy). Discreteness levels of geometry: √γℓPn with integer n.

Time before the big bang – p.7

Scale of discreteness

Dimensional argument: Planck length ℓP = √ G ≈ 10−35m.

[Analogy in quantum mechanics: Bohr radius ∝ 2/mee2]

Precise role of Planck length to be determined from calculations. Loop quantum gravity: √γℓP with γ ≈ 0.238 (from black hole entropy). Discreteness levels of geometry: √γℓPn with integer n. Typical interplay for quantum gravity: − → state more semiclassical for higher excitations, larger n − → higher n implies coarser discreteness Quantum behavior at small n and discreteness at large n gives deviations from classical theory. Leverage to be exploited by observations, despite smallness of ℓP.

Time before the big bang – p.7

Quantum cosmology

Discreteness has dynamical implications, most easily seen for isotropic spaces: |E| = a2/4 (scale factor a), A = ˙ a/2. Orthonormal states A|µ = eiµA/2, µ ∈ R and basic operators

• eiµ′A/2|µ

= |µ+µ′ ˆ E|µ =

1 6γℓ2 Pµ|µ

from ˆ E = − 1

3iγG ∂ ∂A.

Time before the big bang – p.8

Quantum cosmology

Discreteness has dynamical implications, most easily seen for isotropic spaces: |E| = a2/4 (scale factor a), A = ˙ a/2. Orthonormal states A|µ = eiµA/2, µ ∈ R and basic operators

• eiµ′A/2|µ

= |µ+µ′ ˆ E|µ =

1 6γℓ2 Pµ|µ

from ˆ E = − 1

3iγG ∂ ∂A.

Operators follow from full holonomy-flux operators. Representation inequivalent to Wheeler–DeWitt representation:

• eiµ′A/2 not continuous in µ′; ˆ

E with discrete spectrum.

Time before the big bang – p.8

Wave function and dynamics

Wave function ψ(a, φ) in Wheeler–DeWitt. ψµ(φ) in loop quantum cosmology subject to difference equation (Vµ+5 − Vµ+3)ψµ+4(φ) − 2(Vµ+1 − Vµ−1)ψµ(φ) +(Vµ−3 − Vµ−5)ψµ−4(φ) = − ˆ Hmatter(µ)ψµ(φ) with volume eigenvalues Vµ = (γℓ2

P|µ|/6)3/2.

Matter Hamiltonian ˆ Hmatter(µ), well-defined in loop quantization.

Time before the big bang – p.9

Wave function and dynamics

Wave function ψ(a, φ) in Wheeler–DeWitt. ψµ(φ) in loop quantum cosmology subject to difference equation (Vµ+5 − Vµ+3)ψµ+4(φ) − 2(Vµ+1 − Vµ−1)ψµ(φ) +(Vµ−3 − Vµ−5)ψµ−4(φ) = − ˆ Hmatter(µ)ψµ(φ) with volume eigenvalues Vµ = (γℓ2

P|µ|/6)3/2.

Matter Hamiltonian ˆ Hmatter(µ), well-defined in loop quantization. µ with both signs: evolution continues to new branch preceding big bang (at µ = 0), provided automatically. Sign sgn(µ) from handedness of triad, spatial orientation. Implied by search for quantizable index-free objects. Orientation flips while going through classical singularity, universe “turns inside out”.

Time before the big bang – p.9

Role of spatial discreteness

Absence of a singularity: Difference equation uniquely connects wave function at both sides of vanishing volume µ = 0. discrete space − →         

• discrete steps of

wave function evolution

• no diverging

a−3 in matter Hamiltonian          − → time before the big bang

Time before the big bang – p.10

Role of spatial discreteness

Absence of a singularity: Difference equation uniquely connects wave function at both sides of vanishing volume µ = 0. discrete space − →         

• discrete steps of

wave function evolution

• no diverging

a−3 in matter Hamiltonian          − → time before the big bang Inverse in matter Hamiltonian e.g. Hφ = 1

2a−3p2 φ + a3V (φ) for

scalar φ with momentum pφ.

• a−3

bounded

• perator

when expressed through holonomies and fluxes.

2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a l=0 l=1/4 l=1/2 l=3/4 l=0.999

Time before the big bang – p.10

The role of matter

Absence of singularity independent of form of matter: recurrence of difference equation (Vµ+5 − Vµ+3)ψµ+4(φ) − 2(Vµ+1 − Vµ−1)ψµ(φ) +(Vµ−3 − Vµ−5)ψµ−4(φ) = − ˆ Hmatter(µ)ψµ(φ) not affected.

Time before the big bang – p.11

The role of matter

Absence of singularity independent of form of matter: recurrence of difference equation (Vµ+5 − Vµ+3)ψµ+4(φ) − 2(Vµ+1 − Vµ−1)ψµ(φ) +(Vµ−3 − Vµ−5)ψµ−4(φ) = − ˆ Hmatter(µ)ψµ(φ) not affected. But: Orientation reversal key in transition through singularity. Parity violation on matter side ( ˆ Hmatter(µ) = ˆ Hmatter(−µ)) makes equation non-invariant under µ → −µ. State of the universe before and after the big bang must then be

• different. Important to understand origin of the universe before

the big bang.

Time before the big bang – p.11

Effective picture

Difficult to analyze more generally, but effective equations available: Solve dynamics for expectation value ˆ V of volume directly, rather than using wave function with its interpretational

• problems. Scheme:

d dt ˆ V = [ ˆ V , ˆ H] i = · · · in terms of other degrees of freedom such as exp(iA) but also higher moments of the state, in particular fluctuations.

Time before the big bang – p.12

Effective picture

Difficult to analyze more generally, but effective equations available: Solve dynamics for expectation value ˆ V of volume directly, rather than using wave function with its interpretational

• problems. Scheme:

d dt ˆ V = [ ˆ V , ˆ H] i = · · · in terms of other degrees of freedom such as exp(iA) but also higher moments of the state, in particular fluctuations. Infinitely many coupled equations in general, but closely related solvable model available where [ ˆ V , ˆ H] is linear in basic

• perators.

Isotropic model with free, massless scalar: plays the role of the harmonic oscillator for cosmology.

Time before the big bang – p.12

Effective picture

Difficult to analyze more generally, but effective equations available: Solve dynamics for expectation value ˆ V of volume directly, rather than using wave function with its interpretational

• problems. Scheme:

d dt ˆ V = [ ˆ V , ˆ H] i = · · · in terms of other degrees of freedom such as exp(iA) but also higher moments of the state, in particular fluctuations. Infinitely many coupled equations in general, but closely related solvable model available where [ ˆ V , ˆ H] is linear in basic

• perators.

Isotropic model with free, massless scalar: plays the role of the harmonic oscillator for cosmology.

Time before the big bang – p.12

Conclusions

Loop quantum gravity establishes discrete fundamental picture

• f space based on well-defined canonical quantization of gravity.

Space-time dynamics can be analyzed in cosmological models. Direct route from discrete space to non-singular evolution: time before the big bang.

Time before the big bang – p.13

Conclusions

Loop quantum gravity establishes discrete fundamental picture

• f space based on well-defined canonical quantization of gravity.

Space-time dynamics can be analyzed in cosmological models. Direct route from discrete space to non-singular evolution: time before the big bang. Absence of singularity independent of precise form of matter, but details play a role for the precise transition. In particular,

• rientation reversal at the place of the classical singularity.

Thus, parity violating matter affects how pre- and post-big bang physics connect. Yet to be put into equations (torsion coupling), but effective equations allow more straightforward analysis. Show possibility

• f parity asymmetric solutions even from gravity itself: state

spreads during evolution, fluctuations in general not symmetric.

Time before the big bang – p.13