quantum theory in real hilbert space: how the complex
TRANSCRIPT
![Page 1: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/1.jpg)
Quantum theory in real Hilbert space: How the
complex Hilbert space structure emerges from
PoincarƩ 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
![Page 2: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/2.jpg)
Contents
1 Standard Quantum Mechanics
2 A taste of Quantum Logic
3 Real Quantum Mechanics
4 Work in progress
![Page 3: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/3.jpg)
Standard Quantum Mechanics
General setting: Complex Hilbert space (H, (Ā·|Ā·))
(Bounded) Observables: Self-adjoint elements on H
Aā = A ā B(H)
Possibile outcomes: Ļ(A) ā R
States: unit-trace trace-class operators M ā B1(H)
Example
tr(MA) is the expectation value of A = Aā
Let us analyse the situation from a more abstract point of view
![Page 4: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/4.jpg)
Quantum Logic
Let O be the set of observables and B(R) the set of Borelians.
Deļæ½nition
The Logic of a physical system is the set L of statements
P(A)ā = "the value of A falls within ā"
with A ā O and ā ā B(R)
(Strong) physical arguments lead to a structure of complete
orthocomplemented lattice (L,ā¤,ā„) with
logical implication: P ā¤ Q
logical conjuction: P ā§ Q := inf{P,Q}logical disjuntion: P āØ Q := sup{P,Q}logical negation: Pā„
![Page 5: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/5.jpg)
Quantum Logic
Deļæ½nition
A state is a probability measure Āµ : L ā [0, 1]
Āµ(I ) = 1
if (Pi )iāN ā L with Pn ā„ Pm (stat. indep.) then
Āµ
( āāØi=1
Pi
)=āāi=1
Āµ(Pi )
N.B. Construction meaningful for classical and quantum systems
classical: distributive lattice (P ā§ (Q āØ Qā²) = (P ā§ Q) āØ (P ā§ Qā²))
quantum: non-distributive lattice (P ā§ (Q āØ Qā²) 6= (P ā§ Q) āØ (P ā§ Qā²))
ā due to existence of incompatible propositions
![Page 6: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/6.jpg)
Quantum Logic: Piron-SolĆØr Theorem
Piron-SolĆØr Theorem
[+ technical hypotheses] if L is irreducible (elementary particle)
L ā¼= P(H) := {P ā B(H) | Pā = P, PP = P}
where H is a Hilbert space over the ļæ½eld R,C or H
Consequences
1 Gleason Theorem: states must be of the form Āµ = tr(M Ā·)2 O turn out to be identiļæ½ed with selfadjoint operators on H
A ā O ā{P
(A)ā |ā ā B(R)
}ā L is a PVM over H
the complex case corresponds to the traditional theorywhat about the other two of them? What is their meaning?
![Page 7: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/7.jpg)
Real Quantum Mechanics: Algebra of Observables
Focus on the real case:
Strategy
1 Take Piron-SolĆØr thesis simply as a clue on the nature of L2 Relax Piron-SolĆØr thesis: weaken P(H)
Our framework - I Part
Let H be over R andM =Mā²ā² ā B(H) a von Neumann algebra.
Quantum propositions are the elements of PM(H)
Observables are the self-adjoint elements ofMStates are probability measures Āµ : PM(H)ā [0, 1]
![Page 8: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/8.jpg)
Real Quantum Mechanics
The particle being elementary, we can takeM irreducible
1 if H were complex, thanks to Schur's lemma we would get
Mā² = {aI , a ā C} and soM =Mā²ā² = 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 Mā² = {aI , a ā R} ā¼= R2 Mā² = {aI + bJ, a, b ā R} ā¼= C3 Mā² = {aI + bJ + cK + dJK , a, b, c , d ā R} ā¼= H
where J,K are imaginary operators, with JK = āKJ
N.B J ā B(H) is imaginary if Jā = āJ and JJ = āI
![Page 9: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/9.jpg)
Real Quantum Mechanics
Let us consider the second case: Mā² ā¼= C
Deļæ½nition (Complexiļæ½cation)
The real Hilbert space H equipped with (a, b ā R, u, v ā H)
1 (a + jb)v := (aI + bJ)v
2 (u|v)J := (u|v)ā j(u|Jv)
is a complex Hilbert space, denoted by HJ
Proposition
A R-linear operator A is C-linear iļæ½ AJ = JA
1 A is unitary on H iļæ½ it is unitary on HJ
2 A is (anti) selfadjoint on H iļæ½ it is (anti) selfadjoint on HJ
![Page 10: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/10.jpg)
Real Quantum Mechanics
A similar argument holds for the quaternionic case: Mā² ā¼= H
Deļæ½nition (Quaternioniļæ½cation)
The real Hilbert space H equipped with
1 (a + jb + kc + jkd)v := (aI + bJ + cK + dJK )v
2 (u|v)J := (u|v)ā j(u|Jv)ā k(u|Kv)ā jk(u|JKv)
is a quaternionic Hilbert space, denoted by HJK
Proposition
A R-linear operator A is H-linear iļæ½ AJ = JA,AK = KA
1 A is unitary on H iļæ½ it is unitary on HJK
2 A is (anti) selfadjoint on H iļæ½ it is (anti) selfadjoint on HJK
![Page 11: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/11.jpg)
Real Quantum Mechanics
Proposition [reminder]
One and only one of the following holds forM irreducible
1 Mā² = {aI , a ā R} ā¼= R2 Mā² = {aI + bJ, a, b ā R} ā¼= C3 Mā² = {aI + bJ + cK + dJK , a, b, c , d ā R} ā¼= H
where J,K are imaginary operators, with JK = āKJ
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 )
![Page 12: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/12.jpg)
Real Quantum Mechanics: PoincarƩ Symmetry
Our framework - II Part
There is another important contraint on the system: symmetries
The maximal symmetry group for an elementary particle is
PoincarƩ group P :=
space-time translations
space rotations
boosts
We can identity this action as a group representation
P 3 g 7ā Ī±g ā Aut(PM(H)) which is
locally faithful
weak continuous
with no proper ļæ½xed points
![Page 13: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/13.jpg)
Real Quantum Mechanics
Notice: Ī±g ā 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 7ā Ug on H,HJ ,HJK , respectively, s.t.
Ī±g (P) = UgPUāg āP
Important remark: Let V unitary element of Z :=Mā©Mā² then
(VUg )P(VUg )ā = UgPUg
Ug is deļæ½ned up to phases of UZ
![Page 14: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/14.jpg)
Real Quantum Mechanics
We must make another important physical assumption
Our framework - III Part
Elementary particle characterized by its maximal symmetry group
ā its observables must come from the representation Ī± somehow.
How? There is a natural way this can be achieved in
PM(H) ā ć{{Ug | g ā P} āŖ UZ}ćs
= {{Ug | g ā P} āŖ UZ}ā²ā²
Phases must be included: only Ī±g has physical meaning and
Ī±gā¼= āUg up to phasesā
![Page 15: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/15.jpg)
Real Quantum Mechanics
Real commutant: M = B(H), PM(H) = P(H), UZ = {Ā±I}
P(H) ā {Ug | g ā P}ā²ā² ā B(H)ā B(H) = {Ug | g ā P}ā²ā²
Main Result
Let Ugt = etP0 be the time-displacement and P0 = J0|P0| (PD)where Jā0 = āJ0 is a partial isometry. Then it holds
J0J0 = āI and J0 ā {Ug , g ā P}ā²(= B(H)ā²
)CONTRADICTION!
B(H)ā² = RI =ā āJ0 = Jā0 = (aI )ā = aI = J0 =ā J0 = 0
![Page 16: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/16.jpg)
Real Quantum Mechanics: Conclusions
Quaternionic commutant: M = B(HJK ), UZ = {Ā±I}
ā Diļæ½erent 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 Āµ = tr(M Ā·)
Conclusions
we recover the standard complex theory!
the complex structure arises from PoincarƩ invariance
![Page 17: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/17.jpg)
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 algebraMā B(H)3 see what happens...
OBSTRUCTION
What is a quaternionic von Neumann algebra?
Current situation
1 we chose a real subalgebraMā B(H) such thatMā²ā² =M2 mimicked the real theory
3 three mutually exclusive possibilities came out.
4 the extremal possibilities lead to a contradiction
![Page 18: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/18.jpg)
Thank you for the attention!
![Page 19: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/19.jpg)
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 Scientiļæ½c (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)
![Page 20: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/20.jpg)
Appendix: quaternionic commutant case
Quaternionic commutant M = B(HJK ), P(M) = P(HJK)
1 P(HJK ) ā {Ug | g ā P}ā²ā²
2 g 7ā Ug is quaternionic irreducible on HJK
Take J āMā² (or K ) and deļæ½ne HJ the usual way. 1) and 2) imply
Theorem
The map g 7ā Ug is a irreducible strongly-continuous faithfulunitary representation on HJ
![Page 21: Quantum theory in real Hilbert space: How the complex](https://reader033.vdocuments.us/reader033/viewer/2022041609/6252ea9099547049d0402713/html5/thumbnails/21.jpg)
Appendix: quaternionic commutant case
Let us work on HJ : Take the time-translation t 7ā gt
Stone Theorem Ugt = etP0 with Pā0 = āP0
Polar decomposition: P0 = J0|P0|1 |P0| ā„ 0 and |P0|ā = |P0|2 Jā
0= āJ0 and J0 is a partial isometry
Naive result (complex version)
It holds J0 = Ā±iI = Ā±J
Let us go back to H: it still holds Ugt = etP0 and P0 = J0|P0|
Properties of Polar Decomposition
K ā {Ug |g ā P}ā² ā KetP0 = etP0K ā KP0 = P0K ā KJ0 = J0K
IMPOSSIBLE! because JK = āKJ