Quantum Measure Quantum Measure Theory:Theory:

A New InterpretationA New Interpretation

Yousef Ghazi-TabatabaiYousef Ghazi-Tabatabai

IntroductionIntroduction In the histories approach to quantum mechanics, the basic In the histories approach to quantum mechanics, the basic

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

ContentsContents The Histories ApproachThe Histories Approach Classical Stochastic MechanicsClassical Stochastic Mechanics The Quantum MeasureThe Quantum Measure The Syracuse InterpretationThe Syracuse Interpretation Example 1: The Double SlitExample 1: The Double Slit Example 2: The Triple SlitExample 2: The Triple Slit Potential ProblemsPotential Problems In Detail: Problem 2In Detail: Problem 2 Quadratic Co-EventsQuadratic Co-Events Current Research: SolutionsCurrent Research: Solutions ReferencesReferences

MotivationMotivation To achieve a better understanding of the To achieve a better understanding of the

foundations of Quantum Mechanicsfoundations of Quantum Mechanics This is a research area in its own rightThis is a research area in its own right Such an understanding would be useful, or even Such an understanding would be useful, or even

necessary, in the search for a viable theory of Quantum necessary, in the search for a viable theory of Quantum GravityGravity

For example, we hope that a working interpretation For example, we hope that a working interpretation of quantum measure theory will suggest a causality of quantum measure theory will suggest a causality condition to use in place of the “Bell causality condition to use in place of the “Bell causality condition” in the quantisation of causal set theorycondition” in the quantisation of causal set theory

The histories formalism is a spacetime based approach to The histories formalism is a spacetime based approach to QM based upon Feynman’s work on path integralsQM based upon Feynman’s work on path integrals The basic objects are spacetime paths, or histories, so it The basic objects are spacetime paths, or histories, so it

becomes easier to handle covariance & causalitybecomes easier to handle covariance & causality

The Histories ApproachThe Histories Approach The histories approach is an The histories approach is an

alternate formulation of quantum alternate formulation of quantum mechanics that allows for mechanics that allows for generalisationgeneralisation

A history is a potential path from the A history is a potential path from the initial to the final state, represented initial to the final state, represented by a class operator encoding the by a class operator encoding the dynamics:dynamics:

CCαα=P=Pt8t8((αα88)…. P)…. Pt8t8((αα88)|)|ψψinitinit>> The set of all histories is called the The set of all histories is called the

Sample Space, typically denoted Sample Space, typically denoted ΩΩ The dynamics can also be encoded in The dynamics can also be encoded in

the decoherence functional:the decoherence functional: D: D: PP((ΩΩ) x ) x PP((ΩΩ) → C) → C D(A,B)=D(B,A)D(A,B)=D(B,A)†

D(AD(A└┘└┘B,C)=D(A,C)+D(B,C)B,C)=D(A,C)+D(B,C) D(D(ΩΩ, , ΩΩ)=1)=1 D({D({α},{β}) = CCβ

† CCαα

Final State

t2

t3

t4

t5

t6

t7

Initial State

α β

t1

α = α2

β= β2

α = α1

β= β1

Classical Stochastic Classical Stochastic MechanicsMechanics

Sample Space of histories, Sample Space of histories, ΩΩ Dynamics: Probability Measure PDynamics: Probability Measure P

P: P: P P ((ΩΩ) → R) → R P(A) ≥ 0P(A) ≥ 0 P(AP(A└┘└┘B) = P(A) + P(B) Kolmogorov Sum RuleB) = P(A) + P(B) Kolmogorov Sum Rule

Decoherence: P(A) = D(A,A) is a probability measure if D(A,B) Decoherence: P(A) = D(A,A) is a probability measure if D(A,B) = 0 for 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 fStart with frr: : ΩΩ Z Z22, f, frr((αα) =1 iff r = αα Extend to fExtend to frr: : P P (( ΩΩ) ) Z Z22, f, frr(A(A) =1 iff r in A

Preclusive: P(A) = 0 ÞÞ ffrr(A) = 0(A) = 0 {ffrr | r in ΩΩ} = Hom( Bool(} = Hom( Bool(ΩΩ)) , ZZ22 ) )

addition ↔ ∆ addition ↔ ∆ multiplication ↔ ∩multiplication ↔ ∩

The Quantum MeasureThe Quantum Measure P(A) = D(A,A) given decoherence suggests P(A) = D(A,A) given decoherence suggests μμ(A) = D(A,A) in general(A) = D(A,A) in general

μμ : : P P ((ΩΩ) → R) → R μμ(A) ≥ 0(A) ≥ 0 μμ(A(A└┘└┘B) ≠ B) ≠ μμ(A) + (A) + μμ(B) in general - Kolmogorov Sum Rule not obeyed (eg (B) in general - Kolmogorov Sum Rule not obeyed (eg

double slit)double slit)

Generalising the Kolmogorov Sum Rule: The Hierarchy of Generalising the Kolmogorov Sum Rule: The Hierarchy of Interference TermsInterference Terms

II11(A) = |A|(A) = |A|

II22(A,B) = |A (A,B) = |A └┘└┘B| - |A| - |B|B| - |A| - |B|

II33(A,B,C) = |A(A,B,C) = |A└┘└┘BB└┘└┘C| - |AC| - |A└┘└┘B| - |BB| - |B└┘└┘C| - |AC| - |A└┘└┘C| + |A| + |B| + |C|C| + |A| + |B| + |C| EtcEtc

Quantum Mechanics:Quantum Mechanics: |A||A| = = μμ(A) = D(A,A) (A) = D(A,A) II33(A,B,C) = (A,B,C) = 00

Re(D(A,B)) = ½ IRe(D(A,B)) = ½ I22(A,B)(A,B)

The Syracuse The Syracuse InterpretationInterpretation

Hom( Bool(Hom( Bool(ΩΩ)) , ZZ22 ) implies the classical co-events and is too ) implies the classical co-events and is too restrictive for quantum mechanics (Kochen-Specker)restrictive for quantum mechanics (Kochen-Specker)

Generalise to HomGeneralise to Hom++( Bool(( Bool(ΩΩ)) , ZZ22 ), the space of homomorphisms ), the space of homomorphisms preserving addition but not necessarily preserving multiplicationpreserving addition but not necessarily preserving multiplication As before demand preclusivity: As before demand preclusivity: μμ(A) = 0 (A) = 0 ÞÞ f(A) = 0 f(A) = 0 Impose the further condition f(Impose the further condition f(ΩΩ ) = 1 (unitality) to avoid f(A) = f(¬A) ) = 1 (unitality) to avoid f(A) = f(¬A)

= 1= 1 For finite For finite ΩΩ, Hom, Hom++( Bool(( Bool(ΩΩ)) , ZZ22 ) is a group ‘dual’ to ) is a group ‘dual’ to P P ((ΩΩ))

Duality:Duality: Given A in Given A in P P ((ΩΩ), define f), define fAA({({αα}) = 1 iff }) = 1 iff αα in A in A Extend by linearity: fExtend by linearity: fAA(B) = (B) = |A|A∩B| mod 2∩B| mod 2

Group Structure:Group Structure: (f(fA A + f+ fB B )( C ) = f)( C ) = fAA(C) + f(C) + fBB(C) = f(C) = fA+BA+B(C)(C) AssociativeAssociative Identity: fIdentity: føø

Inverse: Inverse: ffA A + f+ fA A = = ffA+AA+A = = fføø

Example 1: The Double Example 1: The Double SlitSlit

A B

We know that f(A) + f(B) = f(A ∆ B) =f(We know that f(A) + f(B) = f(A ∆ B) =f(ΩΩ) = 1) = 1 So f(A) = 1 So f(A) = 1 ÛÛ f(B) = 0 f(B) = 0 and f(B) = 1 and f(B) = 1 ÛÛ f(A) = 0 f(A) = 0 Hence we have two possible co-events:Hence we have two possible co-events:

f(A) = 1, f(B) = 0f(A) = 1, f(B) = 0 f(A) = 0, f(B) = 1f(A) = 0, f(B) = 1

These are the classical co-eventsThese are the classical co-events Both A & B are admittedBoth A & B are admitted

Example 2: The Triple Example 2: The Triple SlitSlit

+ - +

A B

We know that:We know that: f(A) + f(B) + f(C) = f(f(A) + f(B) + f(C) = f(ΩΩ) = 1) = 1 μμ(A∆B) = (A∆B) = μμ(B∆C) = 0 (B∆C) = 0 ÞÞ f(A) + f(B) = f(B) + f(C) = 0 f(A) + f(B) = f(B) + f(C) = 0

Then: Then: f(A) = 0 f(A) = 0 ÞÞ f(B) = 0 f(B) = 0 ÞÞ f(C) = 0 f(C) = 0 Þ Þ f(f(WW) = 0 contradicting unitality) = 0 contradicting unitality f(A) = 1 f(A) = 1 ÞÞ f(B) = 1 f(B) = 1 ÞÞ f(C) = 1 f(C) = 1

This time we have only one co-event, which is not classicalThis time we have only one co-event, which is not classical Not every outcome is allowed – problems?Not every outcome is allowed – problems? Post-selectionPost-selection

C

Potential ProblemsPotential Problems

1.1. Given a decoherence functional, can we Given a decoherence functional, can we always find a non-zero preclusive co-always find a non-zero preclusive co-event?event?

2.2. Can we always find a unital preclusive Can we always find a unital preclusive co-event?co-event?

3.3. Are the (minimal) co-events classical Are the (minimal) co-events classical whenever the measure is classical?whenever the measure is classical?

4.4. Will we forced to make predictions in Will we forced to make predictions in contradiction with quantum mechanics?contradiction with quantum mechanics?

In Detail: Problem 2In Detail: Problem 2 We can not always find a We can not always find a

unital preclusive co-eventunital preclusive co-event μμ(A(Aii)=0 )=0 ÞÞ f(A f(Aii) =0) =0 There is only one non-zero, There is only one non-zero,

preclusive co-event:preclusive co-event: f({af({aii}) = 1}) = 1 Then f(Then f(WW) = 0) = 0

This structure can arise from This structure can arise from the four slit set upthe four slit set up

a2

a0

a3

A2A3

a1

A1

Quadratic Co-EventsQuadratic Co-Events Quadratic co-events: the first attempt at a solution to Quadratic co-events: the first attempt at a solution to

problem 2problem 2 Generalise co-events from HomGeneralise co-events from Hom++( Bool(( Bool(ΩΩ)) , ZZ22 ) by relaxing ) by relaxing

linearitylinearity Replace linearity with the sum rule obeyed by the quantum Replace linearity with the sum rule obeyed by the quantum

measure: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-We know that we can always find a non-zero, preclusive co-event under this definition – we can solve problem 1event under this definition – we can solve problem 1

However, quadratic co-events fail to solve problem 2However, quadratic co-events fail to solve problem 2 There are decoherence functionals that do not admit unital There are decoherence functionals that do not admit unital

quadratic co-eventsquadratic co-events This seems to generalise to higher order sum rulesThis seems to generalise to higher order sum rules

WWe need e need aa different way of generalising from Hom( Bool( different way of generalising from Hom( Bool(ΩΩ)) , ZZ22 ) )

Current Research: Current Research: SolutionsSolutions

Generalise from Hom( Bool(Generalise from Hom( Bool(ΩΩ)) , ZZ22 ) to Hom ) to Hom**( Bool(( Bool(ΩΩ)) , ZZ22 ), the s ), the spapace of homomorphisms preserving ce of homomorphisms preserving multiplication but not necessarily additionmultiplication but not necessarily addition As before demand preclusivity: As before demand preclusivity: μμ(A) = 0 (A) = 0 ÞÞ f(A) = 0 f(A) = 0 f(A)=0 places no restriction on f(B) where B contains Af(A)=0 places no restriction on f(B) where B contains A

Hence since Hence since μμ((ΩΩ) ) ≠≠ 0, we can always find a co-event with f( 0, we can always find a co-event with f(ΩΩ)=1)=1 Thus this generalisation satisfies problems 1, 2 & 3. Does it Thus this generalisation satisfies problems 1, 2 & 3. Does it

satisfy problem 4?satisfy problem 4? Potential problems:Potential problems:

We do get unwanted preclusions in the three-slit experimentWe do get unwanted preclusions in the three-slit experiment However, we get this by “post-conditioning” on the outcomeHowever, we get this by “post-conditioning” on the outcome

We can not (yet) rule out decoherence functionals that do not We can not (yet) rule out decoherence functionals that do not admit “classical” co-events on decoherent setsadmit “classical” co-events on decoherent sets

ReferencesReferences

Quantum Mechanics as Quantum Quantum Mechanics as Quantum Measure TheoryMeasure Theory, R Sorkin, , R Sorkin, gr-qc/9401003gr-qc/9401003

Quantum Dynamics without the Wave Quantum Dynamics without the Wave FunctionFunction, R Sorkin, quant-ph/0610204, R Sorkin, quant-ph/0610204

The Problem of Hidden Variables in The Problem of Hidden Variables in Quantum MechanicsQuantum Mechanics, S Kochen & EP , S Kochen & EP Specker, 1967, Specker, 1967, J Math MechJ Math Mech 1717 59-87 59-87