Cosmological quantum fluctuations from the

Transcription

1 Cosmological quantum fluctuations from the perspectives of Bohmian mechanics and collapse theories Roderich Tumulka Department of Mathematics Sao Paulo, 3 November 2014 Supported by the John Templeton Foundation, project Philosophy of Cosmology, headed by Barry Loewer

2 OQM is problematical when taken literally OQM = orthodox quantum mechanics. Hence the need for better versions of QM. This problem arises even more sharply when talking about cosmology than about laboratory experiments. John S. Bell (1981): When the system in question is the whole world, where is the measurer to be found? Inside, rather than outside, presumably. What exactly qualifies some subsystems to play this role? Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer for a more highly qualified system with a Ph.D.? If the theory is to apply to anything but idealized laboratory operations, are we not obliged to admit that more or less measurement-like processes are going on more or less all the time more or less everywhere?

3 In the same vein: Richard P. Feynman (1959): Does this mean that my observations become real only when I observe an observer observing something as it happens? This is a horrible viewpoint. Do you seriously entertain the thought that without observer there is no reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was there no reality before 9 B.C. before life began? Or are you the observer? Then there is no reality to the world after you are dead? I know a number of otherwise respectable physicists who have bought life insurance.

4 The essence of the problem is this: A measuring device, or an observer, is itself a system S 1 consisting of (a large number of) electrons and nucleons, and can thus be treated as a quantum mechanical system. As S 1 interacts with a measured object (some system S 2 ), the composite system S 1 S 2 has a wave function ψ that evolves unitarily, and in many relevant cases ends up in a superposition of macroscopically different states, ψ = i c iψ i with ψ i = 1. The ψ i could, e.g., correspond to different pointer positions of the measuring device. But according to the collapse rule (measurement postulate) of the quantum formalism, only one pointer position occurs, the i-th with probability c i 2. This is a version of the measurement problem of quantum mechanics, more or less the same as the problem of Schrödinger s cat. Also this facet of the problem becomes more pressing in cosmology, where one considers the whole universe as a system S a quantum mechanical system. There is no outside observer who could collapse the wave function of the universe. But if it evolves unitarily, it will quickly become a superposition of macroscopically different states.

5 A practical side of the problem was emphasized by Daniel Sudarsky and Elias Okon (2013): Structure formation in the early universe becomes a problem in OQM. Structure formation means: A slightly non-uniform distribution of matter in space leads, through the effect of gravity, to clumping of matter to galaxies and stars. This is one of the cosmological quantum fluctuations that the title of this talk refers to.

6 Toy model: Non-relativistic, Newtonian gravity for N particles in a large box Λ = [0, L] 3 with periodic boundary conditions Classically: A near-uniform configuration (say, with initial velocities zero, or random with a Gaussian distribution) evolves to a clumped configuration. Quantum mechanically: A constant wave function ψ 0 on configuration space Λ N gives > 99% weight to near-uniform configurations and evolves, under the Schrödinger equation for a suitable duration t, to a wave function ψ t that gives > 99% weight to clumped configurations but is still invariant under 3-translations (because ψ 0 and the Hamiltonian is). ψ t is a superposition of many clumped states, and not a random clumped state! ψ t contains no information about which clumped configuration is the real one. The quantum average of the mass density at x Λ, ψ t M(x) ψ t with M(x) = N k=1 m k δ 3 (x Q k ) the mass density operator and Q k the position operator of particle, is constant (i.e., independent of x).

7 Boltzmann brains Cosmological quantum fluctuation in the title also refers to another issue where the foundations of QM become relevant to cosmology: the formation of Boltzmann brains. A Boltzmann brain is this: Let M be the present macro-state of your brain. For a classical gas in thermal equilibrium, it has probability 1 that after sufficient waiting time, some atoms will by coincidence (or by fluctuation ) come together in such a way as to form a subsystem in a micro-state belonging to M. That is, this brain comes into existence not by childhood and evolution of life forms, but by coincidence; this brain has memories (duplicates of your present memories), but they are false memories: the events described in the memories never happened to this brain! Boltzmann brains are, of course, very unlikely. But they will happen if the waiting time is long enough, and they will happen more frequently if the system is larger (bigger volume, higher number of particles).

8 The problem of Boltzmann brains The problem is this: If the universe continues to exist forever, and if it reaches universal thermal equilibrium at some point, then the overwhelming majority of brains in the universe will be Boltzmann brains. According to the Copernican principle that we should see what a typical observer sees, the theory predicts that we are Boltzmann brains. But we are not. As emphasized by Sean Carroll (2014), there is good reason to believe that the late universe will be close to de Sitter space-time, and the state of matter will close to the ground state in de Sitter space-time (Bunch-Davies vacuum). This state is stationary (an eigenstate of the Hamiltonian), and the probability distribution it defines on configuration space gives > 99% weight to thermal equilibrium configurations, but positive probability to brain configurations, in fact > 99% probability to configurations containing brains if 3-space is large enough (in particular if infinite). Question: Does this mean there are Boltzmann brains in the Bunch-Davies vacuum? What is the significance of this particular wave function for reality? Does a stationary state mean that nothing happens?

9 More about foundations of QM coming up in cosmology The problem of time in quantum gravity: According to the Wheeler-de Witt equation (the central equation of canonical quantum gravity), the wave function of the universe must be an eigenfunction of the Hamiltonian, and thus time-independent. Again: Does a stationary state mean that nothing happens? (That would take care of the Boltzmann brains, but it would still ruin the theory.) The Wheeler-de Witt wave function is a superposition of various 3-geometries. But we need to talk about 4-geometries. How do these 4-geometries arise? (Proposals are provided by the Bohmian picture, decoherent histories, collapse theories, and perhaps many-worlds. Different proposals lead to rather different conclusions about the 4-geometry [Ward Struyve and Nelson Pinto-Neto 2014].) In the following, I will outline two quantum theories without observers, Bohmian mechanics and GRW theory, and talk about how they solve these problems.

10 Bohmian mechanics

11 Bohmian mechanics For non-relativistic QM: [Slater 1923, de Broglie 1926, Bohm 1952, Bell 1966] Takes particles literally: electrons are material points moving in 3-space and have a definite position Q k (t) at every time t. There is also another physical object, mathematically represented by a wave function ψ t on configuration space R 3N (N = no. of particles). Dynamical laws: dq k dt = Im ψ k ψ ( Q1 m k ψ (t),..., Q N (t) ) (1) ψ i ψ t = Hψ (2) The law of motion (1) is equivalent to dq/dt = j/ρ, where Q = (Q 1,..., Q N ) is the configuration, ρ = ψ 2 is the standard probability density, and j is the standard probability current vector field in configuration space. Quantum equilibrium assumption: At time 0, Q(t = 0) is random with distribution density ψ(t = 0) 2. (3) Quantum non-equilibrium has been explored by Antony Valentini.

12 Example of Bohmian trajectories Q(t): 2-slit experiment Picture: Gernot Bauer (after Chris Dewdney) wave-particle duality (in the literal sense)

13 Equivariance Equivariance theorem: It follows that at any time, Q(t) has distribution ψ(t) 2. John S. Bell (1986): De Broglie showed in detail how the motion of a particle, passing through just one of two holes in the screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. [Speakable and unspeakable in quantum mechanics, page 191]

14 Wave function of a subsystem composite system, Ψ = Ψ(x, y), Q(t) = (X (t), Y (t)) conditional wave function ψ(x) = N Ψ(x, Y ) N = normalization factor = ( dx Ψ(x, Y ) 2 ) 1/2. Time-dependence: ψ t (x) = N t Ψ t (x, Y (t)) Does not, in general evolve according to a Schrödinger eq. Note: conditional probability ρ(x = x Y ) = ψ(x) 2 Absolute uncertainty Inhabitants of a Bohmian universe cannot know a particle s position more precisely than the ψ 2 distribution allows, with ψ the conditional wave function. Q k (t) often called hidden variable better: uncontrollable variable

15 Empirical predictions of Bohmian mechanics Central fact Inhabitants of a Bohmian universe would observe outcomes in agreement with the predictions of quantum mechanics. Inhabitants of a Bohmian universe cannot measure the trajectory of a particle to arbitrary accuracy without influencing it. That is, when the accuracy is high, the trajectory of the particle is not the same as it would have been without interaction with the measuring apparatus. This is a limitation to knowledge.

16 Against positivism Limitations to knowledge may seem strange because they conflict with Positivism The view that a statement is unscientific or even meaningless if it cannot be tested experimentally, that an object is not real if it cannot be observed, and that a variable is not well-defined if it cannot be measured. This form of positivism is exaggerated, it is refuted by the following example: Alice prepares an ensemble of quantum systems, each with a pure state chosen randomly with distribution µ 1 over the unit sphere S(H ) in Hilbert space H. Suppose further that µ 2 µ 1 is another distribution over S(H ) with the same density matrix, ρ µ1 = ρ µ2. Then Bob is unable to determine (even probabilistically) by means of experiments on the systems whether Alice used µ 1 or µ 2, while there is a fact about whether the states actually have distribution µ 1 or µ 2, as Alice knows the pure state of each system. Thus, the predictions of quantum mechanics imply that there are facts in the world which cannot be discovered empirically.

17 Positivism Physicists tend to give arguments based on positivism, such as: The ether is not observable, therefore it doesn t exist. While the conclusion (that the ether doesn t exist) is right, the argument is bad. Here is a better argument: There is a theory (special relativity) that accounts for all empirical facts, that is simple and elegant, and according to which the ether doesn t exist. Thus, the ether is not needed, it only made the theory more complicated.

18 Particle creation in Bohmian mechanics [Bell 1986, Dürr, Goldstein, Tumulka, Zanghì 2003] t t Natural extension of Bohmian mechanics to particle creation: Ψ Fock space = H N, N=0 configuration space of a variable number of particles = R 3N N=0 (a) (a) x x (b) (b) Q(t 2!) jumps (e.g., n-sector (n + 1)- sector) occur in a stochastic way, with rates governed by a further equation of the theory. Q(t 1!) Q(t 2+) Q(t 1+) (c) (d)

19 Bohmian mechanics in relativistic space-time If a preferred foliation (= slicing) of space-time into spacelike hypersurfaces ( time foliation F) is permitted, then there is a simple, convincing analog of Bohmian mechanics, BM F. [Dürr et al. 1999] Without a time foliation, no version of Bohmian mechanics is known that would make predictions anywhere near quantum mechanics. (And I have no hope that such a version can be found in the future.)

20 There is no agreed-upon definition of relativistic theory. Anyway, the possibility seems worth considering that our universe has a time foliation. Simplest choice of time foliation F Let F be the level sets of the function T : space-time R, T (x) = timelike-distance(x, big bang). E.g., T (here-now) = 13.7 billion years Drawing: R. Penrose Alternatively, F might be defined in terms of the quantum state vector ψ, F = F(ψ) [Dürr, Goldstein, Norsen, Struyve, Zanghì 2014] Or, F might be determined by an evolution law (possibly involving ψ) from an initial time leaf.

21 Key facts about BM F Known in the case of N non-interacting Dirac particles, expected to be true also, say, one day, in full QED with photon trajectories: Equivariance Suppose initial configuration is ψ 2 -distributed. Then the configuration of crossing points Q(Σ) = (Q 1 Σ,..., Q N Σ) is ψ Σ 2 -distributed (in the appropriate sense) on every Σ F. Predictions The detected configuration is ψ Σ 2 -distributed on every spacelike Σ. As a consequence, F is invisible, i.e., experimental results reveal no information about F.

22 The GRW collapse theory

23 GRW theories Stochastic wave function evolution ( GRW process ) for non-rel. QM: ψ evolves according to the usual Schrödinger eq for a random duration T with mean N 1 8 years. (N = no. of particles ) At time T, ψ collapses as if an observer outside the universe made an unsharp position measurement with inaccuracy σ = 7 m on a random particle. More precisely: Nature randomly selects i {1... N} and location X R 3 with distribution ψ 2 g with g a Gaussian fct with width σ. ψ(x 1... x N ) gets replaced by ψ = N g(x i X) ψ(x 1... x N ), (N = normalization factor), i.e., localized as in Repeat [Ghirardi, Rimini, Weber Phys. Rev. D 1986]

24 Definition: GRW theories GRWm Matter is continuously distributed in space with density N m(t, x) = m i dx 1 dx N δ R 3 (x x i ) ψ t (x 1... x N ) 2. 3N i=1 ψ evolves stochastically according to the GRW process. GRWf Matter consists of flashes (material points in space-time). Flash at (T, X) for every collapse as ψ evolves according to the GRW process.

25 Empirical predictions Both GRWm and GRWf predict (with tiny deviations) the same probabilities of outcomes as the quantum formalism. An empirical test between GRW theories and QM/Bohm is possible in principle but cannot be carried out with present technology.

26 Empirical tests: Parameter diagram 1 [s ] ERR Adler GRW PUR [m] [s ] ERR Adler GRW PUR [m] ] 1 [s ERR PUR Adler GRW [m] GRWf GRWm CSLm Parameter diagrams (log-log scale). ERR = empirically refuted region (equal), PUR = philosophically unsatisfactory region. From [Feldmann & Tumulka 2012]

27 Relativity and instantaneous collapse Everybody s first idea: If collapse is instantaneous (as opposed to propagating at speed c) then it must violate relativity. That problem is easily avoided For every spacelike hypersurface Σ there is a wave fct ψ Σ H Σ. E.g., H Σ = H N 1, H 1 = L (Σ, 2 C 4, φ ψ = ) Σ d 3 x φ(x)n µ (x)γ µ ψ(x).

28 Relativity and instantaneous collapse Everybody s first idea: If collapse is instantaneous (as opposed to propagating at speed c) then it must violate relativity. That problem is easily avoided For every spacelike hypersurface Σ there is a wave fct ψ Σ H Σ. E.g., H Σ = H N 1, H 1 = L (Σ, 2 C 4, φ ψ = ) Σ d 3 x φ(x)n µ (x)γ µ ψ(x).

29 Relativistic collapse processes (stochastic evolution for ψ) [Diósi 1990, Ghirardi-Grassi-Pearle 1990]: relativistic continuous collapse processes for the state vector of a quantum field theory; however, suffers from divergences. [Bedingham 2011]: a modification that removes the divergences; however, not fully Lorentz invariant. [Tumulka 2006]: a relativistic GRW process for the state vector of N non-interacting spin- 1 2 particles in an external field Given an initial wave function ψ 0 on Σ 0 (and possibly further data), the law for ψ specifies the joint distribution of all ψ Σ with Σ in the future of Σ 0. In situations in which the unitary Schrödinger evolution would lead to a superposition ψ Σ = α c αψ (α) of macroscopically different contributions ψ (α) (with ψ (α) = 1), the law for ψ yields ψ Σ ψ (α) with probability close to c α 2. For any two hypersurfaces Σ, Σ after a local measurement at a space-time point y, ψ Σ and ψ Σ select the same α of that measurement.

30 Still: given a relativistic collapse process for ψ Σ, how do we get facts? [Landau, Peierls 1931; I. Bloch 1967; Hellwig, Kraus 1970; Aharonov, Albert 1980] Problem For Σ = A B with A B =, ρ A = tr B ψ Σ ψ Σ depends on B: ψ A B may be very different from ψ A B (collapses in-between), and tr B ψ A B may be very different from tr B ψ A B. E.g., in an EPR experiment: If Σ lies after the exper. on particle 2 but before that on particle 1, then ρ A will be a pure state. If Σ lies before both exper.s, ρ A will be mixed. Solution Primitive ontology in space-time, such as matter density ontology or flash ontology. BDGGTZ 2014:... We should demand that certain local facts, such as whether a cat is dead or alive, do not depend on the choice of Σ. Fortunately, the macroscopic local situation is practically unambiguous... But the notion of macroscopic is imprecise... The variable that defines local facts need not define a spin state for every particle. But it should define the distribution of matter in space-time and ensure that macroscopic configurations... are unambiguous.

31 Flash ontology Flashes in 2+1-dim space-time forming a binary star.

32 Relativistic GRWf (= rgrwf) [Tumulka 2006] Involves a relativistic version of the GRW process, with a collapse occurring at a random proper time T after the previous collapse for the same particle. A flash occurs at the center of every wave function collapse. There is no fact about who influenced whom (flashes in Paris those in Tokyo or vice versa).

33 Relativistic GRWm (= rgrwm) Nonrelativistic law for m: N m(x, t) = m i d R 3N q δ 3 (x q i ) ψ t (q) 2 = ψ t M(x) ψ t 3N i=1 with mass density operator M(x) = i m iδ 3 (x q i ). Relativistic law for m: [Ghirardi 1999, BDGGTZ 2014] m(x, t) = m(x) = ψ PLC(x) M PLC(x) (x) ψ PLC(x) PLC(x). Examples (i) QFT, M = T µν, m = m µν ; (ii) N Dirac particles, m µ (x) = N i=1 m i δ µ i µ PLC(x) N 1 ( j i ) dσ µ j j (y j ) ψ PLC(x) [γ µ1 γ µn ]ψ PLC(x) with ψ = ψ(y 1,..., y i 1, x, y i+1,..., y N ), measure dσ µ (y) corresponding to the vector-valued 3-form ε µ κλν.

34 Facts about rgrwm Given any of the known relativistic collapse processes for ψ Σ : Suppose that a local measurement is made at a space-time point y. Then the m function in the future light cone of y and the ψ function on any Σ after y agree about the outcome. The empirical predictions of rgrwm agree approximately with those of the quantum formalism. Nonrelativistic limit = nonrelativistic GRWm No signaling (except in a neighborhood of 7 m and 8 s) Microscopic parameter independence (except in that neighborhood) Nonlocal

35 Fluctuations in Bohmian mechanics and GRW theories

36 What about macroscopic superpositions? In Bohmian mechanics, the measurement problem is solved as follows: The wave function after the measurement is a superposition of macroscopically different states, ψ = i c iψ i with ψ i = 1, corresponding to different outcomes, but the actual configuration of particles has the instrument pointer pointing to one particular value: this is the outcome of the experiment. More generally, if ψ is a macroscopic superposition and the contributions ψ i have disjoint supports in configuration space, then the actual configuration Q will lie (because it is ψ 2 -distributed) in one of the supports, thus selecting a particular contribution ψ i ; this happens with probability c i 2. In GRW theories, the measurement problem is solved as follows: A superposition of macroscopically disjoint packets in configuration space usually does not occur, or at least does not persist for more than a fraction of a second. Thus, only one of the contributions ψ i survives with probability approximately c i 2.

37 Again: The toy model of structure formation Schrödinger equation with Newtonian gravity for N particles in a box Λ = [0, L] 3 with periodic boundary conditions. In Bohmian mechanics, if ψ(t) is a superposition of many clumped states, then the actual configuration Q is only one of the clumped configurations thus breaking the translation symmetry. In GRW theories, the collapsed wave function is no longer translation invariant; it has collapsed to (essentially) one of the clumped states.

38 Boltzmann brains in Bohmian mechanics In Bohmian mechanics, the configuration tends to move no more than necessary. Theorem If ψ is a non-degenerate eigenstate of H then the Bohmian configuration does not move. The same is true for Bohmian mechanics with particle creation. Thus, when ψ is the Bunch-Davies vacuum (the ground state in de Sitter space-time, which is non-degenerate), the Bohmian configuration does not move. While there is a positive probability for a brain configuration to occur, this subsystem would not function as a brain because it is frozen. Thus, Boltzmann brains do not occur in Bohmian mechanics in this situation. (They can occur for (thermal) states that are not eigenstates of H.)

39 Boltzmann brains in GRW theories In the GRW process for ψ, the Bunch-Davies vacuum is not stationary. It is a spread-out wave function that will be replaced in collapses by more localized ones in fact, random ones. As a consequence, there is a small probability for a subsystem to be in a Boltzmann brain state, and to evolve like a real brain. As a consequence, there is overwhelming probability that sooner or later, some subsystem of a big region of space filled with gas in the Bunch-Davies state will be a Boltzmann brain. That is, in GRW theory, Boltzmann brains do occur just like in classical physics. Some other reason (involving, e.g., eternal inflation) might keep the Boltzmann brains from occurring.

40 Thank you for your attention