The Logic of Quantum Mechanics · 2015-11-14 · The Logic of Quantum Mechanics. Classical...
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Classical Mechanics Quantum Mechanics Algebraic approah
The Logic of Quantum Mechanics
Nicolo Drago
University of Genova
27 April 2015
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Introduction
Tipical reactions attending the first class in Quantum Mechanics:
Ma che c. . . . sono tutti ’sti simboli a caso?!?S.Murro.
Che figata, non capisco una m. . . !G.Nosari.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Introduction
Aim of the talk:
Introduce the first 4 Axioms of Quantum Mechanics.
Explain the mantra
“Hilbert spaces provide a natural framework for QM”.
Destroy the mantra.
Have a tasty break.
No cats were harmed during the production of these slides.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Phase space
Let S be your favourite physical system (particle, gas, mewingcat. . . ).Fixing a frame I , S is described via the phase space P.
℘loc.= (q1, . . . , qn; p1, . . . pn) ∈ P describes a configuration of S .
Dynamics is ruled by Hamilton’s equations:
R 3 t 7→ ℘(t)loc.= (q1(t), . . . , qn(t); p1(t), . . . pn(t)) ∈ P
dqk
dt=∂H
∂pk,
dpk
dt= − ∂H
∂qk, k = 1, . . . , n,
being H : P→ R the Hamiltonian of the system S .
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Proposition and states on P
A proposition P is realized as a Borel set P ∈ B(P).
P ∨ Q ←→ P ∪ Q
P ∧ Q ←→ P ∩ Q
P ⇒ Q ←→ P ⊆ Q
a P ←→ P \ P
P ←→ tautology
∅ ←→ contradiction.
A state ω is a probability measure on B(P).
ω(P) = probability that P is true if the state of S is ω.
Example: ω = δ℘, ℘ ∈ P, is sharp state.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Proposition and states on P
A proposition P is realized as a Borel set P ∈ B(P).
P ∨ Q ←→ P ∪ Q
P ∧ Q ←→ P ∩ Q
P ⇒ Q ←→ P ⊆ Q
a P ←→ P \ P
P ←→ tautology
∅ ←→ contradiction.
A state ω is a probability measure on B(P).
ω(P) = probability that P is true if the state of S is ω.
Example: ω = δ℘, ℘ ∈ P, is sharp state.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Observables on P
An observable O is a measurable function O : P→ R.Observables are completely characterized by a list of propositionsparametrized by E ∈ B(R):
P(O)E
.= O−1(E )
= The assumed value of O on the system belongs to E .
Observables generate a commutative ∗-algebra A over C.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
First step towards QM
A system S has quantum behaviour if
Energy× Time . ~ = 6.6262 · 10−34Js
1800’s : Two Slit Experiment: light has wave behaviour.
1905 : Photoelectric effect: light is made by particles.
light : What about make your own business?!?
1924 De Broglie: “Particles and waves are always related”.
1926 Schrodinger: “I believe De Broglie.I have ((((((a lovely cat an equation.”
1927 Heisenberg: “Guys, we cannot do better than ∆x∆p = ~”.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
First step towards QM
A system S has quantum behaviour if
Energy× Time . ~ = 6.6262 · 10−34Js
1800’s : Two Slit Experiment: light has wave behaviour.
1905 : Photoelectric effect: light is made by particles.
light : What about make your own business?!?
1924 De Broglie: “Particles and waves are always related”.
1926 Schrodinger: “I believe De Broglie.I have ((((((a lovely cat an equation.”
1927 Heisenberg: “Guys, we cannot do better than ∆x∆p = ~”.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
![Page 10: The Logic of Quantum Mechanics · 2015-11-14 · The Logic of Quantum Mechanics. Classical MechanicsQuantum MechanicsAlgebraic approah First step towards QM A system S hasquantum](https://reader030.fdocuments.us/reader030/viewer/2022041117/5f2d34ba9d015958f65c39b8/html5/thumbnails/10.jpg)
Classical Mechanics Quantum Mechanics Algebraic approah
Pathologies: the crash of logic and probability
There exist incompatible observables A,B i.e.:
measure A on ω at time t =⇒ outcome α:
measure B on ω at time t + ε =⇒ outcome β:
measure A on ω at time t + 2ε =⇒ outcome α′:
Expectancy: α′ → α as ε→ 0+.
Fact: α, α′ are completely uncorrelated.Physical interpretation: measurement of B disturbs themeasurement of A by changing the state of the system.
Consequence: P(A)E ∧ P
(B)E ′ has not physical sense.
Proposition on S are not described by Borel sets.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Pathologies: the crash of logic and probability
There exist incompatible observables A,B i.e.:
measure A on ω at time t =⇒ outcome α:
measure B on ω at time t + ε =⇒ outcome β:
measure A on ω at time t + 2ε =⇒ outcome α′:
Expectancy: α′ → α as ε→ 0+.Fact: α, α′ are completely uncorrelated.Physical interpretation: measurement of B disturbs themeasurement of A by changing the state of the system.
Consequence: P(A)E ∧ P
(B)E ′ has not physical sense.
Proposition on S are not described by Borel sets.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Hilbert space and propositions
Let (H, 〈· | ·〉) be an separable Hilbert space over C.An orthogonal projector P ∈ P(H) is
P : H→ H linear, P2 = P∗ = P.
P proposition ←→ P ∈ P(H)?
QM,Axiom 1
A proposition P is realized as an orthogonal projector P ∈ P(H).
P,Q compatible ←→ [P,Q] = 0
if P,Q are compatible P ∨ Q ←→ P + Q − PQ
if P,Q are compatible P ∧ Q ←→ PQ
if P,Q are compatible P ⇒ Q ←→ P ≤ Q
a P ←→ I − P
I ←→ tautology
0 ←→ contradiction.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Hilbert space and propositions
Let (H, 〈· | ·〉) be an separable Hilbert space over C.An orthogonal projector P ∈ P(H) is
P : H→ H linear, P2 = P∗ = P.
P proposition ←→ P ∈ P(H)?
Observation: P,Q ∈ P(H) ; PQ ∈ P(H)
(PQ)∗ = Q∗P∗
= QP
= PQ ⇔ [Q,P] = 0!
[·, ·] provides a criterion for compatibility.
QM,Axiom 1
A proposition P is realized as an orthogonal projector P ∈ P(H).
P,Q compatible ←→ [P,Q] = 0
if P,Q are compatible P ∨ Q ←→ P + Q − PQ
if P,Q are compatible P ∧ Q ←→ PQ
if P,Q are compatible P ⇒ Q ←→ P ≤ Q
a P ←→ I − P
I ←→ tautology
0 ←→ contradiction.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Hilbert space and propositions
Let (H, 〈· | ·〉) be an separable Hilbert space over C.
QM,Axiom 1
A proposition P is realized as an orthogonal projector P ∈ P(H).
P,Q compatible ←→ [P,Q] = 0
if P,Q are compatible P ∨ Q ←→ P + Q − PQ
if P,Q are compatible P ∧ Q ←→ PQ
if P,Q are compatible P ⇒ Q ←→ P ≤ Q
a P ←→ I − P
I ←→ tautology
0 ←→ contradiction.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
States
A state is a map which associates a probabilistic value to eachproposition.
QM,Axiom 2
A state ω is a map ω : P(H)→ [0, 1] such that:
ω(I ) = 1;
if (Pn)n ⊂ P(H) are such that PnPm = 0 for n 6= m
ω
∨n≥0
Pn
=∑n≥0
ω(Pn).
Example: ψ ∈ H, ‖ψ‖ = 1
ωψ(P) = 〈ψ | Pψ〉 .
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
A bit more about states
ω1, ω2 states ; αω1 + βω2
The state space of S , S(H), is not linear.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
A bit more about states
ω1, ω2 states ⇒ λω1 + (1− λ)ω2, λ ∈ [0, 1]The state space of S , S(H), is convex.
ω is said to be:
pure: if it cannot be decomposed as ω = λω1 + (1− λ)ω2,λ ∈ (0, 1), ω1,2 6= ω;
mixture: if it is not pure.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
A bit more about states
ω1, ω2 states ⇒ λω1 + (1− λ)ω2, λ ∈ [0, 1]The state space of S , S(H), is convex.
ω is said to be:
pure: ψ ∈ H, ‖ψ‖ = 1, ωψ(P) = 〈ψ | Pψ〉 .mixture: (ψn)n ∈ H, 〈ψn | ψm〉 = δn,m, cn ≥ 0,
∑n cn = 1
ω(P) =∑n
cn 〈ψn | Pψn〉 = Tr (TP) ,
being Tφ =∑
n cn 〈ψn | φ〉ψn, Tr(A) =∑
n 〈φn | Aφn〉.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
A bit more about states
Hp: H with finite dimension≥ 3 or infinite dimensional andseparable.
Theorem (Gleason)
For each state ω there exists an positive trace class operator Tsuch that ω(P) = Tr(TP).
Theorem (Kochen-Specker)
There is no state ω taking values in 0, 1.
Quantum Mechanics does not admit sharp states.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Post-measurement states
Let ω be the state of S at a certain time t.
Which state describes S after a positive measure of P?
Positive outcome for P ⇒ ω(P) > 0.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Post-measurement states
Let ω be the state of S at a certain time t.
Which state describes S after a positive measure of P?
Positive outcome for P ⇒ ω(P) > 0.Heuristic argument: assume ω = ωψ.
ωψ → ωPψ
Pψ.
=Pψ
‖Pψ‖ωψ(P) = 〈ψ | Pψ〉 = 〈Pψ | Pψ〉 = ‖Pψ‖2
ωPψ
(Q) =〈Pψ | QPψ〉‖Pψ‖2
=〈ψ | PQPψ〉ωψ(P)
=ωψ(PQP)
ωψ(P).
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Post-measurement states
Let ω be the state of S at a certain time t.
Which state describes S after a positive measure of P?
Positive outcome for P ⇒ ω(P) > 0.
QM,Axiom 3
If S is in a state ω at time t and a proposition P ∈ P(H) is trueafter a measurement at time t, then immediately afterwards thesystem’s state collapses into
ωP(·) =ω(P · P)
ω(P).
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Observables
“Observables are completely characterized by a list of propositionsparametrized by E ∈ B(R).”
QM,Axiom 4
An observable O is a map B(R) 3 E 7→ P(O)E ∈ P(H) such that:
[P(O)E ,P
(O)E ′ ] = 0;
P(O)E∩E ′ = P
(O)E ∧ P
(O)E ′ ;
P(O)R = I ;
P(O)⋃
n En=∨
n P(O)En
.
P(O)E
?= O−1(E )
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Observables
Consider an observable O with discrete spectrumσ(O) = ann ⊂ R.
P(O)n = the measured value of O is precisely an.
Average of O on ω:
∑n
an ω(
P(O)n
)=: Ω
(∑n
an P(O)n
)=: Ω(O)
O : H→ H is self-adjoint.
Observables can be regarded as self-adjoint operators over H.States can be regarded as maps on observables such that
Ω(O) = Average of O on the state Ω.
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The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Observables
Observables can be regarded as self-adjoint operators.States can be regarded as maps on observables.
Theorem (∼ Spectral decomposition)
For each observable O there exists a (possibly unbounded)self-adjoint operator O on H defined as
O.
=
∫σ(O)
λ P(O)λ .
Observables generate a non commutative ∗-algebra A over C.
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Algebraic approach and GNS Theorem
Definition (∗-algebra)
A ∗-algebra A over C is an algebra A = Alg(+, ·) over C with aninvolution ∗ : A → A such that
(a · b)∗ = b∗ · a∗, (αa + βb)∗ = αa∗ + βb∗.
A state ω on a ∗-algebra A is a linear functional ω : A → C whichis:
positive: ω(a∗ · a) ≥ 0;
normalized: ω(1A ) = 1.
Physical observables are self-adjoint elements a = a∗ ∈ A .
Nicolo Drago University of Genova
The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
Algebraic approach and GNS Theorem
Theorem (Gelfand-Naimark-Segal)
Let ω be a state on a ∗-algebra A . There exists a quadruple(Dω,Hω, πω,Ω) such that:
Dω is a dense subspace in Hω;
πω : A → L(Hω) is a ∗-representation of A onto Dω-definedoperator on Hω.
πω(A )Ω = Dω.
ω(a) = 〈Ω | πω(a)Ω〉 .
(Dω,Hω, πω,Ω) is unique up to unitary isomorphism.
ω, ω′ can induce different GNS representations of A !
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The Logic of Quantum Mechanics
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Classical Mechanics Quantum Mechanics Algebraic approah
I would like to make a confession that may seem immoral:I do not believe absolutely in Hilbert spaces anymore.
von Neumann (1935)
Nicolo Drago University of Genova
The Logic of Quantum Mechanics