1. Quantum fermions from classical statistics

2. quantum mechanics can be described by classical statistics !

3. quantum particle from classical probabilities

4. Double slit experiment Is there a classical probability density w(x,t) describing interference ? Suitable time evolution law : local, causal ? Yes ! Bell’s inequalities ? Kochen-Specker Theorem ? Or hidden parameters w(x,α,t) ? or w(x,p,t) ? or w(x,p,t) ?

5. statistical picture of the world basic theory is not deterministic basic theory is not deterministic basic theory makes only statements about probabilities for sequences of events and establishes correlations basic theory makes only statements about probabilities for sequences of events and establishes correlations probabilism is fundamental, not determinism ! probabilism is fundamental, not determinism ! quantum mechanics from classical statistics : not a deterministic hidden variable theory

6. Probabilistic realism Physical theories and laws only describe probabilities

7. Physics only describes probabilities Gott würfelt

8. Physics only describes probabilities Gott würfelt Gott würfelt Gott würfelt nicht

9. Physics only describes probabilities Gott würfelt Gott würfelt Gott würfelt nicht humans can only deal with probabilities

10. probabilistic Physics There is one reality There is one reality This can be described only by probabilities This can be described only by probabilities one droplet of water … 10 20 particles 10 20 particles electromagnetic field electromagnetic field exponential increase of distance between two neighboring trajectories exponential increase of distance between two neighboring trajectories

11. probabilistic realism The basis of Physics are probabilities for predictions of real events

12. laws are based on probabilities determinism as special case : probability for event = 1 or 0 probability for event = 1 or 0 law of big numbers law of big numbers unique ground state … unique ground state …

13. conditional probability sequences of events( measurements ) sequences of events( measurements ) are described by are described by conditional probabilities conditional probabilities both in classical statistics and in quantum statistics

14. w(t 1 ) not very suitable not very suitable for statement, if here and now a pointer falls down :

15. Schrödinger’s cat conditional probability : if nucleus decays then cat dead with w c = 1 (reduction of wave function)

16. classical particle without classical trajectory

17. no classical trajectories also for classical particles in microphysics : trajectories with sharp position and momentum for each moment in time are inadequate idealization ! still possible formally as limiting case

18. quantum particle classical particle particle-wave duality particle-wave duality uncertainty uncertainty no trajectories no trajectories tunneling tunneling interference for double slit interference for double slit particle – wave duality sharp position and momentum classical trajectories maximal energy limits motion only through one slit

19. quantum particle classical particle quantum - probability - amplitude ψ(x) quantum - probability - amplitude ψ(x) Schrödinger - equation Schrödinger - equation classical probability in phase space w(x,p) Liouville - equation for w(x,p) ( corresponds to Newton eq. for trajectories )

20. quantum formalism for classical particle

21. probability distribution for one classical particle classical probability distribution in phase space

22. wave function for classical particle classical probability distribution in phase space wave function for classical particle depends on depends on position position and momentum ! and momentum ! C C

23. wave function for one classical particle real real depends on position and momentum depends on position and momentum square yields probability square yields probability CC similarity to Hilbert space for classical mechanics by Koopman and von Neumann in our case : real wave function permits computation of wave function from probability distribution ( up to some irrelevant signs )

24. quantum laws for observables C C

25. x y p z >0 p z <0 ψ

26. time evolution of classical wave function

27. Liouville - equation describes classical time evolution of classical probability distribution for one particle in potential V(x)

28. time evolution of classical wave function C CC

29. wave equation modified Schrödinger - equation CC

30. wave equation CC fundamenal equation for classical particle in potential V(x) replaces Newton’s equations

31. particle - wave duality wave properties of particles : continuous probability distribution

32. particle – wave duality experiment if particle at position x – yes or no : discrete alternative probability distribution for finding particle at position x : continuous 1 1 0

33. particle – wave duality All statistical properties of classical particles can be described in quantum formalism ! no quantum particles yet ! no quantum particles yet !

34. modification of Liouville equation

35. modification of evolution for classical probability distribution HWHWHWHW HWHWHWHW CC

36. quantum particle with evolution equation all expectation values and correlations for quantum – observables, as computed from classical probability distribution, coincide for all times precisely with predictions of quantum mechanics for particle in potential V CCC

37. classical probabilities – not a deterministic classical theory quantum pa rticle from classical probabilities in phase space !

38. zwitter difference between quantum and classical particles only through different time evolution continuousinterpolation CL QM HWHWHWHW

39. zwitter - Hamiltonian γ=0 : quantum – particle γ=0 : quantum – particle γ=π/2 : classical particle γ=π/2 : classical particle other interpolating Hamiltonians possible !

40. How good is quantum mechanics ? small parameter γ can be tested experimentally zwitter : no conserved microscopic energy static state : or C

41. experiments for determination or limits on zwitter – parameter γ ? lifetime of nuclear spin states > 60 h ( Heil et al.) : γ 60 h ( Heil et al.) : γ < 10 -14

42. fermions from classical statistics

43. Classical probabilities for two interfering Majorana spinors Interferenceterms

44. Ising-type lattice model x : points on lattice n(x) =1 : particle present, n(x)=0 : particle absent n(x) =1 : particle present, n(x)=0 : particle absent

45. microphysical ensemble states τ states τ labeled by sequences of occupation numbers or bits n s = 0 or 1 labeled by sequences of occupation numbers or bits n s = 0 or 1 τ = [ n s ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc. τ = [ n s ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc. s=(x,γ) s=(x,γ) probabilities p τ > 0 probabilities p τ > 0

46. (infinitely) many degrees of freedom s = ( x, γ ) x : lattice points, γ : different species number of values of s : B number of states τ : 2^B

47. Classical wave function Classical wave function q is real, not necessarily positive Positivity of probabilities automatic.

48. Time evolution Rotation preserves normalization of probabilities Evolution equation specifies dynamics simple evolution : R independent of q

49. Grassmann formalism Formulation of evolution equation in terms of action of Grassmann functional integral Formulation of evolution equation in terms of action of Grassmann functional integral Symmetries simple, e.g. Lorentz symmetry for relativistic particles Symmetries simple, e.g. Lorentz symmetry for relativistic particles Result : evolution of classical wave function describes dynamics of Dirac particles Result : evolution of classical wave function describes dynamics of Dirac particles Dirac equation for wave function of single particle state Dirac equation for wave function of single particle state Non-relativistic approximation : Schrödinger equation for particle in potential Non-relativistic approximation : Schrödinger equation for particle in potential

50. Grassmann wave function Map between classical states and basis elements of Grassmann algebra For every n s = 0 : g τ contains factor ψ s Grassmann wave function : s = ( x, γ )

51. Functional integral Grassmann wave function depends on t, since classical wave function q depends on t ( fixed basis elements of Grassmann algebra ) ( fixed basis elements of Grassmann algebra ) Evolution equation for g(t) Evolution equation for g(t) Functional integral Functional integral

52. Wave function from functional integral Wave function from functional integral L(t) depends only on ψ(t) and ψ(t+ε)

53. Evolution equation Evolution equation for classical wave function, and therefore also for classical probability distribution, is specified by action S Evolution equation for classical wave function, and therefore also for classical probability distribution, is specified by action S Real Grassmann algebra needed, since classical wave function is real Real Grassmann algebra needed, since classical wave function is real

54. Massless Majorana spinors in four dimensions

55. Time evolution linear in q, non-linear in p

56. One particle states One –particle wave function obeys Dirac equation : arbitrary static “vacuum” state

57. Dirac spinor in electromagnetic field one particle state obeys Dirac equation complex Dirac equation in electromagnetic field

58. Schrödinger equation Non – relativistic approximation : Non – relativistic approximation : Time-evolution of particle in a potential described by standard Schrödinger equation. Time-evolution of particle in a potential described by standard Schrödinger equation. Time evolution of probabilities in classical statistical Ising-type model generates all quantum features of particle in a potential, as interference ( double slit ) or tunneling. This holds if initial distribution corresponds to one- particle state. Time evolution of probabilities in classical statistical Ising-type model generates all quantum features of particle in a potential, as interference ( double slit ) or tunneling. This holds if initial distribution corresponds to one- particle state.

59. quantum particle from classical probabilities ν

60. what is an atom ? quantum mechanics : isolated object quantum mechanics : isolated object quantum field theory : excitation of complicated vacuum quantum field theory : excitation of complicated vacuum classical statistics : sub-system of ensemble with infinitely many degrees of freedom classical statistics : sub-system of ensemble with infinitely many degrees of freedom

61. i

62. Phases and complex structure introduce complex spinors : complex wave function :

63. h

64. Simple conversion factor for units

65. unitary time evolution ν

66. fermions and bosons ν

67. [A,B]=C

68. non-commuting observables classical statistical systems admit many product structures of observables classical statistical systems admit many product structures of observables many different definitions of correlation functions possible, not only classical correlation ! many different definitions of correlation functions possible, not only classical correlation ! type of measurement determines correct selection of correlation function ! type of measurement determines correct selection of correlation function ! example 1 : euclidean lattice gauge theories example 1 : euclidean lattice gauge theories example 2 : function observables example 2 : function observables

69. function observables

70. microphysical ensemble states τ states τ labeled by sequences of occupation numbers or bits n s = 0 or 1 labeled by sequences of occupation numbers or bits n s = 0 or 1 τ = [ n s ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc. τ = [ n s ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc. probabilities p τ > 0 probabilities p τ > 0

71. function observable

72. s I(x 1 ) I(x 4 ) I(x 2 ) I(x 3 ) normalized difference between occupied and empty bits in interval

73. generalized function observable generalized function observable normalization classicalexpectationvalue several species α

74. position classical observable : fixed value for every state τ

75. momentum derivative observable derivative observable classical observable : fixed value for every state τ

76. complex structure

77. classical product of position and momentum observables classical product of position and momentum observables commutes !

78. different products of observables differs from classical product

79. Which product describes correlations of measurements ?

80. coarse graining of information for subsystems

81. density matrix from coarse graining position and momentum observables use only position and momentum observables use only small part of the information contained in p τ, small part of the information contained in p τ, relevant part can be described by density matrix relevant part can be described by density matrix subsystem described only by information subsystem described only by information which is contained in density matrix which is contained in density matrix coarse graining of information coarse graining of information

82. quantum density matrix density matrix has the properties of a quantum density matrix a quantum density matrix

83. quantum operators

84. quantum product of observables the product is compatible with the coarse graining and can be represented by operator product

85. incomplete statistics classical product is not computable from information which is not computable from information which is available for subsystem ! is available for subsystem ! cannot be used for measurements in the subsystem ! cannot be used for measurements in the subsystem !

86. classical and quantum dispersion

87. subsystem probabilities in contrast :

88. squared momentum quantum product between classical observables : maps to product of quantum operators

89. non – commutativity in classical statistics commutator depends on choice of product !

90. measurement correlation correlation between measurements of positon and momentum is given by quantum product correlation between measurements of positon and momentum is given by quantum product this correlation is compatible with information contained in subsystem this correlation is compatible with information contained in subsystem

91. coarse graining from fundamental fermions at the Planck scale to atoms at the Bohr scale coarse graining from fundamental fermions at the Planck scale to atoms at the Bohr scale p([n s ]) ρ(x, x´)

92. conclusion quantum statistics emerges from classical statistics quantum statistics emerges from classical statistics quantum state, superposition, interference, entanglement, probability amplitude quantum state, superposition, interference, entanglement, probability amplitude unitary time evolution of quantum mechanics can be described by suitable time evolution of classical probabilities unitary time evolution of quantum mechanics can be described by suitable time evolution of classical probabilities conditional correlations for measurements both in quantum and classical statistics conditional correlations for measurements both in quantum and classical statistics

93. end

94. Can quantum physics be described by classical probabilities ? “ No go “ theorems Bell, Clauser, Horne, Shimony, Holt Bell, Clauser, Horne, Shimony, Holt implicit assumption : use of classical correlation function for correlation between measurements implicit assumption : use of classical correlation function for correlation between measurements Kochen, Specker Kochen, Specker assumption : unique map from quantum operators to classical observables assumption : unique map from quantum operators to classical observables