The Logic of Quantum Physics
- 2. Brief Guide to Quantum Mechanics
- 3. Brief Guide to Quantum Mechanics System States Dynamics Observables Measurement Collapse
- 4. Brief Guide to Quantum Mechanics System: Vector Space (over ℂ) States Dynamics Observables Measurement Collapse |↑⟩ |↓⟩ |↑⟩ |↓⟩ |0⟩ |1⟩ |2⟩ |0⟩ |2⟩ |1⟩
- 5. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics Observables Measurement Collapse |↑⟩ |↓⟩ |↑⟩ |↓⟩ |0⟩ |1⟩ |2⟩ |0⟩ |2⟩ |1⟩ |Ψ⟩ = c1|↑⟩ + c2|↓⟩ |Ψ⟩ = c1|0⟩ + c2|1⟩ + c3|2⟩
- 6. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables Measurement Collapse iħ ∂|Ψ⟩/∂t = Ĥ|Ψ⟩
- 7. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse Subspace A subset that is closed under vector addition and scalar multiplication
- 8. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |↑⟩ |↓⟩ |0⟩ |2⟩ |1⟩ |↑⟩ |↓⟩
- 9. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |↑⟩ |↓⟩
- 10. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |↑⟩ |↓⟩ Test of
- 11. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |↑⟩ |↓⟩ Test of Test of
- 12. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |↑⟩ |↓⟩ Test of Test of Test of
- 13. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |0⟩ |2⟩ |1⟩ |0⟩ |1⟩ |2⟩
- 14. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |0⟩ |2⟩ |1⟩ Test of (not |2⟩) |0⟩ |1⟩ |2⟩
- 15. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement Collapse |0⟩ |2⟩ |1⟩ Test of (not |2⟩) |0⟩ |1⟩ |2⟩ Test of (.8|0⟩ + .2|1⟩ + .6|2⟩)
- 16. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse |↑⟩ |↓⟩ |Ψ⟩ Test of
- 17. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse |↑⟩ |↓⟩ |Ψ⟩ Test of { ⟨↑|Ψ⟩ Pr(Test of ↑ returns positive | state is |Ψ⟩) = |⟨↑|Ψ⟩|2
- 18. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse State is |Ψ⟩ = c1|↑⟩ + c2|↓⟩ Test of |↑⟩ Pr(Positive result) = |c1|2 Pr(Negative result) = |c2|2 State is |Ψ⟩ Test of |Φ⟩ Pr(Positive) = |⟨Φ|Ψ⟩|2
- 19. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection |↑⟩ |↓⟩ |Ψ⟩ Test of If test is positive, new state is |↑⟩ If test is negative, new state is |↓⟩
- 20. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection Test of (not |2⟩) If test is positive, new state is |Ψ’⟩ If test is negative, new state is |2⟩ |0⟩ |2⟩ |1⟩ Test of (not |2⟩) |Ψ⟩ |Ψ’⟩
- 21. Brief Guide to Quantum Mechanics System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection
- 22. Contrast to Classical Mechanics Classical System: Set (“Phase space”)System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection
- 23. Contrast to Classical Mechanics Classical System: Set Classical States: Points System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection
- 24. Contrast to Classical Mechanics Classical System: Set Classical States: Points Classical Dynamics: F = ma System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection
- 25. Contrast to Classical Mechanics Classical System: Set Classical States: Points Classical Dynamics: F = ma Classical Observables: Subsets System: Vector Space States: Vectors Dynamics: Schrödinger Equation Observables: Subspaces Measurement: (Dot Product)2 is Probability Collapse: Projection
- 26. The Logic of Classical Physics Experimental Proposition: System S will pass test T with probability 100%
- 27. The Logic of Classical Physics Experimental Proposition: System S will pass test T with probability 100% Experimental Propositions ↔ Subsets of phase space of S
- 28. The Logic of Classical Physics Experimental Proposition: System S will pass test T with probability 100% Experimental Propositions ↔ Subsets of phase space of S P∧Q ↔ Intersection of subsets for P and Q
- 29. The Logic of Classical Physics Experimental Proposition: System S will pass test T with probability 100% Experimental Propositions ↔ Subsets of phase space of S P∧Q ↔ Intersection of subsets for P and Q P∨Q ↔ Union of subsets for P and Q
- 30. The Logic of Classical Physics Experimental Proposition: System S will pass test T with probability 100% Experimental Propositions ↔ Subsets of phase space of S P∧Q ↔ Intersection of subsets for P and Q P∨Q ↔ Union of subsets for P and Q ¬P ↔ Complement of subset for P
- 31. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100%
- 32. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S.
- 33. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q
- 34. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Union of subspaces for P and Q?
- 35. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Union of subspaces for P and Q? NO! |↑⟩ |↓⟩ P Q
- 36. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q P Qspan(P, Q)
- 37. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Complement of subspace for P?
- 38. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Complement of subspace for P? NO! P Pc
- 39. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P |↑⟩ |↓⟩ P ¬P
- 40. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P |0⟩ |1⟩ |2⟩ P ¬P
- 41. ¬P The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P P ↔ ¬¬P Ortho-complement of ortho-complement of P = P |0⟩ |1⟩ |2⟩ P
- 42. ¬P The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P ¬(P ∧ ¬P) Intersection of P and its orthocomplement is {0} Ortho-complement of {0} is entire space |0⟩ |1⟩ |2⟩ P
- 43. ¬P The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P P ∨ ¬P Span of P and its orthocomplement is entire space |0⟩ |1⟩ |2⟩ P
- 44. The Logic of Quantum Systems Experimental Proposition: System S will pass test T with probability 100% Experimental propositions ↔ Subspaces of the vector space of S. P∧Q ↔ Intersection of subspaces for P and Q P∨Q ↔ Span of subspaces for P and Q ¬P ↔ Ortho-complement of subspace for P Distributive Law P ∧ (Q ∨ R) = P (P ∧ Q) ∨ (P ∧ R) = {0} |↑⟩ |↓⟩ P Q R
- 45. The Logic of Quantum Systems So (P ∧ (Q ∨ R)) can be true while ((P ∧ Q) ∨ (P ∧ R)) is false This extends to ∀ (extended conjunction) and ∃ (extended disjunction). The following can all be true: ∀x∀y ¬(P(x) ∧ Q(y)) ¬ ∃x∃y (P(x) ∧ Q(y)) ∃x P(x) ∧ ∃y Q(y)
- 46. Quantum Logic as an Interpretation of QM Putnam: “All so-called ‘anomalies’ in quantum mechanics come down to the non- standardness of the logic.”
- 47. Quantum Logic as an Interpretation of QM Incompatible observables: Let P(x) be “The particle will definitely be measured at position x.” Let Q(y) be “The particle will definitely be measured with momentum y.” Distinct 1d subspaces, so their intersection is {0} I.e. the assertion that a particle has a definite position and momentum is a logical contradiction in quantum logic
- 48. Quantum Logic as an Interpretation of QM Incompatible observables: Let P(x) be “The particle will definitely be measured at position x.” Let Q(y) be “The particle will definitely be measured with momentum y.” ∃x∃y (P(x) ∧ Q(y)) is false ∀x∀y ¬(P(x) ∧ Q(y)) is true But ∃x P(x) ∧ ∃y Q(y) is true!
- 49. Quantum Logic as an Interpretation of QM Let y1, y2, …, yn be a complete list of possible momenta. P(x) ∧ (Q(y1) ∨ Q(y2) ∨ … ∨ Q(yn)) = P(x) Span of Q(y1), Q(y2), …, and Q(yn) is the entire space (P(x) ∧ Q(y1)) ∨ (P(x) ∧ Q(y2)) ∨ … ∨ (P(x) ∧ Q(yn)) = {0} Each (P(x) ∧ Q(yn)) is {0}, and the span of n 0-vectors is still {0}
- 50. Quantum Logic as an Interpretation of QM Putnam: “The world consists of particles. Each of these particles has a position. And each of these particles has a momentum. But it must not be concluded that each of these particles has a position and a momentum!
- 51. Putnam on QL - Logic is empirical, and open to revision in the light of new physics - Logic of quantum mechanics is non-Boolean - “All so-called ‘anomalies’ in quantum mechanics come down to the non- standardness of the logic.” - Quantum logic tells us to take realist interpretation of quantum mechanics - “Measurement only determines what is already the case: it does not bring into existence the observable measured, or cause it to ‘take on a sharp value’ which it did not already possess.” - States are maximally consistent sets of sentences - Dynamics are deterministic, indeterminacy comes from the classical incompleteness of states - Meaning of qor and qnot is the same as or and not
- 52. Quantum Logic as an Interpretation of QM Putnam: “Measurement only determines what is already the case: it does not bring into existence the observable measured, or cause it to ‘take on a sharp value’ which it did not already possess.” Take any observable A, where {aj} is the set of all values A can take on. Then a1 v a2 v a3 v … = ∃n an is a tautology. So for every observable A, there exists some j such that the value of A is aj. No collapse, no hidden variables!