Is the wave function real or not?

Transcription

1 Is the wave function real or not? Lessons from quantum gravity Claus Kiefer Institut für Theoretische Physik Universität zu Köln

2 Contents Quantum Mechanics Quantum Gravity Quantum Cosmology Lessons

3 The superposition principle Let Ψ 1 and Ψ 2 be physical states. Then, αψ 1 + βψ 2 is again a physical state. For more than one degree of freedom, this leads to the entanglement between systems (Verschränkung). Linearity of the Schrödinger equation: the sum of two solutions is again a solution. Classical states form only a tiny subset in the space of all possible states. Erwin Schrödinger 1935: I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ-functions) have become entangled.... Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated...

4 Experimental tests of the superposition principle Binding energies of atoms and molecules (Helium,... ) Interference of fullerenes C 60 and C 70 and more complicated molecules Welcher-Weg-experiments: Entanglement of an intrinsic atomic state with atomic momentum Entanglement of photon pairs over distances of more than 30 km Superposition of macroscopically different currents in SQUIDs Neutrino oscillations, superposition of K-mesons,...

5 A particular example (Vienna experiment) tetraphenylporphyrin (C 44H 30N 4) (left) and fluorofullerene C 60F 48 (right) counts in 40 s spectrometer background level position of 3rd grating (µm) Interference pattern of tetraphenylporphyrin L. Hackermüller et al., Phys. Rev. Lett. 91 (2003)

6 Decoherence Decoherence: Irreversible emergence of classical properties through the unavoidable interaction with the environment. Objects then appear classically, although they are fundamentally described by quantum theory. Important conceptual and quantitative developments in the early years by Zeh (1971, 1973), Kübler and Zeh (1973), Zurek (1981, 1982), Harris and Stodolsky (1981, 1982), Caldeira and Leggett (1983), Joos (1984), Joos and Zeh (1985),... ; experimental tests since 1996

7 Left: Decoherence through particle collisions. Right: Decoherence through heating of fullerenes. From: M. Arndt and K. Hornberger, Quantum interferometry with complex molecules, arxiv: v1

8 Interpretation of quantum theory The superposition principle is universally valid many-worlds interpretation ( Everett interpretation ): the dead and the alive Schrödinger cat indeed exist simultaneously in different Everett branches The current formalism of quantum theory must be modified in order to accommodate a collapse of the wave function such that only the dead or the alive Schrödinger cat exists Change of the kinematical structure (e.g. the de Broglie Bohm theory) Purely operationalistic interpretations (concept of reality?)

9 What about gravity? Richard Feynman 1957:... if you believe in quantum mechanics up to any level then you have to believe in gravitational quantization in order to describe this experiment.... It may turn out, since we ve never done an experiment at this level, that it s not possible... that there is something the matter with our quantum mechanics when we have too much action in the system, or too much mass or something. But that is the only way I can see which would keep you from the necessity of quantizing the gravitational field. It s a way that I don t want to propose....

10 Main approaches to quantum gravity No question about quantum gravity is more difficult than the question, What is the question? (John Wheeler 1984) Quantum general relativity Covariant approaches (perturbation theory, path integrals including spin foams, asymptotic safety,... ) Canonical approaches (geometrodynamics, connection dynamics, loop dynamics,... ) String theory Fundamental discrete approaches (quantum topology, causal sets, group field theory,... ); have partially grown out of the other approaches C. Kiefer, Quantum Gravity (Oxford 2012)

11 Erwin Schrödinger 1926: We know today, in fact, that our classical mechanics fails for very small dimensions of the path and for very great curvatures. Perhaps this failure is in strict analogy with the failure of geometrical optics... that becomes evident as soon as the obstacles or apertures are no longer great compared with the real, finite, wavelength.... Then it becomes a question of searching for an undulatory mechanics, and the most obvious way is by an elaboration of the Hamiltonian analogy on the lines of undulatory optics. 1 1 wir wissen doch heute, daß unsere klassische Mechanik bei sehr kleinen Bahndimensionen und sehr starken Bahnkrümmungen versagt. Vielleicht ist dieses Versagen eine volle Analogie zum Versagen der geometrischen Optik..., das bekanntlich eintritt, sobald die Hindernisse oder Öffnungen nicht mehr groß sind gegen die wirkliche, endliche Wellenlänge.... Dann gilt es, eine undulatorische Mechanik zu suchen und der nächstliegende Weg dazu ist wohl die wellentheoretische Ausgestaltung des Hamiltonschen Bildes.

12 Quantum geometrodynamics (a) John Archibald Wheeler (b) Bryce DeWitt Application of Schrödinger s procedure to general relativity leads to ĤΨ ( 16πG 2 δ 2 G abcd (16πG) 1 h ( (3) R 2Λ ) ) Ψ = 0 δh ab δh cd Wheeler DeWitt equation ˆD a δψ Ψ 2 b = 0 i δh ab quantum diffeomorphism (momentum) constraint

13 Problem of time External time t has vanished from the formalism This holds also for loop quantum gravity and probably for string theory Wheeler DeWitt equation has the structure of a wave equation any may therefore allow the introduction of an intrinsic time Hilbert-space structure in quantum mechanics is connected with the probability interpretation, in particular with probability conservation in time t; what happens with this structure in a timeless situation? What is an observable in quantum gravity?

14 Quantum cosmology Gell-Mann and Hartle 1990: Quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology. A universally valid quantum theory must be applied to the Universe as a whole as the only closed quantum system in the strict sense; need quantum theory of gravity, since gravity dominates on large scales

15 Example Indefinite Oscillator ˆ Hψ(a, χ) ( Ha + Hχ )ψ C.K. (1990) a + χ ψ=0 a2 χ2

16 Born Oppenheimer type of approximation Describe small inhomogeneities by multipoles {x n } around the minisuperspace variables (e.g. a and φ) ( H 0 + ) H n (a, φ, x n ) Ψ(α, φ, {x n }) = 0 n (Halliwell and Hawking 1985) If ψ 0 is of WKB form, ψ 0 C exp(is 0 / ) (with a slowly varying prefactor C), one will get with Ψ = ψ 0 n ψ n, i ψ n t H n ψ n with t S 0 t: WKB time controls the dynamics in this approximation

17 Decoherence in quantum cosmology In quantum cosmology, arbitrary superpositions of the gravitational field and matter states can occur. How can we understand the emergence of an (approximate) classical Universe?

18 Time from symmetry breaking Analogy from molecular physics: emergence of chirality V(z) > > dynamical origin: decoherence through scattering by light or air molecules Quantum cosmology: decoherence between exp(is 0 /G )- and exp( is 0 /G )-components of the wave function through interaction with e.g. weak gravitational waves Example for decoherence ) factor: exp exp ( 10 43) (C.K. 1992) ( πmh2 0 a3 128

19 Decoherence of primordial fluctuations During the inflationary phase (ca after the Big Bang) there is a quantum-to-classical transition for the ubiquitous fluctuations of the inflaton and the metric. The process of decoherence is crucial in understanding this transition (C.K., Lohmar, Polarski, Starobinsky 1998, 2007). The fluctuations then behave like classical stochastic quantities and yield the seeds for the structures in the Universe. Quantum gravity is needed to understand the power spectrum.

20

21 Predictions in quantum cosmology Anthropic interpretation We find ourselves in a decohered branch of the wave function that is suitable for life (cf. landscape picture) Peak in the wave function If the wave function is peaked around particular values of a, φ,..., this corresponds to the prediction that these values occur with high probability; if the wave function vanishes, the corresponding values are not allowed (relevant e.g. for singularity avoidance) Semiclassical interpretation The wave function can only be interpreted in the semiclassical regime, where an approximate WKB time emerges from the timeless Wheeler DeWitt equation. Sharp peak in the wave function as a prediction? Inflation, for example, occurs naturally if Ψ has a peak at a sufficiently large value of the inflaton field φ.

22 No-boundary proposal Time t Imaginary Time τ τ = 0 S. W. Hawking, Vatican conference 1982: There ought to be something very special about the boundary conditions of the universe and what can be more special than the condition that there is no boundary. Ψ[h ab, Φ, Σ] = M ν(m) DgDΦ e S E[g µν,φ] M

23 The CMB spectrum from the PLANCK mission Figure credit: ESA/PLANCK Collaboration

24 First observational test of quantum gravity Within the inflationary scenario, the observed CMB fluctuations can only be understood from quantized metric plus scalar field modes. This is an indirect test of linearized quantum gravity. The observation of primordial B-modes would be an indirect confirmation of the existence of gravitons. The difference in the duration of inflation between the cold spots and the hot spots in the CMB spectrum is only of the order of the Planck time.

25 More direct observations? Next order in the Born Oppenheimer approximation gives Ĥ m Ĥm + 1 m 2 (various terms) P (C.K. and Singh 1991; Barvinsky and C.K. 1998) The anisotropy spectrum of the cosmic background radiation may contain information about quantum gravitational corrections terms to the (functional) Schrödinger equation (C.K. with Krämer, Brizuela, )

26 Interpretation of quantum cosmology Almost all approaches to quantum gravity preserve the linear structure of quantum theory and thus the strict validity of the superposition principle. Main interpretation of quantum cosmology: Everett interpretation (with decoherence as a key ingredient) Bryce S. DeWitt 1967: 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.

27 At the fundamental level of quantum gravity, there is no need for a probability interpretation, since there exist neither time nor observers. Time and observers appear only in the semiclassical limit; classical properties follow through decoherence. The probability interpretation is thus needed only in this limit and can perhaps be derived in the sense of Zurek (2005). The origin of the direction of time can be understood in this framework, at least in principle.

28 Lessons In quantum mechanics, the wave function determines many structural issues; it has dynamical consequences and thus cannot be pure information; this is even more important in quantum gravity/cosmology, where probabilistic questions are less relevant; according to modern cosmology, all structure in the Universe can be traced back to an early quantum state; this state undergoes a quantum-to-classical transition, but gives imprints in the CMB (and other systems). The wave function is real!