General Relativity without paradigm of space-time covariance: sensible quantum gravity and resolution of the “problem of time” Hoi-Lai YU Institute of Physics, Academia Sinica, Taiwan. 2, March, 2012 Co-author: Chopin Soo Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 1 / 22
Overview Introduction The purposes of this talk: Main Themes of this work: Theory of gravity with only spatial covariance, construction of local Hamil- tonian for dynamical evolution and resolution of “problem of time” Any sensible quantum theory of time has to link quantum time devel- opments to passage of time measured by physical clocks in classical space-times! Where/what is physical time in Quantum Gravity? Outlines of this talk: 1 Hints/Ingredients for a sensible theory of Quantum Gravity 2 Theory of gravity without full space-time covariance General framework, and quantum theory Emergence of classical space-time Paradigm shift and resolution of “problem of time” Improvements to the quantum theory 3 Further discussions Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 2 / 22
Overview Introduction Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 3 / 22
Overview Introduction Hints/Ingredients for a sensible theory of quantum gravity 1 Dynamics of spacetime doesn’t make sense, geometrodynamics evolves in Superspace 2 QG wave functions are generically distributional, ∴ concept of a particular spacetime cannot be fundamental, then why 4D covariance? 3 GR cannot enforce full 4D spacetime covariance off-shell , fundamental symmetry (classical and quantum) is 3D diffeomorphism invariance(arena = Superspace) 4 The local Hamiltonian should not be the generator of symmetry, but determines only dynamics 5 Local Hamiltonian constraint H = 0 replaced by Master Constraint; [ H ( x )] 2 � √ M := q ( x ) = 0, not just math. trick but Paradigm shift Σ 6 DeWitt supermetric has one -ive eigenvalue ⇒ intrinsic time mode 7 A theory of QG should be described by a S-eqt first order in intrinsic time with +ive semi-definite probability density Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 4 / 22
Theory of gravity without full space-time covariance General framework General Framework 1 1 Decomposition of the spatial metric q ij = q 3 ¯ q ij on Σ 1 1 2 Symplectic potential, � π ij δ q ij = � π ij δ ¯ 3 ⇒ ln q 3 and ¯ ˜ ¯ q ij + πδ ln q q ij π ij and traceless are respectively conjugate to π = q ij ˜ π ij = q 1 π ij − q ij π 3 [˜ ¯ 3 ] parts of the original momentum variable 3 Non-trivial Poisson brackets are 1 π ij ( x ′ ) = P ij � � kl δ ( x , x ′ ) , � 3 ( x ) , π ( x ′ ) � = δ ( x , x ′ ) q kl ( x ) , ¯ ¯ ln q P ij k δ j l δ j kl := 1 k ) − 1 2 ( δ i l + δ i q ij ¯ 3 ¯ q kl ; trace-free projector depends on ¯ q ij 1 4 Separation carries over to the quantum theory, the ln q 3 d.o.f separate from others to be identified as temporal information carrier. However, 1 3 can be consistently physical time intervals associated with δ ln q realized only when dogma of full spacetime covariance is relinquished Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 5 / 22
Theory of gravity without full space-time covariance General framework 5 DeWitt supermetric, G ijkl = 1 λ 2 ( q ik q jl + q il q jk ) − 3 λ − 1 q ij q kl , has signature [ sgn ( 1 3 − λ ) , + , + , + , + , +], and comes equipped with 1 3 provided λ > 1 intrinsic temporal intervals δ ln q 3 6 The Hamiltonian constraint is of the general form, √ q π ij ˜ π kl + V ( q ij ) 0 = 2 κ H = G ijkl ˜ 1 3(3 λ − 1) π 2 + ¯ π ij ¯ π kl + V [¯ = − G ijkl ¯ q ij , q ] = − β 2 π 2 + ¯ H 2 [¯ π ij , ¯ q ij , q ] = − ( βπ − ¯ H )( βπ + ¯ H ) � q ij , q ] , β 2 := ¯ π ij ¯ π kl + V [¯ 1 1 π ij , ¯ H [¯ q ij , q ] = 2 (¯ q ik ¯ q jl + ¯ q il ¯ q jk )¯ 3(3 λ − 1) q Einstein’s GR ( λ = 1 and V [¯ q ij , q ] = − (2 κ ) 2 ( R − 2Λ eff )) is a particular realization of a wider class of theories, all of which factorizes marvelously as in the last step Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 6 / 22
Theory of gravity without full space-time covariance General framework Note several important features: Only spatial diffeomorphism is intact H 2 / √ q = 0 equivalently enforces the local � Master constraint, M = constraint and its physical content. H determines dynamical evolution but not generates symmetry. M decouples from H i is attained, paving the road for quantization ⇒ For, theories with only spatial diff. inv. will have physical dynamics dictated by H , but encoded in M M itself does not generate dynamical evolution, but only spatial diff.; { q ij , m ( t ) M + H k [ N k ] }| M =0 ⇔ H =0 ≈ { q ij , H k [ N k ] } = L � N q ij . Therefore, true physical evolution can only be w.r.t to an intrinsic time extracted from the WDW eqt only ( βπ + ¯ H ) = 0 is all that is needed to recover the classical content of H = 0. This is a breakthrough 1 3 , therefore semiclassical HJ eqt is first order (i) π is conjugate to ln q in intrinsic time with consequence of completeness (ii) QG will now be dictated by a corresponding WDW eqt which is a S-eqt first order in intrinsic time Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 7 / 22
Theory of gravity without full space-time covariance Quantum Gravity Quantum Gravity 1 Consistent quantum theory of gravity starts with spatial diff. inv. H ) 2 / √ q . ( βπ + ¯ � M | Ψ � = 0; M := Positive-semi-definite inner product for | Ψ � will equivalently imply π + ˆ ¯ q ]) | Ψ � = 0 and ˆ H [ˆ π ij , ˆ ( β ˆ ¯ q ij , ˆ ¯ H i | Ψ � = 0 π ij = � i P ij 2 In metric representation; ˆ π = 3 � δ ln q , ˆ δ δ ¯ q lk operates on i lk δ ¯ i S � are: Ψ[¯ q ij , q ]; S-eqt and HJ-eqt for semi-classical states Ce δ Ψ = ¯ H [ˆ π ij , q ij ]Ψ i � β ¯ 1 δ ln q 3 δ S δ S π ij = P ij + ¯ β H [¯ ; ¯ q ij ln q ] = 0 1 kl δ ¯ q kl δ ln q 3 ∇ j δ Ψ δ q ij = 0 enforces spatial diffeomorphism symmetry 1 3 True Hamiltonian ¯ 1 H generating intrinsic time evolution w.r.t. β δ ln q 3 Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 8 / 22
Theory of gravity without full space-time covariance Emergence of classical spacetime Emergence of classical spacetime 1 The first order HJ equation bridges quantum and classical regimes, has complete solution S = S ( (3) G ; α ) 2 Constructive interference; S ( (3) G ; α + δα ) = S ( (3) G ; α ); S ( (3) G + δ (3) G ; α + δα ) = S ( (3) G + δ (3) G ; α ) δ S ( (3) G ; α ) ⇒ δ � � � δ q ij = 0 subject to M = H i = 0. δα δ q ij 0 = δ � � ( π ij δ q ij + δ N i H i ) + δ m M � δα q ij + q ij = δ � � 1 3 δ N i ∇ j π + q − 1 3 + ¯ π ij δ ¯ 3 δ N i ∇ j ¯ π ij ) � ( πδ ln q δα δ [¯ ¯ δ ¯ q ij ( x ) − L � Ndt ¯ q ij ( x ) π mn H ( y ) /β ] G klmn ¯ = P kl = P kl δ ( x − y ) ij ij β ¯ 1 1 δ ¯ π kl ( x ) H 3 ( y ) − L � 3 ( y ) δ ln q Ndt ln q M generates no evolution w.r.t unphysical coordinate time, but w.r.t 1 3 through constructive interferences at deeper level intrinsic time ln q Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 9 / 22
Theory of gravity without full space-time covariance Emergence of classical spacetime 4 EOM relates mom. to coord. time derivative of the metric which can be interpreted as extrinsic curvature to allow emergence of spacetime 1 1 3 − L � N q ij ) , Ndt := δ ln q Ndt ln q 2 κ π kl = 1 2 N ( dq ij 3 √ q G ijkl ˜ dt − L � H / √ q ) (4 βκ ¯ In Einstein’s GR with arbitrary lapse function N , the EOM is, dq ij � = 2 N π kl + L � d 3 x [ NH + N i H i ] � � dt = q ij , √ q (2 κ ) G ijkl ˜ N q ij This relates the extrinsic curvature to the momentum by 2 N ( dq ij √ q β ¯ 1 N q ij ) = 2 κ π kl ⇒ 1 3 Tr ( K ) = 2 κ K ij := dt − L � √ q G ijkl ˜ H proves that the lapse function and intrinsic time are precisely related (a posteriori by the EOM) by the same formula in the above for reconstruction of spacetime 5 For theories with full 4-d diff. invariance(i.e. GR), this relation is an identity which does not compromise the arbitrariness of N Soo & Yu (arXiv:1201.3164) Sensible Quantum Gravity 3, March, 2012; APS2012 10 / 22
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