M2010 c© Dr J.F. Geurdes 2010, Probability and quantum foundation J.F. Geurdes Abstract: A classical probabilistics explanation for a typical quantum effect in Hardy's paradox is demonstrated. 1. Classical probability The common claim is that a classical probability triple, (Ω,F, P ) cannot explain quantum effects. Here the sample space Ω is any non-empty set. The σ-field, F is obtained from the set of all subsets, P(Ω) = 2Ω, of Ω. F is called a σ-field [1] if, (i) Ω ∈ F, (ii) E ∈ F ⇒ Ec = (Ω − E) ∈ F, (iii) E,F, ... ∈ F ⇒ E ∪ F ∪ .... ∈ F. The triple is completed with a probability measure P , such that, (∀ : X ∈ F) (0 ≤ P (X) ≤ 1), P (Ω) = 1. 2. Pre-measurement characteristics, numerals and algebra Let us inspect the possibilities of classical probability for Hardy's paradox [2] where quantum particles like electron and positron can be measured after mutual annihilation. This appears to reject the possibility of pre-measurement characteristics [3]. Apart from zero, unity and two the numerals of von Neuman and of Zermelo [4] are disjoint. This fact may represent mutual exclusion of electron and positron. We have, D0 = C0 = ∅. Von Neuman numerals are (n = 0, 1, 2, 3, ...) Cn+1 = {C0, C1, ....., Cn} .(1) Hence, C1 = {∅}, C2 = {∅, {∅}}, C3 = {∅, {∅} , {∅, {∅}}}, etc. Zermelo's system is Dn+1 = {Dn} .(2) Hence, D1 = {∅}, D2 = {{∅}}, D3 = {{{∅}}}, etc. We establish mutual exclusion (annihilation) with C3, modeling particle 1 and D3 modeling particle 2, for instance, as C3 ∩D3 = ∅. The sample space equals Ω = C3 ∪D3, or Ω = {∅, {∅} , {∅, {∅}} , {{∅}}} .(3) This entails the σ-field F = P(Ω) = 2Ω. Explicitly: F = {Ω, ∅, {∅} , {∅, {∅}} , {∅, {∅, {∅}}} , {∅, {{∅}}} , {{∅} , {∅, {∅}}} , {{∅} , {{∅}}}} ∪ {{{∅, {∅}} , {{∅}}} , {∅, {∅} , {∅, {∅}}} , {∅, {∅} , {{∅}}} , {{∅} , {∅, {∅}} , {{∅}}}} ∪ {{∅, {∅, {∅}} , {{∅}}} , {{∅}} , {{∅, {∅}}} , {{{∅}}}} Note, D3, C3 ∈ F. The probability measure for Ω is P (X) = |X|/|Ω|, with, |X| the cardinality of X, i.e. P ∼ Uniform(Ω). (Ω,F, P ) establishes classical probability. Finally, let us introduce the 'union of a set' [5], [6] operation, ∪Z = {x|(∃ : y ∈ Z)(x ∈ y)}(4) 1 imsart-lnms ver. 2007/12/10 file: CognMath_5.tex date: September 21, 2010 2 3. Application to Hardy's physics For set A = {C2} and B = D3 we see A ⊂ C = C3 and B = D = D3, hence, B ⊂ D3. Obviously, A∩B = ∅. The A and B represent disjoint parts of the electron and positron. Now, C2 ∈ A and that means, {∅, {∅}} ∈ A. Hence (eq. 4), x = ∅ and x = {∅} in C2 giving ∪A = C2 = {∅, {∅}} ∈ F. Identically, D2 ∈ B = D3. Hence, x = {∅} = D1, such that ∪B = D2 = {{∅}} ∈ F. Now, C2 ∩ D2 = {{∅}} ⇒ P (C2 ∩ D2) 6= 0. There exist subsets of C3 and D3 that, after taking the union, allows for simultaneous probability 6= 0. Hence, classical probability can do something similar to quantum mechanics if the ∪ on disjoint (sub)sets in annihilation processes is physical. 4. Conclusion A classical probabilistics explanation for a typical quantum behavior, similar to tunneling, has been found. If ∪ cannot be excluded from physics it may represent a quantum physical process and establishes a classical explanation. A possible physical picture for ∪ can perhaps be associated to a 'dark' mirror-matter sector [7], [8], [9] that may arise as a consequence of the experimentally established weak interaction parity non-invariance [10], [11]. References [1] Rosenthal, J. (2006). A first look at Rigorous Probability Theory World Scientific, Singapore [2] Hardy, L. (1992). Quantum mechanics, local realistic theories and Lorentzinvariant local realistic theories Phys. Rev. Lett. 68 2981-2984. [3] Einstein, A. Podolsky, B. Rosen, N. (1935). Can quantum-mechanical description of reality be considered complete? Phys. Rev. 47 777-780. [4] Jaquette, D. (2002) Philosophy of mathematics: An anthology Blackwell Publ., Oxford UK. [5] Randal Holmes, M. (2009). Elementary set theory with a universal set Volume 10 of the cahiers du centre de logique Academia Louvain-laNeuve(Belgium). [6] Hajnal, A. & Hamburger, P. (1999). Set Theory London Mathematical Society Student Texts 48, Camb. Univ. Press Cambridge (UK). [7] Foot, R. (2004). Mirror matter-type dark matter Int. J. Mod. Phys. D13, 2161-2192, arXiv:astro-ph/0407623v1. [8] Okun, B. (2007). Mirror particles and mirror matter: 50 years of speculation and search Physics-Uspekhi 50(4) 380-391, arXiv:hep-ph/0606202v2. [9] Foot, R. (2007) Mirror dark matter arXiv:hep-ph/07062694v1. [10] Lee, T.D. & Yang, C.N. (1956). Question of Partity conservation in weak interaction Phys. Rev.104(1), 254-258. [11] Ambler, E., Hayward, R.W., Hopes, D.D., Hudson, R.R., & Wu, C.S. (1956). Experimental test of parity conservation in beta decay Phys. Rev.105, 1413-1414. imsart-lnms ver. 2007/12/10 file: CognMath_5.tex date: September 21,