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

Classical Mechanics Quantum Mechanics Algebraic approah

The Logic of Quantum Mechanics

Nicolo Drago

University of Genova

27 April 2015

Nicolo Drago University of Genova



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.




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.




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 .




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.




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.




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.




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 = ~”.




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 = ~”.




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.




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.




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.





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 P ←→ I − P

I ←→ tautology






Let (H, 〈· | ·〉) be an separable Hilbert space over C.

QM,Axiom 1






a P ←→ I − P

I ←→ tautology





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〉 .




A bit more about states

ω1, ω2 states ; αω1 + βω2

The state space of S , S(H), is not linear.





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.





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〉.





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.




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.







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).







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).




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 )




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.




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.




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 .




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 !




I would like to make a confession that may seem immoral:I do not believe absolutely in Hilbert spaces anymore.

von Neumann (1935)



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

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