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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
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:Proposition2. — 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
3
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