Post on 10-Apr-2022
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