Prépublication de l'Institut Fourier no

Stéphane Attal

Description

by Stéphane ATTAL Prépublication de l'Institut Fourier no 495 (2000) ?fi fl?fi ? !fi?_$? fi fi? ?_$fl%& fl ' ( $ Abstract. — We show how the toy Fock space can be embedded into the usual Fock space of quantum stochastic calculus. This embedding gives rise to a rigorous discrete approximation of the Fock space and its natural noise operators. We recover the quantum Ito table from the discrete one. We finally show that the quantum Brownian motion and Poisson process can be simultaneously approached by quantum Bernoulli random walks. I. The toy Fock space. Let us realise a Bernoulli random walk on its canonical space. Let? ) *0, 1+, and - be the?-field generated by finite cylinders. One denotes by ?n the coordinate mapping : ?n .?/ ) ?n, for all n01 . Let p 0 20, 1 3 and q ) 14p. Let µp be the probability measure on .?, - / which makes the sequence . ?n /n5 , to be a sequence of independent, identically distributed Bernoulli random variables with law p?1 6 q?0. Let 7 p 3 8 2 denote the expectation with respect to µp . We have 7 p 3?n 2 ) 7 p 3?2n 2 ) p.

• fock space


• ?t


• bernoulli random


• thus ameasurable function


• space ?


• now define


• finite subsets


• quantum stochastic

Sujets

• Fock space

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 8
• Langue: English

Extrait

byStéphane ATTAL
o
Prépublicationdel’InstitutFouriern 495(2000)
Abstract. — We show how the toy Fock space can be embedded into the usual
Fockspaceofquantumstochasticcalculus. Thisembeddinggivesrisetoarigorous discrete
approximationoftheFockspaceanditsnaturalnoiseoperators. WerecoverthequantumIto
tablefromthediscreteone. WefinallyshowthatthequantumBrownianmotionandPoisson
processcanbesimultaneouslyapproachedbyquantumBernoullirandomwalks.
I. ThetoyFockspace.
LetusrealiseaBernoullirandomwalkonitscanonicalspace. LetΩ 0,1 and
betheσ-fieldgeneratedbyfinitecylinders. Onedenotesbyν thecoordinatemapping:n
ν ω ω ,foralln .n n
Let p 0,1 and q 1 p. Letμ be the probability measure on Ω, whichp
makes the sequence ν to be a sequence of independent, identically distributedn n
Bernoulli random variables with law pδ qδ . Let denote the expectation with1 0 p
2respecttoμ . Wehave ν ν p. Thustherandomvariablesp p n p n
ν pn
X ,n
pq
satisfythefollowing:
i) theX areindependent,n
ii) X takesthevalue q/pwithprobabilityp and p/q withprobabilityq,n
2iii) X 0and X 1.p n p n
2Let Φ bethespaceL Ω, ,μ . Wedefineparticularelementsof Φ byp p p
X , inthesenseX ω 1forallω Ω
X X X ifA i ,...,i isanyfinitesubsetof .A i i 1 n1 n
Keywords: Fockspaces;creation,annihilationandconservationprocesses;Bernoullirandomwalks.
Math. classification: 81S25.
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Let denote the set of finite subsets of . From i) and iii) above it is clearf
X ; A isanorthonormalsetofvectorsof Φ .A f p
Proposition1. — Thefamily X ; A isanorthonormalbasisof Φ .A f p
Proof. — We just have to prove that X , A forms a total set in Φ . InA f p
thesamewayasfortheX ,defineA
ν
ν ν ν for A i ,...,i .A i i 1 n1 n
Itissufficienttoprovethattheset ν ; A istotal.A f
Thespace Ω, ,μ canbeidentifiedto 0,1 , 0,1 ,μ˜ forsomeprobabilityp p
measureμ˜ ,viathebase2decompositionofrealnumbers. Notethatp
1 if ω 1n
ν ω ωn n
0 if ω 0n
thusν ω . Consequentlyν ω . Nowlet f Φ besuchn ω 1 A ω 1 ω 1 pn i in1
n nthat f ,ν 0 for all A . Let I k2 , k 1 2 be a dyadic interval withA f
n nk < 2 . Thebase2decompositionofk2 isoftheform α ,...,α ,0,0,... . Thus1 n
f ω dμ˜ ω f ω dμ˜ ω .ω α ω αp n n p1 1
I 0,1
The function canbeclearly writtenasalinearcombination oftheν .ω α ω α A1 1 n n
Thus f dμ˜ 0.Theintegraloff vanishesoneverydyadicinterval,thusonallintervals.pI
Itisnoweasytoconcludethat f 0.
Wehaveprovedthateveryelement f Φ admitsauniquedecompositionp
f f A X 1A
A f
with
2 2f f A < . 2
A f
We can now define the toy Fock space. The toy Fock space is the separable Hilbert space
Φ whose orthonormal basis is chosen to be indexed by . Let X ; A bef A f
thisbasis. Asaconsequencethereisanaturalisomorphism between Φand Φ . Foreachp
p 0,1 ,thespace Φ iscalledthep-probabilisticinterpretationof Φ.p
Theonlypropertythatallowstomakeadifferencebetween Φand Φ ,orbetweenp
2different Φ ’s,istheproduct. Indeed,as Φ isaL spaceitadmitsanaturalproduct. Thep p
waywe havechosenthe basisof Φ makesthe productbeingdeterminedbythe value ofp
2X ,n .n
2
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0Proposition2. — In Φ wehavep
2X 1 c Xp nn
q pwherec .p pq
Proof.
1 12 2 2 2X ν p 2pν p 1 2p νn nn npq pq
21 p qp q p2p q p ν 1 νn n
pq qp qp
pc c ν pp p n
1 ν 1 c .n p
pq pq pq
The product that the p-probabilistic interpretation Φ determines in Φ is calledp
p-product.
On Φ,onedefinesthecreation,annihilationandconservationoperatorsby
a X XA A n n/An
a X XA A n n An
a X X .A A n An
Notethata ,a ,a arecompletelydeterminedbyn n n
i) theirvalueon andX ,n
ii) thefacttheyacttrivialyonX ,m n.m
Whatwemeanexactlyisthefollowing. IfH denotestheclosedsubspacegeneratedbyn
andX ,thenthereexistsanaturalisomorphismbetween Φand H (wherethecount-n n
n
able tensor product is understood to be associated to the stabilizing sequence un n
suchthatu foralln)givenbyn
X X X if A i ,...,i .A i i 1 n1 2
The definitions of a , a , a show that these operators act only on H and act as thenn n n
ηεidentity everywhere else. In particular a commutes with a for all n m and allmn
ηεε, η , ,0 . The compositions a a are given by the following discrete quantum Itonn
table.
ηεProposition3. — Theproductsa a aregivenbynn
η
an
ε a a an n nan
a 0 a 0n n
a I a 0 an n n
a a 0 a .n n n
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