Does Time Exist in Quantum Gravity?

Transcription

1 Does Time Exist in Quantum Gravity? Claus Kiefer Institut für Theoretische Physik Universität zu Köln

2 Contents The Problem of Time Quantum Gravity Quantum Cosmology Arrow of Time

3 The problem of time Absolute time in quantum theory: i ψ t = Ĥψ Dynamical time in general relativity: R µν 2 g µνr = 8πG c 4 T µν QUANTUM GRAVITY?

4 Wolfgang Pauli (955): Es scheint mir..., daß nicht so sehr die Linearität oder Nichtlinearität Kern der Sache ist, sondern eben der Umstand, daß hier eine allgemeinere Gruppe als die Lorentzgruppe vorhanden ist.... BACKGROUND INDEPENDENCE Matvei Bronstein (936): The elimination of the logical inconsistencies connected with this requires a radical reconstruction of the theory, and in particular, the rejection of a Riemannian geometry dealing, as we see here, with values unobservable in principle, and perhaps also the rejection of our ordinary concepts of space and time, modifying them by some much deeper and nonevident concepts. Wer s nicht glaubt, bezahlt einen Taler.

5 Main approaches to quantum gravity No question about quantum gravity is more difficult than the question, What is the question? (John Wheeler 984) Quantum general relativity Covariant approaches (perturbation theory, path integrals,... ) Canonical approaches (geometrodynamics, connection dynamics, loop dynamics,... ) String theory Other approaches (Quantization of topology, causal sets... ) (See e.g. C. Kiefer, Quantum Gravity (Oxford 27))

6 Max Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der königlich-preußischen Akademie der Wissenschaften zu Berlin, phys.-math. Klasse, Seiten 44 8 (899)

7 Planck Units l P = t P = l P c = m P = l P c = G c 3,62 35 m G c 5 5,4 44 s c G 2,7 8 9 GeV kg,22 c 2 Max Planck (899): Diese Grössen behalten ihre natürliche Bedeutung so lange bei, als die Gesetze der Gravitation, der Lichtfortpflanzung im Vacuum und die beiden Hauptsätze der Wärmetheorie in Gültigkeit bleiben, sie müssen also, von den verschiedensten Intelligenzen nach den verschiedensten Methoden gemessen, sich immer wieder als die nämlichen ergeben.

8 Canonical Quantum Gravity Central equations are constraints: ĤΨ = Oldest approach: Quantum geometrodynamics (Wheeler DeWitt equation): Ψ depends only on the threedimensional metric, but is invariant under coordinate transformations no external time parameter is present Wheeler DeWitt equation is (locally) hyperbolic and thereby defines an intrinsic time; it should at least be valid away from the Planck scale (if not at any scale) More recently: Loop quantum gravity B. S. DeWitt (967); J. A. Wheeler (968)

9 Quantization of a Friedmann Universe Closed Friedmann Lemaître universe with scale factor a, containing a homogeneous massive scalar field φ (two-dimensional minisuperspace) ds 2 = N 2 (t)dt 2 + a 2 (t)dω 2 3 The Wheeler DeWitt equation reads (with units 2G/3π = ) ( ( 2 2 a 2 a ) 2 2 ) a a a 3 φ 2 a + Λa3 3 + m2 a 3 φ 2 ψ(a,φ) = Factor ordering chosen in order to achieve covariance in minisuperspace

10 Determinism in classical and quantum theory Classical theory Quantum theory a φ Ú º º Ö Ò Ø Ð ÓÒ Ø ÓÒ give initial conditions on a=constant Recollapsing part is deterministic successor of expanding part Recollapsing wave packet must be present initially

11 Example Indefinite Oscillator ( ) 2 Ĥψ(a,χ) ( H a + H χ )ψ a 2 2 χ 2 a2 + χ 2 ψ = (C.K. 99)

12 Validity of Semiclassical Approximation? Closed universe: Final condition ψ a wave packets in general disperse WKB approximation not always valid Solution: Decoherence (see below)

13 Introduction of inhomogeneities Describe small inhomogeneities by multipoles {x n } around the minisuperspace variables (e.g. a and φ) ( H + ) H n (a,φ,x n ) Ψ(α,φ, {x n }) = n (Halliwell and Hawking 985) If ψ is of WKB form, ψ C exp(is / ) (with a slowly varying prefactor C), one gets from ψ (α,φ) n ψ n(α,φ,x n ): i ψ n t H n ψ n with t S t: WKB time controls the dynamics in this approximation (Temps retrouvé)

14 Often: WKB time t is correlated with the scale factor a as long as the latter is monotonic But what happens with superpositions such as Ψ = C exp(is / ) n ψ n (α, φ, x n ) +C exp( is / ) n ψ n (α, φ, x n)?

15 Decoherence Irreversible emergence of classical properties through the unavoidable interaction with the environment (irrelevant degrees of freedom) t t t (a) (b) (c) without decoherence medium decoherence strong decoherence

16 Decoherence in quantum cosmology Quantum gravity superposition of different metrics Decoherence? System : Global degrees of freedom (radius of Universe, inflaton field,... ) Environment : Density fluctuations, gravitational waves, other fields (Zeh 986, C.K. 987) Example: Scale factor a of de Sitter space (a e H It ) ( system ) is decohered by gravitons ( environment ) according to ρ (a,a ) ρ (a,a )exp ( CH 3 I a(a a ) 2), C > The Universe assumes classical properties at the beginning of the inflationary phase (Barvinsky, Kamenshchik, C.K. 999)

17 Time from Symmetry Breaking Analogy from molecular physics: emergence of chirality 4 V(z) 2 3 > > dynamical origin: decoherence due to scattering with light or air molecules quantum cosmology: decoherence between exp(is / )- and exp( is / )-part of wave function through interaction with multipoles one example for decoherence factor: exp exp ` 43 (C. K. 992) πmh2 a3 28

18 Decoherence of primordial fluctuations The modes for the inflaton field and the gravitons evolve into a squeezed quantum state during inflation (r > ) They decohere through coupling to other fields (pointer basis = field basis) Decoherence time is given by t d H I (C.K., Lohmar, Polarski, Starobinsky 998, 27) 34 s Fluctuations assume classical properties during inflation

19 Interpretation of quantum cosmology Both quantum general relativity and string theory preserve the linear structure for the quantum states = strict validity of the superposition principle only interpretation so far: Everett interpretation (with decoherence as an essential part) B. S. DeWitt 967: Everett s view of the world is a very natural one to adopt in the quantum theory of gravity, where one is accustomed to speak without embarassment of the wave function of the universe. It is possible that Everett s view is not only natural but essential.

20 Quantum cosmology with big brake Classical model: Equation of state p = A/ρ, A >, for a Friedmann universe with scale factor a(t) and scalar field φ(t) with potential (24πG = ) ) V (φ) = V (sinh ( φ ) ; sinh ( φ ) develops pressure singularity (only ä(t) becomes singular) Quantum model: Normalizable solutions of the Wheeler DeWitt equation vanish at the classical singularity 5 Ψ(τ,φ) φ a 4 τ φ 5 (Kamenshchik, C. K., Sandhöfer 27)

21 The Arrows of Time Although almost all the fundamental laws of physics do not distinguish a direction of time, most phenomena are irreversible Second Law of Thermodynamics : Entropy never decreases in a closed system Paul Cézanne, Nature morte au crâne, 895/9 (The Barnes Foundation, Pennsylvania)

22 How special is the Universe? Penrose (98): Entropy of the observed part of the Universe is maximal if all its mass is in one black hole; the probability for our Universe would then be ( exp S kb) exp ( Smax k B ) exp ( 88) exp ( 23 ) exp ( 23) updated version (C. K. 29): exp(s) exp(s max ) exp(3. 4 ) exp( ) exp( )

23 Origin of time direction Fundamental asymmetry with respect to intrinsic time : ĤΨ = 2 α 2 + n 2 x 2 + V n (α,x n ) n }{{} for α Is compatible with simple boundary condition: Ψ = Ψ α ψ (α) n ψ n (x n ) Entropy increases with increasing α, since entanglement with other degrees of freedom increases defines time direction Is the expansion of the Universe a tautology?

24 Big Bang Big Crunch black holes Hawking radiation black holes Radius zero Radius zero Hawking radiation Hawking radiation maximal extension (C.K. and Zeh 995)

25 Does Time Exist in Quantum Gravity? Time and spacetime do not exist at the fundamental level. The standard concept of time can be recovered as an approximative concept in special situations. Quantum Gravity offers the means to understand the origin of irreversibility. Reference: C.K., arxiv: [gr-qc]