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 Gott würfelt Gott würfelt nicht

8. Physics only describes probabilities Gott würfelt

9. but nothing beyond classical statistics is needed for quantum mechanics !

10. fermions from classical statistics

11. Classical probabilities for two interfering Majorana spinors Interferenceterms

12. 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

13. 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

14. (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

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

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

17. 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

18. 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 )

19. 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

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

21. 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

22. Massless Majorana spinors in four dimensions

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

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

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

26. 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.

27. quantum particle from classical probabilities ν

28. 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

29. i

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

31. h

32. Simple conversion factor for units

33. unitary time evolution ν

34. fermions and bosons ν

35. [A,B]=C

36. 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

37. function observables

38. 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

39. function observable

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

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

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

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

44. complex structure

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

46. different products of observables differs from classical product

47. Which product describes correlations of measurements ?

48. coarse graining of information for subsystems

49. 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

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

51. quantum operators

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

53. 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 !

54. classical and quantum dispersion

55. subsystem probabilities in contrast :

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

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

58. 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

59. 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´)

60. quantum mechanics from classical statistics probability amplitude probability amplitude entanglement entanglement interference interference superposition of states superposition of states fermions and bosons fermions and bosons unitary time evolution unitary time evolution transition amplitude transition amplitude non-commuting operators non-commuting operators violation of Bell’s inequalities violation of Bell’s inequalities

61. 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

62. zwitters

63. zwitters no different concepts for classical and quantum particles no different concepts for classical and quantum particles continuous interpolation between quantum particles and classical particles is possible continuous interpolation between quantum particles and classical particles is possible ( not only in classical limit ) ( not only in classical limit )

64. 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 particles sharp position and momentum classical trajectories maximal energy limits motion only through one slit

65. quantum particle from classical probabilities in phase space probability distribution in phase space for probability distribution in phase space for o ne particle o ne particle w(x,p) w(x,p) as for classical particle ! as for classical particle ! observables different from classical observables observables different from classical observables time evolution of probability distribution different from the one for classical particle time evolution of probability distribution different from the one for classical particle

66. 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

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

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

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

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

71. 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

72. 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