Quantum theory in real Hilbert space: How the

complex Hilbert space structure emerges from

Poincaré symmetry

Marco Oppio

University of Trento (Italy)

June 16, 2017

INFN BELL Fundamental problems of quantum physics

Based on the paper arXiv:1611.09029 in collaboration with V. Moretti

Reviews in Mathematical Physics in print

Contents

1 Standard Quantum Mechanics

2 A taste of Quantum Logic

3 Real Quantum Mechanics

4 Work in progress

Standard Quantum Mechanics

General setting: Complex Hilbert space (H, (·|·))

(Bounded) Observables: Self-adjoint elements on H

A∗ = A ∈ B(H)

Possibile outcomes: σ(A) ⊂ R

States: unit-trace trace-class operators M ∈ B1(H)

Example

tr(MA) is the expectation value of A = A∗

Let us analyse the situation from a more abstract point of view

Quantum Logic

Let O be the set of observables and B(R) the set of Borelians.

De�nition

The Logic of a physical system is the set L of statements

P(A)∆ = "the value of A falls within ∆"

with A ∈ O and ∆ ∈ B(R)

(Strong) physical arguments lead to a structure of complete

orthocomplemented lattice (L,≤,⊥) with

logical implication: P ≤ Q

logical conjuction: P ∧ Q := inf{P,Q}logical disjuntion: P ∨ Q := sup{P,Q}logical negation: P⊥

Quantum Logic

De�nition

A state is a probability measure µ : L → [0, 1]

µ(I ) = 1

if (Pi )i∈N ⊂ L with Pn ⊥ Pm (stat. indep.) then

µ

( ∞∨i=1

Pi

)=∞∑i=1

µ(Pi )

N.B. Construction meaningful for classical and quantum systems

classical: distributive lattice (P ∧ (Q ∨ Q′) = (P ∧ Q) ∨ (P ∧ Q′))

quantum: non-distributive lattice (P ∧ (Q ∨ Q′) 6= (P ∧ Q) ∨ (P ∧ Q′))

⇒ due to existence of incompatible propositions

Quantum Logic: Piron-Solèr Theorem

Piron-Solèr Theorem

[+ technical hypotheses] if L is irreducible (elementary particle)

L ∼= P(H) := {P ∈ B(H) | P∗ = P, PP = P}

where H is a Hilbert space over the �eld R,C or H

Consequences

1 Gleason Theorem: states must be of the form µ = tr(M ·)2 O turn out to be identi�ed with selfadjoint operators on H

A ∈ O ⇒{P

(A)∆ |∆ ∈ B(R)

}⊂ L is a PVM over H

the complex case corresponds to the traditional theorywhat about the other two of them? What is their meaning?

Real Quantum Mechanics: Algebra of Observables

Focus on the real case:

Strategy

1 Take Piron-Solèr thesis simply as a clue on the nature of L2 Relax Piron-Solèr thesis: weaken P(H)

Our framework - I Part

Let H be over R andM =M′′ ⊂ B(H) a von Neumann algebra.

Quantum propositions are the elements of PM(H)

Observables are the self-adjoint elements ofMStates are probability measures µ : PM(H)→ [0, 1]

Real Quantum Mechanics

The particle being elementary, we can takeM irreducible

1 if H were complex, thanks to Schur's lemma we would get

M′ = {aI , a ∈ C} and soM =M′′ = B(H)

2 but H is real and Schur's lemma holds in a weaker form

Theorem (Schur's Lemma for real Hilbert spaces)

One and only one of the following facts holds

1 M′ = {aI , a ∈ R} ∼= R2 M′ = {aI + bJ, a, b ∈ R} ∼= C3 M′ = {aI + bJ + cK + dJK , a, b, c , d ∈ R} ∼= H

where J,K are imaginary operators, with JK = −KJ

N.B J ∈ B(H) is imaginary if J∗ = −J and JJ = −I

Real Quantum Mechanics

Let us consider the second case: M′ ∼= C

De�nition (Complexi�cation)

The real Hilbert space H equipped with (a, b ∈ R, u, v ∈ H)

1 (a + jb)v := (aI + bJ)v

2 (u|v)J := (u|v)− j(u|Jv)

is a complex Hilbert space, denoted by HJ

Proposition

A R-linear operator A is C-linear i� AJ = JA

1 A is unitary on H i� it is unitary on HJ

2 A is (anti) selfadjoint on H i� it is (anti) selfadjoint on HJ

Real Quantum Mechanics

A similar argument holds for the quaternionic case: M′ ∼= H

De�nition (Quaternioni�cation)

The real Hilbert space H equipped with

1 (a + jb + kc + jkd)v := (aI + bJ + cK + dJK )v

2 (u|v)J := (u|v)− j(u|Jv)− k(u|Kv)− jk(u|JKv)

is a quaternionic Hilbert space, denoted by HJK

Proposition

A R-linear operator A is H-linear i� AJ = JA,AK = KA

1 A is unitary on H i� it is unitary on HJK

2 A is (anti) selfadjoint on H i� it is (anti) selfadjoint on HJK

Real Quantum Mechanics

Proposition [reminder]

One and only one of the following holds forM irreducible

1 M′ = {aI , a ∈ R} ∼= R2 M′ = {aI + bJ, a, b ∈ R} ∼= C3 M′ = {aI + bJ + cK + dJK , a, b, c , d ∈ R} ∼= H

where J,K are imaginary operators, with JK = −KJ

Proposition

It holds respectively

1 M = B(H) and PM(H) = P(H)

2 M = B(HJ) and PM(H) = P(HJ)

3 M = B(HJK ) and PM(H) = P(HJK )

Real Quantum Mechanics: Poincaré Symmetry

Our framework - II Part

There is another important contraint on the system: symmetries

The maximal symmetry group for an elementary particle is

Poincaré group P :=

space-time translations

space rotations

boosts

We can identity this action as a group representation

P 3 g 7→ αg ∈ Aut(PM(H)) which is

locally faithful

weak continuous

with no proper �xed points

Real Quantum Mechanics

Notice: αg ∈ Aut(PM(H)) = Aut(P(K)) with K = H,HJ ,HJK

Theorem [Bargmann, Pontryagin, Varadarajan, Wigner]

There exists an irreducible strongly-continuous locally faithful

unitary representation g 7→ Ug on H,HJ ,HJK , respectively, s.t.

αg (P) = UgPU∗g ∀P

Important remark: Let V unitary element of Z :=M∩M′ then

(VUg )P(VUg )∗ = UgPUg

Ug is de�ned up to phases of UZ

Real Quantum Mechanics

We must make another important physical assumption

Our framework - III Part

Elementary particle characterized by its maximal symmetry group

⇒ its observables must come from the representation α somehow.

How? There is a natural way this can be achieved in

PM(H) ⊂ 〈{{Ug | g ∈ P} ∪ UZ}〉s

= {{Ug | g ∈ P} ∪ UZ}′′

Phases must be included: only αg has physical meaning and

αg∼= ”Ug up to phases”

Real Quantum Mechanics

Real commutant: M = B(H), PM(H) = P(H), UZ = {±I}

P(H) ⊂ {Ug | g ∈ P}′′ ⊂ B(H)⇒ B(H) = {Ug | g ∈ P}′′

Main Result

Let Ugt = etP0 be the time-displacement and P0 = J0|P0| (PD)where J∗0 = −J0 is a partial isometry. Then it holds

J0J0 = −I and J0 ∈ {Ug , g ∈ P}′(= B(H)′

)CONTRADICTION!

B(H)′ = RI =⇒ −J0 = J∗0 = (aI )∗ = aI = J0 =⇒ J0 = 0

Real Quantum Mechanics: Conclusions

Quaternionic commutant: M = B(HJK ), UZ = {±I}

⇒ Di�erent treatment same conclusion: CONTRADICTION!

Only the complex commutant case survives

1 HJ complex Hilbert space

2 observables are self-adjoint elements over HJ

3 Gleason Theorem: states must be of the form µ = tr(M ·)

Conclusions

we recover the standard complex theory!

the complex structure arises from Poincaré invariance

Work in progress

What about the quaternionic case suggested by Piron-Soler?

STRATEGY: mimic the discussion done for the real case

1 take a quaternionic Hilbert space H2 consider an irreducible von Neumann algebraM⊂ B(H)3 see what happens...

OBSTRUCTION

What is a quaternionic von Neumann algebra?

Current situation

1 we chose a real subalgebraM⊂ B(H) such thatM′′ =M2 mimicked the real theory

3 three mutually exclusive possibilities came out.

4 the extremal possibilities lead to a contradiction

Thank you for the attention!

And to conclude, some bibliography...

K. Engesser, D.M. Gabbay, D. Lehmann (editors): Handbookof Quantum Logic and Quantum Structures. Elsevier,Amsterdam (2009)

R. Kadison, J.R. Ringrose: Fundamentals of the Theory of

Operator Algebras, (Vol. I, II, III, IV) Graduate Studies inMathematics, AMS (1997)

B. Li: Real Operator Algebras. World Scienti�c (2003)

V. Moretti: Spectral Theory and Quantum Mechanics, With an

Introduction to the Algebraic Formulation. Springer, 2013

V.S. Varadarajan, The Geometry of Quantum Mechanics. 2ndEdition, Springer (2007)

Appendix: quaternionic commutant case

Quaternionic commutant M = B(HJK ), P(M) = P(HJK)

1 P(HJK ) ⊂ {Ug | g ∈ P}′′

2 g 7→ Ug is quaternionic irreducible on HJK

Take J ∈M′ (or K ) and de�ne HJ the usual way. 1) and 2) imply

Theorem

The map g 7→ Ug is a irreducible strongly-continuous faithfulunitary representation on HJ

Appendix: quaternionic commutant case

Let us work on HJ : Take the time-translation t 7→ gt

Stone Theorem Ugt = etP0 with P∗0 = −P0

Polar decomposition: P0 = J0|P0|1 |P0| ≥ 0 and |P0|∗ = |P0|2 J∗

0= −J0 and J0 is a partial isometry

Naive result (complex version)

It holds J0 = ±iI = ±J

Let us go back to H: it still holds Ugt = etP0 and P0 = J0|P0|

Properties of Polar Decomposition

K ∈ {Ug |g ∈ P}′ ⇒ KetP0 = etP0K ⇒ KP0 = P0K ⇒ KJ0 = J0K

IMPOSSIBLE! because JK = −KJ