Boson-Fermion unification implemented by Wick calculus 4 0 John Gough 0 2 Department of Computing & Mathematics n Nottingham-Trent University, Burton Street, a Nottingham NG1 4BU, United Kingdom. J [email protected] 5 1 2 Abstract v 4 We construct a transformation between Bose Fock space Γ+(h) and 2 FermiFockspaceΓ−(h)thatissuper-symmetricinthesensethatitcon- 0 vertsBoson fields intoFermi fieldsovera fixed one-particle space h. The 9 transformation the spectral splitting of h into a continuous direct inte- 0 03 gFroaclkosfpiancteesrnovaelrspLa2ce(skω(k)ωω),ωR.+W,deωprtehsaetntisaatnhaetourryalogfeinnetreaglriazatitoionnoonftthhee theory of quantum stochastic calculus and which we refer to as a Wick / (cid:0) (cid:1) h calculus. p - Keywords: second quantization, quantum stochastics, supersymmetry. h t a 1 Introduction m : v The formalism of second quantization is fundamental to modern physics [1] Xi and admits a natural functional calculus. A very specialized version of this is quantum stochastic calculus. Quantum stochastic processes, parameterized by r a time t, are families of operators on Fock space over Hilbert spaces of the type L2(k,R ,dt). Thatis, kis a fixedHilbert space,termedthe internal space, and + φ L2(k,R ,dt) is square-integrable k valued function. As is well-known, we + ha∈ve the natural isomorphism L2(k,R ,dt) = k L2(R ,dt). The stochastic + ∼ ⊗ + calculus has been developed in the Boson case [2], generalizing the classical Itoˆ theory, and Fermion setting [3], generalizing the Clifford-Itˆo theory [4]. The Bose and Fermi theories have been unified by means of a continuous version of the Jordan-Wigner transformation [5]. Here we wish to consider second quantizations of Hilbert spaces of the type L2((k ) ,R ,dω) where now we work with families (k ) of internal spaces ω ω + ω ω indexed by parameter(interpreted here as frequency) ω > 0. The study of Wiener-Itoˆ integrals (in the time domain) on Fock spaces over direct integral Hilbert spaces was first considered by Sunder [6]. 1 Our motivation comes from modelling physical quantum reservoirs. In such cases, the k arise as the mass shell Hilbert spaces for a fixed energy ω. ω An infinitely extended quantum reservoir can be considered as the second quantization of particle having one-particle space h and having one-particle HamiltonianH 0whichisafixedself-adjointoperatorH onh. Specifically,we ≥ takeH tohaveabsolutelycontinuousspectrum. Therethenexistsanorthogonal projection valued measure Π[.] concentrated on [0, ) such that ∞ H ωΠ[dω]. (1.1) ≡ Z[0,∞) Given Ω>0 we consider the subspace h such that Ω D ⊂ +∞ dt f exp i(H Ω)t g < (1.2) |h | { − } i| ∞ Z−∞ whenever f,g . On this domain we define the sesquilinear form Ω ∈D +∞ (g f) := f exp i(H Ω)t g dt (1.3) | Ω h | { − } i Z−∞ and it is convenient to consider the Hilbert space k obtained by factoring out Ω from the null elements = f :(f f) =0 and taking the Hilbert space DΩ NΩ { | Ω } completion with respect to the (..) -norm. Formally we have | Ω 2π f Π[dω] g (g f) dω (1.4) h | i≡ | ω We then obtain the continuous direct integral decomposition ⊕ h∼= dω kω. (1.5) Z[0,∞) Foreachω 0,k is a Hilbertspace withinner product (..) andwemay con- ≥ ω | ω siderhtoconsistsofallvectorsφ=(φ ) ,whereφ k and (φ φ ) ω ω≥0 ω ∈ ω [0,∞) ω| ω ω <+ . The inner product on h may be represented by ∞ R φψ = dω (φ ψ ) . (1.6) h | i ω| ω ω Z[0,∞) We may write h as L2((k ) ,R ,dω). It is natural,in the light ofthe develop- ω ω + ment ofquantum stochastic calculus,to considerthe space L2 ((k ) ,R ,dω) loc ω ω + of locally square-integrable objects φ = (φ ) where now we only require ω ω dω (φ φ ) <+ for any compact Borel subset B. B ω| ω ω ∞ Example: Thebasicsituationwehaveinmind[9]isareservoirparticlemoving R in ν-dimensions and having the spectrum of elementary excitations ω = ω(k) where k= (k ,...,k ) are the momenta coordinate. We take h =L2(Rν,dνk) 1 ν and (Hf)(k) = ω(k)f(k). We shall take it that the spectrum foliates the momentum space into the mass shells M := k:ω(k)=ω and that the ω { } 2 Lesbegue measure on Rν can be decomposed locally as dνk = dω dσ where ω × ω = ω(k) and dσ is surface measure on M . (For instance, if H corresponds ω ω to ∆ in the position representationthen ω(k) is radialcoordinate and dσ ω − will be surface measure on the sphere of radius √ω.) The individual Hilbert p spaces are k =L2(M ,dσ ). ω ω ω Note that we can deal with quasi-free gauge-invariantstates on Γ (h) pro- ± vided only that the covariance matrix Q commutes with the filtration of h in- ducedbyH. SincewehaveQ=cothβH−µ (Bose)andQ=tanhβH−µ (Fermi) 2 2 for the free particle Gibbs states, this construction arises naturally. 2 Mathematical Notations and Preliminaries LetΓ(h):= ∞ h⊗n be the “Full”Fock spaceoverafixedcomplex separable n=0 Hilbert space h. The (anti)-symmetrization operators Π are define through ± linear extensLion of the relations Π f f := 1 ( )σ f ± 1 ⊗ ··· ⊗ n n! σ∈Sn ± σ(1) ⊗ f , with f h, S denotes the permutation group on 1,...,n and σ(n) j n (···1⊗)σ is the parity∈of the permutation σ. The Bose FoPck spa{ce Γ (h}) and + − the Fermi Fock space Γ (h) having h as one-particle space are then defined as − the subspaces Γ (h) := Π Γ(h). As usual, we distinguish the Fock vacuum ± ± Φ=(1,0,0...), though we shall write Φ for emphasis. ± Let g h, U unitary and H self-adjoint on h. We define the following ∈ operators on the Full Fock space A+(h) f f : =√n+1h f f ; 1 n 1 n ⊗···⊗ ⊗ ⊗···⊗ 1 A−(h) f f : = hf f f ; 1⊗···⊗ n √n h | 1i 2⊗···⊗ n Γ(U) f f : =(Uf ) (Uf ); 1 n 1 n ⊗···⊗ ⊗···⊗ γ(H) f f : = f (Hf ) f . (2.1) 1 n 1 j n ⊗···⊗ ⊗···⊗ ⊗···⊗ j X Bose creation and annihilation fields are then defined on Γ (h) as + B±(h):=Π A±(h) Π (2.2) + + while Fermi creation and annihilation fields are defined on Γ (h) as + F±(h):=Π A±(h) Π . (2.3) − − Using the traditional conventions [A,B]=AB BA and A,B =AB+BA, − { } we have the canonical (anti)-commutation relations B−(f),B+(g) = f g ; F−(f),F+(g) = f g . (2.4) h | i h | i Second quan(cid:2)tization operato(cid:3)rs are define(cid:8)d as Γ (U) :=(cid:9)Π Γ(U) Π and dif- ± ± ± ferentialsecondquantizationoperatorsasγ (U):=Π γ(U) Π . Wehavethe ± ± ± relation exp itγ (H) =Γ eitH . (2.5) ± ± (cid:8) (cid:9) (cid:0) (cid:1) 3 More generally we may take the argument of the differential second quanti- zations to be bounded: for the rank-one operator H = f g described in | ih | standard Dirac bra-ket notation, we have γ (f g ) B+(f)B−(g) and + | ih | ≡ γ (f g ) F+(f)F−(g).The following relations will be useful − | ih | ≡ Γ (U) B±(φ) Γ U† =B±(Uφ); Γ (U) F±(φ) Γ U† =F±(Uφ) + + − − (2.6) (cid:0) (cid:1) (cid:0) (cid:1) In the Bose case, the exponential vector map ε : h Γ (h) is introduced + 7→ as 1 ε(f)= ∞ f⊗n (2.7) ⊕n=0√n! with f⊗n the n-fold tensor product of f with itself. The Fock vacuum is, in particular, given by Φ=ε(0). The set ε(h) is total in Γ (h) and we note that + ε(f) ε(g) = exp f g . The next result is the basis for the development of a h | i h | i calculus of second quantized fields, see [7] for proofs. Lemma (2.1): The operations of Bose or Fermi second quantization have the natural functorial property Γ±(h1⊕h2)∼=Γ±(h1)⊗Γ±(h2) . (2.8) 3 Spectral Processes and Wick Calculus Let h be a separable complex Hilbert space admitting the continuous direct integral decomposition ⊕ h= dω k . (3.1) ω Z[0,∞) That is, for each ω 0, k is a Hilbert space with inner product (..) and we ≥ ω | ω have that h consists of all vectorsφ=(φ ) , where φ k and we have the ω ω≥0 ω ∈ ω inner product on h φψ = dω (φ ψ ) . (3.2) h | i ω| ω ω Z[0,∞) ⊕ For 0 Ω <Ω , let k = dω k , then in these notations ≤ 1 2 [Ω1,Ω2] [Ω1,Ω2] ω R h=k k (3.3) ∼ [0,Ω]⊕ (Ω,∞) for each Ω > 0. This leads to the following continuous tensor product decom- position for Fock space Γ (h)=Γ k Γ k . (3.4) + ∼ + [0,Ω] ⊗ + (Ω,∞) (cid:0) (cid:1) (cid:0) (cid:1) We define an absolutely continuous, orthogonal projection valued measure Π on [0, ) by ∞ (Π φ) :=1 (ω)φ (3.5) A ω A ω 4 where A is any Borel set and 1 its characteristic function. A Next of all, fix an initial Hilbert space H and set 0 H:=H Γ (h); H :=H Γ k ; H :=Γ k . (3.6) 0 + Ω] 0 + [0,Ω] (Ω + (Ω,∞) ⊗ ⊗ (cid:0) (cid:1) (cid:0) (cid:1) Definition (3.1): A family (X ) of operators on H is said to be spectrally- Ω Ω≥0 adapted if, for each Ω > 0, the operator X is the algebraic ampliation to Ω H ε k ε k of an operator on H with domain H ε k . We 0 [0,Ω) (Ω,∞) Ω] 0 [0,Ω) ⊗ ⊗ ⊗ also de(cid:0)mand(cid:1)the(cid:0)existenc(cid:1)e of an adjoint process XΩ† having th(cid:0)e sam(cid:1)e am- Ω≥0 pliation structure. (Here denotes the algebraic(cid:16)ten(cid:17)sor product.) ⊗ Definition(3.2): Let φ h, wedefine the(Bosonic) creation and annihilation ∈ spectral processes on Γ (h) to be + B±(Ω):=B± Π φ (3.7) φ [0,Ω] and the conservation spectral process to be(cid:0) (cid:1) Λ(Ω):=γ Π (3.8) + [0,Ω] for each Ω 0. (cid:0) (cid:1) ≥ The operators B±(Ω), Λ(Ω) are spectrally-adapted in this sense. φ Let (X (Ω)) be piecewise constant, spectrally-adapted processes for jk Ω≥0 j,k 0,1 . The Wick integral ∈{ } X = X Λ(dω)+X B+(dω)+X B−(dω)+X dω Ω 11⊗ 10⊗ φ 01⊗ ψ 00⊗ Z[0,Ω](cid:16) (cid:17) (3.9) is defined in such a way that u ε(f) Xv ε(g) is interpreted as h ⊗ | ⊗ i Ω u ε(f) (X (f g ) +X (f φ ) +X (ψ g ) +X ) v ε(g) dω h ⊗ | 11 ω| ω ω 10 ω| ω ω 01 ω| ω ω 00 ⊗ i Z0 (3.10) for all u,v H and f,g h. Formally we write this as X X(dω). ∈ 0 ∈ Ω ≡ [0,Ω] Similarly, we set X† = (X Λ(dω) +X B−(dω) +X B+(dω) Ω [0,Ω] 11 ⊗ 10⊗ φ 0R1⊗ ψ +X dω). 00⊗ R Lemma (3.3): Let X be the stochastic integral with piecewise constant, spec- Ω trally adapted coefficients as above, then Ω XΩu ε(f) 2 dω exp Ω ω+3 (fω′ fω′)dω′ k ⊗ k ≤Z[0,Ω] ( − Zω | ) 3(f f ) X (ω) u ε(f) 2+3(φ φ ) X (ω) u ε(f) 2 × ω| ω ωk 11 ⊗ k ω| ω ωk 10 ⊗ k h +(ψ ψ ) X (ω) u ε(f) 2+ X (ω) u ε(f) 2 . (3.11) ω| ω ωk 01 ⊗ k k 00 ⊗ k i 5 Proof. This is a generic type of estimate in quantum stochastic calculus and in our case is a straightforward adaptation of section 2 of [8] and we omit it. Let (X (Ω)) be spectrally adapted processes that are weakly measur- jk Ω≥0 able and satisfy the following locally square-integrability conditions (for arbi- trary u H , f h) 0 ∈ ∈ dω (f f ) X (ω) u ε(f) 2 < ; ω| ω ωk 11 ⊗ k ∞ Z[0,Ω] dω X (ω) u ε(f) 2 < , otherwise. jk k ⊗ k ∞ Z[0,Ω] Then the Wick integral X = X Λ(dω)+X B+(dω)+X B−(dω)+X dω Ω 11⊗ 10⊗ φ 01⊗ ψ 00⊗ Z[0,Ω](cid:16) (cid:17) exists and is well-defined. It can be understood as the limit of an approx- imating sequence X(n) , each one constructed using piecewise continu- Ω Ω≥0 ous coefficients: th(cid:16)e app(cid:17)roximating coefficients X(n) should be chosen so that j 2 dω X X(n) 0 and the limit process will be independent of [0,Ω] jk− jk → the appro(cid:13)x(cid:16)imating sequ(cid:17)en(cid:13)ce used. R (cid:13) (cid:13) Itisus(cid:13)efultousethed(cid:13)ifferentialnotationXΩ ≡ [0,Ω]X(dω)withX(dω)= X Λ(dω) +X B+(dω) +X B−(dω) +X dω. We can readily 11 ⊗ 10 ⊗ φ 01 ⊗ ψ R 00 ⊗ obtain the integral relation Ω Ω u ε(f) X Y X Y X Y (dω) X(dω)Y v ε(g) Ω Ω 0 0 ω ω * ⊗ | " − −Z0 −Z0 # ⊗ + = u ε(f) [X Y (f g ) +X Y (f φ ) ] h ⊗ | 11 11 ω| ω ω 11 10 ω| ω ω Z[0,Ω] +X Y (ψ g ) +X Y ] v ε(g) dω (3.12) 01 11 ω| ω ω 01 10 ⊗ i Here we encounter a familiar problem from quantum mechanics: the product ofWickorderedexpressionsis notimmediately Wick ordered. The non-Leibniz term in (3.12) is the result of putting to Wick order, in quantum stochastic calculus it would be called the Itoˆ correction, and for bounded coefficients we have the Itoˆ product formula (XY)(dω)=X(dω)Y (ω)+X(ω)Y (dω)+X(dω)Y (dω) (3.13) whichisevaluatedbytherulethatthefundamentaldifferentialsΛ(dω),B±(dω) φ anddωcommutewithspectrallyadaptedprocessesandbythequantumspectral 6 Itoˆ table: Λ(dω) B+(dω) B−(dω) dω φ φ Λ(dω) Λ(dω) B+(dω) 0 0 φ B+(dω) 0 0 0 0 (3.14) ψ B−(dω) B−(dω) (ψ φ ) dω 0 0 ψ ψ ω| ω ω dω 0 0 0 0 4 Super-symmetric Spectral Transformations Definition (4.1): We define the spectral parity processes to be J :=Γ Π +Π . (4.1) Ω + [0,Ω] (Ω,∞) − (cid:0) (cid:1) With respect to the decomposition (3.4) we have J ( 1)Λ(Ω) 1 . Ω (Ω ≡ − ⊗ Lemma(4.2): Theprocess (J ) isaunitary,self-adjoint,frequency-adapted Ω Ω≥0 process satisfying the properties 1. [Jω,Jω′]=0; 2. J Φ =Φ ; ω + + 3. dJ = 2J Λ(dω),J =1. ω ω 0 − ⊗ Proof. Property 1 follows is immediate from the observation that JωJω′ = Γ Π Π +Π where a = ω ω′ and b = ω ω′. Property 2 is + [0,a) [a,b] (b,∞) − ∧ ∨ evident from the fact that Φ =ε(0). + (cid:0) (cid:1) Next from the rule Λ(dω)Λ(dω)=Λ(dω), we have df(Λ(ω))=[f(Λ(Ω)+1) f(Λ(ω))] Λ(dω) − ⊗ for analytic functions f. Setting f(λ)=exp iπλ gives property 3. { } Definition(4.3): Let φ h, wedefinetheFermioniccreation andannihilation ∈ spectral processes to be F±(Ω):= J B±(dω). (4.2) φ ω ⊗ φ Z[0,Ω] Lemma (4.4): For each Ω 0 the Fermionic processes F±(Ω) anti-commute ≥ φ with the parity operator J . Ω Proof. Wefirstnotethattheexponentialvectorsareastabledomainforthe parity operator and the Bosonic, and hence Fermionic, processes. In particular 7 we have ε(f) J ,F−(Ω) ε(g) = ε(f) J F−(Ω) ε(g) + ε(f) F−(Ω)J ε(g) | Ω φ | Ω φ | φ Ω D n o E D E D E = dω (φ g ) ε Π f +Π f ε Π g+Π g ω| ω ω − [0,Ω] (Ω,∞) | − [0,ω] (ω,∞) Z[0,Ω] (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11) dω (φ g ) ε(f) ε Π g+Π g − ω| ω ω | − [0,Ω] (Ω,∞) Z[0,Ω] (cid:10) (cid:0) (cid:1)(cid:11) = dω (φ g ) ε(f) (J J J J ) ε(g) =0 ω| ω ω h | Ω ω− ω Ω i Z[0,Ω] andsowededucethat J ,F−(Ω) =0. Theproofoftherelation J ,F+(Ω) Ω φ Ω φ =0 is similar. n o n o Theorem (4.5): The Fermionic processes F±(Ω) are bounded and satisfy the φ canonical anti-commutation relations F−(Ω),F+(Ω) = dω (φ ψ ) ; (4.3a) φ ψ ω| ω ω n o Z[0,Ω] F−(Ω),F−(Ω) = 0= F+(Ω),F+(Ω) . (4.3b) φ ψ φ ψ n o n o Proof. Using quantum spectral calculus we have d F−(ω),F+(ω) = J ,F+(ω) B−(dω)+ F−(ω),J B+(dω) φ ψ ω ψ ⊗ φ φ ω ⊗ ψ n o +nJ2 B−(doω)B+(dω) n o ω⊗ φ ψ = (φ ψ ) dω ω| ω ω which can be integrated to obtain (4.3a). Next, since dF−(ω)=J B+(dω), the Itoˆ formula (3.13) gives φ ω⊗ φ 2 dF−(ω) = J F−(ω)+F−(ω)J B−(dω)=0 φ ω φ φ ω ⊗ φ (cid:16) (cid:17) (cid:16) (cid:17) again by virtue of the previous lemma. Since F−(0) = 0, the equation can be φ integrated to show that F−(ω)2 =0. Likewise, we can show that F+(ω)2 =0. φ φ We then obtain (4.3b) through polarization. Lemma (4.6): For each Ω 0 and φ ,...,φ h we have the identity ≥ 1 n ∈ F+ (Ω) F+ (Ω) Φ = ( 1)σ B+ (dω ) B+ (dω ) Φ φ1 ··· φn + − φσ(1) 1 ··· φσ(n) n + σX∈Sn 0<ω1<Z···ωn<Ω where ( 1)σ is the parity of the permutation σ S . n − ∈ 8 Proof. The proofwillbe one byinduction. The identity is triviallytrue for n=1.The rules of the spectral quantum stochastic calculus yields d F+ (ω) F+ (ω) = φ1 ··· φn+1 h i n+1 \ ( 1)n−k F+ (ω) F+ (ω) F+ (ω)J B+ (dω) − φ1 ··· φk ··· φn+1 ω ⊗ φk kX=1 h i therefore ε(f) F+ (Ω) F+ (Ω) Φ = | φ1 ··· φn + Ω n+1 D \ E dω ( 1)n−k ε(f) F+ (ω) F+ (ω) F+ (ω) Φ (f φ (ω)) ; Z0 Xk=1 − D | φ1 ··· φk ··· φn+1 +E ω| k ω therefore,if the identity holds up to the n-th order,then it holds for n+1 also. Corollary (4.7): For each Ω 0 and φ ,...,φ h we have the identity ≥ 1 n ∈ B+ (Ω) B+ (Ω) Φ = F+ (dω ) F+ (dω ) Φ . φ1 ··· φn + σX∈SnZ0<ω1<···ωn<Ω φσ(1) 1 ··· φσ(n) n + Theorem (4.8): There exists a unique unitary mapping Ξ : Γ (h) Γ (h) + − 7→ with the properties 1. ΞΦ =Φ ; + − 2. ΞF±(Ω) Ξ−1 =F± Π φ . φ [0,Ω] Proof. Firstofall,observethatanyn-parti(cid:0)clevecto(cid:1)rΠ f f canbe + 1 n ⊗···⊗ written as B+(f ) B+(f )Φ and so any vector in Γ (h) can be obtained 1 n + + ··· a sums of products of creators acting on the Fock vacuum. In other words Φ + is cyclic in Γ (h) for the B+(.) fields. Likewise, Φ is cyclic for the Fermionic + + creator fields F+(Ω) too. φ We next of all note that the Fermionic annihilator fields F±(Ω) annihilate φ Φ . We consider the mapping Ξ : Γ (h) Γ (h) defined by linear exten- + + − 7→ sion from Ξ(Φ ) = Φ and Ξ F+ (Ω)...F+ (Ω)Φ = Π Π φ + − φ1 φn − [0,Ω] 1 ⊗···⊗ Π[0,Ω]φn . (cid:16) (cid:17) (cid:0) (cid:1) It is readily seen that Ξ is a densely defined isometry and so extends to a (cid:0) (cid:1) unitary. Theorem (4.9): The unitary mapping Ξ : Γ (h) Γ (h) has the following + − 7→ covariance for the differential second quantizations Ξγ Π Ξ−1 =γ Π . (4.4) + [0,Ω] − [0,Ω] (cid:0) (cid:1) (cid:0) (cid:1) 9 Proof. Actually,(4.4)isthedifferentialversionoftherelationΞΓ (U ) Ξ−1 + t =γ (U )where U isthe unitaryexp itΠ =eitΠ +Π . This will − t t [0,Ω] [0,Ω] (Ω,∞) follow from the fact that Γ+(Ut) F±((cid:8)φ,Ω) Γ+(cid:9)Ut† =F±(Utφ,Ω) and this is established on the domain of exponential vector(cid:16)s: (cid:17) ε(f) Γ (U ) F±(Ω) Γ U† ε(g) = ε U†f F±(Ω) ε U†g | + t φ + t t | φ t D Ω (cid:16) (cid:17) E D (cid:16) (cid:17) (cid:16) (cid:17)E = dω ε U†f J B±(dω) ε U†g t | ω⊗ φ t Z0 D (cid:16) (cid:17) (cid:16) (cid:17)E = ε(f) F± (Ω) ε(g) | Utφ D E and we stress the importance of the fact that U is diagonal in our spectral t decomposition. References [1] Berezin,F.A.,MethodofSecondQuantization,AcademicPress,NewYork, (1996) [2] Hudson, R.L., Parthasarathy, K.R.: Quantum Itoˆ’s formula and stochastic evolutions. Commun.Math.Phys. 93, 301-323 (1984) [3] Applebaum, D.B., Hudson, R.L.: Fermion Itoˆ’s formula and stochastic evo- lutions. Commun.Math.Phys. 96, 473-496 (1984) [4] Barnett, C., Streater, R.F., Wilde, I., The Itoˆ-Clifford Integral I, J. Funct. 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