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QUANTUM LOGIC AND NON-COMMUTATIVE GEOMETRY
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1. QUANTUM LOGIC AND NON-COMMUTATIVE GEOMETRY P.A. Marchetti Universita’ diPadova SISSA 2011 P.A. M., R. Rubele, Int. Jour. Theor. Phys. 46 (2007) 49

2. Underlying idea • To a physical system it is intrinsically and not “a priori” associated a “logic” of the propositions concerning its properties. This “logic” reveals a kind of “coarse-grained” structure of the system independent of a class of its details. • Quoting the logician Girard “ instead of teaching logic to nature it is reasonable to learn from her”.

3. Mathematical description of a physical system Basic entities • Observables : physical quantities that can be measured (position, momentum,….spin) • States : they characterize the knowledge on the system; states ofmaximal knowledge are called pure states (e.g. initial conditions in CM and ray vectors in QM)

4. Mathematical description of a physical system • Probability rules yielding the probability that a measure of an observable (a) measured in a state (φ ) gives a result in a borel set (Δ) of R:Paφ (Δ) • Comparison theory/experiment: (frequentist interpretation) Paφ (Δ)=limN→∞N(Δ)/N • Proposition (or Question): observable describing a property of the system, whose possible values in a measurement are only 0 and 1 corresponding to NO and YES results (e.g. is the spin up? )

5. Aim and plan • We propose a general scheme for the "logic" of propositions of physical systems in the framework given by Non Commutative Geometry, encompassing both classical and quantum cases, both with finite and infinite degrees of freedom. Plan of the talk: • How Non-Commutative Geometry is related to the Logic of physical systems • The standard (W*) approach and its problems • The new (Baire*) approach with some examples

6. Non-Commutative Geometry A key idea of NCG is that one can generalize many branches of functional analysis, such as measure theory, topology and differential geometry, by replacing the commutative algebras of functions with some degree of regularity over a space X, by suitable non-commutative (NC) algebras which may in a sense be interpreted as the “algebras of functions over a non-commutative space”. (e.g. in Quantum Mechanics a non-commutative phase space) • As in the commutative case one can consider degrees of regularity ranging from measurable to continuous to smooth

7. Non-commutative generalizations NC Algebras C* (norm complete)→ Topology Baire*(monotonically σ- complete) → Topological measure W* (weakly complete) →Measure(top. lost) pre-C* (holomorphic- ally closed) Complex functions • Continuous bounded C(X) • Measurable bounded B(X) • Measurable essentially bounded L∞(X) • Smooth C ω(X)

8. States on C*-algebras • The non-commutative analogue of probability measures on C(X) are the algebraic states: the linear positive (a*a is positive) normalized functionals on a C*-algebra. • In particular Dirac measures with support on one point in X are generalized by pure states, i.e. states that cannot be written as convex combinations of other states.

9. Projectors • Elements of a NC algebra satisfying a²=a=a* are called projectors; they are the NC generalization of characteristic functions χ [Δ] of a set Δin the space X (for which * is complex conjugation). For x ∈ X, χ [Δ] (x)=1 if x ∈ Δ, 0 otherwise. • The set of characteristic functions is naturally endowed with a complete orthomodular lattice (L) structure. U

10. Lattices • A lattice is an algebraic structure L with operations meet ∧ and join ∨(think as intersection and union, or AND and OR) satisfying associativity, commutativity, idempotence for meet ∧ and join ∨ and the adsorbing property: a ∧ (a ∨ b)=a, a ∨ (a ∧ b)=a • Apartial ordering ≤ is defined by a ≤ b iff(a ∧ b)=a • L is distributive if ∧ is distributive w.r.t. ∨ and viceversa • The orthomodular lattices have also neutral elements for ∧, denoted 1, and for ∨, denoted 0, and an involutive map, the orthocomplement⊥ (think as NOT) satisfying a ⊥ ∧ a=0, a ⊥ ∨ a=1, a ≤ b→ b ⊥ ≤ a ⊥ • orthomodularity : a, b ∈ L and a ≤ b→ b = a ∨ (a ⊥ ∧ b) • L is atomic if it has minimal (non 0) elements (atoms)

11. The lattice of commutative projectors • An example of lattice L is given by the subsets of a space X, with the meet as set-intersection and the join as set-union. L is distributive, atomic, with points as atoms. • Replacing the sets by their characteristic functions (commutative projectors) one obtains a distributive L with orthocomplement⊥, meet ∧ and join ∨ operations defined by: χ [Δ] ⊥ = 1 − χ [Δ]= χ [X-Δ] χ [Δ1] ∧ χ [Δ2]= χ [Δ1] χ [Δ2]= χ [Δ1∩ Δ2] , χ [Δ1] ∨ χ [Δ2]= (χ [Δ1] ⊥ ∧ χ [Δ2] ⊥) ⊥ = χ [Δ1U Δ2]

12. Standard Quantum Logic • Relation with quantum logic: Birkhoff and von Neumann → each question pertaining to a physical system can be expressed in terms ofpropositions ( with only YES-NO answer e.g. is the spin up?) . • The set of propositions of quantum mechanics can be represented as the complete orthomodular lattice of closed subspaces of a separable complex Hilbert space H, called standard quantum logic.

13. Standard Quantum Logic • Vectors in a subspace of H identified by a proposition correspond to the states where the proposition has YES answer (⊥=NOT, ∧ =AND, ∨=OR) • Such lattice can be characterized also algebraically in terms of the associated orthogonal projectors, p, in H , with the definitions of orthocomplement⊥, meet ∧ , join ∨ operations given by: p⊥ = 1 − p, p1 ∧ p2=limn→∞ (p1p2)n, p1∨ p2 = (p1 ⊥ ∧ p2 ⊥ )⊥

14. Logic and NCG: first attempt (W*) • The set of projectors of any W*-algebra W has the structure of a complete orthomodular lattice → it has been proposed to identify as a model for the propositional lattice of physical systems the lattice of projectors of a W*-algebra. (Cirelli-Gallone…) • One can introduce (Jauch-Piron) the logical states (=logical variables) as maps from the lattice of propositions to [0,1], yielding the probability that the proposition (e.g. is the spin up?) is true in that state (truth-value of the proposition on the logical variable).

15. Logic and NCG: first attempt (W*) • In the W* approach the logical states are given by the restrictionφL of the algebraic states φ on Wto the propositional lattice of projectors of W, P(W), with pure algebraic states describing maximal knowledge on the system. Lattice of propositions Measurable ess. bounded observables P(W) → W Logical states φL↘ ↙φAlgebraic states [0,1] →C Truth value of value of Expectation value of the propositions the observables

16. Logic and NCG: first attempt (W*) • QM of a particle (assumed without spin) in this setting is given in terms of the W* algebra of bounded operators on a separable infinite Hilbert space H, B(H), generated by Heisenberg commutation relations; propositions are given by standard quantum logic. • CM of a particleinstead is given in terms of a commutative W*-algebra L∞(Ω), where Ω is the phase space; the corresponding lattice of propositions is therefore distributive (commutativity of classical observables in L∞(Ω)→ distributivity for classical propositions)

17. W* problems-classical • In classical systems points in phase space are of zero measure and hence “invisible” to L∞(Ω) → it is not naturally defined the pure state of classical mechanics corresponding to a single point in Ωselecting “initial conditions” of the system. • Points in Ωcan be support of Dirac measures naturally defined as states on C0(Ω ), the C* (but not W*) algebra of bounded continuous functions on Ωvanishing at infinity, generated by the commutation relations [q,p]=0, i.e CCR with h=0. • However C0(Ω) does not contain non-trivial projectors, since these are characteristic functions which are not continuous.

18. W* problems-quantum • Analogously the pure states on the W* algebra of bounded operators B(H ) include also unphysical “improper states” (Dirac δ-like). • Instead the pure states on the C*-algebra (but not W*) of compact operators on a separable Hilbert space K(H),generated by Heisenberg CCR [q,p]=iħ, are exactly in correspondence with the rays of H, as physically required. • HoweverK(H) does not contain a lattice of projectors even σ-complete, i.e. stable under a countable number of meet and join operations.

19. Logic and NCG: Baire*-algebras • The above problems suggest that a natural framework to embed an algebraic model of elementary propositions is a “NC space” in general larger then the C*-algebra of “continuous bounded observables” A, but smaller than the W*-algebra of “essentially bounded measurable observables”, and containing a σ-complete (necessary for a logical “sharp” interpretation) orthomodular lattice of projectors. • This “space” is the Baire*-algebra of measurable bounded observables .

20. Logic and NCG: Baire*-algebras • A commutative Baire* algebra is the space of measurable bounded complex functions, B(X), on some topological space X . Itis identified when the topology , encoded by C(X), of the space X is given. • Analogously in NCG the Baire*algebra of bounded measurable observables is chosen as a “closure” of the C*-algebra A of continuous observables, which in the NCG approach identifies the topology of the non-commutative phase space and it is taken as the basic algebra, identifying the space of physical states. We denote such Baire* algebra (the Baire*enveloping algebra of A) by B(A).

21. Logic of physical systems • We characterize the system by the C*-algebra A of its continuous observables. • We identify the σ-complete lattice P(B(A)) of projectors of the algebra of bounded measurable observables, B(A), as a model for the lattice of propositions of the physical system described by A • We identify the logical states φL as the restriction to P(B(A)) of thelift φ˜ to B(A) of algebraic states φ on A.

22. Logic and NCG: Baire*-algebras Lattice of Measurable bounded Continuous bounded propositions observables observables P(B(A)) → B(A) ← A LogicalAlgebraic statesφL↘ Lift φ ̃↘ ↙φstates [0,1] → C Truth value of value of Expectation value of the propositions the observables

23. Logic of physical systems • If a ∈ B(A), thenφ˜(a) is the expectation value of the measurable observable a in the state φ˜ and in particular if p is a projector in B(A), then φ˜(p) =φL(p) ∈ [0, 1] yields the probability that the proposition represented by p is true (truth value) in the logical state φL. • The lift to B(A) of pure states on A identify the states of maximal knowledge on the system

24. Consequences of our proposal • Consequence of this proposal is that the lattice of elementary propositions of a physical system, although always orthomodularσ-complete it is not always complete, nor atomic, nor Hilbertian (i.e. isomorphic to all the orthogonal subspaces of a separable Hilbert space, as in standard QL). • These specific features are encoded in the C*-algebra of “continuous bounded observables” A of the system. • More obviously, for classical systems A is abelian and this implies a distributive property for the lattice of propositions.

25. Examples: CM of particle • Systems in classical mechanics with phase space Ω A = C0(Ω) the continuous functions (vanishing at ∞) and B(A) = B(Ω) the measurable bounded functions. The states on A are the probability measures on C0(Ω) which have a unique extension to B(Ω). Pure states are Dirac measures with support on one point in phase space, hence corresponding as desired to an “initial condition” (points are now visible! they weren’t in the W* L∞). The lattice of propositions P(B(Ω)) is both atomic and distributive, σ-complete but not complete (B(A) is not W*).

26. Examples: QM of particle in Rn • The algebra A is the C*-algebra generated by the Heisenberg commutation relations and it is isomorphic to K(H) with H separable infinite dimensional; B(A) ≃ B(H ); the states correspond to the statistical matrices and pure states are rays. • P(B(A)) is atomic (atoms= rays) and Hilbertian • In the Baire approach are naturally excluded the “improper” (Dirac δ-like) states of the W* approach, since the states are lifted from K(H) which doesn’t admit “improper” states.

27. Examples: QM of particle on S1 • A is generated by the Weyl commutation relations on S1 einϕeiβp = e iβp einϕeinβ φ= angle parametrizing the circle S1, n ∈ Z, β∈ [0, 2π]/ħ. • space of states = ⊕θHθdirect sum of the so-called θ -sectors , Hθ, θ ∈ [0, 2π) describing the Aharonov-Bohm phase (H≃ Hθseparable infinite dimensional). • A ≃ ⊕θK(Hθ) ≃ C(S1, K(H)) , the continuous functions on S1, K(H)-valued ; B(A) ≃ B(S1,B(H )), the bounded measurable functions on S1,B(H)-valued. • P(B(A)) is atomic, coincides with the lattice of closed subspaces of ⊕θHθ , but is not the usual Hilbert lattice of standard Quantum Logic, since ⊕θHθis not separable, so that in particular the lattice P(B(A))) is not complete.

28. Examples: RQFT • Local observable algebras in massive RQFT in a double cone O • A(O) can be identified as the “space of bounded continuous observables with support in O” • A deep result of RQFT with mass gap is that B(A(O)) is a type III1 von Neumann (W*) algebra and the associated lattice of propositions is non-atomic, but complete.

29. Baire* algebra • A C*-algebra A is called monotonically sequentially complete if every bounded monotone sequence of the self-adjoint part of A, Asa, possesses a limit in Asa. • A state φ over a monotonically sequentially complete C*-algebra A is called σ-normal if for every bounded monotone sequence {xn} n∈Nin Asawehave φ (∨nxn)= ∨n φ(xn). Definition (Pedersen)A C*-algebra B is called a Baire∗-algebra if it is monotonically sequentially complete and it admits a separating family F (φ (a)=0 F of σ-normalstates.

30. Preliminary: Atomicrepresentation • Given a C*-algebra A, let ^A, be its spectrum, i.e. the set of (equivalence classes of unitarily equivalent) irreducible representations of A. Let φ be a (representative) pure state corresponding to a point of ˆA, and πφthe corresponding representation. • The atomic representation of A is given by πa= ⊕φ∈ ˆA πφand it is a faithful representation of A.

31. Baire* enveloping algebra • Given a C*-algebra, A, we define the monotone sequential closure of its self-adjoint part Asa as the smallest subset of the representation πa(A)”, containing πa(Asa) , and the limit of every monotone sequence of elements of πa(Asa) and we denote it by B(Asa). • The Baire* enveloping algebra of A (Pedersen), is given by B(A) ≡ B(Asa) + iB(Asa). • B(A) is a Baire∗-algebra with the family of σ-normal states given by the unique extension of the states on A to B(A).

32. Possibility of completion • In the definition of enveloping Baire∗-algebra we can replace the atomic representation πawith the universal representation πu = ⊕φ∈S(A)πφ, whereS(A) is the set of states on A and the corresponding B(A) is isomorphic to the one defined via πa. • Then B(A) ⊂ πu(A)”, which is the universal enveloping von Neumann algebra of A. • Therefore the σ-complete orthomodular lattice of B(A) describing the propositions of the system characterized by A can be embedded in the completeorthomodularlattice of πu(A)”; thisembedding in a complete lattice, itisnottrue for a genericorthomodular lattice, althoughitistrue for distributive ones.

33. Summary • We propose that the lattice of propositions of physical systems can be represented as the σ–complete orthomodular lattice of projectors of the space of “ measurable bounded observables ” (Baire* algebra) on a generally “non-commutative space X” in the sense of Non Commutative Geometry. This algebrais obtained as a closure of the C*-algebra of “continuous bounded observables” on X. • The propositional logic depends on the physical system, but it captures only a very “coarse grained” structure of it. E.g. it is able to identify the classical or quantum nature of the system, countable versus non-countable set of superselection sectors.

34. Future developments • Here we first characterize the physical system through the C* algebra of its continuous bounded observables Aand then derive the logic of its propositions as P(B(A)). However it would be more satisfactory from the logical point of view, to start from the structure of the logic and then derive its embedding in an algebraic structure describing its observables and derive, not assume, the Born-von Neumann-Lueders rule giving the probability structure , summarized in the present approach by the algebraic-logical states. But we need an extension of the lattice structure, a justification of the complex algebra…

35. Conclusion • The Baire*-algebras for bounded measurable observables make transparent the interpretation of quantum mechanics as a “theory of quantum probability” on a Topological NC phase space • As Streater pointed out: “Though the classical axioms were yet to be written down by Kolmogorov, Heisenberg, with help of the Copenhagen interpretation, invented a generalisation of the concept of probability, and physicists showed that this was the model of probability chosen by atoms and molecules.” • One can hope to say similar words about the logical foundations behind the probability structure of nature…the above approach might be seen as a step in this direction…