To be pub. in Quantum [Un]Speakables II (50th anniversary of Bell):

Transcription

1 To be pub. in Quantum [Un]Speakables II (50th anniversary of Bell): Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

2 To be pub. in Quantum [Un]Speakables II (50th anniversary of Bell): Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

3 To be pub. in Quantum [Un]Speakables II (50th anniversary of Bell): Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

4 Ensembles of Bohmian trajectories: Real, Surreal, and Hyper-real Howard M. Wiseman & ( Michael J. Hall & Dirk-André Deckert ) & ( Dylan Mahler & Lee Rozema & Kent Fisher & Lydia Vermeyden & Kevin J. Resch & Aephraim Steinberg ) Centre for Quantum Dynamics Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

5 Outline 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

6 Orthodox Quantum Mechanics: who could ask for anything more? Outline 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

7 Orthodox Quantum Mechanics: who could ask for anything more? The Problems with Quantum Theory Quantum Theory has a lot of problems HardcoreGamer.com Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

8 Orthodox Quantum Mechanics: who could ask for anything more? The Problems with Quantum Theory Quantum Theory has a lot of problems HardcoreGamer.com Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

9 Orthodox Quantum Mechanics: who could ask for anything more? Problems: The Problems with Quantum Theory The mathematical formalism is remote from the everyday world. Bell s theorem: it cannot be replaced by a local realistic (causal) process in space-time. Quantum Theory has a lot of problems HardcoreGamer.com Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

10 Orthodox Quantum Mechanics: who could ask for anything more? Problems: The Problems with Quantum Theory The mathematical formalism is remote from the everyday world. Bell s theorem: it cannot be replaced by a local realistic (causal) process in space-time. Attitudes: Operationalism: QT describes only what we (macroscopic observers) expect to happen. But how can we be real if we are made up of quantum particles? Realism: Macroscopic observers are real because there is a reality for all quantum systems. Quantum Theory has a lot of problems HardcoreGamer.com Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

11 Orthodox Quantum Mechanics: who could ask for anything more? Types of Realism All about the quantum state Ψ(t) or wavefunction Ψ(q, t): For simplicity consider a D-dimensional universe comprising P scalar nonrelativistic distinguishable particles, and no fields. e.g. for D = 3 the pth particle has position (q 3p 2, q 3p 1, q 3p ), so the total configuration variable q = {q 1,, q K }, K = DP. [ Then i t Ψ(q, t) = V (q) ( ) ] 2 K k=1 2 2m k q k Ψ(q, t). Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

12 Orthodox Quantum Mechanics: who could ask for anything more? Types of Realism All about the quantum state Ψ(t) or wavefunction Ψ(q, t): For simplicity consider a D-dimensional universe comprising P scalar nonrelativistic distinguishable particles, and no fields. e.g. for D = 3 the pth particle has position (q 3p 2, q 3p 1, q 3p ), so the total configuration variable q = {q 1,, q K }, K = DP. [ Then i t Ψ(q, t) = V (q) ( ) ] 2 K k=1 2 2m k q k Ψ(q, t). 1 Ψ(q, t) or Ψ(t), despite its remoteness from the everyday world, is all that is real. Many Worlds Interpretation Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

13 Orthodox Quantum Mechanics: who could ask for anything more? Types of Realism All about the quantum state Ψ(t) or wavefunction Ψ(q, t): For simplicity consider a D-dimensional universe comprising P scalar nonrelativistic distinguishable particles, and no fields. e.g. for D = 3 the pth particle has position (q 3p 2, q 3p 1, q 3p ), so the total configuration variable q = {q 1,, q K }, K = DP. [ Then i t Ψ(q, t) = V (q) ( ) ] 2 K k=1 2 2m k q k Ψ(q, t). 1 Ψ(q, t) or Ψ(t), despite its remoteness from the everyday world, is all that is real. Many Worlds Interpretation 2 Ψ(q, t) or Ψ(t), and something else more connected with the everyday world, is real. Hidden Variables Interpretations Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

14 Orthodox Quantum Mechanics: who could ask for anything more? Types of Realism All about the quantum state Ψ(t) or wavefunction Ψ(q, t): For simplicity consider a D-dimensional universe comprising P scalar nonrelativistic distinguishable particles, and no fields. e.g. for D = 3 the pth particle has position (q 3p 2, q 3p 1, q 3p ), so the total configuration variable q = {q 1,, q K }, K = DP. [ Then i t Ψ(q, t) = V (q) ( ) ] 2 K k=1 2 2m k q k Ψ(q, t). 1 Ψ(q, t) or Ψ(t), despite its remoteness from the everyday world, is all that is real. Many Worlds Interpretation 2 Ψ(q, t) or Ψ(t), and something else more connected with the everyday world, is real. Hidden Variables Interpretations 3 Only something else, which is connected with, but not limited to, the everyday world, is real. e.g. Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

15 Orthodox Quantum Mechanics: who could ask for anything more? Realism Type 2: Hidden Variables Interpretations e.g. the de Broglie Bohm interpretation. Ψ(q, t) is a real wave in configuration space. In addition there is a single real configuration x(t). It is piloted by Ψ(q, t) (de Broglie, 1927): ẋ k (t) = m k Im x k Ψ(x; t) Ψ(x; t) The a priori probability distribution for x at t = t 0 is P(x; t 0 ) = Ψ(x; t 0 ) 2. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

16 Bohmian Mechanics: who could ask for anything more? Outline 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

17 Bohmian Mechanics: who could ask for anything more? Outline Issue 1 with Bohmian mechanics: why believe it is real? 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

18 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Non-uniqueness of Bohmian mechanics Experientially adequate hidden variable theories can be formulated for all sorts of variables, not just position x. In general they have to be stochastic (Bell, 1984). Even restricting to position x and deterministic dynamics, ẋ = v ψ(t) (x), there are infinitely many functional expressions for v ( ) : P ψ(t) (x)/ t + [P ψ(t) (x; t)v ψ(t) (x)] = 0, with P ψ(t) (x) = ψ(t) x x ψ(t). = why believe x and its Bohmian dynamics is real? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

19 Bohmian Mechanics: who could ask for anything more? Outline Addressing issue 1: theory and experiment 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

20 Bohmian Mechanics: who could ask for anything more? Addressing issue 1: theory and experiment Motivating Bohmian mechanics If we require determinism then the HV must have a continuous spectrum like ˆq or ˆp. If we assume the ability to do weak and strong measurements, and define (HMW, NJP, 2007) v ψ(t) (x) = lim τ 1 E ψ(t) [q strong (t + τ) q weak (t) q strong (t + τ) = x] τ 0 or = lim E ψ(t) [(dq/dt) weak (t) q strong (t + τ) = x], τ 0 then we get the standard Bohmian expression for v ψ(t) (x)... Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

21 Bohmian Mechanics: who could ask for anything more? Addressing issue 1: theory and experiment Motivating Bohmian mechanics If we require determinism then the HV must have a continuous spectrum like ˆq or ˆp. If we assume the ability to do weak and strong measurements, and define (HMW, NJP, 2007) v ψ(t) (x) = lim τ 1 E ψ(t) [q strong (t + τ) q weak (t) q strong (t + τ) = x] τ 0 or = lim E ψ(t) [(dq/dt) weak (t) q strong (t + τ) = x], τ 0 then we get the standard Bohmian expression for v ψ(t) (x)... as long as Ĥ is at most quadratic in operators canonically conjugate to ˆx... Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

22 Bohmian Mechanics: who could ask for anything more? Addressing issue 1: theory and experiment Motivating Bohmian mechanics If we require determinism then the HV must have a continuous spectrum like ˆq or ˆp. If we assume the ability to do weak and strong measurements, and define (HMW, NJP, 2007) v ψ(t) (x) = lim τ 1 E ψ(t) [q strong (t + τ) q weak (t) q strong (t + τ) = x] τ 0 or = lim E ψ(t) [(dq/dt) weak (t) q strong (t + τ) = x], τ 0 then we get the standard Bohmian expression for v ψ(t) (x)... as long as Ĥ is at most quadratic in operators canonically conjugate to ˆx... which is actually a feature because it forces us to choose ˆx = ˆq rather than ˆx = ˆp. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

23 can be gained on a Our quantum particles are single photons Bohmian Mechanics: who could ask for anything more? Addressing issue 1: theory and experiment r hand, there may be emitted by a liquid helium-cooled InGaAs quantum dot (23, [Kocsis 24) embedded & al. & in Steinberg a GaAs/AlAs mi- (Science, 2011)] imparted toexperiment the sysuently postselect the cropillar cavity. The dot is optically pumped by a ate. Postselecting on CW laser at 810 nm and emits single photons at... and one can measure it (even as a naive experimentalist ) 4 Downloaded fro Transverse coordinate[mm] Note that it is not possible to follow an individual particle. These trajectories are created by patching together little increments inferred from the weak velocities Propagation distance[mm] d, and the measured weak momentum value k x at this point found. This Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

24 Bohmian Mechanics: who could ask for anything more? Outline Issue 2 with Bohmian mechanics: it seems surreal 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

25 Bohmian Mechanics: who could ask for anything more? Surreal Trajectories Issue 2 with Bohmian mechanics: it seems surreal Englert, Scully, Süssman, and Walther (1992). Three Q. systems: 1 particle 1 (x 1 and z 1 = ct), 2 the WWM device ( spin H / V ), 3 particle 2 (x 2 ), the pointer. In BM, the WWM information is not real until it has moved the pointer. A WWM readout B WWM readout pointer x 1 z 1 =ct pointer x 1 z 1 =ct Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

26 Bohmian Mechanics: who could ask for anything more? Outline Addressing issue 2: theory and experiment 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

27 Bohmian Mechanics: who could ask for anything more? Addressing issue 2: theory and experiment Surreal trajectories, and nonlocality v1 (x1, x2 ; t) = right vleft 1 (x1 ;t)+v1 2 (x1 ;t) = weak-valued velocity of particle 1 alone. s2 (x1, x2 ; t) = s2 (x1 ; t) = weak-(or strong-)valued spin of particle 2. Wiseman et al. (Griffith University) { B) A) 4 Transverse coordinate[mm] With delayed readout, Bohmian theory says H > > D Propagation distance[mm] Ensembles of Bohmian trajectories 6000 EmQM, Vienna, / 38

28 Bohmian Mechanics: who could ask for anything more? Addressing issue 2: theory and experiment Real Nonlocality ( Setting Dependence ) With immediate readout, Bohmian theory says v 1 (x 1, x 2 ; t) = v 1 (x 1, outcome; t) = 1 s w.v. velocity conditioned on readout of 2 s polarization. To best see nonlocality, use two quantum eraser readouts i.e. Θ / Θ + π, where Φ H eiφ V, for Θ = 0 and Θ = π/ A) B) Transverse coordinate[mm] Propagation distance[mm] Transverse coordinate[mm] v 1/c x Transverse coordinate [mm] φ = 0 φ = π/2 φ = π φ = 3π/2 v 1 /c x Transverse coordinate [mm] Propagation distance[mm] Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

29 Many Worlds Interpretation: who could ask for anything more? Outline 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

30 Many Worlds Interpretation: who could ask for anything more? Realism Type 1: the Many Worlds Interpretation Ψ(q, t) is highly structured, and at any time t, Ψ(q, t), when smoothed out a bit, has local maxima at a vast number of macroscopically different configurations { q 1, q 2,...}. These configurations, { q} are the many worlds (de Witt, 1973) As time increases, each local maximum is liable to split into one or more local maxima a branching or splitting (Everett, 1957). Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

31 Many Worlds Interpretation: who could ask for anything more? Issues with the Many Worlds Interpretation (MWI) 1 It is not clear exactly what is real. Is it Ψ(q; t)? But in an abstract sense it is just a vector in Hilbert state Ψ(t), so how can it have any structure? Is it the local maxima { q} of the coarse-grained Ψ(q, t), the worlds? But this coarse-graining is vague; the number of worlds is not defined; and the timing of the splitting is also not defined. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

32 Many Worlds Interpretation: who could ask for anything more? Issues with the Many Worlds Interpretation (MWI) 1 It is not clear exactly what is real. Is it Ψ(q; t)? But in an abstract sense it is just a vector in Hilbert state Ψ(t), so how can it have any structure? Is it the local maxima { q} of the coarse-grained Ψ(q, t), the worlds? But this coarse-graining is vague; the number of worlds is not defined; and the timing of the splitting is also not defined. 2 Some worlds are more real than others. When a world splits, some daughter worlds are bigger than others: Ψ(t) α Ψ 1 (t + τ) + β Ψ 2 (t + τ) ; α > β. But each world will feel equally real to its respective inhabitants (our future selves, at time t + τ). So we should I care more about my future self in the world 1, and in particular why should I care in the ratio α 2 : β 2? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

33 Many Worlds Interpretation: who could ask for anything more? Issues with the Many Worlds Interpretation (MWI) 1 It is not clear exactly what is real. Is it Ψ(q; t)? But in an abstract sense it is just a vector in Hilbert state Ψ(t), so how can it have any structure? Is it the local maxima { q} of the coarse-grained Ψ(q, t), the worlds? But this coarse-graining is vague; the number of worlds is not defined; and the timing of the splitting is also not defined. 2 Some worlds are more real than others. When a world splits, some daughter worlds are bigger than others: Ψ(t) α Ψ 1 (t + τ) + β Ψ 2 (t + τ) ; α > β. But each world will feel equally real to its respective inhabitants (our future selves, at time t + τ). So we should I care more about my future self in the world 1, and in particular why should I care in the ratio α 2 : β 2? 3 If the worlds really split, why not just postulate that the branches I don t experience get pruned? They have no effect on anything! Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

34 Bohmian mechanics and Many Worlds: better together? Outline 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

35 Bohmian mechanics and Many Worlds: better together? Outline Issue 3 with Bohmian mechanics: it should be hyper-real 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

36 Bohmian mechanics and Many Worlds: better together? Empty Waves Issue 3 with Bohmian mechanics: it should be hyper-real Bohmian mechanics in the light of the MWI If Ψ(q, t) is real in Bohmian mechanics, isn t it just Many Worlds in denial? The true configuration x(t) will typically be near one of the MWI-worlds q(t). What about all of the empty MWI-worlds? Won t their denizens still feel real even with no x? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

37 Bohmian mechanics and Many Worlds: better together? Empty Waves Issue 3 with Bohmian mechanics: it should be hyper-real Bohmian mechanics in the light of the MWI If Ψ(q, t) is real in Bohmian mechanics, isn t it just Many Worlds in denial? The true configuration x(t) will typically be near one of the MWI-worlds q(t). What about all of the empty MWI-worlds? Won t their denizens still feel real even with no x? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

38 Bohmian mechanics and Many Worlds: better together? Empty Waves Issue 3 with Bohmian mechanics: it should be hyper-real Bohmian mechanics in the light of the MWI If Ψ(q, t) is real in Bohmian mechanics, isn t it just Many Worlds in denial? The true configuration x(t) will typically be near one of the MWI-worlds q(t). What about all of the empty MWI-worlds? Won t their denizens still feel real even with no x? Also, Probability Why should the wavefunction play this dual role of pilot wave and defining the a priori probability distribution for x? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

39 Bohmian mechanics and Many Worlds: better together? Outline Addressing issue 3 (&...): theory of Many Interacting Worlds 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

40 Bohmian mechanics and Many Worlds: better together? Repulsive trajectories? Addressing issue 3 (&...): theory of Many Interacting Worlds Suggestive of trajectories for a bunch of particles which repel one another. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

41 Bohmian mechanics and Many Worlds: better together? Repulsive trajectories? Addressing issue 3 (&...): theory of Many Interacting Worlds Suggestive of trajectories for a bunch of particles which repel one another. Why not take this literally? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

42 Bohmian mechanics and Many Worlds: better together? Repulsive trajectories? Addressing issue 3 (&...): theory of Many Interacting Worlds Suggestive of trajectories for a bunch of particles which repel one another. Why not take this literally? Bill Poirier, Bohmian mechanics without pilot waves Chem. Phys. (2010) developed this theory for a continuous ensemble of particles. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

43 Bohmian mechanics and Many Worlds: better together? Repulsive trajectories? Addressing issue 3 (&...): theory of Many Interacting Worlds Suggestive of trajectories for a bunch of particles which repel one another. Why not take this literally? Bill Poirier, Bohmian mechanics without pilot waves Chem. Phys. (2010) developed this theory for a continuous ensemble of particles. We think it is clearer to imagine a finite (but very large) ensemble: Phys. Rev. X 4, (2014). Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

44 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds A single particle; many Bohmian worlds Consider a world comprising a single nonrelativistic particle of mass m, in one spatial dimension with potential V (q). In Bohm s 1952 formulation a particle obeys a modified force law, mẍ(t) = [V (q) + Q(q)] q, q=x(t) with quantum potential Q(q) = ψ(q, t) 1 2 2m ψ(q, t). Let there be N 1 worlds {x n } N n=1 : x n < x n+1, for all n. Say the x n (t 0 ) are arranged evenly according to the distribution P(x; t 0 ) = ψ(x; t 0 ) 2 and the ẋ n (t 0 ) obey de Broglie s formula ẋ n (t 0 ) = m Im ψ (q; t 0 ) ψ(q; t 0 ) q=x n (t 0 ) Then by Bohm s force law, the x n (t) will remain arranged evenly according to P(x; t) = ψ(x; t) 2 for all times. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

45 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds A single particle; many interacting worlds Our idea: If we approximate ψ(q, t) 2 by the local density of worlds, we can replace the quantum potential Q(q) by a function of the positions of the nearby worlds. e.g. toy model: the world-positions evolve via Newton s equations mẍ n (t) = x n [ V (x n ) + n Q uip 3 ( x n 1, x n, x n +1 )]. where the 3-body (3-world) local potential can be chosen as Q uip 3 (x n 1, x n, x n+1) [ ] = m x n+1 x n x n x n 1 As N, we should recover the (virtual) Bohmian ensemble. However, there is no wavefunction in the ontology of our theory; our ensemble of worlds is real, not virtual. this is necessary because our worlds interact. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

46 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds Realism Type 3: Many Interacting Worlds db-b virtual ensemble, guided by real wavefunction. MIW real ensemble, reconstructed wavefunction. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

47 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds What else have we done? Suggested a conservative inter-world potential that may work for the many-particles case. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

48 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds What else have we done? Suggested a conservative inter-world potential that may work for the many-particles case. Using some generic properties of such inter-world potentials, given a qualitative explanation for quantum tunneling. derived Ehrenfest s theorem, as in CM and QM, for all N. derived quadratic-in-time wave-packet spreading. Wiseman et al. (Griffith University) V0 ½mv02 Ensembles of Bohmian trajectories x1 x2 EmQM, Vienna, / 38

49 Bohmian mechanics and Many Worlds: better together? Addressing issue 3 (&...): theory of Many Interacting Worlds What else have we done? Suggested a conservative inter-world potential that may work for the many-particles case. Using some generic properties of such inter-world potentials, given a qualitative explanation for quantum tunneling. derived Ehrenfest s theorem, as in CM and QM, for all N. derived quadratic-in-time wave-packet spreading. V0 ½mv02 xn x1 x2 4 2 Developed and tested an algorithm for finding ground states of a particle in a 1D potential. Wiseman et al. (Griffith University) st Ensembles of Bohmian trajectories EmQM, Vienna, / 38

50 Bohmian mechanics and Many Worlds: better together? Open questions Addressing issue 3 (&...): theory of Many Interacting Worlds 1 Is the dynamics stable for excited state distributions and (more generally) distributions with nodes? 2 When is it useful as a numerical tool? 3 Can we deal with spin? Interacting spins (a quantum computer)? 4 Can we explain Bell-nonlocality with a simple, few-world model? 5 Can we deal with relativistic QM? Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

51 Bohmian mechanics and Many Worlds: better together? Recapitulation Addressing issue 3 (&...): theory of Many Interacting Worlds 1 Orthodox Quantum Mechanics: who could ask for anything more? 2 Bohmian Mechanics: who could ask for anything more? Issue 1 with Bohmian mechanics: why believe it is real? Addressing issue 1: theory and experiment Issue 2 with Bohmian mechanics: it seems surreal Addressing issue 2: theory and experiment 3 Many Worlds Interpretation: who could ask for anything more? 4 Bohmian mechanics and Many Worlds: better together? Issue 3 with Bohmian mechanics: it should be hyper-real Addressing issue 3 (&...): theory of Many Interacting Worlds Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

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53 Ontology and Epistemology All worlds are equally real. Your consciousness supervenes on only one of the worlds. Just as (here and in classical physics) your consciousness supervenes only on one part (i.e. you) of a world. There is no wavefunction and hence no collapse of the wavefunction. Effective wavefunction collapse is just Bayesian updating by some consciousness about which world it is likely to supervene upon. Agreement with standard QM emerges (in the N limit) much the same as in de Broglie Bohm. All quantum effects are a consequence of interaction between worlds so they are observable! For finite N deviations from QM may be observable. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

54 Analytical results from E = [mẋ 2 + V + Q 3 ] Ehrenfest s theorem, as in CM and QM, for all N, d dt x = 1 m mẋ, d dt mẋ = V (x) for the (real!) ensemble averages e.g. x N 1 N n=1 x n. Ensemble spreading V t [x] = V 0 [x] + 2t m Cov 0[x, mẋ] + t2 [2 E m ẋ 2] m as in QM and CM, for all N. Qualitative explanation for nonclassical barrier transmission and nonclassical reflection, via the quantum repulsion for N > 1. The harmonic oscillator ground configuration has an energy E = N 1 ω N 2, as in CM for N = 1 and as in QM in the limit N. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

55 MBER 14 PHYSICAL REVIEW LETTERS How the Result of a Measurement of a Component of the Spin of a Spin- 2 Particle Can Turn Out to be 100 Yakir Aharonov, David Z. Albert, and Lev Vaidman Physics Department, University of South Carolina, Columbia, South Carolina 29208, and School of Physics and AstronomyT, el Av-iv UniversityR, amat Aviv 69978, Israel PRL 60, 1351 (1988). (Received 30 June 1987) Consider an arbitrary e have found that the usual measuring procedure for preselected and postselected ensembles system observable A. tum systems gives unusual results. Under some natural conditions of weakness of the measuremen result consistently Assume defines a probe a newith kind of value for a quantum variable, which we call the weak val escription of the measurement of the weak value of a component of a spin for an ensemble [ˆq, ˆp] = i, initially in a elected and postselected spin- 2 particles is presented. MUS (minimum S numbers: uncertainty Bz state). The probe state is defined by σp in, p in, and q in = 0. ill describe Assume an experiment (von Neumann) which meaponent of Ĥ = δ(t)â ˆq, so that a spin-, particle and yields a a ˆpf ˆp in = Â. a;, the final probability distribution will be ' Gaussian with the spread hatt. The center far frombythemeasuring range of "allowed" p f can values. estimate iana will as A(p be at f ) = thep f mean p in value. of A: (A with a brief description of the standard One measurement like this will give no in Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

56 Initial and Final States. For initial system state ψ in, we can obtain, by repeating the experiment, E[A(p f ) ψ in ] = ψ in  ψin. Now consider a final strong measurement on the system too. Consider the sub-ensemble where the final result corresponds to projecting onto state φ f. Then we can consider the post-selected average E[A(p f ) ψ in, φ f ]. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38

57 The Weak Measurement Limit In the weak measurement limit, σ p, E[A(p f ) ψ in, φ f ] φ f A w ψ in Re φf  ψin φ f ψ in. Q Why is this the weak measurement limit? A Because very little information in any individual result and (ˆp in p in ) 2 = σ 2 p. A(ˆp f ) =  + (ˆpin p in ) A Because weak (not no) disturbance: ŝ f = ŝ in i[ŝ in, Â] ˆqin and (ˆq in ) 2 = 1/(2σ p ) 2 0 in this limit. Note: the weaker the measurement, the larger the number of repetitions required to obtain a reliable average. Wiseman et al. (Griffith University) Ensembles of Bohmian trajectories EmQM, Vienna, / 38