1. Fermions and non-commuting observables from classical probabilities

2. quantum mechanics can be described by classical statistics !

3. 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

4. Probabilistic realism Physical theories and laws only describe probabilities

5. Physics only describes probabilities Gott würfelt

6. fermions from classical statistics

7. 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

8. Grassmann functional integral action : partition function :

9. Grassmann wave function

10. observables representation as functional integral

11. particle numbers

12. time evolution

13. d=2 quantum field theory

14. time evolution of Grassmann wave function

15. Lorentz invariance

16. 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

17. one - particle wave function from coarse graining of microphysical classical statistical ensemble non – commutativity in classical statistics

18. 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

19. function observable

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

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

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

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

24. complex structure

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

26. different products of observables differs from classical product

27. Which product describes correlations of measurements ?

28. coarse graining of information for subsystems

29. 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

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

31. quantum operators

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

33. 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 !

34. classical and quantum dispersion

35. subsystem probabilities in contrast :

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

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

38. 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

39. 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´)

40. 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

41. end