Generalizing Everett’s Quantum Mechanics for Quantum Cosmology James Hartle University of California, Santa Barbara
Quantum Cosmology • If the universe is a quantum predictions. Ψ system, it has a quantum state. • A theory of this state and calculations of its observable predictions are the objectives of quantum cosmology. • Such a theory is a necessary part of any final theory. Otherwise there are no
Why Extrapolate Quantum Mechanics to Cosmology? • The ever expanding domain of success of Zeilinger, grp quantum theory on laboratory scales. • The remarkable lack of 14000 12000 alternative ideas. All counts in 40 s 10000 current fundamental 8000 theories are quantum 6000 mechanical. 4000 spectrometer background level 58 59 60 61 62 63 position of 3rd grating ( µ m)
Quantum Mechanics Permits a Simple Fundamental Theory of the Universe’s Initial State. • Were the laws deterministic, present complexity would have to be encoded in the fundamental initial condition. • But in quantum mechanics, present complexity can arise from the quantum accidents of past history. Hubble Ultradeep Field
Example of a Current Question in Quantum Cosmology T. Hertog, S.W. Hawking, J.H. • Assume the no-boundary theory of the initial quantum state. • Assume a matter field and a positive Λ . • What it the probability that the universe behaved classically in the past and bounced at a minimum radius R?
What Quantum Cosmology Requires from Quantum Mechanics • Probabilities for alternative coarse-grained histories of geometry and matter fields. • Coarse-grained alternatives defined in four- dimensional, diffeomorphism invariant terms. • Alternatives for the past as well as future history.
The Past in Cosmology and in Quantum Mechanics • Reconstruction of the past in cosmology is essential to understand our present and simplify the prediction of the future. • Decoherent histories quantum theory allows a coherent discussion of the past in quantum mechanics through probabilities for past histories conditioned on present data and the initial condition of the universe
Cosmology is the Killer App for Everett Quantum Mechanics
Everett’s Quantum Mechanics • The textbook quantum mechanics of measurements and observers has to be generalized to apply to cosmology. • Everett’s key idea was to take quantum mechanics seriously for the universe. • Understanding quantum mechanics for cosmology helps understand how it applies in the laboratory.
Decoherent Histories QM • Everett’s ideas were extended and clarified by many. • The modern synthesis of decoherent histories quantum theory is adequate for the model cosmology of fields in a box when quantum gravity is neglected. • But we don’t live in a box, and quantum gravity is not negligible in cosmology. • A further generalization is needed.
Quantum Mechanics and Spacetime Familiar quantum theory assumes a fixed spacetime: • To define the “t” in the Schroedinger equation: i ¯ hd | Ψ � /dt = H | Ψ � • To define the spacelike surfaces on which the wave function is reduced on measurement or on which alternatives are defined in decoherent histories: | Ψ � → P | Ψ � / || P | Ψ �|| α n ( t n ) · · · P 1 | Ψ α � = P n α 1 ( t α 1 ) | Ψ � • But in quantum gravity spacetime geometry is fluctuating and without definite value so a generalization of these laws of evolution is needed.
Quantum Mechanics and Spacetime Familiar quantum theory assumes a fixed spacetime: • To define the “t” in the Schroedinger equation: i ¯ hd | Ψ � /dt = H | Ψ � • To define the spacelike surfaces on which the wave function is reduced on measurement or on which alternatives are defined in decoherent histories: | Ψ � → P | Ψ � / || P | Ψ �|| α n ( t n ) · · · P 1 | Ψ α � = P n α 1 ( t α 1 ) | Ψ � • But in quantum gravity spacetime geometry is fluctuating and without definite value so a generalization of these laws of evolution is needed.
The Simplicity of Everett QM • The conceptual simplicity of the Everett formulations provide a springboard for generalizations and extensions, because they are free from a fundamental dependence on complex physical phenomena such as measurements, observers, consciousness, etc. • Measurements, observers, consciousness can be understood within quantum mechanics, but a detailed understanding is not necessary to understand quantum mechanics or its generalizations.
Generalized Quantum Theory (Gell-Mann, Isham, Linden, .....) 1. The sets of fine-grained histories. 2. The sets of coarse grained histories. (Generally partitions of the sets of fine-grained histories into classes { α }). 3. A decoherence functional D defining the interference between coarse-grained Superposition Princ. If { β } is a histories and satisfying i) coarse graining of { α }: Hermiticty, ii) normalization, iii) positivity, and iv) the � � D ( β � , β ) = D ( α � , α ) principle of superposition. → α � ∈ β α ∈ β
Generalized Quantum Theory (cont’d) • Decoherence: D ( β � , β ) ≈ δ β � β p ( β ) • The probabilities p( β ) so defined are consistent as a consequence of decoherence. � p ( β ) = p ( α ) α ∈ β • The decoherence functional of DH is one way of satisfying the axioms but not the only way. D ( β � , β ) ≡ � Ψ β � | Ψ β � β n ( t n ) · · · P 1 | Ψ β � = P n β 1 ( t 1 ) | Ψ � • Therein lies the possibility of generalization.
Key Idea about Histories: Histories need not describe evolution in spacetime but can describe evolution of spacetime.
Fine Grained Histories of Spacetime 4d metrics Simplicial Spin foams with matter geometries fields.
Fine Grained Histories of Spacetime 4d metrics Simplicial Spin foams with matter geometries fields.
Coarse Graining Every assertion that can be made about the universe corresponds to a partition of the fine-grained histories in the class where it is true and the class where it is false. Example: Bounce Problem: • A partition in to the class C that are classical (to some approx.) and the class (NC) that are not. • A partition of C into the class CB which bounce and the class CS which are singular.
Measure of Interference (Schematic) • Branch State Vectors (e.g for classical bounce, CB) � | Ψ CB � = δgδφ exp ( iS [ g, φ ]) | Ψ no bound � CB • Decoherence functional: D ( α � , α ) ≡ � Ψ � α | Ψ α � • Decoherence and probabilities: D ( α � , α ) ≈ δ α � α p ( α ) p ( CB ) = || | Ψ CB �|| 2 Ask speaker for details afterwards.
Example of a Current Question in Quantum Cosmology T. Hertog, S.W. Hawking, J.H. • Assume the no-boundary theory of the initial quantum state. • Assume a matter field is the solution to p ( CB ) = || Ψ CB || 2 this problem and a positive Λ . • What it the probability that the universe behaved classically in the past and bounced at a minimum radius R?
What’s Real? • Whether one set of histories is more real than other sets, • Or whether one history in that set is real and the others are not, • Or whether all the sets and all the histories are equally real. Doesn’t seem to have much to do with the calculation of the probability that the universe bounces at a small radius or its interpretation.
A Fully Four-Dimensional Formulation • Fine grained histories: 4d histories of spacetime geometry and matter fields. • Coarse grainings: partitions of the fine grained histories into 4d diffeomorphism invariant classes. • Measure of Interference: decoherence functional defined by 4d sums over histories. Is there an equivalent 3+1 formulation in terms of the evolution of states on spacelike surfaces?
3+1 From 4-d Non-Relativistic QM factorization of path integrals across spacelike surfaces B B T T Feynman ‘48 = dx t x 0 0 A A � � � B, T | A, 0 � ≡ δx exp( iS [ x ( t )] / ¯ h ) = dxψ ∗ B ( x, t ) ψ A ( x, t ) [ A,B ] � where ψ A ( x, t ) ≡ δx exp( iS [ x ( t )] / ¯ h ) [ A,x ] We derive states on spacelike surfaces, their inner products, and their unitary evolution idψ A /dt = Hψ A
Requirements for a 3+1 Formulation • Fine-grained histories that are single valued in a time variable. • Alternatives at a moment of time. But, in quantum gravity: • Histories of spacetime geometry are not single valued in any time variable. • There are no diffeo invariant alternatives at a moment of time. There is a 4-d formulation of quantum mechanics but not a 3+1 formulation.
Recovering States Approximately when Geometry is Approximately Classical • For coarse-grainings defining geometry well above the Planck Scale and for particular initial conditions | Ψ � the semiclassical approximation to the sum over geometries may be adequate: � � | Ψ α � ≡ α δgδφ exp( iS [ g, φ ]) | Ψ � ≈ α δφ exp( iS [ˆ g, φ ])) | Ψ � • This defines a quantum field theory on a background spacetime which gives: ˆ g • well defined time(s). • states on spacelike surfaces • alternatives at a moment of time • unitary evolution
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