1. Quantum Measure Theory: A New Interpretation Yousef Ghazi-Tabatabai

2. Introduction In the histories approach to quantum mechanics, the basic object is a sample space of histories, typically in configuration space, leading from an initial to a final state. In classical stochastic mechanics the dynamics of this system can be embodied in a probability measure mapping the power set of the sample space to R, and the classical interpretation can be viewed as a map from the sample space to {0,1} sending the `real’ history to 1 and all other histories to 0 – which can be extended to a map from the power set of the sample space to {0,1}, with a set mapped to one iff it contains the `real’ history. Quantum measure theory attempts to replicate this approach for quantum mechanics, encoding the dynamics in a more generalized `quantum measure’ that does not obey the Kolmogorov sum rule. This talk will focus on recent developments in the interpretation of quantum measure theory, generalizing the classical interpretation to a wider class of functions from the power set of the sample space to {0,1}. In the histories approach to quantum mechanics, the basic object is a sample space of histories, typically in configuration space, leading from an initial to a final state. In classical stochastic mechanics the dynamics of this system can be embodied in a probability measure mapping the power set of the sample space to R, and the classical interpretation can be viewed as a map from the sample space to {0,1} sending the `real’ history to 1 and all other histories to 0 – which can be extended to a map from the power set of the sample space to {0,1}, with a set mapped to one iff it contains the `real’ history. Quantum measure theory attempts to replicate this approach for quantum mechanics, encoding the dynamics in a more generalized `quantum measure’ that does not obey the Kolmogorov sum rule. This talk will focus on recent developments in the interpretation of quantum measure theory, generalizing the classical interpretation to a wider class of functions from the power set of the sample space to {0,1}.

3. Contents The Histories Approach The Histories Approach Classical Stochastic Mechanics Classical Stochastic Mechanics The Quantum Measure The Quantum Measure The Syracuse Interpretation The Syracuse Interpretation Example 1: The Double Slit Example 1: The Double Slit Example 2: The Triple Slit Example 2: The Triple Slit Potential Problems Potential Problems In Detail: Problem 2 In Detail: Problem 2 Quadratic Co-Events Quadratic Co-Events Current Research: Solutions Current Research: Solutions References References

4. Motivation To achieve a better understanding of the foundations of Quantum Mechanics To achieve a better understanding of the foundations of Quantum Mechanics This is a research area in its own right This is a research area in its own right Such an understanding would be useful, or even necessary, in the search for a viable theory of Quantum Gravity Such an understanding would be useful, or even necessary, in the search for a viable theory of Quantum Gravity For example, we hope that a working interpretation of quantum measure theory will suggest a causality condition to use in place of the “Bell causality condition” in the quantisation of causal set theory For example, we hope that a working interpretation of quantum measure theory will suggest a causality condition to use in place of the “Bell causality condition” in the quantisation of causal set theory The histories formalism is a spacetime based approach to QM based upon Feynman’s work on path integrals The histories formalism is a spacetime based approach to QM based upon Feynman’s work on path integrals The basic objects are spacetime paths, or histories, so it becomes easier to handle covariance & causality The basic objects are spacetime paths, or histories, so it becomes easier to handle covariance & causality

5. The Histories Approach The histories approach is an alternate formulation of quantum mechanics that allows for generalisation The histories approach is an alternate formulation of quantum mechanics that allows for generalisation A history is a potential path from the initial to the final state, represented by a class operator encoding the dynamics: A history is a potential path from the initial to the final state, represented by a class operator encoding the dynamics: C α =P t8 (α 8 )…. P t8 (α 8 )|ψ init > C α =P t8 (α 8 )…. P t8 (α 8 )|ψ init > The set of all histories is called the Sample Space, typically denoted Ω The set of all histories is called the Sample Space, typically denoted Ω The dynamics can also be encoded in the decoherence functional: The dynamics can also be encoded in the decoherence functional: D: P (Ω) x P (Ω) → C D: P (Ω) x P (Ω) → C D(A,B)=D(B,A) D(A,B)=D(B,A) † D(A └┘ B,C)=D(A,C)+D(B,C) D(A └┘ B,C)=D(A,C)+D(B,C) D(Ω, Ω)=1 D(Ω, Ω)=1 D({C C α D({α},{β}) = C β † C α Final State t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 Initial State α β t1t1 α = α 2 β= β 2 α = α 1 β= β 1

6. Classical Stochastic Mechanics Sample Space of histories, Ω Sample Space of histories, Ω Dynamics: Probability Measure P Dynamics: Probability Measure P P: P (Ω) → R P: P (Ω) → R P(A) ≥ 0 P(A) ≥ 0 P(A └┘ B) = P(A) + P(B) Kolmogorov Sum Rule P(A └┘ B) = P(A) + P(B) Kolmogorov Sum Rule Decoherence: P(A) = D(A,A) is a probability measure if D(A,B) = 0 for A ≠ B Decoherence: P(A) = D(A,A) is a probability measure if D(A,B) = 0 for A ≠ B Interpretation: Exactly one history, r, is ‘real’ Interpretation: Exactly one history, r, is ‘real’ Each history can be represented by a ‘co-event’ Each history can be represented by a ‘co-event’ Start with f r : Ω  Z 2, f r (αα Start with f r : Ω  Z 2, f r (α) =1 iff r = α Extend to f r : P ( Ω)  Z 2, f r (A Extend to f r : P ( Ω)  Z 2, f r (A) =1 iff r in A Þ f r (A) = 0 Preclusive: P(A) = 0 Þ f r (A) = 0 f r Ω} = Hom( Bool(Ω)Z 2 ) {f r | r in Ω} = Hom( Bool(Ω), Z 2 ) addition ↔ ∆ addition ↔ ∆ multiplication ↔ ∩ multiplication ↔ ∩

7. The Quantum Measure P(A) = D(A,A) given decoherence suggests μ(A) = D(A,A) in general P(A) = D(A,A) given decoherence suggests μ(A) = D(A,A) in general μ : P (Ω) → R μ : P (Ω) → R μ(A) ≥ 0 μ(A) ≥ 0 μ(A └┘ B) ≠ μ(A) + μ(B) in general - Kolmogorov Sum Rule not obeyed (eg double slit) μ(A └┘ B) ≠ μ(A) + μ(B) in general - Kolmogorov Sum Rule not obeyed (eg double slit) Generalising the Kolmogorov Sum Rule: The Hierarchy of Interference Terms Generalising the Kolmogorov Sum Rule: The Hierarchy of Interference Terms I 1 (A) = |A| I 1 (A) = |A| I 2 (A,B) = |A └┘ B| - |A| - |B| I 2 (A,B) = |A └┘ B| - |A| - |B| I 3 (A,B,C) = |A └┘ B └┘ C| - |A └┘ B| - |B └┘ C| - |A └┘ C| + |A| + |B| + |C| I 3 (A,B,C) = |A └┘ B └┘ C| - |A └┘ B| - |B └┘ C| - |A └┘ C| + |A| + |B| + |C| Etc Etc Quantum Mechanics: Quantum Mechanics: |A| = μ(A) = D(A,A) |A| = μ(A) = D(A,A) I 3 (A,B,C) = 0 I 3 (A,B,C) = 0 Re(D(A,B)) = ½ I 2 (A,B) Re(D(A,B)) = ½ I 2 (A,B)

8. The Syracuse Interpretation Hom( Bool(Ω)Z 2 ) implies the classical co-events and is too restrictive for quantum mechanics (Kochen-Specker) Hom( Bool(Ω), Z 2 ) implies the classical co-events and is too restrictive for quantum mechanics (Kochen-Specker) Generalise to Hom + ( Bool(Ω)Z 2 ), the space of homomorphisms preserving addition but not necessarily preserving multiplication Generalise to Hom + ( Bool(Ω), Z 2 ), the space of homomorphisms preserving addition but not necessarily preserving multiplication As before demand preclusivity: μ(A) = 0 Þ f(A) = 0 As before demand preclusivity: μ(A) = 0 Þ f(A) = 0 Impose the further condition f(Ω ) = 1 (unitality) to avoid f(A) = f(¬A) = 1 Impose the further condition f(Ω ) = 1 (unitality) to avoid f(A) = f(¬A) = 1 For finite Ω, Hom + ( Bool(Ω)Z 2 ) is a group ‘dual’ to P (Ω) For finite Ω, Hom + ( Bool(Ω), Z 2 ) is a group ‘dual’ to P (Ω) Duality: Duality: Given A in P (Ω), define f A ({α}) = 1 iff α in A Given A in P (Ω), define f A ({α}) = 1 iff α in A Extend by linearity: f A (B) = |A ∩B| mod 2 Extend by linearity: f A (B) = |A ∩B| mod 2 Group Structure: Group Structure: (f A + f B )( C ) = f A (C) + f B (C) = f A+B (C) (f A + f B )( C ) = f A (C) + f B (C) = f A+B (C) Associative Associative Identity: f ø Identity: f ø Inverse: f A + f A = f A+A = f ø Inverse: f A + f A = f A+A = f ø

9. Example 1: The Double Slit AB We know that f(A) + f(B) = f(A ∆ B) =f(Ω) = 1 So f(A) = 1 Û f(B) = 0 and f(B) = 1 Û f(A) = 0 Hence we have two possible co-events: f(A) = 1, f(B) = 0 f(A) = 0, f(B) = 1 These are the classical co-events Both A & B are admitted

10. Example 2: The Triple Slit +-+ AB We know that: f(A) + f(B) + f(C) = f(Ω) = 1 μ(A∆B) = μ(B∆C) = 0 Þ f(A) + f(B) = f(B) + f(C) = 0 Then: f(A) = 0 Þ f(B) = 0 Þ f(C) = 0 Þ f( W ) = 0 contradicting unitality f(A) = 1 Þ f(B) = 1 Þ f(C) = 1 This time we have only one co-event, which is not classical Not every outcome is allowed – problems? Post-selection C

11. Potential Problems 1. Given a decoherence functional, can we always find a non-zero preclusive co-event? 2. Can we always find a unital preclusive co- event? 3. Are the (minimal) co-events classical whenever the measure is classical? 4. Will we forced to make predictions in contradiction with quantum mechanics?

12. In Detail: Problem 2 We can not always find a unital preclusive co-event We can not always find a unital preclusive co-event μ(A i )=0 Þ f(A i ) =0 μ(A i )=0 Þ f(A i ) =0 There is only one non-zero, preclusive co-event: There is only one non-zero, preclusive co-event: f({a i }) = 1 f({a i }) = 1 Then f( W ) = 0 Then f( W ) = 0 This structure can arise from the four slit set up This structure can arise from the four slit set up a2a2 a0a0 a3a3 A2A2 A3A3 a1a1 A1A1

13. Quadratic Co-Events Quadratic co-events: the first attempt at a solution to problem 2 Quadratic co-events: the first attempt at a solution to problem 2 Generalise co-events from Hom + ( Bool(Ω)Z 2 ) by relaxing linearity Generalise co-events from Hom + ( Bool(Ω), Z 2 ) by relaxing linearity Replace linearity with the sum rule obeyed by the quantum measure: Replace linearity with the sum rule obeyed by the quantum measure: f(A∆B∆C) = f(A∆B) + f(B∆C) + f(A∆C) + f(A) + f(B) + f(C) f(A∆B∆C) = f(A∆B) + f(B∆C) + f(A∆C) + f(A) + f(B) + f(C) We know that we can always find a non-zero, preclusive co-event under this definition – we can solve problem 1 We know that we can always find a non-zero, preclusive co-event under this definition – we can solve problem 1 However, quadratic co-events fail to solve problem 2 However, quadratic co-events fail to solve problem 2 There are decoherence functionals that do not admit unital quadratic co-events There are decoherence functionals that do not admit unital quadratic co-events This seems to generalise to higher order sum rules This seems to generalise to higher order sum rules We need a different way of generalising from Hom( Bool(Ω)Z 2 ) We need a different way of generalising from Hom( Bool(Ω), Z 2 )

14. Current Research: Solutions Generalise from Hom( Bool(Ω)Z 2 ) to Hom * ( Bool(Ω)Z 2 ), the space of homomorphisms preserving multiplication but not necessarily addition Generalise from Hom( Bool(Ω), Z 2 ) to Hom * ( Bool(Ω), Z 2 ), the space of homomorphisms preserving multiplication but not necessarily addition As before demand preclusivity: μ(A) = 0 Þ f(A) = 0 As before demand preclusivity: μ(A) = 0 Þ f(A) = 0 f(A)=0 places no restriction on f(B) where B contains A f(A)=0 places no restriction on f(B) where B contains A Hence since μ(Ω) ≠ 0, we can always find a co-event with f(Ω)=1 Hence since μ(Ω) ≠ 0, we can always find a co-event with f(Ω)=1 Thus this generalisation satisfies problems 1, 2 & 3. Does it satisfy problem 4? Thus this generalisation satisfies problems 1, 2 & 3. Does it satisfy problem 4? Potential problems: Potential problems: We do get unwanted preclusions in the three-slit experiment We do get unwanted preclusions in the three-slit experiment However, we get this by “post-conditioning” on the outcome However, we get this by “post-conditioning” on the outcome We can not (yet) rule out decoherence functionals that do not admit “classical” co-events on decoherent sets We can not (yet) rule out decoherence functionals that do not admit “classical” co-events on decoherent sets

15. References Quantum Mechanics as Quantum Measure Theory, R Sorkin, gr-qc/9401003 Quantum Mechanics as Quantum Measure Theory, R Sorkin, gr-qc/9401003 Quantum Dynamics without the Wave Function, R Sorkin, quant-ph/0610204 Quantum Dynamics without the Wave Function, R Sorkin, quant-ph/0610204 The Problem of Hidden Variables in Quantum Mechanics, S Kochen & EP Specker, 1967, J Math Mech 17 59-87 The Problem of Hidden Variables in Quantum Mechanics, S Kochen & EP Specker, 1967, J Math Mech 17 59-87