Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
A Few Good Examples why to Refute Classical Logic in Quantum Mechanics Lovre Grisogono Mateo Paulišić University of Zagreb Faculty of Science Department of Physics Split, July 7-8 2014 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Classical Logic Classical logic is a type of formal logic which is most intensively studied and most widely used on all levels of education. Classical logic is defined by a set of properties. Classical logic generally includes propositional logic and first-order logic. Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Definition of CL We define classical logic with the following set of properties: Binarity: Set of truth values - V(P) ∈ {0, 1} Commutativity: P ∧ Q ≡ Q ∧ P, P ∨ Q ≡ Q ∨ P Distributivity: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) Principle of excluded middle: V(P ∨ ¬P) = 1 Principle of non-contradiction: V(¬(P ∧ ¬P)) = 1, alternatively V(P ∧ ¬P) = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Quantum Mechanics Branch of physics Atomar and subatomar scale Commutative and non-commutative operators Uncertainty principle Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM But the Question Is? In 1936 Birkoff and von Neumann wrote the article “The Logic of Quantum Mechanics”. Birkoff and von Neumann wanted to find the logical structure in quantum mechanics which did not conform to classical logic. Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM But the Question Is? In 1936 Birkoff and von Neumann wrote the article “The Logic of Quantum Mechanics”. Birkoff and von Neumann wanted to find the logical structure in quantum mechanics which did not conform to classical logic. Should we refute classical logic for QM? Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Definitions Quantum mechanical proposition is a projection operator (generally observable) P - experiment outcome prediction Px = x = 1 · x for x ∈ LP Py = 0 = 0 · y for y ∈ L⊥ P P ∧ Q - operator projecting on LP∧Q = LP ∩ LQ P ∨ Q - operator projecting on LP + LQ ¬P = 1 − P P ∧ Q = lim (PQ)n n→∞ P ∨ Q = ¬[¬P ∧ ¬Q] Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Examples Example 1: Sz = ( ~2 )P1 − ( ~2 )P2 P1 = |↑i h↑| P2 = |↓i h↓| 1 0 Matrix notation: |↑i = |↓i = 0 1 P1 = 1 0 0 0 0 0 P2 = 0 1 For Sz : Sz = ~ 2 1 0 0 −1 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Using previous definitions: P1 ∧ P2 = lim (P1 P2 )n n→+∞ n 1 0 0 0 = lim · 0 0 0 1 n→+∞ n 0 0 = lim =0 n→+∞ 0 0 P1 ∨ P2 = 1 − [(1 − P1 ) ∧ (1 − P2 )] = 1 − [P2 ∧ P1 ] =1 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Using previous definitions: P1 ∧ P2 = lim (P1 P2 )n n→+∞ n 1 0 0 0 = lim · 0 0 0 1 n→+∞ n 0 0 = lim =0 n→+∞ 0 0 P1 ∨ P2 = 1 − [(1 − P1 ) ∧ (1 − P2 )] = 1 − [P2 ∧ P1 ] =1 Calculations were simplified due to [P1 , P2 ] = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Example of non-commuting observables: Q1 − ~2 Q2 , it is easily seen that: 1 1 1 −1 1 1 Q1 = 2 Q2 = 2 1 1 −1 1 1 1 are projectors projecting on subspaces , 1 −1 Operators Q1 and Q2 do not commute with P1 and P2 P1 ∧ Q1 = lim (P1 Q1 )n n→+∞ 1 n 1 1 = lim ( 2 ) =0 0 0 n→+∞ Sx = ~ 2 P1 ∨ Q1 = 1 − ((1 − P1 ) ∧ (1 − Q1 )) = 1 − (P2 ∧ Q2 ) =1−0=1 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Non-commutativity 6= zero conjunction Example: particle with spin 1 1 0 0 B = 2~12 [S 2 − Sz2 ] + 2~12 Sz = 0 1 0 0 0 0 1 1 0 2 0 2 1 2 D = ~2 Sz = 0 1 0 , [B, D] 6= 0 but 1 is a 1 0 12 0 2 common eigenvector. n 0 12 0 0 0 D ∧ B = lim 0 1 0 = 0 1 0 n→+∞ 1 0 12 0 0 0 2 1 2 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of distributivity: Easily calculated as before: P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of distributivity: Easily calculated as before: P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0 Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 ) Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of distributivity: Easily calculated as before: P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0 Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 ) Q1 ∧ 1 ≡ 0 ∨ 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of distributivity: Easily calculated as before: P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0 Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 ) Q1 ∧ 1 ≡ 0 ∨ 0 Q1 = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of distributivity: Easily calculated as before: P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0 Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 ) Q1 ∧ 1 ≡ 0 ∨ 0 Q1 = 0 Comment: Modular law holds: C →A A ∧ (B ∨ C ) ≡ (A ∧ B) ∨ (A ∧ C ) ≡ (A ∧ B) ∨ C Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of continuity - example using polarization let P = cos2 cos sin cos sin sin2 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of continuity - example using polarization let P = cos2 cos sin cos sin sin2 lim P = P1 →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of continuity - example using polarization let P = cos2 cos sin cos sin sin2 lim P = P1 →0 for 6= 0, P ∧ P1 = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of continuity - example using polarization let P = cos2 cos sin cos sin sin2 lim P = P1 →0 for 6= 0, P ∧ P1 = 0 lim (P ∧ P1 ) = 0 →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Failure of continuity - example using polarization let P = cos2 cos sin cos sin sin2 lim P = P1 →0 for 6= 0, P ∧ P1 = 0 lim (P ∧ P1 ) = 0 →0 (lim P ) ∧ P1 = P1 ∧ P1 = P1 →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization Figure: Polarization example P1 light bulb unpolarized light vertical polarizing filter P2 filter polarizing at direction b Polarizing filter = Mechanical analogue to the projection operator Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization Figure: Polarization example a P1 P2 I1 b (a) a c a P3 P1 φ P2 c I1 b I1cos2φsin2φ (b) Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization H - horizontal filter, V - vertical filter, D - diagonal filter H n = H, V n = V , HV = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization H - horizontal filter, V - vertical filter, D - diagonal filter H n = H, V n = V , HV = 0 VHD 6= VDH Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization H - horizontal filter, V - vertical filter, D - diagonal filter H n = H, V n = V , HV = 0 VHD 6= VDH H ∧ V = lim (HV )n = 0 n→∞ V ∧ D = lim (VD)n = 0 n→∞ D ∨ V = ¬[¬D ∧ ¬V ] = ¬[Danti ∧ H] = ¬0 = 1 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization H - horizontal filter, V - vertical filter, D - diagonal filter H n = H, V n = V , HV = 0 VHD 6= VDH H ∧ V = lim (HV )n = 0 n→∞ V ∧ D = lim (VD)n = 0 n→∞ D ∨ V = ¬[¬D ∧ ¬V ] = ¬[Danti ∧ H] = ¬0 = 1 Failure of distributivity H ∧ (D ∨ V ) ≡ H ∧ 1 ≡ H Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Polarization H - horizontal filter, V - vertical filter, D - diagonal filter H n = H, V n = V , HV = 0 VHD 6= VDH H ∧ V = lim (HV )n = 0 n→∞ V ∧ D = lim (VD)n = 0 n→∞ D ∨ V = ¬[¬D ∧ ¬V ] = ¬[Danti ∧ H] = ¬0 = 1 Failure of distributivity H ∧ (D ∨ V ) ≡ H ∧ 1 ≡ H (H ∧ D) ∨ (H ∧ V ) ≡ 0 ∨ 0 ≡ 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Non-continuity of conjunction D - polarizing filter inclined at to the horizontal lim D = H →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Non-continuity of conjunction D - polarizing filter inclined at to the horizontal lim D = H →0 D ∧ H = 0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Non-continuity of conjunction D - polarizing filter inclined at to the horizontal lim D = H →0 D ∧ H = 0 lim (D ∧ H) = 0 →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Non-continuity of conjunction D - polarizing filter inclined at to the horizontal lim D = H →0 D ∧ H = 0 lim (D ∧ H) = 0 →0 (lim D ) ∧ H = H →0 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Two slit experiment - failure of distributivity A1 - electron passes through slit 1 A2 - electron passes through slit 2 P(A1 , R) - probablity that electron hits region R, after A1 P(A2 , R) - probablity that electron hits region R, after A2 Classically, one would expect P(R) = 12 P(A1 , R) + 21 P(A1 , R) Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Two slit experiment - failure of distributivity A1 - electron passes through slit 1 A2 - electron passes through slit 2 P(A1 , R) - probablity that electron hits region R, after A1 P(A2 , R) - probablity that electron hits region R, after A2 Classically, one would expect P(R) = 12 P(A1 , R) + 21 P(A1 , R) Quantum formula P(R) = p 1 P(A1 , R) + 12 P(A1 , R) + P(A1 , R)P(A2 , R) cos δ 2 Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM P((A1 ∨ A2 ) · R) P(A1 ∨ A2 ) P((A1 · R) ∨ (A2 ∨ R)) = P(A1 ∨ A2 ) P(A1 · R) P(A2 · R) = + P(A1 ∨ A2 ) P(A1 ∨ A2 ) P(A1 ∨ A2 , R) = Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM P((A1 ∨ A2 ) · R) P(A1 ∨ A2 ) P((A1 · R) ∨ (A2 ∨ R)) = P(A1 ∨ A2 ) P(A1 · R) P(A2 · R) = + P(A1 ∨ A2 ) P(A1 ∨ A2 ) P(A1 ∨ A2 , R) = P(A1 ) = P(A2 ) P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 ) Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM P((A1 ∨ A2 ) · R) P(A1 ∨ A2 ) P((A1 · R) ∨ (A2 ∨ R)) = P(A1 ∨ A2 ) P(A1 · R) P(A2 · R) = + P(A1 ∨ A2 ) P(A1 ∨ A2 ) P(A1 ∨ A2 , R) = P(A1 ) = P(A2 ) P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 ) P(A1 · R) P(A1 · R) = = P(A1 ∨ A2 ) 2P(A1 ) P(A2 · R) P(A2 · R) = = P(A1 ∨ A2 ) 2P(A2 ) Lovre Grisogono & Mateo Paulišić 1 P(A1 , R) ; 2 1 P(A2 , R) 2 A Few Good Examples why to Refute Classical Logic in QM P((A1 ∨ A2 ) · R) P(A1 ∨ A2 ) P((A1 · R) ∨ (A2 ∨ R)) = P(A1 ∨ A2 ) P(A1 · R) P(A2 · R) = + P(A1 ∨ A2 ) P(A1 ∨ A2 ) P(A1 ∨ A2 , R) = P(A1 ) = P(A2 ) P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 ) P(A1 · R) P(A1 · R) = = P(A1 ∨ A2 ) 2P(A1 ) P(A2 · R) P(A2 · R) = = P(A1 ∨ A2 ) 2P(A2 ) 1 P(A1 , R) ; 2 1 P(A2 , R) 2 P(R) = 12 P(A1 , R) + 21 P(A1 , R) Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Conclusion The question was: Should we refute classical logic for QM? And the anwser is: Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Conclusion The question was: Should we refute classical logic for QM? And the anwser is: We should. Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM Literature ’Quantum mechanics’ [Auletta, G., Fortunato, M., Parisi, G. Quantum mechanics, Cambridge University Press, 2009] ’Quantum logic’ [Adler, C. G., Wirth, J. F. Quantum Logic. In American Journal of Physics 51(5): 1983, pp. 412-417.] ’Is logic empirical’ [Putnam, H. Is Logic Empirical? In Boston Studies in the Philosophy of Science 5, R. S. Cohen, M. W. Wartofsky, ur. Dodrecht: D. Reidel, 1968, pp. 174-197.] Lovre Grisogono & Mateo Paulišić A Few Good Examples why to Refute Classical Logic in QM