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Consistent Quantum Histories and the Probability for Singularity Resolution Consistent Quantum Histories and the Probability for Singularity Resolution Parampreet Singh Department of Physics & Astronomy Louisiana State University Fundamental Problems in Quantum Physics ICTS, Bengaluru (Based on: David Craig, PS, Phys. Rev. D 82, 123526 (2010); Found.Phys. 41, 371 (2011), Class. Quantum Grav. 30, 205008 (2013), arXiv:1603.09671 (World Scientific (2017)), and work to appear soon) 1 / 21 In the last decade, rigorous quantization of homogeneous cosmological spacetimes has been performed using techniques of loop quantum gravity. Physical Hilbert space, inner product and observables are known for various loop quantized spacetimes. Singularity resolution has been understood in terms of expectation values of observables such as volume and energy density. How do we extract the probability of singularity resolution? What are the implications for Wheeler-DeWitt quantum universes? Outline: Introduction and Consistent Histories approach Wheeler-DeWitt theory and singularities Loop quantum cosmology and bounce Exactly solvable loop quantum cosmology Probabilities in Wheeler-DeWitt and loop quantum cosmologies. Towards a covariant generalization Summary 2 / 21 Additional difficulties in quantum gravitational models. Problem of time. Quite hard to obtain physical Hilbert space, inner product, and expectation values of observables. How do we assign probabilities in a closed quantum system corresponding to a quantum gravitational spacetime? Consistent histories formulation: (Gell-Mann, Griffiths, Hartle, Halliwell, Omnes, ...) If predictions are to be extracted from quantum system, then it must satisfy a consistency condition. Branch wavefunctions of alternative quantum histories must have no interference. When the interference between the histories vanishes, consistent probabilities can be assigned. Introduction In a closed system, conventional notion of measurements in the quantum theory is inadequate. No external observers/environment. 3 / 21 How do we assign probabilities in a closed quantum system corresponding to a quantum gravitational spacetime? Consistent histories formulation: (Gell-Mann, Griffiths, Hartle, Halliwell, Omnes, ...) If predictions are to be extracted from quantum system, then it must satisfy a consistency condition. Branch wavefunctions of alternative quantum histories must have no interference. When the interference between the histories vanishes, consistent probabilities can be assigned. Introduction In a closed system, conventional notion of measurements in the quantum theory is inadequate. No external observers/environment. Additional difficulties in quantum gravitational models. Problem of time. Quite hard to obtain physical Hilbert space, inner product, and expectation values of observables. 3 / 21 Consistent histories formulation: (Gell-Mann, Griffiths, Hartle, Halliwell, Omnes, ...) If predictions are to be extracted from quantum system, then it must satisfy a consistency condition. Branch wavefunctions of alternative quantum histories must have no interference. When the interference between the histories vanishes, consistent probabilities can be assigned. Introduction In a closed system, conventional notion of measurements in the quantum theory is inadequate. No external observers/environment. Additional difficulties in quantum gravitational models. Problem of time. Quite hard to obtain physical Hilbert space, inner product, and expectation values of observables. How do we assign probabilities in a closed quantum system corresponding to a quantum gravitational spacetime? 3 / 21 Introduction In a closed system, conventional notion of measurements in the quantum theory is inadequate. No external observers/environment. Additional difficulties in quantum gravitational models. Problem of time. Quite hard to obtain physical Hilbert space, inner product, and expectation values of observables. How do we assign probabilities in a closed quantum system corresponding to a quantum gravitational spacetime? Consistent histories formulation: (Gell-Mann, Griffiths, Hartle, Halliwell, Omnes, ...) If predictions are to be extracted from quantum system, then it must satisfy a consistency condition. Branch wavefunctions of alternative quantum histories must have no interference. When the interference between the histories vanishes, consistent probabilities can be assigned. 3 / 21 Coarse grained histories: Physically meaningful partitions of the fine grained histories. Examples: Only diffeomorphism invariant partitions allowed in a covariant quantization of gravity; which slit the particle went through? Many physical questions tied to such histories. Decoherence functional: Measures the quantum mechanical interference between alternative coarse grained histories and assigns probabilities if interference vanishes. Sets of histories with negligible interference between all the members of the set decohere. Probabilities can be assigned to these consistent histories. Consistent histories approach Fine grained histories: Most refined descriptions of the Universe. Examples: Individual paths in a path integral formulation; detailed time history of a particle in a two-slit experiment. 4 / 21 Decoherence functional: Measures the quantum mechanical interference between alternative coarse grained histories and assigns probabilities if interference vanishes. Sets of histories with negligible interference between all the members of the set decohere. Probabilities can be assigned to these consistent histories. Consistent histories approach Fine grained histories: Most refined descriptions of the Universe. Examples: Individual paths in a path integral formulation; detailed time history of a particle in a two-slit experiment. Coarse grained histories: Physically meaningful partitions of the fine grained histories. Examples: Only diffeomorphism invariant partitions allowed in a covariant quantization of gravity; which slit the particle went through? Many physical questions tied to such histories. 4 / 21 Consistent histories approach Fine grained histories: Most refined descriptions of the Universe. Examples: Individual paths in a path integral formulation; detailed time history of a particle in a two-slit experiment. Coarse grained histories: Physically meaningful partitions of the fine grained histories. Examples: Only diffeomorphism invariant partitions allowed in a covariant quantization of gravity; which slit the particle went through? Many physical questions tied to such histories. Decoherence functional: Measures the quantum mechanical interference between alternative coarse grained histories and assigns probabilities if interference vanishes. Sets of histories with negligible interference between all the members of the set decohere. Probabilities can be assigned to these consistent histories. 4 / 21 Histories, class operators and branch wavefunctions Consider a system with a Hilbert space H, self-adjoint Hamiltonian α α H and a family of observables A with eigenvalues ak . −iH(t −t )= Propagator: U(ti; tj) = e i j ~ Heisenberg projections: P α (t) = U y(t)P α U(t) ak ak Sets of alternative histories defined by giving a time sequence of 1 n sets of events, such as fPα1 (t1)g:::fPα1 (tn)g Example of a coarse grained history: Observer decides to measure which slit electron went through at t1, electron passed through lower slit at t2, and strikes the screen at point yn at t3 3 2 1 Pyn (t3)PL(t2)Pobs(t1) 5 / 21 Each fine-grained history h can be represented by a class operator α1 α2 αn X Ch = P (t1)P (t2) ··· P (tn) with Ch = 1 ak1 ak2 akn h Coarse-grained histories correspond to coarse-grainings of the projections at some or all of the times ti: X X X α α αn C = ··· P 1 (t1)P 2 (t2) ··· P (tn) h ak1 ak2 akn a 2∆a¯ a 2∆a¯ a 2∆a¯ k1 k1 k2 k2 kn kn (involves a union of alternatives at any time ti) An exclusive, exhaustive set of fine-grained histories for the system may be regarded as the set of sequences of eigenvalues (taken at all times ti) fhg = f(aα1 ; aα2 ; : : : ; aαn )g k1 k2 kn 6 / 21 Coarse-grained histories correspond to coarse-grainings of the projections at some or all of the times ti: X X X α α αn C = ··· P 1 (t1)P 2 (t2) ··· P (tn) h ak1 ak2 akn a 2∆a¯ a 2∆a¯ a 2∆a¯ k1 k1 k2 k2 kn kn (involves a union of alternatives at any time ti) An exclusive, exhaustive set of fine-grained histories for the system may be regarded as the set of sequences of eigenvalues (taken at all times ti) fhg = f(aα1 ; aα2 ; : : : ; aαn )g k1 k2 kn Each fine-grained history h can be represented by a class operator α1 α2 αn X Ch = P (t1)P (t2) ··· P (tn) with Ch = 1 ak1 ak2 akn h 6 / 21 An exclusive, exhaustive set of fine-grained histories for the system may be regarded as the set of sequences of eigenvalues (taken at all times ti) fhg = f(aα1 ; aα2 ; : : : ; aαn )g k1 k2 kn Each fine-grained history h can be represented by a class operator α1 α2 αn X Ch = P (t1)P (t2) ··· P (tn) with Ch = 1 ak1 ak2 akn h Coarse-grained histories correspond to coarse-grainings of the projections at some or all of the times ti: X X X α α αn C = ··· P 1 (t1)P 2 (t2) ··· P (tn) h ak1 ak2 akn a 2∆a¯ a 2∆a¯ a 2∆a¯ k1 k1 k2 k2 kn kn (involves a union of alternatives at any time ti) 6 / 21 0 0 Decoherence functional: d(h; h ) = h hj hi When histories do not interfere, d(h; h0) = 0 for h 6= h0. Probability for a particular history: 2 p(h) = d(h; h) = kChj ik P with h p(h) = 1. Quantum state of the system corresponding to a history h: y αn α1 j hi ≡ C j i = P (tn) ··· P (t1)j i h akn ak1 (Branch wave function) A pure initial state can be resolved in to branch wave functions: X j i = j hi h 7 / 21 Quantum state of the system corresponding to a history h: y αn α1 j hi ≡ C j i = P (tn) ··· P (t1)j i h akn ak1 (Branch wave function) A pure initial state can be resolved in to branch wave functions: X j i = j hi h 0 0 Decoherence functional: d(h; h ) = h hj hi When histories do not interfere, d(h; h0) = 0 for h 6= h0. Probability for a particular history: 2 p(h) = d(h; h) = kChj ik P with h p(h) = 1.
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