We prove that the quadratic second quantization of an operator p on
$L^2(\mathbb{R}^d)\cap L^\infty (\mathbb{R}^d)$ is an orthogonal projection on
the quadratic Fock space if and only if p =MI, where MI is a multiplication
operator by a characteristic function I.Comment: 1

Dhahri, Ameur

English

arXiv

We prove that the quadratic second quantization of an operator p on $L^2(\mathbb{R}^d)\cap L^\infty (\mathbb{R}^d)$ is an orthogonal projection on the quadratic Fock space if and only if p =MI, where MI is a multiplication operator by a characteristic function I.ou

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On the quadratic Fock functor

We prove that the quadratic second quantization of an operator p on L-2(R-d) boolean AND L-infinity(R-d) is an orthogonal projection on the quadratic Fock space if and only if p is a multiplication operator by a characteristic function chi(I), I subset of R-d

AMEUR DHAHRI

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July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012Infinite Dimensional Analysis, Quantum Probabilityand Related TopicsVol. 15, No. 2 (2012) 1250012 (8 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219025712500129ON THE QUADRATIC FOCK FUNCTORAMEUR DHAHRICentro de Análisis Estocástico y Aplicaciones,Pontificia Universidad Católica de Chile, Chileadhahri@puc.clReceived 4 November 2010Published 27 July 2012Communicated by L. AccardiWe prove that the quadratic second quantization of an operator p on L2(Rd)∩L∞(Rd)is an orthogonal projection on the quadratic Fock space if and only if p is a multiplicationoperator by a characteristic function χI , I ⊂ Rd.Keywords: Quadratic Fock functor; quadratic second quantization; orthogonal projec-tion; interacting Fock space; multiplication operator.AMS Subject Classification: 461. IntroductionThe renormalized square of white noise (RSWN) was first introduced by Accardi–Lu–Volovich in Ref. 1. Later, Sniady introduced the free RSWN white noise(cf. Ref. 9). Subsequently, its relation with the Lévy processes on real Lie alge-bras was established in Ref. 7.Recently, in Refs. 3 and 4, the authors constructed the quadratic Fock functor.In particular, they characterized the operators on the one-particle Hilbert algebrawhose quadratic second quantization is isometric (respectively, unitary). A sufficientcondition for the contractivity of the quadratic second quantization was derived too.It is well known that the first-order second quantization Γ1(p) of an operator p,defined on the usual Fock space, is an orthogonal projection if and only if p isan orthogonal projection (see Ref. 8). In the present paper, it is shown that theset of orthogonal projections p, whose quadratic second quantization Γ2(p) is anorthogonal projection, is quite reduced. More precisely, we prove that Γ2(p) is anorthogonal projection if and only if p is a multiplication operator by a characteristicfunction χI , I ⊂ Rd.This paper is organized as follows. In Sec. 2, we recall some basic properties ofthe quadratic Fock functor. The main result is proved in Sec. 3.1250012-1Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012A. Dhahri2. Quadratic Fock FunctorThe algebra of the renormalized square of white noise (RSWN) with test functionHilbert algebraA := L2(R) ∩ L∞(R)is the ∗-Lie-algebra, with central element denoted one, generators B+f , Bh, Ng, f, g,h ∈ L2(Rd) ∩ L∞(Rd)}, involution(B+f )∗ = Bf , N∗f = Nf̄and commutation relations[Bf , B+g ] = 2c〈f, g〉 + 4Nf̄g, [Na, B+f ] = 2B+af ,[B+f , B+g ] = [Bf , Bg] = [Na, Na′ ] = 0,(1)for all a, a′, f , g ∈ L2(Rd) ∩ L∞(Rd).The Fock representation of the RSWN is characterized by a cyclic vector ΦsatisfyingBfΦ = NgΦ = 0for all f, g ∈ L2(Rd) ∩ L∞(Rd) (see Refs. 2 and 7).2.1. Quadratic Fock spaceIn this subsection, we recall some basic definitions and properties of the quadraticexponential vectors and the quadratic Fock space. We refer the interested reader toRefs. 3 and 4 for more details.The quadratic Fock space Γ2(L2(Rd) ∩ L∞(Rd)) is the closed linear span of{B+nf Φ, n ∈ N, f ∈ L2(Rd) ∩ L∞(Rd)}, where B+0f Φ = Φ, for all f ∈ L2(Rd) ∩L∞(Rd).Moreover, the scalar product between two n-particle vectors is given by thefollowing (see Ref. 3).Proposition 1. For all f, g ∈ L2(Rd) ∩ L∞(Rd), one has〈B+nf Φ, B+ng Φ〉 = cn−1∑k=022k+1n!(n − 1)!((n − k − 1)!)2 〈fk+1, gk+1〉· 〈B+(n−k−1)f Φ, B+(n−k−1)g Φ〉.The quadratic exponential vector of an element f ∈ L2(Rd) ∩ L∞(Rd), if itexists, is given byΨ(f) =∑n≥0B+nf Φn!,1250012-2Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012On the Quadratic Fock Functorwhere by definitionΨ(0) = B+0f Φ = Φ. (2)It is proved in Ref. 3 that the quadratic exponential vector Ψ(f) exists if ‖f‖∞ < 12 .Furthermore, the scalar product between two exponential vectors, Ψ(f) and Ψ(g),is given by〈Ψ(f), Ψ(g)〉 = e− c2RRdln(1−4f̄(s)g(s))ds. (3)Now, we refer to Ref. 3 for the proof of the following theorem.Theorem 1. The quadratic exponential vectors are linearly independents.Moreover, the set of quadratic exponential vectors is a total set in Γ2(L2(Rd) ∩L∞(Rd)).Finally, from Ref. 4 it follows that Γ2(L2(Rd) ∩L∞(Rd)) is an interacting Fockspace.Theorem 2. There is a natural isomorphism between the quadratic Fock spaceΓ2(L2(Rd)∩L∞(Rd)) and the interacting Fock space⊕∞n=0⊗nsymm{L2(Rd), 〈·, ·〉n},with scalar products :〈f⊗n, g⊗n〉n =∑i1+2i2+···+kik=n22n−1(n!)2ci1+···+iki1! · · · ik!2i2 · · · kik 〈f, g〉i1〈f2, g2〉i2 · · · 〈fk, gk〉ik .2.2. Quadratic second quantizationFor all linear operator T on L2(Rd)∩L∞(Rd), we define its quadratic second quan-tization, if it is well defined, byΓ2(T )Ψ(f) = Ψ(Tf),for all f ∈ L2(Rd)∩L∞(Rd). Note that in Ref. 4, the authors have proved thatΓ2(T ) is well defined if and only if T is a contraction on L2(Rd) ∩ L∞(Rd) withrespect to the norm ‖·‖∞. Moreover, they have given a characterization of operatorsT on L2(Rd) ∩ L∞(Rd) whose quadratic second quantization is isometric (respec-tively, unitary). The contractivity of Γ2(T ) was also investigated.3. Main ResultGiven a contraction p on L2(Rd) ∩ L∞(Rd) with respect to ‖ · ‖∞, the aim of thissection is to prove under which condition Γ2(p) is an orthogonal projection onΓ2(L2(Rd) ∩ L∞(Rd)).Lemma 1. Let p be a contraction on L2(Rd) ∩ L∞(Rd) with respect to the norm‖ · ‖∞. If Γ2(p) is an orthogonal projection on Γ2(L2(Rd) ∩ L∞(Rd)), then one has〈(p(f))n, (p(g))n〉 = 〈fn, (p(g))n〉 = 〈(p(f))n, gn〉, (4)for all f, g ∈ L2(Rd) ∩ L∞(Rd) and all n ≥ 1.1250012-3Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012A. DhahriProof. Let f, g ∈ L2(Rd)∩L∞(Rd), then there exists δ > 0 such that 2√δ‖f‖∞ <1 and 2√δ‖g‖∞ < 1. Note that Γ2(p) is an orthogonal projection on Γ2(L2(Rd) ∩L∞(Rd)), then for all ≤ t ≤ δ on has〈Ψ(√tp(f)), Ψ(√tg)〉 = 〈Ψ(√tf), Ψ(√tp(g))〉 = 〈Ψ(√tp(f)), Ψ(√tp(g))〉This implies thate−c2RRdln(1−4tp(f)(s)g(s))ds = e−c2RRdln(1−4tf̄(s)p(g)(s))ds= e−c2RRdln(1−4tp(f)(s)p(g)(s))ds.It follows that∫Rdln(1 − 4tp(f)(s)g(s))ds =∫Rdln(1 − 4tf̄(s)p(g)(s))ds=∫Rdln(1 − 4tp(f)(s)p(g)(s))ds. (5)Puths(t) = ln(1 − 4tp(f)(s)g(s)), h(t) =∫Rdln(1 − 4tp(f)(s)g(s))ds.Then, the nth derivative (in t) of hs(t) is given byh(n)s (t) = 22n(−1)n(n − 1)!(p(f)(s))n(g(s))n(1 − 4tp(f)(s)g(s))−n.Hence, uniformly for t ≤ δ, one has|h(n)s (t)| ≤22n(n − 1)!|p(f)(s)|n|g(s)|n(1 − 4‖p(f)‖∞‖g‖∞)n . (6)Thus, the left-hand side of (6) is integrable in s andh(n)(t) =∫Rdh(n)s (t)ds.This yieldsh(n)(0) = 22n(−1)n(n − 1)!〈(p(f))n, gn〉. (7)Finally, the result of the above lemma follows from identities (5) and (7).Note that the set of contractions p on L2(Rd) ∩L∞(Rd) with respect to ‖ · ‖∞,such that Γ2(p) is an orthogonal projection on Γ2(L2(Rd)∩L∞(Rd)), is reduced tothe following.Lemma 2. Let p be a contraction on L2(Rd) ∩ L∞(Rd) with respect to the norm‖ · ‖∞. If Γ2(p) is an orthogonal projection on Γ2(L2(Rd) ∩ L∞(Rd)), thenp(f̄) = p(f),for all f ∈ L2(Rd) ∩ L∞(Rd).1250012-4Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012On the Quadratic Fock FunctorProof. Let p be a contraction on L2(Rd)∩L∞(Rd) with respect to the norm ‖ · ‖∞such that Γ2(p) is an orthogonal projection on Γ2(L2(Rd) ∩ L∞(Rd)). Then, fromLemma 1 it is clear that p = p∗ = p2 (taking n = 1 in (4)). Moreover, for allf1, f2, g1, g2 ∈ L2(Rd) ∩ L∞(Rd), one has〈(p(f1 + f2))2, (g1 + g2)2〉 = 〈(p(f1 + f2))2, (p(g1 + g2))2〉.It follows that〈(p(f1))2, g21 + g22〉 + 〈(p(f2))2, g21 + g22〉 + 4〈p(f1)p(f2), g1g2〉= 〈(p(f1))2, (p(g1))2 + (p(g2))2〉 + 〈(p(f2))2, (p(g1))2 + (p(g2))2〉+ 4〈p(f1)p(f2), p(g1)p(g2)〉. (8)Then, using (4) and (8) to obtain〈p(f1)p(f2), g1g2〉 = 〈p(f1)p(f2), p(g1)p(g2)〉, (9)for all f1, f2, g1, g2 ∈ L2(Rd) ∩ L∞(Rd). Now, denote by Ma the multiplicationoperator by the function a ∈ L2(Rd) ∩ L∞(Rd). Then, identity (9) implies that〈Mp(f2)ḡ2p(f1), g1〉 = 〈Mp(f2)p(g2)p(f1), p(g1)〉,for all f1, f2, g1, g2 ∈ L2(Rd) ∩ L∞(Rd). This gives thatMp(f2)ḡ2p = pMp(f2)p(g2)p, (10)for all f2, g2 ∈ L2(Rd) ∩ L∞(Rd). Taking the adjoint in (10), one getspMp(f2)g2 = pMp(f2)p(g2)p. (11)Note that, for all f, g ∈ L2(Rd) ∩ L∞(Rd), identity (10) implies thatMp(f)ḡp = pMp(f)p(g)p. (12)Moreover, from (11), one haspMp(f)p(g)p = pMfp(g). (13)Therefore, identities (12) and (13) yieldMp(f)ḡp = pMfp(g), (14)for all f, g ∈ L2(Rd) ∩ L∞(Rd). Hence, for all f, g, h ∈ L2(Rd) ∩ L∞(Rd), identity(14) givesp(fp(g)h) = p(f)ḡp(h). (15)1250012-5Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012A. DhahriTaking f = g = p(h) in (14) to getM|p(h)|2p = pM|p(h)|2 . (16)Then, if we put f = h = p(g) in (15), one hasp(p(g)p(g)2) = p(|p(g)|2p(g)) = (p(g))2ḡ.But, from (16), one hasp(|p(g)|2p(g)) = (pM|p(g)|2)(p(g)) = M|p(g)|2p(p(g)) = |p(g)|2p(g).Hence, one obtains|p(g)|2p(g) = (p(g))2ḡ, (17)for all g ∈ L2(Rd)∩L∞(Rd). Now, let g be a real function in L2(Rd)∩L∞(Rd). So,the polar decomposition of p(g) is given byp(g) = |p(g)|eiθp(g) .Thus, identity (17) implies that|p(g)|3e−iθp(g) = |p(g)|2g.This proves that for all x ∈ Rd, θp(g)(x) = kxπ, kx ∈ Z. Therefore, p(g) is areal function. Now, taking f = f1 + if 2 ∈ L2(Rd) ∩ L∞(Rd), where f1, f2 are realfunctions on Rd. It is clear thatp(f̄) = p(f1 − if 2) = p(f1) + ip(f2) = p(f).This completes the proof of the above lemma.As a consequence of Lemmas 1 and 2 we prove the following theorem.Theorem 3. Let p be a contraction on L2(Rd)∩L∞(Rd) with respect to the norm‖ · ‖∞. Then, Γ2(p) is an orthogonal projection on Γ2(L2(Rd)∩L∞(Rd)) if and onlyif p = MχI , where MχI is a multiplication operator by a characteristic functionχI , I ⊂ Rd.Proof. Note that if p = MχI , I ⊂ Rd, then from identity (3) it is clear thate−c2RIln(1−4f̄(s)g(s))ds = 〈Ψ2(p(f)), Ψ2(g)〉= 〈Ψ2(f), Ψ2(p(g))〉= 〈Ψ2(p(f)), Ψ2(p(g))〉,for all f, g ∈ L2(Rd) ∩ L∞(Rd) such that ‖f‖∞ < 12 and ‖g‖∞ < 12 . Hence, Γ2(p)is an orthogonal projection on Γ2(L2(Rd) ∩ L∞(Rd)).1250012-6Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012On the Quadratic Fock FunctorNow, suppose that Γ2(p) is an orthogonal projection on Γ2(L2(Rd) ∩ L∞(Rd)).Then, Lemma 1 implies that〈(p(f))n, (p(g))n〉 = 〈fn, (p(g))n〉 = 〈(p(f))n, gn〉,for all f, g ∈ L2(Rd) ∩ L∞(Rd) and all n ≥ 1. In particular, if n = 2 one has〈(p(f1 + f̄2))2, g2〉 = 〈(f1 + f̄2)2, (p(g))2〉,for all f1, f2, g ∈ L2(Rd) ∩ L∞(Rd). This gives〈(p(f1))2, g2〉 + 2〈p(f1)p(f̄2), g2〉 + 〈(p(f̄2))2, g2〉= 〈f21 , (p(g))2〉 + 2〈f1f̄2, (p(g))2〉 + 〈(f̄2)2, (p(g))2〉. (18)Using identity (18) and Lemma 1 to get〈p(f1)p(f̄2), g2〉 = 〈f1f̄2, (p(g))2〉.This yields ∫Rdf̄1(x)f2(x)(p(g))2(x)dx =∫Rdp(f1)(x)p(f2)(x)g2(x)dx. (19)But, from Lemma 2, one has p(f) = p(f̄), for all f ∈ L2(Rd) ∩ L∞(Rd). Then,identity (19) implies that〈f1, M(p(g))2f2〉 = 〈f1, (pMg2p)f2〉,for all f, g ∈ L2(Rd) ∩ L∞(Rd). Hence, one obtainsM(p(g))2 = pMg2p. (20)In particular, for g = χI where I ⊂ Rd, one hasM(p(χI ))2 = pMχI p. (21)If I tends to Rd, the operator MχI converges to id (identity of L2(Rd) ∩ L∞(Rd))for the strong topology. From (21), it follows thatp(f) = p2(f) = limI↑RdM(p(χI))2f,for all f ∈ L2(Rd)∩L∞(Rd). But, the set of multiplication operators is a closed setfor the strong topology. This proves that p = Ma, where a ∈ L∞(Rd). Note thatp = p2 is a positive operator. This implies that a is a positive function. Moreover,one has pn = p for all n ∈ N∗. This gives Man = Ma for all n ∈ N∗. It followsthat an = a for all n ∈ N∗. Therefore, the operator a is necessarily a characteristicfunction on Rd.AcknowledgmentsI gratefully acknowledge stimulating discussions with Eric Ricard. I would likealso to thank Rolando Rebolledo for his hospitality, for reading the paper and1250012-7Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.July 19, 2012 9:5 WSPC/S0219-0257 102-IDAQPRT 1250012A. Dhahrifor interesting comments. Research partially supported by PBCT-ADI 13 Grant“Laboratorio de Análisis y Estocástico”, Chile.References1. L. Accardi, Y. G. Lu and I. V. Volovich, White noise approach to classical and quantumstochastic calculi, Centro Vito Volterra, Università di Roma “Tor Vergata,” preprint375 (1999).2. L. Accardi, G. Amosov and U. Franz, Second quantization automorphisms of the renor-malized square of white noise (RSWN) algebra, Infin. Dimens. Anal. Quantum Probab.Relat. Top. 7 (2004) 183–194.3. L. Accardi and A. Dhahri, Quadratic exponential vectors, J. Math. Phys. 50 (2009)122103.4. L. Accardi and A. Dhahri, The quadratic Fock functor, J. Math. Phys. 51 (2010)022105.5. L. Accardi, A. Dhahri and M. Skeide, Extension of quadratic exponential vectors, Pro-ceedings of the 29th Conference on Quantum Probability and Related Topics, Vol. XXV,pp. 262–266.6. L. Accardi and M. Skeide, On the relation of the square of white noise and the finitedifference algebra, IDAQP Infin. Dimens. Analy., Quantum Probab. Relat Top. 3 (2000)185–189, Volterra Preprint No. 386 (1999).7. L. Accardi, U. Franz and M. Skeide, Renormalized squares of white noise and othernon-Gaussian noises as Lévy processes on real Lie algebras, Commun. Math. Phys. 228(2002) 123–150.8. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (BirkhäuserVerlag, Basel, Boston, Berlin).9. P. Sniady, Quadratic bosonic and free white noise, Commun. Math. Phys. 3 (2002)615–628.1250012-8Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2012.15. Downloaded from www.worldscientific.comby Dr ameur dhahri on 01/13/15. For personal use only.

ON THE QUADRATIC FOCK FUNCTOR

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On the quadratic Fock functor

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