The Quantum to Classical Transition in Inflationary Cosmology

Transcription

1 The Quantum to Classical Transition in Inflationary Cosmology C. D. McCoy Department of Philosophy University of California San Diego Foundations of Physics Munich, 31 July 2013

2 Questions to Address 1. How does the measurement problem arise in cosmological structure formation? 2. Which proposed solutions of the measurement problem are applicable to structure formation? 3. How does the Everett interpretation fare in explaining this quantum to classical" transition?

3 A History of the Universe

4 The Cosmic Microwave Background Radiation Θ(n ) δt (n ) X = a`m Y`m (n ) T0 `,m

5 The Power Spectrum of the Cosmic Microwave Background C l = 1 2l + 1 alm, a lm = 2 π m k 2 dk P Φ (k) 2 l (k)

6 The Power Spectrum of the Cosmic Microwave Background C l = 1 2l + 1 alm, a lm = 2 π m k 2 dk P Φ (k) 2 l (k)

7 The Power Spectrum of the Cosmic Microwave Background C l = 1 2l + 1 alm, a lm = 2 π m k 2 dk P Φ (k) 2 l (k)

8 The Power Spectrum of the Cosmic Microwave Background C l = 1 2l + 1 alm, a lm = 2 π m k 2 dk P Φ (k) 2 l (k)

9 Primordial Perturbations Assume that the primordial spectrum follows a power law: P(k) = A k ns 1 Then the CMB data suggest that the spectrum is approximately scale invariant (P(k) const); 2. small amplitude (A << 1); 3. and approximately Gaussian (n s 1);

10 Primordial Perturbations Assume that the primordial spectrum follows a power law: P(k) = A k ns 1 Then the CMB data suggest that the spectrum is approximately scale invariant (P(k) const); 2. small amplitude (A << 1); 3. and approximately Gaussian (n s 1);...but a naive retrodiction of this spectrum (classical general relativity and relativistic hydrodynamics) results in all modes being generated beyond the horizon (acausal) at early times/high densities.

11 Origin of the Primordial Spectrum

12 Origin of the Primordial Spectrum Inexplicable acausal correlations Horizon Problem!

13 Origin of the Primordial Spectrum

14 Origin of the Primordial Spectrum Inflation solves horizon problem Inflation implemented by a scalar field with a certain potential

15 Quantum Fluctuations as the Origin of the Primordial Spectrum* Einstein-Hilbert Action and Action of a Scalar Field (Inflaton). S = 1 d 4 x R g + d 4 x [ ] 1 φ φ g gµν 16π 2 x µ x ν V (φ) Linearize metric and scalar field, separating homogeneous part from perturbation. g µν (x, t) = ḡ µν (t) + δg µν (x, t) φ(x, t) = φ(t) + δφ(x, t) ds 2 = a 2 (η) [ (1 + 2Φ) dη 2 + (1 2Ψ) γ ij dx i dx j ] *Mukhanov, Feldman, and Brandenberger, Phys Rep 1992 *Mukhanov, Physical Foundations of Cosmology 2005

16 Quantum Fluctuations as the Origin of the Primordial Spectrum Introduce a gauge-invariant combination of the scalar field and metric perturbations, the Mukhanov-Sasaki variable v: v(η, x) = a(δφ + φ φ Ψ) z = a H H And write the action in terms of v and z: S 2 = Ldη d 3 x = 1 ) (v 2 v,i v,i + z 2 z v 2 dη d 3 x v k + (k 2 z z )v k = 0

17 Quantizing the Inflaton Define the momentum π canonically conjugate to v as π = v. Promote v and π to operators obeying the standard commutation relations. Decompose solutions into adiabatic modes with creation and annihilation operators and define the vacuum as â k 0 = 0. Compute the power spectrum: P Φ (k) = Φ k 2 2π 2 = 1 2π 2 ˆΦ(x, η)ˆφ (y, η)

18 Quantizing the Inflaton Define the momentum π canonically conjugate to v as π = v. Promote v and π to operators obeying the standard commutation relations. Decompose solutions into adiabatic modes with creation and annihilation operators and define the vacuum as â k 0 = 0. Compute the power spectrum: P Φ (k) = Φ k 2 2π 2 = 1 2π 2 ˆΦ(x, η)ˆφ (y, η) Quantum correlation function to ensemble average? 0 ˆΦ(x, η)ˆφ(y, η) 0 ˆΦ(x, η)ˆφ (y, η)

19 Quantum Theory without Observers Quantum theory applied to cosmology must be a quantum theory without observers.

20 Quantum Theory without Observers Quantum theory applied to cosmology must be a quantum theory without observers. 1. Collapse Theories Martin, Vennin, and Peter, Phys. Rev. D 2012; Lochan, Das, and Bassi, Phys. Rev. D 2012; Cañate, Pearle, and Sudarsky, Phys, Rev. D 2013

21 Quantum Theory without Observers Quantum theory applied to cosmology must be a quantum theory without observers. 1. Collapse Theories Martin, Vennin, and Peter, Phys. Rev. D 2012; Lochan, Das, and Bassi, Phys. Rev. D 2012; Cañate, Pearle, and Sudarsky, Phys, Rev. D Bohmian Theories Valentini, Phys. Rev. D 2010; Pinto-Neto, Santos, and Struyve, Phys. Rev. D 2012

22 Quantum Theory without Observers Quantum theory applied to cosmology must be a quantum theory without observers. 1. Collapse Theories Martin, Vennin, and Peter, Phys. Rev. D 2012; Lochan, Das, and Bassi, Phys. Rev. D 2012; Cañate, Pearle, and Sudarsky, Phys, Rev. D Bohmian Theories Valentini, Phys. Rev. D 2010; Pinto-Neto, Santos, and Struyve, Phys. Rev. D Everett interpretation???

23 Classicality Condition for Many Worlds (Wallace) A history is a sequence of time-indexed PVMs. The set of such histories generated from some such sequence of PVMs is a history space P. The history operator Ĉα of the history α is Ĉ α = ˆα(n)... ˆα(0). The decoherence functional is a complex function on pairs of histories (relative to a choice of state ψ ) given by D(α, β) = ψ Ĉ αĉβ ψ. A history space is said to satisfy the decoherence condition or to be decoherent if the decoherence functional between any two incompatible histories is zero. If P satisfies the decoherence condition, it is a coarse-graining of a (decoherent) history space which has a branching structure relative to ψ.

24 Decoherence in the Early Universe Decoherence is essentially the suppression of interference with respect to a system s macroscopic degrees of freedom via interactions with microscopic degrees of freedom (internal or external environment").

25 Decoherence in the Early Universe Decoherence is essentially the suppression of interference with respect to a system s macroscopic degrees of freedom via interactions with microscopic degrees of freedom (internal or external environment"). Decoherence does not solve the measurement problem, but does it place a constraint on quantum theories of structure formation?

26 Decoherence in the Early Universe Decoherence is essentially the suppression of interference with respect to a system s macroscopic degrees of freedom via interactions with microscopic degrees of freedom (internal or external environment"). Decoherence does not solve the measurement problem, but does it place a constraint on quantum theories of structure formation? In the literature on the quantum to classical transition during cosmological structure formation, decoherence has been shown to occur in many ways: external scalar fields, short wavelength modes, non-linear coupling between modes, spatial entanglement, thermal fluctuations after reheating...

27 Decoherence in the Early Universe Decoherence is essentially the suppression of interference with respect to a system s macroscopic degrees of freedom via interactions with microscopic degrees of freedom (internal or external environment"). Decoherence does not solve the measurement problem, but does it place a constraint on quantum theories of structure formation? In the literature on the quantum to classical transition during cosmological structure formation, decoherence has been shown to occur in many ways: external scalar fields, short wavelength modes, non-linear coupling between modes, spatial entanglement, thermal fluctuations after reheating... What attitude should one take to these demonstrations?

28 Attitudes Toward Structure Formation and the Everett Interpretation 1. The ease of generating decoherence in the inflationary scenario is further evidence that decoherence is ubiquitous, thus by requiring decoherence the Everett interpretation places a minimal constraint on theories of structure formation. What accounts for the ubiquity of decoherence and therefore the direction of branching?

29 Attitudes Toward Structure Formation and the Everett Interpretation 1. The ease of generating decoherence in the inflationary scenario is further evidence that decoherence is ubiquitous, thus by requiring decoherence the Everett interpretation places a minimal constraint on theories of structure formation. What accounts for the ubiquity of decoherence and therefore the direction of branching? Initial Conditions?

30 Attitudes Toward Structure Formation and the Everett Interpretation 1. The ease of generating decoherence in the inflationary scenario is further evidence that decoherence is ubiquitous, thus by requiring decoherence the Everett interpretation places a minimal constraint on theories of structure formation. What accounts for the ubiquity of decoherence and therefore the direction of branching? Initial Conditions? 2. The variety of ways to decohere fields in the early universe that people have proposed raises theoretical worries about inflation or the Everett interpretation. Too little known about the inflaton and its interactions with other physical fields There is no principled division" of the universe into system and environment.

31 Attitudes Toward Structure Formation and the Everett Interpretation 1. The ease of generating decoherence in the inflationary scenario is further evidence that decoherence is ubiquitous, thus by requiring decoherence the Everett interpretation places a minimal constraint on theories of structure formation. What accounts for the ubiquity of decoherence and therefore the direction of branching? Initial Conditions? 2. The variety of ways to decohere fields in the early universe that people have proposed raises theoretical worries about inflation or the Everett interpretation. Too little known about the inflaton and its interactions with other physical fields. Effective theory? There is no principled division" of the universe into system and environment.

32 Attitudes Toward Structure Formation and the Everett Interpretation 1. The ease of generating decoherence in the inflationary scenario is further evidence that decoherence is ubiquitous, thus by requiring decoherence the Everett interpretation places a minimal constraint on theories of structure formation. What accounts for the ubiquity of decoherence and therefore the direction of branching? Initial Conditions? 2. The variety of ways to decohere fields in the early universe that people have proposed raises theoretical worries about inflation or the Everett interpretation. Too little known about the inflaton and its interactions with other physical fields. Effective theory? There is no principled division" of the universe into system and environment. Need to show consistency!

33 Conclusions 1. The standard theory of structure formation is a quantum theory with the appearance of classical outcomes thus the quantum measurement problem appears in cosmology as it does elsewhere in physics. 2. A solution of the measurement problem must be a quantum theory without observers to be applicable to cosmology. 3. The Everett interpretation (modulo concerns over probability) is applicable to cosmology, but is conservative and unlikely to raise issues in this application still, work is required to demonstrate satisfaction of the decoherence condition. 4. Further work on the quantum measurement problem is of relatively little importance to cosmology; of far more importance is understanding the initial state of the universe and the nature of the inflaton.

34 Structure Formation and the Measurement Problem Mukhanov, Physical Foundations of Cosmology When we look at the sky we see the galaxies in certain positions. If these galaxies originated from initial quantum fluctuations, a natural question arises: how does a galaxy, e.g. Andromeda, find itself at a particular place if the initial vacuum state was translational-invariant with no preferred position in space? Quantum mechanical unitary evolution does not destroy translational invariance and hence the answer to this question must lie in the transition from quantum fluctuations to classical inhomogeneities.

35 Structure Formation and the Measurement Problem Decoherence is a necessary condition for the emergence of classical inhomogeneities and can easily be justified for amplified cosmological perturbations. However, decoherence is not sufficient to explain the breaking of translational invariance. It can be shown that as a result of unitary evolution we obtain a state which is a superposition of many macroscopically different states, each corresponding to a particular realization of galaxy distribution.

36 Structure Formation and the Measurement Problem Most of these realizations have the same statistical properties. Such a state is a close cosmic analog of the Schrödinger cat." Therefore, to pick an observed macroscopic state from the superposition we have to appeal either to Bohr s reduction postulate or to Everett s many-worlds interpretation of quantum mechanics.

37 Quantum Measurement Problem Today The quantum measurement problem is not a psuedo-problem. It is a conceptual stumbling block to the universal application of quantum theory, e.g. in cosmology.

38 Quantum Measurement Problem Today The quantum measurement problem is not a psuedo-problem. It is a conceptual stumbling block to the universal application of quantum theory, e.g. in cosmology. The quantum measurement problem is not solved by decoherence. Decoherence does not destroy superpositions.

39 Quantum Measurement Problem Today The quantum measurement problem is not a psuedo-problem. It is a conceptual stumbling block to the universal application of quantum theory, e.g. in cosmology. The quantum measurement problem is not solved by decoherence. Decoherence does not destroy superpositions. The quantum measurement problem is not solved. Conceptual problems are not solved". There are many approaches, some of which are coherent, of interest, etc. Others are not.

40 Quantum Measurement Problem Today The quantum measurement problem is not a psuedo-problem. It is a conceptual stumbling block to the universal application of quantum theory, e.g. in cosmology. The quantum measurement problem is not solved by decoherence. Decoherence does not destroy superpositions. The quantum measurement problem is not solved. Conceptual problems are not solved". There are many approaches, some of which are coherent, of interest, etc. Others are not. The quantum measurement problem is not a severe difficulty.