* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download What is and to which end does one study Bohmian Mechanics?
Particle in a box wikipedia , lookup
Coherent states wikipedia , lookup
Density matrix wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum field theory wikipedia , lookup
Hydrogen atom wikipedia , lookup
Renormalization wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Bell test experiments wikipedia , lookup
Schrödinger equation wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
De Broglie–Bohm theory wikipedia , lookup
Scalar field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Erwin Schrödinger wikipedia , lookup
Quantum entanglement wikipedia , lookup
Renormalization group wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Dirac equation wikipedia , lookup
Elementary particle wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Bell's theorem wikipedia , lookup
Atomic theory wikipedia , lookup
Quantum state wikipedia , lookup
History of quantum field theory wikipedia , lookup
Path integral formulation wikipedia , lookup
EPR paradox wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Double-slit experiment wikipedia , lookup
Identical particles wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Matter wave wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Probability amplitude wikipedia , lookup
Wave function wikipedia , lookup
Canonical quantization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Hidden variable theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
What is and to which end does one study Bohmian Mechanics? Detlef Dürr Mathematisches Institut LMU München Feb. 2015 What is Bohmian Mechanics? What is Bohmian Mechanics? • Matter is described by point particles in physical space, i.e. an N-particle universe is described by Q1 , ..., QN , Qi ∈ R3 particle positions What is Bohmian Mechanics? • Matter is described by point particles in physical space, i.e. an N-particle universe is described by Q1 , ..., QN , Qi ∈ R3 particle positions • particles move The LAW of motion The LAW of motion • respects Galilean symmetry but is non-Newtonian. It is a mathematically consistent simplification of the Hamilton Jacobi idea of mechanics: Q(t) = (Q1 (t), ..., QN (t)), ∇= ∂ ∂q configuration obeys (time reversal invariance in ”first order” theory achieved by complex conjugation) dQ Ψ∗ ∇Ψ = v Ψ (Q(t), t) = αIm ∗ (Q(t), t) guiding equation dt Ψ Ψ α is a dimensional parameter the guiding field is The LAW of motion • respects Galilean symmetry but is non-Newtonian. It is a mathematically consistent simplification of the Hamilton Jacobi idea of mechanics: Q(t) = (Q1 (t), ..., QN (t)), ∇= ∂ ∂q configuration obeys (time reversal invariance in ”first order” theory achieved by complex conjugation) dQ Ψ∗ ∇Ψ = v Ψ (Q(t), t) = αIm ∗ (Q(t), t) guiding equation dt Ψ Ψ α is a dimensional parameter the guiding field is • the ”universal” wave function Ψ : R3N × R 7→ C(n) (q = (q1 , . . . , qN ), t) 7→ Ψ(q, t) Ψ Ψ • solves the Schrödinger equation i H=− Pn ∂Ψ (q, t) = HΨ(q, t) “Schrödinger“ equation ∂t α k=1 2 ∆k + W (Galilean invariant operator) Bohmian motion encompasses Newtonian motion Bohmian motion encompasses Newtonian motion i • write Ψ in polar form Ψ(q, t) = R(q, t)e ~ S(q,t) with R, S real functions and ~ an (action-) dimensional constant Bohmian motion encompasses Newtonian motion i • write Ψ in polar form Ψ(q, t) = R(q, t)e ~ S(q,t) with R, S real functions and ~ an (action-) dimensional constant • S satisfies a Hamilton Jacobi type of equation Bohmian motion encompasses Newtonian motion i • write Ψ in polar form Ψ(q, t) = R(q, t)e ~ S(q,t) with R, S real functions and ~ an (action-) dimensional constant • S satisfies a Hamilton Jacobi type of equation • v = m−1 ∇S for a Newtonian particle with mass m Bohmian motion encompasses Newtonian motion i • write Ψ in polar form Ψ(q, t) = R(q, t)e ~ S(q,t) with R, S real functions and ~ an (action-) dimensional constant • S satisfies a Hamilton Jacobi type of equation • v = m−1 ∇S for a Newtonian particle with mass m • vΨ = α ~ ∇S potential” =⇒ identify α = ~ m and W ~ =: V as the ”Newtonian • Newtonian Bohmian motion for ”Quantum Potential” ~2 ∆R 2m R ≈0 Bohmian mechanics with Newtonian identification of parameters1 dQ Ψ∗ ∇Ψ = v Ψ (Q(t), t) = ~m−1 Im ∗ (Q(t), t) dt Ψ Ψ where m is a diagonal matrix with mass entries mk ! n X ~2 ∂Ψ i~ (q, t) = − ∆k + V (q) Ψ(q, t) ∂t 2mk k=1 1 Analogy: Boltzmann’s constant kB relates thermodynamics to Newtonian mechanics, ~ relates Newtonian mechanics to Bohmian Mechanics simplest way to Bohmian mechanics simplest way to Bohmian mechanics • R 2 = |Ψ|2 satifies ∂t |Ψ|2 = −∇ · (v Ψ |Ψ|2 ) =: −∇ · j Ψ the quantum flux equation simplest way to Bohmian mechanics • R 2 = |Ψ|2 satifies ∂t |Ψ|2 = −∇ · (v Ψ |Ψ|2 ) =: −∇ · j Ψ the quantum flux equation • vΨ = jΨ |Ψ|2 (Pauli 1927, J.S. Bell 1964) ”Bohmian Mechanics agrees with Quantum Predictions” ”Bohmian Mechanics agrees with Quantum Predictions” • quantum flux equation means ρ(t) = |Ψ(t)|2 is equivariant: Assume Q is distributed according to ρ = |Ψ|2 then Q(t) at any other time is distributed according to ρ(t) = |Ψ(t)|2 Historical Criticisms Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated • solution: configuration space of identical particles is (like in classical mechanics) the manifold R3N /SN −→ Fermion-Boson-Alternative Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated • solution: configuration space of identical particles is (like in classical mechanics) the manifold R3N /SN −→ Fermion-Boson-Alternative • spin cannot be described in BM Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated • solution: configuration space of identical particles is (like in classical mechanics) the manifold R3N /SN −→ Fermion-Boson-Alternative • spin cannot be described in BM • solution: read Ψ∗ ∇Ψ Ψ∗ Ψ as inner product in spinor space Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated • solution: configuration space of identical particles is (like in classical mechanics) the manifold R3N /SN −→ Fermion-Boson-Alternative • spin cannot be described in BM • solution: read Ψ∗ ∇Ψ Ψ∗ Ψ as inner product in spinor space • particle creation and annihilation contradicts the existence of particles Historical Criticisms • |ψ|2 is a probability, probability is subjective, hence the Bohmian motion is guided by ignorance of the observer • In BM the position plays a distinguished role: The unitary symmetry of Hilbert-space is violated • In BM particles have definite positions: The indistinguishability of quantum particles is violated • solution: configuration space of identical particles is (like in classical mechanics) the manifold R3N /SN −→ Fermion-Boson-Alternative • spin cannot be described in BM • solution: read Ψ∗ ∇Ψ Ψ∗ Ψ as inner product in spinor space • particle creation and annihilation contradicts the existence of particles • solution: standard birth and death process Einstein’s criticism: Ψ on configuration space2 cannot be physical 2 1080 dimensional Einstein’s criticism: Ψ on configuration space2 cannot be physical • Ψ as function of configuration is called ”entanglement” of the wave function 2 1080 dimensional Einstein’s criticism: Ψ on configuration space2 cannot be physical • Ψ as function of configuration is called ”entanglement” of the wave function • k−th particle’s trajectory Qk (t) dQk (t) ~ Ψ∗ ∇ k Ψ = Im (Q1 , (t) . . . , QN (t), t), dt mk Ψ∗ Ψ 2 1080 dimensional Einstein’s criticism: Ψ on configuration space2 cannot be physical • Ψ as function of configuration is called ”entanglement” of the wave function • k−th particle’s trajectory Qk (t) dQk (t) ~ Ψ∗ ∇ k Ψ = Im (Q1 , (t) . . . , QN (t), t), dt mk Ψ∗ Ψ for entangled wave function influenced by all particles at t =⇒ manifestly not local, against the spirit of relativity 2 1080 dimensional Einstein’s criticism answered by John S. Bell Einstein’s criticism answered by John S. Bell • the derivation of Bell’s inequalities and the experimental results establish that nature is nonlocal (Jean Bricmont’s talk) Einstein’s criticism answered by John S. Bell • the derivation of Bell’s inequalities and the experimental results establish that nature is nonlocal (Jean Bricmont’s talk) • Ψ is that nonlocal agent, which produces nonlocal correlations Einstein’s criticism answered by John S. Bell • the derivation of Bell’s inequalities and the experimental results establish that nature is nonlocal (Jean Bricmont’s talk) • Ψ is that nonlocal agent, which produces nonlocal correlations • BM is just what the doctor ordered. universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis • Boltzmann’s statistical analysis of BM (ρ = |ϕ|2 ) based on typicality measure dPΨ = |Ψ|2 dq 3N which is equivariant (cf. quantum flux equation) Bohmian flow TtΨ : Q 7→ Q(t) commutes with Schrödinger evolution Ψ 7→ Ψt : dPΨ ◦ (TtΨ )−1 = dPΨt universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis • Boltzmann’s statistical analysis of BM (ρ = |ϕ|2 ) based on typicality measure dPΨ = |Ψ|2 dq 3N which is equivariant (cf. quantum flux equation) Bohmian flow TtΨ : Q 7→ Q(t) commutes with Schrödinger evolution Ψ 7→ Ψt : dPΨ ◦ (TtΨ )−1 = dPΨt universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis • Boltzmann’s statistical analysis of BM (ρ = |ϕ|2 ) based on typicality measure dPΨ = |Ψ|2 dq 3N which is equivariant (cf. quantum flux equation) Bohmian flow TtΨ : Q 7→ Q(t) commutes with Schrödinger evolution Ψ 7→ Ψt : dPΨ ◦ (TtΨ )−1 = dPΨt • Anology: Stationarity of microcanonical measure (Liouville equation) on phase space in Hamiltonian Mechanics universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis • Boltzmann’s statistical analysis of BM (ρ = |ϕ|2 ) based on typicality measure dPΨ = |Ψ|2 dq 3N which is equivariant (cf. quantum flux equation) Bohmian flow TtΨ : Q 7→ Q(t) commutes with Schrödinger evolution Ψ 7→ Ψt : dPΨ ◦ (TtΨ )−1 = dPΨt • Anology: Stationarity of microcanonical measure (Liouville equation) on phase space in Hamiltonian Mechanics • ρ is the empirical density in an ensemble of subsystems universal Ψ and subsystem’s ϕ S. Goldstein, N. Zanghì and I started this analysis 25 years ago here at IHES • BM is a complete quantum theory, notions like measurement or observer are not fundamental notions for defining the theory • the empirical import of BM comes solely from mathematical analysis • Boltzmann’s statistical analysis of BM (ρ = |ϕ|2 ) based on typicality measure dPΨ = |Ψ|2 dq 3N which is equivariant (cf. quantum flux equation) Bohmian flow TtΨ : Q 7→ Q(t) commutes with Schrödinger evolution Ψ 7→ Ψt : dPΨ ◦ (TtΨ )−1 = dPΨt • Anology: Stationarity of microcanonical measure (Liouville equation) on phase space in Hamiltonian Mechanics • ρ is the empirical density in an ensemble of subsystems • ϕ is wave function of subsystem conditional wave function ϕ of subsystem X = (X1 , . . . , Xn ) system’s particles Q = (X , Y ) splitting in system and rest of universe Y ⇓ ϕ (x) := Ψ(x, Y )/kΨ(Y )k normalized conditional wave function of subsystem guides X conditional wave function ϕ of subsystem X = (X1 , . . . , Xn ) system’s particles Q = (X , Y ) splitting in system and rest of universe Y ⇓ ϕ (x) := Ψ(x, Y )/kΨ(Y )k normalized conditional wave function of subsystem guides X crucial ”conditional measure” formula PΨ (X ∈ dx|ϕY = ϕ) = |ϕ(x)|2 dx Autonomous subsystem: effective wave function If wave function of universe Ψ(x, y ) = ϕ(x)Φ(y ) + Ψ(x, y )⊥ where suppΦ ∩ suppΨ⊥ = ∅ macroscopically disjoint and if Y ∈ suppΦ e.g. preparation of ϕ ⇓ Autonomous subsystem: effective wave function If wave function of universe Ψ(x, y ) = ϕ(x)Φ(y ) + Ψ(x, y )⊥ where suppΦ ∩ suppΨ⊥ = ∅ macroscopically disjoint and if Y ∈ suppΦ e.g. preparation of ϕ ⇓ ϕY = ϕ is effective wave function for system Autonomous subsystem: effective wave function If wave function of universe Ψ(x, y ) = ϕ(x)Φ(y ) + Ψ(x, y )⊥ where suppΦ ∩ suppΨ⊥ = ∅ macroscopically disjoint and if Y ∈ suppΦ e.g. preparation of ϕ ⇓ ϕY = ϕ is effective wave function for system decoherence sustains disjointness of supports ⇓ Schrödinger equation for ϕ for some time n i~ X ~2 ∂ϕ (x, t) = − ∆k ϕ(x, t) + V (x)ϕ(x, t) ∂t 2mk k=1 macroscopically disjoint Y - supports X N = 10 N =10 Y 24 Bohmian Subsystem (X , ϕ) physical variables dX ϕ∗ ∇ϕ = v ϕ (X (t), t) = ~m−1 Im ∗ (X (t), t) guiding equation dt ϕ ϕ n i~ X ~2 ∂ϕ (x, t) = − ∆k ϕ(x, t)+V (x)ϕ(x, t) Schrödinger equation ∂t 2mk k=1 Born’s law ρ = |ϕ|2 Born’s law ρ = |ϕ|2 • Consider an ensemble of subsystems each having effective wave function ϕ Born’s law ρ = |ϕ|2 • Consider an ensemble of subsystems each having effective wave function ϕ • Theorem: PΨ -typically the empirical distribution ρ of X -values is ≈ |ϕ|2 Born’s law ρ = |ϕ|2 • Consider an ensemble of subsystems each having effective wave function ϕ • Theorem: PΨ -typically the empirical distribution ρ of X -values is ≈ |ϕ|2 • In short: Quantum Equilibrium holds! Hydrogene ground state: ρ = |ψ0 |2 , 0 v ψ0 = 0 two slit experiment, computed trajectories computer simulation of Bohmian trajectories by Chris Dewdney 1 kx/k 0.5 −0.5 −1 m −3 x 10 1 two slit experiment: weak measurement of phase, trajectories reconstructed kx/k 0.5 0 momentum of the photons reaching any particular position in the image plane, and, by repeating this procedure in a series of planes, we can reconstruct trajectories over that range. In this sense, weak measurement finally allows us to speak about what happens to an ensemble of particles inside an interferometer. Our quantum particles are single photons emitted by a liquid helium-cooled InGaAs quantum dot (23, 24) embedded in a GaAs/AlAs micropillar cavity. The dot is optically pumped by a CW laser at 810 nm and emits single photons at −0.5 −1 5 ate [mm] photon counts) of omponents of |y〉, d and blue curves), m values calculated imaging planes at z = 5.6 m, and (D) ata points are the background subentum values were fect of Poissonian he magenta curve ned from enforcing between adjacent e-grained averags, the probabilityre not as sensitive um values to highern with steep mo- Fig. 3. The reconstructed average trajectories of an 4 ensemble of single photons in the double-slit apparatus. The trajectories are re2 constructed over the range 2.75 T 0.05 to 8.2 T 0.1 m by using the momentum data 0 (black points in Fig. 2) from 41 imaging planes. Here, −2 80 trajectories are shown. To reconstruct a set of trajectories, we determined the −4 weak momentum values for the transverse x positions at the initial plane. On the basis −6 of this initial position and momentum information, the 3000 4000 5000 6000 7000 8000 x position on the subsequent Propagation distance[mm] imaging plane that each trajectory lands is calculated, and the measured weak momentum value kx at this point found. This process is repeated until the final imaging plane is reached and the trajectories are traced out. If a trajectory lands on a point that is not the center of a pixel, then a cubic spline interpolation between neighboring momentum values is used. Transverse coordinate[mm] 0 to the initial uncertainty in pointer position, so that significant information is acquired in a single shot. However, this implies that the pointer momentum must be very uncertain, and it is this uncertainty that creates the uncontrollable, irreversible disturbance associated with measurement. In a “weak” measurement, the pointer shift is small and little information can be gained on a single shot; but, on the other hand, there may be arbitrarily little disturbance imparted to the system. It is possible to subsequently postselect the system on a desired final state. Postselecting on Downloaded from www.science −3 x 10 S.Kocsis et al: Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer. Science 2011 www.sciencemag.org SCIENCE VOL 332 3 JUNE 2011 1171 operational analysis of BM: PVM’s system (X , ϕ) and apparatus (Y , Φ) with pointer positions Yα pointing towards value α. Suppose then for ϕ = =⇒ P α cα ϕα , ϕΦ ϕα Φ P Schrödinger evolution α −→ |cα |2 = 1 Schrödinger evolution −→ Ψ= ϕα Φα X α cα ϕα Φα operational analysis of BM: PVM’s system (X , ϕ) and apparatus (Y , Φ) with pointer positions Yα pointing towards value α. Suppose then for ϕ = P α cα ϕα , ϕΦ ϕα Φ P Schrödinger evolution α −→ |cα |2 = 1 Schrödinger evolution −→ Ψ= ϕα Φα X cα ϕα Φα α =⇒ • If Y ∈ suppΦβ then ϕβ is new effective wave function for system (effective wave function collapse) operational analysis of BM: PVM’s system (X , ϕ) and apparatus (Y , Φ) with pointer positions Yα pointing towards value α. Suppose then for ϕ = P α cα ϕα , ϕΦ ϕα Φ P Schrödinger evolution α −→ |cα |2 = 1 Schrödinger evolution −→ Ψ= ϕα Φα X cα ϕα Φα α =⇒ • If Y ∈ suppΦβ then ϕβ is new effective wave function for system (effective wave function collapse) • the ϕα ’s form an orthogonal family (⇒ PVM) operational analysis of BM: PVM’s system (X , ϕ) and apparatus (Y , Φ) with pointer positions Yα pointing towards value α. Suppose then for ϕ = P α cα ϕα , ϕΦ ϕα Φ P Schrödinger evolution α −→ |cα |2 = 1 Schrödinger evolution −→ Ψ= ϕα Φα X cα ϕα Φα α =⇒ • If Y ∈ suppΦβ then ϕβ is new effective wave function for system (effective wave function collapse) • the ϕα ’s form an orthogonal family (⇒ PVM) • Probϕ (β) = Probϕ (Y ∈ suppΦβ ) = |cβ |2 = |hϕ|ϕβ i|2 operational analysis of BM: PVM’s system (X , ϕ) and apparatus (Y , Φ) with pointer positions Yα pointing towards value α. Suppose then for ϕ = P α cα ϕα , ϕΦ ϕα Φ P Schrödinger evolution α −→ |cα |2 = 1 Schrödinger evolution −→ Ψ= ϕα Φα X cα ϕα Φα α =⇒ • If Y ∈ suppΦβ then ϕβ is new effective wave function for system (effective wave function collapse) • the ϕα ’s form an orthogonal family (⇒ PVM) • Probϕ (β) = Probϕ (Y ∈ suppΦβ ) = |cβ |2 = |hϕ|ϕβ i|2 • PVM ⇒ self adjoint  = the experiment P α|ϕα ihϕα | encodes all relevant data for operational analysis: POVMs Suppose not ϕα Φ Schrödinger evolution −→ ϕα Φα but apparatus (Y , ψ) with values F (Y ) = λ ∈ Λ then probability for pointer position if system’s wave function is ϕ Probϕ (A) := PΦT (F −1 (A)) , A ⊂ Λ can be written as = hϕ| Z A d λ|φλ ihφλ ||ϕi where in general hφλ |φν i = 6 δλ,ν (overcomplete set) Z d λ|φλ ihφλ |, A ⊂ Λ A is called POVM or generalised observable Heisenberg’s uncertainty relation follows from BM Equivariance of ρ = |ϕ|2 ∂|ϕ(x, t)|2 = −divv ϕ (x, t)|ϕ(x, t)|2 =⇒ ∂t Eϕ (f (X (t)) = Eϕ(t) (f (X )) Heisenberg’s uncertainty relation follows from BM Equivariance of ρ = |ϕ|2 ∂|ϕ(x, t)|2 = −divv ϕ (x, t)|ϕ(x, t)|2 =⇒ ∂t Eϕ (f (X (t)) = Eϕ(t) (f (X )) Heisenberg’s uncertainty relation follows from BM Equivariance of ρ = |ϕ|2 ∂|ϕ(x, t)|2 = −divv ϕ (x, t)|ϕ(x, t)|2 =⇒ ∂t Eϕ (f (X (t)) = Eϕ(t) (f (X )) ⇓ by analysis m m X (t) L V∞ := lim t→∞ ~ ~ t is distributed according to |ϕ̂|2 Heisenberg’s uncertainty relation follows from BM Equivariance of ρ = |ϕ|2 ∂|ϕ(x, t)|2 = −divv ϕ (x, t)|ϕ(x, t)|2 =⇒ ∂t Eϕ (f (X (t)) = Eϕ(t) (f (X )) ⇓ by analysis m m X (t) L V∞ := lim t→∞ ~ ~ t P̂ = Z is distributed according to |ϕ̂|2 ⇓ dkk|kihk| momentum observable empirical import: (X (t), ϕ) for interesting ϕ empirical import: (X (t), ϕ) for interesting ϕ • classical limit empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ measurement |ϕ|2 through measuring X , phase by weak empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ measurement |ϕ|2 through measuring X , phase by weak • statistics of (arrival) time empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ measurement |ϕ|2 through measuring X , phase by weak • statistics of (arrival) time empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ measurement |ϕ|2 through measuring X , phase by weak • statistics of (arrival) time empirical import: (X (t), ϕ) for interesting ϕ • classical limit Bohmian trajectories approximately Newtonian • measurement of ϕ measurement |ϕ|2 through measuring X , phase by weak • statistics of (arrival) time for good wave functions good statistics arrival time statistics G supp ! 0 arrival time statistics G supp ! 0 when and where does a counter click? time statistics for Bohmian flow Pψ (X (τ ) ∈ dS, τ ∈ dt) = v ψ |ψ|2 · dSdt = j ψ · dSdt scattering formalism and scattering cross section Born’s scattering formula for single particle 16.7 The Scattering Cross-Section 373 #R BEAM ! R x1 # A "# x3 V k1 k0 k3 x2 supp ! k2 Fig. 16.6 Schematic representation of a scattering experiment. Also shown at the bottom is the support of the Fourier transform of the wave function y , which is a member of a beam of identical wave functions impinging on the target. See the text for further explanation it to compute the probability of scattering into the cone CS (see Remark 16.2). But since we already have the the crucial formula (16.56) for the crossing probability, which directly connects with the clicks in the detector, we shall now go on with that scattering formalism and scattering cross section Born’s scattering formula for single particle 16.7 The Scattering Cross-Section 373 #R BEAM ! R x1 # A "# x3 V k1 k0 k3 x2 supp ! k2 Fig. 16.6 Schematic representation of a scattering experiment. Also shown at the bottom is the R large support of the Fourier transform of the wave function y , which is a member of a beam of identical ψ 2 wave functions impinging R on the target. See the text for further explanation in P (X (τ ) ∈ Σ , τ ∈ [0, ∞)) ≈ R CΣ dkhk|Sψ i it to compute the probability of scattering into the cone CS (see Remark 16.2). But since we already have the the crucial formula (16.56) for the crossing probability, which directly connects with the clicks in the detector, we shall now go on with that which in the limit R → ∞ should become (1). In Bohmian mechanics the joint probability (2) is readily defined, where P is the measure Pψ on the particles’ initial positions with density |ψ|2 . By virtue of the continuity of Bohmian trajectories, first B crossing times through boundaries of regions in space are well defined. Let texl,R be the first exit time of Bl,R 3 the lth particle from the ball xl < R then X ψ (x , t ) ∈ R is the position of the lth particle at this 0 ex l time (see Figure 1). We shall show that # # $ $2 ! " Bl,R $% $ (3) lim Pψ X ψ ... $ψout (k)$ d3 k1 · · · d3 kN . l (x0 , tex ) ∈ RΣl ∀ l ∈ {1, . . . , N } = many particle scattering R→∞ C Σ1 C ΣN RΣ 1 B 1,R Xψ 1 (x0 , tex ) Σ1 supp ψ x0,2 x0,1 S2 Target RS 2 Σ2 RΣ 2 B 2,R Xψ 2 (x0 , tex ) Figure 1. Sketch of the scattering situation for N = 2. The idea for proving (3) is rather simple: Far away from the scattering center, the particles’ trajectories become more or less straight lines directed along the asymptotic velocity vψ ∞ (x0 ) := lim t→∞ % 2 1 X ψ (x0 , t) , t which in the limit R → ∞ should become (1). In Bohmian mechanics the joint probability (2) is readily defined, where P is the measure Pψ on the particles’ initial positions with density |ψ|2 . By virtue of the continuity of Bohmian trajectories, first B crossing times through boundaries of regions in space are well defined. Let texl,R be the first exit time of Bl,R 3 the lth particle from the ball xl < R then X ψ (x , t ) ∈ R is the position of the lth particle at this 0 ex l time (see Figure 1). We shall show that # # $ $2 ! " Bl,R $% $ (3) lim Pψ X ψ ... $ψout (k)$ d3 k1 · · · d3 kN . l (x0 , tex ) ∈ RΣl ∀ l ∈ {1, . . . , N } = many particle scattering R→∞ C Σ1 C ΣN RΣ 1 B 1,R Xψ 1 (x0 , tex ) Σ1 supp ψ x0,2 x0,1 S2 Target RS 2 Σ2 RΣ 2 B 2,R Xψ 2 (x0 , tex ) Figure 1. Sketch of the scattering situation for N = 2. ”genuine” Bohmian analysis The idea for proving (3) is rather simple: Far away from the scattering center, the particles’ trajectories become more or less straight lines directed along the asymptotic velocity vψ ∞ (x0 ) := lim t→∞ % 2 1 X ψ (x0 , t) , t Gretchen Frage: Wie hältst du es mit der Relativität? Relativistic Bohmian Theory Weinberg’s challenge Gretchen Frage: Wie hältst du es mit der Relativität? Relativistic Bohmian Theory Weinberg’s challenge It does not seem possible to extend Bohm’s version of quantum mechanics to theories in which particles can be created and destroyed, which includes all known relativistic quantum theories. (Steven Weinberg to Shelly Goldstein, 1996) Why should it be problematic to have particles created or destroyed? Why should it be problematic to have particles created or destroyed? • philosophically not possible? Why should it be problematic to have particles created or destroyed? • philosophically not possible? • technically, i.e. mathematically not possible? Why should it be problematic to have particles created or destroyed? • philosophically not possible? • technically, i.e. mathematically not possible? • not possible in a deterministic theory of particles in motion? Creation and Annihilation, the configuration space S∞ Q: configuration space Q = n=0 Q(n) (disjoint union) a) Q(0) no particle b) Q(1) one particle c) Q(2) two particles d) Q(3) three particles (a) (b) Q(t2−) Q(t1−) Q(t2+) (c) Q(t1+) (d) The LAW: equivariant Markov Process The LAW: equivariant Markov Process • guiding field Ψ ∈ F, a Fock space The LAW: equivariant Markov Process • guiding field Ψ ∈ F, a Fock space • P(dq): positive-operator-valued measure (POVM) on Q acting on F so that the probability that the systems particles in the state Ψ are in dq at time t is Pt (dq) = hΨt |P(dq)|Ψt i The LAW: equivariant Markov Process • guiding field Ψ ∈ F, a Fock space • P(dq): positive-operator-valued measure (POVM) on Q acting on F so that the probability that the systems particles in the state Ψ are in dq at time t is Pt (dq) = hΨt |P(dq)|Ψt i • For a Hamiltonian H (e.g. quantum field Hamiltonian) ∂Ψt = HΨt −→ ∂t d Pt (dq) 2 = Im hΨt |P(dq)H|Ψt i . dt ~ i~ The LAW: equivariant Markov Process • guiding field Ψ ∈ F, a Fock space • P(dq): positive-operator-valued measure (POVM) on Q acting on F so that the probability that the systems particles in the state Ψ are in dq at time t is Pt (dq) = hΨt |P(dq)|Ψt i • For a Hamiltonian H (e.g. quantum field Hamiltonian) ∂Ψt = HΨt −→ ∂t d Pt (dq) 2 = Im hΨt |P(dq)H|Ψt i . dt ~ • Find ”minimal” generator so that (rewrite left hand side, so that) i~ d Pt (dq) = Lt Pt (dq) . dt (Minimal) Markovian Process: Flow, (No) Diffusion, (Only as much as necessary) Jumps Quantum field Hamiltonians provide rates for configuration jumps Generator for pure Jump-Process Z (Lρ)(dq) = σ(dq|q 0 )ρ(dq 0 ) − σ(dq 0 |q)ρ(dq) q 0 ∈Q H = H0 + HI L = L0 + LI HI is often an Integral-Operator −→ Jump-Generator given by rates σ(dq|q 0 ) = [(2/~) Im hΨ|P(dq)HI P(dq 0 )|Ψi]+ . hΨ|P(dq 0 )|Ψi The tension with relativity challenge: Einstein’s criticism of QM The tension with relativity challenge: Einstein’s criticism of QM Nature is nonlocal, the wave function is the nonlocal agent, Bohmian Mechanics takes the wave function seriously: it needs for its formulation a simultaneity structure, e.g. a foliation F which seems to be against the spirit of relativity The tension with relativity challenge: Einstein’s criticism of QM Nature is nonlocal, the wave function is the nonlocal agent, Bohmian Mechanics takes the wave function seriously: it needs for its formulation a simultaneity structure, e.g. a foliation F which seems to be against the spirit of relativity Possible relief: The foliation F Ψ is given by the wave function, e.g. defined by a time like vector field induced by the wave function. Covariance is expressed by the commutative diagram Ψ −−−−→ F Ψ Λ Ug y y g (1) 0 Ψ0 −−−−→ F Ψ . Here the natural action Λg on the foliation is the action of Lorentzian g on any leaf Σ of the foliation F Ψ . to which end... to which end... • non commutativity of observables is a simple consequence of BM to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms • the challenge BM offers: Dirac divided the difficulties of quantum theory in two classes to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms • the challenge BM offers: Dirac divided the difficulties of quantum theory in two classes • First class difficulties are those which have to do with the measurement problem–how do facts arise? to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms • the challenge BM offers: Dirac divided the difficulties of quantum theory in two classes • First class difficulties are those which have to do with the measurement problem–how do facts arise? • Second class difficulties are those which have to do with singularities in field theories (self energy, Dirac vacuum, ...) to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms • the challenge BM offers: Dirac divided the difficulties of quantum theory in two classes • First class difficulties are those which have to do with the measurement problem–how do facts arise? • Second class difficulties are those which have to do with singularities in field theories (self energy, Dirac vacuum, ...) • BM solves first class difficulties – it encourages the search for relativistic interacting theories which are mathematically coherent from the start to which end... • non commutativity of observables is a simple consequence of BM • nonlocality of nature is manifest in BM • BM is in Bell’s sense a theory like Maxwell-theory of electromagnetism: theorems instead of a cornucopia of axioms • the challenge BM offers: Dirac divided the difficulties of quantum theory in two classes • First class difficulties are those which have to do with the measurement problem–how do facts arise? • Second class difficulties are those which have to do with singularities in field theories (self energy, Dirac vacuum, ...) • BM solves first class difficulties – it encourages the search for relativistic interacting theories which are mathematically coherent from the start • a guiding example is Gauss-Weber-Tetrode-Fokker-Schwarzschild-Wheeler-Feynman direct interaction theory. Fully relativistic and without fields (my friends Shelly and Nino are not enthusiastic about that theory, my young friends are and future is theirs) the end: perhaps more on the solutions of second class difficulties in 25 years