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- ometry. The full space-time met- ric g µν is thus the physical object to be quantized non-perturbatively, rather than using perturbations h µν on a background space-time η µν . Time before the big bang – p.2
Canonical quantum gravity Combine general relativity and quantum mechanics: → ( q ab , p ab ) where q ab is the spatial metric “Substitutions” ( q, p ) − appearing in the line element d s 2 = − N 2 d t 2 + q ab (d x a + N a d t )(d x b + N b d t ) for a dynamical, curved space-time, and p ab its momentum related geometrically to extrinsic curvature 1 K ab = 2 N ( L t q ab − 2 D ( a N b ) ) . Time before the big bang – p.3
Canonical quantum gravity Combine general relativity and quantum mechanics: → ( q ab , p ab ) where q ab is the spatial metric “Substitutions” ( q, p ) − appearing in the line element d s 2 = − N 2 d t 2 + q ab (d x a + N a d t )(d x b + N b d t ) for a dynamical, curved space-time, and p ab its momentum related geometrically to extrinsic curvature 1 K ab = 2 N ( L t q ab − 2 D ( a N b ) ) . Wave function: ψ ( q ) − → ψ [ q ab ] , subject to i � ∂ ∂tψ ( q ) = ˆ Hψ [ q ab ] = 0 , ˆ ˆ Hψ ( q ) − → Dψ [ q ab ] = 0 Wheeler–DeWitt equation Time before the big bang – p.3
Space-time structure Difficulty: tensor fields such as q ab subject to transformation laws, but theory must be coordinate invariant. In generally covariant systems, non-linear change of coordinates q ab �→ ∂x ′ a ′ ∂x ′ b ′ ∂x b q a ′ b ′ ∂x a 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 q ab subject to transformation laws, but theory must be coordinate invariant. In generally covariant systems, non-linear change of coordinates q ab �→ ∂x ′ a ′ ∂x ′ b ′ ∂x b q a ′ b ′ ∂x a 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 i K ab , | det( e j ( q ab , p ab ) �→ ( − ǫ ijk e b j ( ∂ [ a e k b ] + 1 2 e c k e l a ∂ [ c e l b ] ) + e b b ) | e a i ) using co-triad e i a (three co-vector fields) such that e i a e i b = q ab . and its inverse e a i . More compactly: ( q ab , p cd ) �→ ( A i a , E b j ) with Ashtekar connection A i a and densitized triad E b 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 i K ab , | det( e j ( q ab , p ab ) �→ ( − ǫ ijk e b j ( ∂ [ a e k b ] + 1 2 e c k e l a ∂ [ c e l b ] ) + e b b ) | e a i ) using co-triad e i a (three co-vector fields) such that e i a e i b = q ab . and its inverse e a i . More compactly: ( q ab , p cd ) �→ ( A i a , E b j ) with Ashtekar connection A i a and densitized triad E b 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 A i e a d t d 2 yn a E a h e ( A ) = P exp a τ i ˙ , F S ( E ) = i τ i e S e a , co-normal n a and Pauli matrices τ i . with tangent vector ˙ Time before the big bang – p.5
Representation Holonomies give connection, can be used as non-tensorial configuration variables: wave functions ψ [ h e ( A i a )] . Take values in SU(2): compact. [Quantization on compact space, e.g. circle: wave functions ψ n ( φ ) = exp( inφ ) with integer n . Momentum eigenvalues discrete: ˆ pψ n ( φ ) = − i � d ψ n ( φ ) / d φ = � nψ n ( 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 [ h e ( A i a )] . Take values in SU(2): compact. [Quantization on compact space, e.g. circle: wave functions ψ n ( φ ) = exp( inφ ) with integer n . Momentum eigenvalues discrete: ˆ pψ n ( φ ) = − i � d ψ n ( φ ) / d φ = � nψ n ( φ ) .] Fluxes canonically conjugate, become derivative operators on SU(2), analogous to angular momentum: discrete spectra. → densitized triad E a Construction: spatial metric q ab − i − → flux operator. 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 √ G � ≈ 10 − 35 m . Dimensional argument: Planck length ℓ P = [Analogy in quantum mechanics: Bohr radius ∝ � 2 /m e e 2 ] Precise role of Planck length to be determined from calculations. Loop quantum gravity: √ γℓ P with γ ≈ 0 . 238 (from black hole entropy). Discreteness levels of geometry: √ γℓ P n with integer n . Time before the big bang – p.7
Scale of discreteness √ G � ≈ 10 − 35 m . Dimensional argument: Planck length ℓ P = [Analogy in quantum mechanics: Bohr radius ∝ � 2 /m e e 2 ] Precise role of Planck length to be determined from calculations. Loop quantum gravity: √ γℓ P with γ ≈ 0 . 238 (from black hole entropy). Discreteness levels of geometry: √ γℓ P n 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 ℓ P . Time before the big bang – p.7
Quantum cosmology Discreteness has dynamical implications, most easily seen for isotropic spaces: | E | = a 2 / 4 (scale factor a ), A = ˙ a/ 2 . Orthonormal states � A | µ � = e iµA/ 2 , µ ∈ R and basic operators � e iµ ′ A/ 2 | µ � | µ + µ ′ � = ˆ 1 6 γℓ 2 E | µ � = P µ | µ � from ˆ E = − 1 3 i � γG ∂ ∂A . Time before the big bang – p.8
Quantum cosmology Discreteness has dynamical implications, most easily seen for isotropic spaces: | E | = a 2 / 4 (scale factor a ), A = ˙ a/ 2 . Orthonormal states � A | µ � = e iµA/ 2 , µ ∈ R and basic operators � e iµ ′ A/ 2 | µ � | µ + µ ′ � = ˆ 6 γℓ 2 1 E | µ � = P µ | µ � from ˆ E = − 1 3 i � γG ∂ ∂A . Operators follow from full holonomy-flux operators. Representation inequivalent to Wheeler–DeWitt representation: � e iµ ′ 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. ψ µ ( φ ) in loop quantum cosmology subject to difference equation ( V µ +5 − V µ +3 ) ψ µ +4 ( φ ) − 2( V µ +1 − V µ − 1 ) ψ µ ( φ ) +( V µ − 3 − V µ − 5 ) ψ µ − 4 ( φ ) = − ˆ H matter ( µ ) ψ µ ( φ ) with volume eigenvalues V µ = ( γℓ 2 P | µ | / 6) 3 / 2 . Matter Hamiltonian ˆ H matter ( µ ) , well-defined in loop quantization. Time before the big bang – p.9
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