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Transcript
```QUANTUM LOGIC AND
NON-COMMUTATIVE GEOMETRY
P.A. Marchetti
Universita’ di Padova
SISSA 2011
P.A. M., R. Rubele, Int. Jour. Theor. Phys. 46 (2007) 49
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”.
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 of maximal
knowledge are called pure states (e.g. initial
conditions in CM and ray vectors in QM)
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:
Pa φ (Δ)
• Comparison theory/experiment: (frequentist
interpretation) Pa φ (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? )
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
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
Non-commutative generalizations
Complex functions
• Continuous bounded C(X)
• Measurable bounded B(X)
• Measurable essentially
bounded L∞ (X)
• Smooth C ω(X)
NC Algebras
C* (norm complete)→
Topology
Baire*(monotonically
σ- complete) →
Topological measure
W* (weakly complete)
→Measure(top. lost)
pre-C* (holomorphically closed)
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.
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 Δ in 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.
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
• A partial 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)
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 [Δ1 U Δ2]
Standard Quantum Logic
• Relation with quantum logic: Birkhoff and von
Neumann → each question pertaining to a
physical system can be expressed in terms of
propositions ( 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.
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 ⊥ )⊥
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).
Logic and NCG: first attempt (W*)
• In the W* approach the logical states are given by
the restriction φL of the algebraic states φ on W to 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)
→
Logical states φL↘
W
↙φ
Algebraic states
[0,1] → C
Truth value of value of
the propositions
Expectation value of
the observables
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 particle instead 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)
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 Ω
selecting “initial conditions” of the system.
• Points in Ω can be support of Dirac measures
naturally defined as states on C0(Ω ), the C* (but
not W*) algebra of bounded continuous functions
on Ω vanishing 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.
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.
• However K(H) does not contain a lattice of
projectors even σ-complete, i.e. stable under a
countable number of meet and join operations.
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 .
Logic and NCG: Baire*-algebras
• A commutative Baire* algebra is the space of
measurable bounded complex functions, B(X), on
some topological space X . It is 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).
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 the lift φ˜ to B(A) of
algebraic states φ on A.
Logic and NCG: Baire*-algebras
Lattice of
Measurable bounded
propositions
observables
P(B(A)) →
B(A)
Continuous bounded
observables
←
A
Logical
states
Algebraic
φL↘
Lift
[0,1]
Truth value of value of
the propositions
φ↘
̃
↙φ
→
C
Expectation value of
the observables
states
Logic of physical systems
• If a ∈ B(A), then φ˜(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
Consequences of our proposal
• Consequence of this proposal is that the lattice of
elementary propositions of a physical system,
although always orthomodular σ-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.
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*).
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.
Examples: QM of particle on S1
• A is generated by the Weyl commutation relations on S1
einϕ eiβp = e iβp einϕ ei nβ
φ= angle parametrizing the circle S1, n ∈ Z, β∈ [0, 2π]/ħ.
• space of states = ⊕θHθ direct sum of the so-called θ sectors , Hθ, θ ∈ [0, 2π) describing the Aharonov-Bohm
phase (H ≃ Hθ separable 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 ⊕θHθ is not separable, so
that in particular the lattice P(B(A))) is not complete.
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 nonatomic, but complete.
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∈N in Asa we have
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 ∀φ ∈ F → a=0)
of σ-normal states.
Preliminary: Atomic representation
• 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 π φ the
corresponding representation.
• The atomic representation of A is given by
πa = ⊕φ∈ ˆA πφ
and it is a faithful representation of A.
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).
Possibility of completion
• In the definition of enveloping Baire∗-algebra we
can replace the atomic representation πa with the
universal representation πu = ⊕φ∈S(A)πφ, where
S(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
complete orthomodular lattice of πu(A)”; this
embedding in a complete lattice, it is not true for
a generic orthomodular lattice, although it is true
for distributive ones.
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 algebra is 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.
Future developments
• Here we first characterize the physical system
through the C* algebra of its continuous bounded
observables A and 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…
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…
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