Normally Ordered Disentanglement of Multi- Dimensional Schrödinger Algebra Exponentials
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1 Communications on Stochastic Analysis Volume Number 3 Article 5-9 Normally Ordered Disentanglement of Multi- Dimensional Schrödinger Algebra Exponentials Luigi Accardi Universitá di Roma Tor Vergata, Via di Torvergata, Roma, Italy, accardi@volterra.mat.uniroma.it Andreas Boukas Universitá di Roma Tor Vergata, Via di Torvergata, Roma, Italy, boukas.andreas@ac.eap.gr Follow this and additional works at: Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Accardi, Luigi and Boukas, Andreas 9) "Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials," Communications on Stochastic Analysis: Vol. : No. 3, Article 5. DOI:.339/cosa..3.5 Available at:
2 Communications on Stochastic Analysis Vol., No. 3 8) Serials Publications NORMALLY ORDERED DISENTANGLEMENT OF MULTI-DIMENSIONAL SCHRÖDINGER ALGEBRA EXPONENTIALS LUIGI ACCARDI AND ANDREAS BOUKAS Abstract. We derive a normally ordered disentanglement or splitting formula for exponentials of the infinite-dimensional Schrödinger Lie algebra generators. As an application we compute the vacuum characteristic function of a quantum random variable defined as a self-adjoint finite sum of Fock space operators, satisfying the multi-dimensional Schrödinger Lie algebra commutation relations.. Introduction The Lie algebra sl, C) is a building block of the Virasoro algebra, which is generated by a countable set of copies of sl, C) with non trivial commutation relations among them see [3]). At the moment a family of Fock type unitary representations of the Virasoro algebra is known. They are built on a countable tensor product of copies of the usual mode Boson Fock space where the elements of the Virasoro algebra are represented as infinite quadratic expressions of usual Boson creation and annihilation operators. Approximating these infinite quadratic expressions by finite sums naturally lea to the study of quadratic expressions in the first order) Boson creation and annihilation operators. It is known that these quadratic expressions are a Lie algebra, in fact this is the algebra of derivations on the Heisenberg Lie algebra described in section below. The structure of this algebra, that in the physics literature is known under the name of multi dimensional Schrödinger algebra, will be briefly recalled in section below. Thus the Virasoro algebra can be realized as a Lie sub algebra of an dimensional version of the Schrödinger algebra. In a series of papers [], [3], we have studied the problem of determining the vacuum distributions of the Virasoro algebra and, using a finite dimensional truncated version of the boson representation of this algebra, we have computed the vacuum characteristic functions of the Virasoro fiel in some very special cases [3]. In these cases we find a product of Gamma functions, but the product is nonhomogenous. Moreover, even for these simple fiel, the limit as N + of these characteristic functions does not exist. From the above discussion it follows that Received 8--; Communicated by the editors. Mathematics Subject Classification. Primary E3, 47C5, 8R, Secondary 46L6, 6B5. Key wor and phrases. Schrödinger algebra, Boson quadratic forms, disentanglement, splitting lemma, Fock space, quantum random variable, vacuum characteristic function. 83
3 84 LUIGI ACCARDI AND ANDREAS BOUKAS to realize the program of determining the vacuum distributions of the Virasoro fiel, one has to study exponentials of the skew adjoint elements of Schrodd), which can be interpreted as exponentials of quadratic forms in the first order Boson creators and annihilators. The study of vacuum expectations of exponentials of quadratic expressions in the boson Fock operators e iquadratic form of a i aj).) has a long history starting from Friedrichs [9] see also [5]), who first found a disentanglement formula that allows to deduce an explicit form for them. The operators in the exponent of the left hand side of.) are skew Hermitian elements of the homogeneous Schrödinger algebra Schrodd) discussed in section below and the expectations in.) are the Fourier transforms of the vacuum spectral measures of these operators. The above mentioned papers derive some expressions for these vacuum expectations, however the existing formulas are not explicit and using them it is practically impossible to find the explicit form of these characteristic functions with the exception of a few very special cases. For this reason, since Friedrichs s paper and book [9] several publications have been devoted to this problem, for example [7]. We mention in particular the paper [6] which is an important step towar the problem of determining the canonical forms of quadratic Boson expressions and contains a detailed bibliography on the physics literature on this issue. In the present paper we develop a brute force attack to the problem in which, rather than to look for explicit formulas which involve implicit quantities we look for explicit equations on explicit quantities, thus reducing the problem to finding explicit solutions of a system of Riccati type equations. We know from the previously mentioned results that a solution of this system always exists and the results of [6] may be interpreted in the sense that sometimes uniqueness may fail. We are convinced that any attempt to find really explicit forms for the characteristic functions.) will end up facing the problem of solving a system of Riccati type equations. As shown in the following text, the deduction of these equations is not an easy task. The problem of canonical forms now arises also for these equations as a preliminary step towar their solutions and will be considered in a future paper.. Definitions and Notation In this paper all algebras and Lie algebras will be on the complex numbers unless stated otherwise. By a set of generators of a Lie algebra we mean a linear basis of it. Throughout this paper we use the notation for the commutator of x and y. [x, y] := xy yx.. The mode Heisenberg algebra. The mode Heisenberg algebra, or Heis), is the 3 dimensional Lie algebra with generators {a, a, }, is a central
4 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 85 element, satisfying the commutation and duality relations [a, a ] := ; a ) = a, =, with all other commutators vanishing. Such a pair of generators, like a an a, will also be called a Boson pair. The universal enveloping algebra of Heis) is called the mode Full Oscillator Algebra and denoted Pa, a). It can be identified with the polynomial algebra in a, a and has a natural structure of Lie algebra with generators Bk n := a n a k + δ nk, ; n, k, N, K N,.) involution given by Bk n ) = Bn k, and commutation relations see []) In these notations we have [B n k, B N K ] = k N k N) B n+n k+k..) B = a ; B = a ; B = a ; B = a ; B = a a + ; B =. The mode oscillator algebra, or Osc), is the Lie algebra generated by and its commutation relations are: {a, a, a a +, } [B, B ] = ; [B, B ] = B ; [B, B ] = B... The d mode Heisenberg algebra. With each i N \ {} := N we associate a copy of the mode Heisenberg algebra, denoted Heis) i with generators {a i, a i} and with common central element. The corresponding Full Oscillator Algebra is denoted Pa, a) i. For d N {+ }, the d dimensional Heisenberg algebra denoted Heisd) is the Lie algebra with generators {a i, a i, : i N, i d} and additional relations induced by [a i, a j ] = δ i,j := δ i,j ; [a i, a j ] = [a i, a j ] = ; i, j N, i, j d. The corresponding polynomial algebra is denoted and its generators P d a i, a j) := {polynomials in a i, a j, i, j N, i, j d} B n i)b kj) = a i n a k j ; B i) = B j) := ; i, j N satisfy the commutation relations [B n i)b kj), B N I)B KJ)] =δ i,j knb n i)b N I)B k j)b KJ) In particular δ i,j nkb n i)b N I)B kj)b K J). [B n i)b kj), B N i)b Kj)] = δ i,j k N k N) B n+n i)b k+k j)
5 86 LUIGI ACCARDI AND ANDREAS BOUKAS and the commutation relations of the are Bk n i) := a n i k ai + δ nk, = Bn i)bki) + δ nk, ; n, k, N, K N [B n k i), B N K j)] = δ i,j k N k N) B n+n k+k i)..3. The mode Schrödinger algebra. The mode Schrödinger algebra, or Schrod), is the Lie sub algebra of Pa, a) generated by {a, a, a, a, a a +, }, where n, k, N, K {,, } with n + k and N + K. In the notation.) the generators of Schrod) take the form {B n k : n, k, N, K {,, }, n + k, N + K }..3) The commutation relations among the generators of Schrod) have the general form.) with n, k, N, K satisfying the constraints in.3). The concise formula.) for the commutation relations among the generators of Schrod), in particular the inclusion of [B, B] = 4 B, was the reason behind the introduction of the additive term in the definition of the generators.)..4. The d mode Schrödinger algebra. The d mode Schrödinger algebra, or Schrodd), is the Lie sub algebra of P d a i, a j) generated by {a h, a k, a i a j, a ia j, a h a k +, : h, k, i, j {,..., d}, i j} ={B h), B k), B i)b j), B i)b j), B h)b k) +, : h, k, i, j {,..., d}, i j}. The elements of Schrodd) consist of all quadratic expressions in the generators a i, a j of the Heisenberg algebra Heisd). Two Lie sub algebras of Schrodd) will be important: The diagonal sub algebra Schrodd) diag generated by {a h,a k, a i a i, a ia i, a h a h +, : h, k, i, j {,..., d}, i j} = {Bi), Bi), Bi), Bi), Bh), : i {,..., d}, i j} and the canonical sub algebra, or Schrodd) can, generated by {a h, a k, a i a j, a ia j, a h a h +, : h, k, i, j {,..., d}, i j} = {B h), B k), B i)b j), B i)b j), B h)b k) +, : h, k, i, j {,..., d}, i j}.
6 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS The Disentanglement problem. In Schrodd) we consider a finite sum of the form, where summation from to d is understood on repeated indices: c i, j)bi)b j) + c i, j)bi)b j) + c i, j) Bh)B k) + ) +c i, j)b i) + c i, j)b i), where c n k i, j) C and we wish to split the corresponding group element e c i,j)b i)b j)+c i,j)b i)b j)+c i,j)b h)b k)+ )+c i,j)b i)+c i,j)b i) as a product of normally ordered exponentials, i.e. of the form e c i)b i)+c i,j)b i)b j) e c i,j)b h)b k)+ ) e c i)b i)+c i,j)b i)b j). In the diagonal sub algebra Schrodd) diag we consider a finite sum of the form c i)bi) + c i)bi) + c i)bi) + c i)bi) + c i)bi) ), where c n k i) C and we wish to split the corresponding group element e N c i)b i)+c i)b i)+c i)b i)+c i)b i)+c i)b i)) as a product of normally ordered exponentials, meaning that exponentials containing creators a i are listed on the left and exponentials containing annihilators a i are listed on the right. That would be a normally ordered version of the splitting formula for the multi-dimensional Heisenberg algebra given in [7]. 3. The Case of Schrodd) diag Lemma 3.. Let a and a be a Boson pair and let f be an analytic function. Then afa a) =fa a + )a 3.) fa a)a =a fa a + ) 3.) [a, fa )] =f a ). 3.3) Proof. The proof is obtained from standard Heisenberg algebra commutation formulas see [8], Propositions.4. and..) for D = a, x = a and h =.
7 88 LUIGI ACCARDI AND ANDREAS BOUKAS Lemma 3.. For all λ C: [B, e λ B ] =4 λ B e λ B + 4 λ e λ B B, 3.4) [B, e λ B ] =λb e λ B, 3.5) [B, e λ B ] =λ B e λ B, 3.6) [B, e λ B ] = λ B e λ B, 3.7) e λ B B =e λ Be λ B, 3.8) [B, e λ B ] =λe λ B, 3.9) e λ B B =e λ Be λ B, 3.) Be λ B =e λ e λ B B, 3.) [e λ B, B ] =λ e λ B λb e λ B, 3.) [e λ B, B ] =4λ Be λ B 4λB e λ B, 3.3) [e λ B, B ] =λe λ B. 3.4) Proof. The proof of 3.4)-3.7) can be found in []. The proof of 3.8) is obtained from 3.) and of 3.9) from 3.3). The proof of 3.) is obtained from 3.) and of 3.) from 3.) by taking adjoints and then replacing λ by λ throughout what you find. To prove 3.) we have e λa a =e λa aa = [e λa, a] + ae λa ) a = λe λa + ae λa ) a = λe λa a + ae λa a = λ λe λa + ae λa ) + a λe λa + ae λa ) =λ e λ a λae λ a + a e λa. Equation 3.3) is obtained by computing e λ B B with the use of 3.4) and then computing e λ B B with the use of 3.7). Finally, 3.4) is the dual of 3.9). Lemma 3.3. For n, k, N, K {,, } with n + k and N + K let Ek n s) := e wn k j;s)bn k j), j=
8 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 89 where wk n j; s) R. Then, for each i {,,..., N}, [B i), E s)] = w i; s)b i)e s), 3.5) [B i), E s)] = w i; s)b i)e s), 3.6) E s)b i) = e w i;s) B i)e s), 3.7) [B i), E s)] = w i; s)e s), 3.8) [B i), E s)] = w i; s)b i)e s), 3.9) E s)b i) = e w i;s) B i)e s), 3.) [E s), B i)] = w i; s) ) E s) w i; s)b i)e s), 3.) [E s), B i)] = 4 w i; s) ) B i)e s) 4w i; s)b i)e s), 3.) [E s), B i)] = w i; s)e s), 3.3) E s)b i) = e w i;s) B i)e s). 3.4) Proof. To prove 3.5) we notice that E s)b i) =e w ;s)b ) e w i ;s)b i ) e w i;s)b i) B i) By Lemma 3. e w i+;s)b i+) e w N;s)B N). so B i) e w i;s)b i) = w i; s)b i) e w i;s)b i) + e w i;s) B i) B i), E s)b i) =e w ;s)b ) e w i ;s)b i ) B i) e w i;s)b i) wi; s)bi) e i)) w i;s)b e w i+;s)b i+) e w N;s)B N) =Bi)E s) wi; s)bi)e s), from which 3.5) follows immediately. Similarly, Es)B i) =e w ;s)b ) e w i ;s)b i ) e w i;s)b i) Bi) e w i+;s)b i+) e w N;s)B N), from which 3.6) follows with the use of 3.7) since e w i;s)b i) Bi) = Bi)e w i;s)b i) wi; s)bi)e w i;s)b i). The proof of 3.7) follows from Es)B i) =e w ;s)b ) e w i ;s)b i ) e w i;s)b i) Bi) e w i+;s)b i+) e w N;s)B N) and the fact that, by 3.8), e w i;s)b i) Bi) = e w i;s) Bi)e w i;s)b i).
9 9 LUIGI ACCARDI AND ANDREAS BOUKAS Similarly, 3.8) follows from Es)B i) =e w ;s)b ) e w i ;s)b i ) e w i;s)b i) Bi) e w i+;s)b i+) e w N;s)B N), and the fact that, by 3.9), e w i;s)b i) Bi) = Bi)e w i;s)b i) wi; s)e w i;s)b i). The proofs of 3.9)-3.4) are along the same lines. Theorem 3.4. For s R and N =,,... let Then F N := c i)bi) + c i)bi) + c i)bi) + c i)bi) + c i)bi) ). e s F N N =e w s) e w i;s)b i) e w i;s)b i) N e w i;s)b i) e w i;s)b i) e w i;s)b i), where, for each i {,,...}, w i; s) is the solution of the Riccati initial value problem and dw i; s) w i; s) =4c i) w i; s) =c i) c i)w i; s) 4c i) w i; s) ) = c i), w i; ) =, w i; s) =e 4c i)w i;t)+c w i; s) = w s) = Proof. Let w i; t) dt + c i) s, 3.5) e w i;t) dt, 3.6) i)) dt e t 4c i)w i;w)+c i)) dw c i) + c i)w i; t) ) dt, 3.7) c i) + c i)w i; t) ) e w i;t) dt, 3.8) c i) w i; t) ) N dt + c i) w i; t) dt. 3.9) Es) := e s N c i)b i)+c i)b i)+c i)b i)+c i)b i)+c i)b i)).
10 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 9 Differentiating with respect to s we find des) = c i)bi) + c i)bi) + c i)bi) + c i)bi) 3.3) +c i)b i) ) Es). Let Ek n s) := e wn k i;s)bn k i). Then, de n k s) = N dw n k ) i; s) Bk n i) Ek n s) 3.3) and by the right-hand side of the formula postulated in the statement of this theorem we have Es) = e w s) E s)e s)e s)e s)e s). 3.3) Applying the differentiation rule f f f n ) = f f f n + f f f n f f f n
11 9 LUIGI ACCARDI AND ANDREAS BOUKAS to 3.3) and using 3.3) and Lemma 3.3 we obtain des) N ) = dw s) dw Es) + i; s) B i) Es) 3.33) N ) dw + i; s) B i) e w s) Es)E s)e s)e s)e s) N ) + e w s) Es)E s) dwi; s) B i) Es)E s)e s) N ) + e w s) Es)E s)e s) dwi; s) B i) Es)E s) N ) + e w s) Es)E s)e s)e s) dwi; s) B i) Es) = dw s) Es) = dw s) Es) + + dwi; s) B i)es) + dwi; s) B i)es) dwi; s) e w s) E s)es)b i)e s)e s)e s) dwi; s) e w s) E s)e s)e s)b i)e s)e s) dwi; s) e w s) E s)e s)e s)e s)b i)e s) dw i; s) dwi; s) B i)es) + B i)es) dwi; s) w i; s)bi)es) dwi; s) B i)es) dwi; s) w i; s)bi)es)
12 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS dwi; s) e w s) B i)es) dwi; s) e w s) w i; s)bi)es) dwi; s) e w s) w i; s)es) dw i; s) w i; s) ) e w i;s) Es) dwi; s) w i; s)e w i;s) Bi)Es) dwi; s) w i; s)wi; s)e w i;s) Bi)Es) dwi; s) e w i;s) B i)es) dw i; s) = dw s) Es) + w i; s) ) e w i;s) B i)es) dwi; s) w i; s)e w i;s) Bi)Es) dwi; s) e w s) E s)es)b i)e s)e s)e s) dwi; s) e w s) E s)e s)e s)b i)e s)e s) dwi; s) e w s) E s)e s)e s)e s)b i)e s) dwi; s) B i)es) + dwi; s) B i)es)
13 94 LUIGI ACCARDI AND ANDREAS BOUKAS dw i; s) B i)es) dwi; s) w i; s)bi)es) dwi; s) e w s) B i)es) dwi; s) e w s) w i; s)bi)es) dwi; s) e w s) w i; s)es) dw i; s) w i; s) ) e w i;s) Es) dwi; s) w i; s)e w i;s) Bi)Es) dwi; s) w i; s)bi)es) dwi; s) w i; s)wi; s)e w i;s) Bi)Es) dwi; s) e w i;s) B i)es) dw i; s) w i; s) ) e w i;s) B i)es) dwi; s) w i; s)e w i;s) Bi)Es). Comparing 3.33) and 3.3) and equating, for each pair n, k), the coefficients of the Bk ne terms we find that the wn k s must satisfy the differential equations: dw s) = dw i; s) w i; s) ) e w i;s) + dw i; s) e w i;s) w i; s), 3.34)
14 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 95 and for each i =,,..., N c i) = dw i; s) dw i; s) w i; s) + 4 dw i; s) w i; s) ) e w i;s), 3.35) c i) = dw i; s) e w i;s), 3.36) c i) = dw i; s) dw i; s) + 4 dw i; s) w i; s)wi; s)e w i;s), c i) = dw i; s) c i) = dw i; s) wi; s) dw i; s) e w s) w i; s) 3.37) e w i;s) dw i; s) w i; s)e w i;s), 3.38) 4 dw i; s) w i; s)e w i;s). 3.39) We require that the wk n i; s) s satisfy the initial condition Solving equation 3.36) for dw i;s) dw i; s) and solving equation 3.39) for dw i;s) dw i; s) w n k i; ) =. we find = c i)e w i;s) 3.4) we find = 4c i)w i; s) + c i). 3.4) Substituting in 3.35) we find that w i; s) is the solution of the Riccati initial value problem dw i; s) and, by 3.4) and 3.4), c i)w i; s) 4c i) w i; s) ) = c i), w i; ) =, w i; s) =4c i) w i; s) =c i) Solving equation 3.38) for dw i;s) 3.4), we find which implies dw i; s) w i; s) = c i) w i; t) dt + c i) s, e w i;t) dt. and substituting, in what we get, dw i;s) = c i)e w i;s) + c i)w i; s)e w i;s), 3.4) e w i;t) dt + c i) w i; t)e w i;t) dt. by
15 96 LUIGI ACCARDI AND ANDREAS BOUKAS Substituting 3.4), 3.4) and 3.4) in 3.37) we find that w i; s) is the solution of the initial value problem dw i; s) so 4c i)w i; s) + c i) ) w i; s) = c i) + c i)w i; s), w i; ) =, wi; s) =e 4c i)w i;t)+c i)) dt Finally, 3.34) and 3.4) imply w s) = + = = + c i) c i) e t 4c i)w i;w)+c i)) dw c i) + c i)w i; t) ) dt. dw i; t) dt w i; t) ) e w i;t) dt dwi; t) e w i;t) w dt i; t) dt e w i;t) w i; t) ) e w i;t) dt c i)e w i;t) + c i)w i; t)e w i;t)) e w i;t) w i; t) dt w i; t) ) N dt + c i) w i; t) dt. Proposition 3.5. The solution of the Riccati initial value problem dw i; s) c i)w i; s) 4c i) w i; s) ) = c i), w i; ) =, 3.43) is where wi; c s) = i) Bi) coth Bi)s) c, 3.44) i) Bi) = c i) 4c i)c i), 3.45) provided that the coefficients of 3.43) are such that and 3.45) and 3.44) make sense, in which case w i; ) = lim s w i; s) =. Moreover, in the notation of
16 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 97 Theorem 3.4, w i; s) = log Bi) log Bi) cosh Bi)s) c i) sinh Bi)s) ), 3.46) w i; s) = c i)c i) coshbi)s) + c i)bi) sinhbi)s) c i)c i) c i) + Bi) 4c i)c i) coshbi)s)), 3.47) wi; s) = Bi)c Bi) cothbi)s) c i)bi) i) 3.48) ) ) c i) Bi) c i)c i) c i)c i)) cothbi)s) cschbi)s)), wi; s) = α 4 i) + α 5 i) cothbi)s) α i) + α i) cothbi)s)) 3.49) cschbi)s) c i)α 3 i) arctan c i) c i)c ) i) Bi) coth Bi)s c i) +α 3 i) arctan c i) ) ) c i)c i) Bi)c i) tanh Bi)s ) 4c i)c i) Bi)c i)3 tanh cothbi)s)), Bi)s ws) c = i) 4c α 6i) + α 7 i)s 3.5) i)c i)bi) +α 8 i) arctan 4c i)c i) ) c i) Bi) coth Bi)s +α 9 i) log Bi) coshbi)s) c i) sinhbi)s) )) c + i) Bi) cothbi)s) c i) α i) + α i)s )) + α i) arctan c i)c i) c i) Bi) tanh Bi)s ) Bi) c i) + Bi)c i) tanh Bi)s + α 3 i) cothbi)s) + α 4 i)s cothbi)s) + α 5 i) arctan c i)c ) i) cothbi)s) Bi) coth c i) Bi)s
17 98 LUIGI ACCARDI AND ANDREAS BOUKAS + α 6 i)cschbi)s) + α 7 i) log cothbi)s)) + α 8 i) log + cothbi)s)) + α 9 i) cothbi)s) log cothbi)s)) + α i) cothbi)s) log + cothbi)s)) + α i) log c i) Bi) cothbi)s) ) + α i) cothbi)s) log c i) Bi) cothbi)s) ) + α 3 i) log Bi) coshbi)s) c i) sinhbi)s) ) + α 4 i) cothbi)s) log Bi) coshbi)s) c i) sinhbi)s) ), where the formulas for the coefficients α k i), k =,..., 4, are given in the Appendix. Proof. For each i, a constant solution Ai) of the differential equation is Letting dw i; s) c i)w i; s) 4c i) w i; s) ) = c i), Ai) = c i) 4c i)c i) c i) 4c i). w i; s) := Ai) + ui; s), substituting in 3.43) we find that ui; s) satisfies the linear first order ODE so Since we find that dui; s) + c i) + 8c i)ai) ) ui; s) = 4c i), 4c ui; s) = i) c i) + + c 8c e c i)ai) w i; ) = = ui; ) = Ai), c = and, after simplifications, we obtain w i; s) = 4c i) c i) + 8c i)ai) Ai), i)+4c i)ai))s. c i) c i) 4c i)c i) coth c i) 4c i)c i) s ) c i). Substituting in 3.5)-3.9) we obtain 3.46)-3.5).
18 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 99 Remark 3.6. We could have proved Theorem 3.4 by noticing that, by the commutativity of the exponents for different values of i, = e s N c i)b i)+c i)b i)+c i)b i)+c i)b i)+c i)b i)) e s i)) c i)b i)+c i)b i)+c i)b i)+c i)b i)+c i)b, and then work separately on each copy of the Schrödinger algebra. Such work was done in [], but acting on the Fock vacuum vector Φ. The approach followed in the proof of Theorem 3.4 is a good, necessary, preparation for the non-diagonal case presented in the next section. 3.. The case d = and a single quadratic Hamiltonian. The splitting formula for Schrod) is given in the following Corollary to Theorem 3.4. Corollary 3.7. For s R e s c B +c B +c B +c B +c B ) =e w s) e w s)b e w s)b i) e w s)b e w s)b e w s)b, where w s) is the solution of the Riccati initial value problem and w s) =4c w s) =c dw s) c w s) 4c w s) ) = c, w ) =, w t) dt + c s, e w t) dt, w s) =e 4c w t)+c ) dt w s) = w s) =c c + c w t) ) e w t) dt, w t) ) dt + c wt) dt. Proof. The proof follows from Theorem 3.4 for N =. e t 4c w w)+c ) dw c + c w t) ) dt, Corollary 3.8. For s R e s c B +c B +c B ) = e w s)b e w s)b e w s)b, where w s) is the solution of the Riccati initial value problem dw s) c w s) 4c w s) ) = c, w ) =,
19 3 LUIGI ACCARDI AND ANDREAS BOUKAS and w s) =4c w s) =c w t) dt + c s, e w t) dt. Proof. The proof follows from Corollary 3.7 for c = c =, since then ws) = ws) = ws) =. Proposition 3.9. Quadratic Hamiltonian) Suppose that Φ is a Fock vacuum vector such that Φ = Φ, Φ = and aφ =. Then, for s R, Φ, e is c a ) +c a +c a) a Φ = B e ic s B cosh ibs) c sinh ibs) ), 3.5) where B = c 4c c. Proof. By Corollary 3.8 and Proposition 3.5, using the fact that B = a ), B = a, B = a a +, and B ) = B, we have e w s)b Φ = Φ, e w s)b Φ = e w s) Φ Φ, e is c a ) +c a +c a) a Φ = e isc e w is). Replacing w is) by 3.46) with is in place of s, we obtain 3.5). Remark 3.. Simplified versions of 3.5), for less general coefficients, can be found in [] and [4]. 4. The Disentanglement Formula in the General Non-diagonal Case We will use the notation µi) := wi, j; s)bj) ; νi) = wi, j; s)bj), and ξj) = E n k s) := G s) = j= j i j= j<i wi, j; s)bi) ; λi) := ξi) + νi), i>j e wn k j;s)bn k j) ; G s) = j= e w i,j;s)b i)b j) ; G s) = i,j= i>j e w i,j;s)b i)b j), i,j= i j e w i,j;s)b i)b j). i,j= i>j
20 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 3 We assume that and w i, i; s) =, i =,,... 4.) w i, j; s) = w i, j; s) =, i j. 4.) Lemma 4.. For i =,,... [B i), G s)] =µi)b i)g s) µi) G s), 4.3) [B i), G s)] =µi)g s), 4.4) [B i), G s)] =λi)b i)g s) ξi) + 3νi) )G s), 4.5) [B i), G s)] =λi)g s), 4.6) [B i), G s)] =λi)b i)g s). 4.7) Proof. To prove 4.3) we notice that G s)b i) = = e w I,j;s)B I)B j) Bi) I,j= I j N e w i,j;s)b i)b j) j= j i j= j i I= I i = e w i,j;s)b i)b j) Bi) By 3.) with λ = µi) we have j= j i e w I,i;s)B I)B i) I= I i N I,j= j i,i I j e w I,i;s)B I)B i) e w I,j;s)B I)B j) B i) I,j= j i,i I j e w i,j;s)b i)b j) Bi) = e B i) N j= w i,j;s)b j) j i Bi) = e µi) B i) B i) Similarly, to prove 4.4) we notice that e w I,j;s)B I)B j). = B i)e µi) B i) + µi) e µi) B i) µi)b i)e µi) B i). G s)bi) = e w i,j;s)b i)b j) Bi) j= j i I,j= I i,j i,i j I= I i e w I,j;s)B I)B j). e w I,i;s)B I)B i)
21 3 LUIGI ACCARDI AND ANDREAS BOUKAS By 3.) with λ = µi) we have j= j i e w i,j;s)b i)b j) Bi) = e B i) N j= w i,j;s)b j) j i Bi) = e µi) B i) B i) = B i)e µi) B i) µi)e µi) B i). Thus G s)bi) = B i)e µi) B i) µi)e µi) i)) N B I,j= I i,j i,i j I= I i e w I,i;s)B I)B i) e w I,j;s)B I)B j) = B i)g s) µi)g s). For 4.5) we have G s)b i) = = I,j= I>j,j=i I,j= I>j,j>i I= I>i e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,i;s)B I)B i) e w i,j;s)b i)b j) Bi) j= j<i I,j= I>j,j>i e w I,j;s)B I)B j). I,j= I>j,j<i I,j= I>j,I>i,j<i e w I,j;s)B I)B j) B i) I,j= I>j,I<i,j<i e w I,j;s)B I)B j) e w I,j;s)B I)B j) As explained above, e νi) B i) B i) = B i)e νi) B i) + νi) e νi) B i) νi)b i)e νi) B i).
22 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 33 Thus = G s)b i) = B i) I,j= I>j,j>i j= j<i + νi) j= j<i I= I>i e w I,i;s)B I)B i) e w i,j;s)b i)b j) e w I,j;s)B I)B j) I= I>i e w I,i;s)B I)B i) e w i,j;s)b i)b j) I,j= I>j,j>i νi) B i) e w I,j;s)B I)B j) I= I>i j= j<i I,j= I>j,I<i,j<i N I,j= I>j,I<i,j<i e w I,i;s)B I)B i) e w i,j;s)b i)b j) e w I,j;s)B I)B j) e w I,i;s)B I)B i) Bi) I= I>i j= j<i e w i,j;s)b i)b j) + νi) I= I>i I,j= I>j,I<i,j<i I,j= I>j,I>i,j<i I,j= I>j,I>i,j<i e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) I,j= I>j,I>i,j<i I,j= I>j,j>i I,j= I>j,I>i,j<i I,j= I>j,I<i,j<i e w I,i;s)B I)B i) e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) I,j= I>j,I>i,j<i I,j= I>j,j>i e w I,j;s)B I)B j) e w I,j;s)B I)B j)
23 34 LUIGI ACCARDI AND ANDREAS BOUKAS j= j<i νi) j= j<i e w i,j;s)b i)b j) I,j= I>j,I<i,j<i e w I,i;s)B I)B i) Bi) I= I>i e w i,j;s)b i)b j) By 3.9) with λ = νi), and by 3.) with λ = ξi), I,j= I>j,I<i,j<i e w I,j;s)B I)B j) I,j= I>j,I>i,j<i e w I,j;s)B I)B j) I,j= I>j,j>i e w I,j;s)B I)B j) I,j= I>j,j>i e νi) B i) B i) = B i)e νi) B i) νi)e νi) B i), e w I,j;s)B I)B j) e w I,j;s)B I)B j). e ξi) B i) B i) = B i)e ξi) B i) + ξi) e ξi) B i) ξi)b i)e ξi) B i). Thus, using the fact that we obtain ξi) + νi) = λi), G s)b i) = B i)g s) + ξi) + 3νi) )G s) λi)b i)g s). For 4.6) we have = G s)b i) = I,j= I>j,j>i I= I>i I,j= I>j,j=i e w I,j;s)B I)B j) e w I,i;s)B I)B i) e w I,j;s)B I)B j) e w i,j;s)b i)b j) Bi) j= j<i I,j= I>j,j>i e w I,j;s)B I)B j). I,j= I>j,I>i,j<i I,j= I>j,I<i,j<i I,j= I>j,j<i e w I,j;s)B I)B j) e w I,j;s)B I)B j) B i) e w I,j;s)B I)B j) Since, as pointed out above, 3.9) with λ = νi) implies e νi) B i) B i) = B i)e νi) B i) νi)e νi) B i),
24 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 35 we have G s)b i) = = = I,j= I>j,j=i I,j= I>j,j>i I= I>i e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,i;s)B I)B i) B i) I,j= I>j,I<i,j<i j= j<i N I,j= I>j,j<i I,j= I>j,I>i,j<i e w i,j;s)b i)b j) νi) e w I,j;s)B I)B j) e w I,i;s)B I)B i) Bi) I= I>i j= j<i e w i,j;s)b i)b j) I,j= I>j,j>i I,j= I>j,j>i I,j= I>j,I>i,j<i I,j= I>j,I<i,j<i e w I,j;s)B I)B j) νi)g s) and, using 3.9) with λ = ξi) we obtain e w I,j;s)B I)B j) B i) e w I,j;s)B I)B j) j= j<i e w i,j;s)b i)b j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) e w I,j;s)B I)B j) G s)b i) = B i)g s) ξi)g s) νi)g s) = B i)g s) λi)g s). Finally, for 4.7) we have G s)b i) = I= I>i e w I,i;s)B I)B i) e w i,j;s)b i)b j) Bi) j= j<i I,j= I>j,j>i e w I,j;s)B I)B j). I,j= I>j,I>i,j<i I,j= I>j,I<i,j<i e w I,j;s)B I)B j) e w I,j;s)B I)B j)
25 36 LUIGI ACCARDI AND ANDREAS BOUKAS By 3.6) with λ = νi), Thus G s)b i) = j= j<i e νi) B i) B i) = B i)e νi) B i) νi) B i) e νi) B i). e w I,i;s)B I)B i) Bi) I= I>i e w i,j;s)b i)b j) νi)b i)g s). I,j= I>j,I<i,j<i I,j= I>j,I>i,j<i e w I,j;s)B I)B j) e w I,j;s)B I)B j) I,j= I>j,j>i e w I,j;s)B I)B j) As above, using 3.6) with λ = ξi) to commute B i) past the product to its left we obtain G s)b i) =B i)g s) ξi)b i)g s) νi)b i)g s) = B i)g s) λi)b i)g s). Lemma 4.. Let Es) = e w s) E s)g s)e s)e s)e s)g s)g s)e s). Then, e w s) E s)g s)e s)b i)e s)e s)g s)g s)e s) 4.8) = B i) w i; s)b i) λi)b i) w i; s)b i) ) Es), e w s) Es)G s)e s)e s)b i)e s)g s)g s)es) 4.9) =e w i;s) Bi) wi; s)bi) λi) wi; s) ) Es), and e w s) Es)G s)e s)e s)e s)g s)g s)bi)e s) 4.) =e w i;s) Bi) + 4 wi; s) ) B i) 4wi; s)bi) λi)bi) + 4wi; s)λi)bi) + ξi) + 3νi) + wi; s) ) w i; s)bi) +4wi; s)wi; s)bi) + wi; s)λi) ) Es) e w i;s) N j= j i e w j;s) w i, j; s) B i)b j) w j; s)b i)b j) w i; s)b i)b j)
26 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS w i; s)w j; s)b i)b j) B j)λi) + w j; s)b j)λi) λj)b i) + w i; s)b i)λj) + λi)λj) w i; s)b j) + w i; s)w j; s)b j) + w i; s)λj) w j; s)b i) + w j; s)w i; s)b i) +wj; s)λi) + wi; s)wj; s) ) Es) + wi, j; s)wi, J; s)e w j;s) e w J;s) j= J= j i J i B J)B j) w j; s)b J)B j) w J; s)b J)B j) + 4w J; s)w j; s)b J)B j) B j)λj) + w j; s)b j)λj) λj)b J) + w J; s)b J)λj) + λj)λj) w J; s)b j) + w J; s)w j; s)b j) + w J; s)λj) w j; s)b J) + w j; s)w J; s)b J) +w j; s)λj) + w J; s)w j; s) ) Es). Moreover, for i j we have e w s) Es)G s)e s)e s)e s)b i)b j)g s)g s)es) 4.) = e w i;s) e w j;s) Bi)B i j) wj; s)e w i;s) e w j;s) Bi)B j) e w i;s) e w j;s) B i)λj) e w i;s) e w j;s) w j; s)b i) + w i; s)e w j;s) B j) w j; s)w i; s)e w j;s) B j) w i; s)e w j;s) λj) w i; s)w j; s)e w j;s)) Es), and for i > j we have e w s) Es)G s)e s)e s)e s)g s)bi)b j)g s)es) 4.) = e w i;s) Bi) wi; s)bi) λi) wi; s)) wi, j ; s)e w j ;s) Bj ) wj ; s)bj ) λj ) wj ; s)) j = j i e w j;s) B j) w j; s)b j) λj) w j; s)) wj, j ; s)e w j ;s) Bj ) wj ; s)bj ) j = j j λj ) w j ; s)) ) Es). Proof. The idea in all cases is to use Lemmas 3., 3.3 and 4. to move the B s and G s all the way to the front of the left hand sides of 4.8)-4.). Keeping in mind
27 38 LUIGI ACCARDI AND ANDREAS BOUKAS that terms like E n, B k and G m commute. The same is true for E n, B k and G m. To prove 4.8) we use 3.5), 4.7) and 3.6). We have: e w s) E s)g s)e s)b i) =e w s) E s)g s)b i)e s) w i; s)b i)e s)) =e w s) E s)g s)b i)e s) w i; s)b i)e w s) E s)g s)e s) =e w s) E s)b i)g s) λi)b i)g s))e s) w i; s)b i) e w s) E s)g s)e s) =e w s) B i)e s) w i; s)b i)e s))g s)e s) λi)b i)e w s) E s)g s)e s) w i; s)b i)e w s) E s)g s)e s), from which 4.8) follows after multiplying both sides of the above from the right by E s)e s)g s)g s)e s). For 4.9) we use 3.7), 3.8), 3.9)and 4.6). We have: e w s) E s)g s)e s)e s)b i) =e w i;s) e w s) E s)g s)e s)b i)e s) =e w i;s) e w s) E s)g s) B i)e s) w i; s)e s) ) E s) =e w i;s) e w s) E s) B i)g s) λi)g s) ) E s)e s) w i; s)e w i;s) e w s) E s)g s)e s)e s) =e w i;s) e w s) B i)e s) w i; s)b i)e s) ) G s)e s)e s) e w i;s) λi)e w s) E s)g s)e s)e s) w i; s)e w i;s) e w s) E s)g s)e s)e s), and 4.9) follows after multiplying the above by E s)g s)g s)e s). For 4.), using 4.3) to commute B i) past G s), we have e w s) E s)g s)e s)e s)e s)g s)g s)b i)e s) =e w s) E s)g s)e s)e s)e s)b i)g s)g s)e s) e w s) E s)g s)e s)e s)e s)µi)b i)g s)g s)e s) + e w s) E s)g s)e s)e s)e s)µi) G s)g s)e s). We will compute the three terms appearing on the right hand side of the above equation separately: Using 3.), 3.), 4.5), 4.6), 3.) and 3.9) to commute B i) and all the resulting B s to the left of e w s) E s)g s)e s)e s)e s)b i)
28 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 39 and then multiplying both sides of the resulting equation by G s)g s)e s), we find e w s) Es)G s)e s)e s)e s)b i)g s)g s)es) =e w i;s) Bi) + 4wi; s)) Bi) 4wi; s)bi) λi)bi) + 4wi; s)λi)bi) + ξi) + 3νi) + wi; s)) wi; s)bi) +4wi; s)wi; s)bi) + wi; s)λi) ) Es). Similarly, using 3.7), 3.8), 4.6) and 3.9) to commute µi)b i) and all the resulting B s to the left of e w s) E s)g s)e s)e s)e s)µi)b i) and then multiplying both sides of the resulting equation by G s)g s)e s), we find e w s) E s)g s)e s)e s)e s)µi)b i)g s)g s)e s) N = e w i;s) e w j;s) wi, j; s) j= j i B i)b j) w j; s)b i)b j) w i; s)b i)b j) + 4w i; s)w j; s)b i)b j) B j)λi) + w j; s)b j)λi) λj)b i) + w i; s)b i)λj) + λi)λj) w i; s)b j) + w i; s)w j; s)b j) + w i; s)λj) w j; s)b i) + w j; s)w i; s)b i) +w j; s)λi) + w i; s)w j; s) ) Es). For the third term, writing µi) as µi)µi) and replacing the µi) s by their definition, with summation indices j and J respectively, using 3.7) twice to commute B j)b J) past E s) and noticing that the resulting term e w s) E s)g s)e s)b j)b J)E s)e s)
29 3 LUIGI ACCARDI AND ANDREAS BOUKAS has already been previously computed, we obtain e w s) Es)G s)e s)e s)e s)µi) G s)g s)es) = wi, j; s)wi, J; s)e w j;s) e w J;s) j= J= j i J i B J)B j) w j; s)b J)B j) w J; s)b J)B j) + 4w J; s)w j; s)b J)B j) B j)λj) + w j; s)b j)λj) λj)b J) + w J; s)b J)λj) + λj)λj) w J; s)b j) + w J; s)w j; s)b j) + w J; s)λj) w j; s)b J) + w j; s)w J; s)b J) +w j; s)λj) + w J; s)w j; s) ) Es). Combining the above three equations we obtain 4.). For 4.), we can use 3.3), 3.4), 3.7), 3.8), 4.6) and 3.9) to gradually move B i)b j) to the left of e w s) E s)g s)e s)e s)e s)b i)b j), where i j, and obtain e w s) Es)G s)e s)e s)e s)b i)b j) = e w i;s) e w j;s) Bi)B i j) wj; s)e w i;s) e w j;s) Bi)B j) e w i;s) e w j;s) B i)λj) e w i;s) e w j;s) w j; s)b i) + w i; s)e w j;s) B j) w j; s)w i; s)e w j;s) B j) w i; s)e w j;s) λj) w i; s)w j; s)e w j;s)) e w s) E s)g s)e s)e s)e s), which upon multiplication by G s)g s)e s) gives 4.). Finally, for 4.), using 4.4) to switch B i) past G s), and the Definition of µi) with summation index j we find e w s) E s)g s)e s)e s)e s)g s)b i)b j) The terms =e w s) Es)G s)e s)e s)b i)e s)g s)bj) wi, j ; s)e w s) Es)G s)e s)e s)b j )Es)G s)bj). j = j i e w s) E s)g s)e s)e s)b i), e w s) E s)g s)e s)e s)b j ) are similar. By 3.7), 3.8), 4.6) and 3.9) e w s) E s)g s)e s)e s)b i) =e w i;s) B i) w i; s)b i) λi) w i; s) ) e w s) E s)g s)e s)e s)
30 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 3 Thus e w s) Es)G s)e s)e s)e s)g s)bi)b j) = e w i;s) Bi) wi; s)bi) λi) wi; s) ) wi, j ; s)e w j ;s) Bj ) wj ; s)bj ) λj ) wj ; s) ) j = j i e w s) E s)g s)e s)e s)e s)g s)b j). Replacing e w s) E s)g s)e s)e s)e s)g s)b j) on the right hand side of the above equation with the above equation without the B j) factor on the right, and with i replaced by j, we obtain e w s) Es)G s)e s)e s)e s)g s)bi)b j) = e w i;s) Bi) wi; s)bi) λi) wi; s) ) wi, j ; s)e w j ;s) Bj ) wj ; s)bj ) λj ) wj ; s) ) j = j i e w j;s) Bj) wj; s)bj) λj) wj; s) ) wj, j ; s)e w j ;s) Bj ) wj ; s)bj ) j = j j λj ) w j ; s) )) e w s) E s)g s)e s)e s)e s)g s). Multiplying both sides by G s)e s) we obtain 4.). Note: All summation indices in the statement as well as in the proof of the following Theorem 4.3, run from to N and equations 4.) and 4.) should be taken into account in the interpretation of all sums. Theorem 4.3. For s R and N =,,... let F N := c i)bi) + c i)bi) + c i)bi) + c i)bi) + c i)bi) ) + c i, j)bi)b j) + c i, j)bi)b j) ) + i,j= i>j c i, j)bi)b j). i,j= i j
31 3 LUIGI ACCARDI AND ANDREAS BOUKAS Then e s F N =e w s) e w i;s)b i) e w i;s)b i) i,j= i j i,j= i>j e w i,j;s)b i)b j) e w i,j;s)b i)b j) N e w i,j;s)b i)b j) e w i;s)b i), i,j= i>j e w i;s)b i) e w i;s)b i) where, for each i, w i; s) is the solution of the Riccati initial value problem dw i; s) c i)w i; s) 4c i) w i; s) ) = c i) ; w i; ) =, and w i; s) =c i)s + 4c i) w i; s) =c i) e w i;u) du. w i; u) du, For i, j) {,,..., N} {,,..., N}, the 3N unknowns w i, j; s), w i, j; s) and w i, j; s) are determined by their initial value w i, j; ) = w i, j; ) = w i, j; ) =,
32 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 33 and the system of 3N first order differential equations 4.3)-4.5): c i, j) = dw i, j; s) + b w i, j; s) + wj, i; s) ) + b wi, j; s) 4.3) dw + b i, j; s) dw + b i, j; s) + b3 wi, k; s)wj, k; s) + b 4 wk, i; s)wk, j; s) k + b 5 wk, i; s) + b 7 wi, k; s) ) wk, j; s) + wj, k; s) ) +b 6 wk, j; s)wk, i; s) ) + k,m b8 w k, m; s) w k, i; s)w m, j; s) + w k, i; s)w j, m; s) +w i, k; s)w m, j; s) + w i, k; s)w j, m; s) ) + b 9 w k, j; s)w k, m; s) w m, i; s) + w i, m; s) ) +b wk, i; s)wk, m; s) wm, j; s) + wj, m; s) )) + dw b i, k; s) 3 w k, j; s) + wj, k; s) ) k dw + b k, j; s) 4 w dw k, i; s) + b i, k; s) 5 w k, j; s) dw + b k, i; s) 6 w k, j; s) + wj, k; s) ) dw +b i, k; s) 7 w k, j; s) + wj, k; s) )) + dwk, m; s) b8 w k, i; s)wm, j; s) k,m + b 9 w k, j; s) + wj, k; s) ) + b w k, i; s) w m, j; s) + w j, m; s) ) + b w k, i; s)w m, j; s) + w k, i; s)w j, k; s) +w i, k; s)w m, j; s) + w i, k; s)w j, k; s) ) + b w k, m; s) w m, j; s) + w j, m; s) ) + b 3 wk, m; s) wm, j; s) + wj, m; s) )) + b4 wk, m; s)wk, l; s) wn, i; s)wm, j; s) k,m,l +w n, i; s)w j, m; s) + w i, n; s)w m, j; s) + w i, n; s)w j, m; s) ) + dw k, m; s) b5 w k, l; s)w m, i; s) w l, j; s) + w j, l; s) )
33 34 LUIGI ACCARDI AND ANDREAS BOUKAS + b 6 w k, l; s) w k, i; s)w l, j; s) +w k, i; s)w j, l; s) + w i, k; s)w l, j; s) + w i, k; s)w j, l; s) ) + b 7 w k, i; s)w m, l; s) w l, j; s) + w j, l; s) ) + b 8 wm, l; s) wk, i; s)wl, j; s) +wk, i; s)wj, l; s) + wi, k; s)wl, j; s) + wi, k; s)wj, l; s) ))) + dw b k, m; s) 9 w k, l; s)wm, n; s) wl, i; s)wn, j; s) k,m,l,n +w l, i; s)w j, n; s) + w i, l; s)w n, j; s) + w i, l; s)w j, n; s) ), c i, j) =p wi, dw j; s) + p i, j; s) 4.4) + p 3 wk, j; s)w dw k, i; s) + p i, k; s) 4 w k, j; s) k dw ) +p k, j; s) 5 w k, i; s) + dw p k, m; s) 6 w k, i; s)wm, j; s), k,m c i, j) =q wj, i; s) + wi, j; s)) 4.5) + q wj, dw i; s) + q 3 + q i, j; s) dw 4 + q i, j; s) 5 + q6 wk, i; s) + wi, k; s)) + q 7 wk, i; s)wk, j; s) k dw + q j, k; s) 8 w k, i; s) + wi, k; s)) dw + q j, k; s) 9 w dw k, i; s) + q i, k; s) w k, j; s)
34 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 35 + q dw k, j; s) dw +q k, j; s) 3 w k, i; s) wk, i; s) + wi, dw k; s)) + q k, i; s) w k, i; s) ) + k,m q4 w k, j; s)w k, m; s)w m, i; s) + w i, m; s)) + q 5 w k, m; s)w k, i; s)w m, i; s) + w i, m; s)) dw + q j, m; s) 6 w m, k; s)wk, i; s) + wi, k; s)) dw + q k, j; s) 7 w k, m; s)wm, i; s) + wi, m; s)) dw + q k, m; s) 8 w m, j; s)wk, i; s) + wi, k; s)) dw + q k, m; s) 9 w k, j; s)wm, i; s) + wi, m; s)) dw + q k, m; s) w k, j; s)wm, i; s) dw ) +q k, m; s) w k, i; s)wm, j; s) + dw q k, m; s) w k, j; s)wm, l; s) + wk, l; s)wm, j; s)) k,m,l w l, i; s) + w i, l; s)), and for i {,,..., N}, the N unknowns w i; s) and w i; s) are determined by their initial value w i; ) = w i; ) =, and the system of N first order differential equations 4.6)-4.7): c dw i) =r i; s) + r w i; s) + r3 wk; s) + r 4 wk; s) ) 4.6) k + r5 wm; s) + r 6 wk; s) ) + r 7 wl; s), k,m k,m,l c i) =u + dw i; s) + k + k,j dw u k; s) 3 + u w i; s) + u dw i; s) ) + u 4 wk; s) + u 5 wk; s) u6 wk; s) + u 7 wk; s) ) + u 8 wm; s) + k,j,m k,j,m,l 4.7) u 9 w l; s), where the coefficients b, p, q, r, u appearing in 4.3)-4.7) depend on the various indices and previously terms and can be found in the Appendix. Finally, w s) is
35 36 LUIGI ACCARDI AND ANDREAS BOUKAS given by s ws) dw = i; t) e w i;t) w dt i; t) 4.8) i dwi; t) e w i;t) w dt i; t) ) e w i;t) i e w j;t) wi, j; t)wi; t)wj; t) j + wi, j; t)wi, m; t)e w j;t) e w m;t) wm; t)wj; t) j,m + i,j i,j dwi, j; t) w dt i; t)wj; t)e w j;t) dwi, j; t) e w i;t) w dt i; t) + k e w j;t) w j; t)) + m w i, k; t)e w k;t) w k; t) w j, m; t)e w m;t) w m; t)) )) dt. ) Proof. As in the proof of Theorem 3.4, let Es) = e sf N. Then, des) = F N Es). 4.9) Since Es) = e w s) E s)g s)e s)e s)e s)g s)g s)e s), we also have that des) + i,j = dw s) Es) + i dwi, j; s) B i)bj)es) dwi; s) B i)es) + i dwi; s) B i)es) + i dwi; s) e w s) E s)g s)es)b i)e s)e s)g s)g s)es)
36 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 37 + i + i + i,j + i,j dwi; s) e w s) E s)g s)e s)e s)b i)e s)g s)g s)es) dwi; s) e w s) E s)g s)e s)e s)e s)g s)g s)bi)e s) dwi, j; s) e w s) E s)g s)e s)e s)e s)b i)b j)g s)g s)es) dwi, j; s) e w s) E s)g s)e s)e s)e s)g s)bi)b j)g s)es), which by Lemma 4. becomes des) + i,j = dw s) Es) + i dwi, j; s) B i)bj)es) dwi; s) B i)es) + i dwi; s) B i)es) 4.) + i + i + i dw i; s) B i) w i; s)b i) λi)b i) w i; s)b i) ) Es) dwi; s) e w i;s) B i) wi; s)bi) λi) wi; s) ) Es) dw i; s) e w i;s) B i) + 4 w i; s) ) B i) 4w i; s)b i) λi)b i) + 4w i; s)λi)b i) + ξi) + 3νi) + w i; s) ) w i; s)b i) +4w i; s)w i; s)b i) + w i; s)λi) ) e w i;s) j e w j;s) w i, j; s) B i)b j) w j; s)b i)b j) w i; s)b i)b j) + 4w i; s)w j; s)b i)b j) B j)λi) + w j; s)b j)λi) λj)b i) + w i; s)b i)λj) + λi)λj) w i; s)b j) + w i; s)w j; s)b j) + w i; s)λj) w j; s)b i) + w j; s)w i; s)b i) +w j; s)λi) + w i; s)w j; s) ) + wi, j; s)wi, m; s)e w j;s) e w m;s) j,m
37 38 LUIGI ACCARDI AND ANDREAS BOUKAS B m)b j) w j; s)b m)b j) w m; s)b m)b j) + 4w m; s)w j; s)b m)b j) B j)λm) + w j; s)b j)λm) λj)b m) + w m; s)b m)λj) + λm)λj) w m; s)b j) + w m; s)w j; s)b j) + w m; s)λj) w j; s)b m) + w j; s)w m; s)b m) +w j; s)λm) + w m; s)w j; s) )) Es) + i,j dw i, j; s) e w i;s) e w j;s) B i)b j) w j; s)e w i;s) e w j;s) B i)b j) e w i;s) e w j;s) B i)λj) e w i;s) e w j;s) w j; s)b i) + w i; s)e w j;s) B j) w j; s)w i; s)e w j;s) B j) w i; s)e w j;s) λj) w i; s)w j; s)e w j;s)) Es) + i,j dw i, j; s) e w i;s) B i) w i; s)b i) λi) w i; s)) wi, j ; s)e w j ;s) Bj ) wj ; s)bj ) λj ) wj ; s)) j e w j;s) Bj) wj; s)bj) λj) wj; s)) j Es). wj, j ; s)e w j ;s) Bj ) wj ; s)bj ) λj ) wj ; s)) Equating coefficients of B i), B i) and B i) in 4.9) and 4.), for each i =,,..., N, we obtain the equations c i) = dw i; s) w i; s) dw i; s) + 4e w i;s) dw i; s) w i; s) ), 4.) c i) =e w i;s) dw i; s), 4.) c i) = dw i; s) 4w i; s)e w i;s) dw i; s). 4.3) Putting 4.) in 4.3) we obtain dwi; s) = c i) + 4wi; s)c i), 4.4) while putting 4.) in 4.) we obtain dw i; s) w i; s) dw i; s) + 4c i) w i; s) ) = c i). 4.5)
38 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 39 Substituting 4.4) in 4.5) we find that w i; s) satisfies the Riccati ODE dw i; s) c i)w i; s) 4c i) w i; s) ) = c i), which along with the initial condition wi; ) = completely determines wi; s). Equation 4.4) then implies while 4.) implies w i; s) = c i)s + 4c i) w i; s) = c i) w i; u) du, e w i;u) du. Equating coefficients of B i)b j), B i)b j), B i)b j), B i) and B i) in 4.9) and 4.), we find that for each i, j, w i, j; s), w i, j; s), w i, j; s), w i; s) and w i; s) satisfy equations 4.3)-4.7). Equating coefficients of the constant terms in 4.9) and 4.) we find that w s) is determined by equation 4.8). Remark 4.4. The formulas for w i; s), w i; s) and w i; s) are the same as in Proposition Vacuum Characteristic Function Theorem 5.. Suppose that Φ is a Fock vacuum vector such that Φ = Φ, Φ =, BΦ = BΦ = and BΦ = Φ. Then the vacuum characteristic function of the quantum random variable F N of Theorems 3.4 and 4.3, is given by Φ, e is F N Φ = e w is) e N I= w I;is) 5.) where w is) and w I; is) are obtained from w s) and w I; s) of Theorems 3.4 and 4.3, respectively, after replacing s by is. Proof. Using the expressions for e s F N, with s replaced by is, given in Theorems 3.4 and 4.3, we notice that, unless n = k =, all exponentials of the general form e Bn k act on Φ as the identity operator, either directly if n < k) or after flipping to the other side of the inner product if n > k). In the case when n = k = we have e w I;is)B I) Φ = e w I;is)B I)B I)+ ) Φ I= = I= e w I;is) Φ I= =e N I= w I;is) Φ. The only other surviving exponential is e w is). Since Φ =, we obtain 5.).
39 3 LUIGI ACCARDI AND ANDREAS BOUKAS Corollary 5.. In the notation of Theorems 4.3 and 5., let G N = I= + ) c I)a I ) + c I)a I ) + c I)a I + c I)a I + c I)a I a I I,J= I>J c I, J)a I a J + c I, J)a I a J ) + c I, J)a I a J. I,J= I J Then Φ, e is G N Φ = e is N I= c I) e w is) e N I= w I;is). Proof. The proof follows from Theorem 5., and the fact that B I) = a I ), B I) = a I ), B I) = a I, B I) = a I, B I) = a I a I +, B I)B J) = a I a J, B I)B J) = a I a J, B I)B J) = a I a J and F N = G N + c I). I= 6. Appendix: Coefficient Formulas 6.. Coefficients in Proposition 3.5. The coefficients α k i), k =,..., 4, appearing in Proposition 3.5 are given by α i) =Bi)c i)bi) c i)) Bi) + c i) + ) i)/bi)) c c i)c i) ) ) c i)/bi) c i)c i), α i) =c i)bi) c i)) Bi) + c i) Bi) c i) ) c i)/bi) c i) c i)c i) c i)c i) )), α 3 i) = 4 + ) i)/bi)) c Bi) c i) Bi) c i) c i)c i) c i)c i) ), α 4 i) = Bi)Bi) c i)) c i)bi) + c i)) 3/, α 5 i) =Bi) Bi) c i)) Bi) + c i)) 3/, α 6 i) =Bi) c i) ) c i)/bi) c i)c i)c i) c i)c i))) log B, ) α 7 i) =Bi) ) c i)/bi) c i)c i)) + ) c i)/bi) )c i)c i), α 8 i) = ) c i)/bi) c i)c i) c i)c i)), α 9 i) =Bi) c i) ) c i)/bi) c i)c i)c i) c i)c i)),
40 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 3 α i) = Bi) Bi) c i)) Bi) + c i)) Bi)c i) c i)bi) + c i)) log /Bi)) + Bi)c i) Bi) c i)) c i) log/bi)) + Bi) 4 c i) ) c i)/bi) c i) c i)c i) c i)c i)) + Bi) 4 ) c i)/bi) c i)c i)c i)c i) + 4 ) c i)/bi) c i) c i) + c i) c i) + ) c i)/bi) + π)) + ) c i)/bi) c i)c i)c i) c i)c i)) Bi) c i) + c i) + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i))) log Bi))), α i) = Bi) Bi) c i)) Bi) + c i)) ) c i)/bi) c i)c i)c i) c i)c i)) ) c i)/bi) c i) c i)c i) c i)c i)) + Bi) 4 + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i)))), α i) = Bi) Bi) c i))3/ Bi) + c i))3/ 4 ) c i)/bi) c i)c i)c i) c i)c i)) + ) c i)/bi) ) c i)c i) ) c i)/bi) c i)c i))), α 3 i) = Bi) 3 Bi) c i)) Bi) + c i)) Bi) 3 c i) Bi) + c i)) log /Bi)) Bi) 3 c i) Bi) c i)) log/bi)) 4 ) c i)/bi) c i) 3 c i)c i) c i)c i)) + ) c i)/bi) Bi) c i)c i)c i) c i)c i)) + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i)) Bi) 4 c i) ) c i)/bi) c i)c i) + c i)c i) ) c i)/bi) π)) + ) c i)/bi) Bi) c i)c i) c i)c i)) Bi) c i) + c i) + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i))) log Bi))),
41 3 LUIGI ACCARDI AND ANDREAS BOUKAS α 4 i) = Bi)Bi) c i)) Bi) + c i)) ) c i)/bi) c i)c i) c i)c i)) ) c i)/bi) c i) c i)c i) c i)c i)) + Bi) 4 + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i)))), α 5 i) = Bi)Bi) c i))3/ Bi) + c i))3/ 4 ) c i)/bi) c i)c i) c i)c i)) + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i))), α 6 i) = Bi) 3 Bi) c i))bi) + c i)) ) c i)/bi) c i)c i) c i)c i)) Bi) c i) + ) c i)/bi) c i)c i)c i) c i)c i)))), c α 7 i) = i) c i) Bi)Bi) c, i)) α 8 i) = α 9 i) = c i) c i) Bi)Bi) + c i)), c i) Bi) c i)), c α i) = i) Bi) + c, i)) α i) = c i) c i) Bi) c i)) Bi) + c i)), Bi)c α i) = i) c i) Bi) c i)) Bi) + c, i)) α 3 i) = Bi) Bi) c i)) Bi) + c )c i)/bi) c i)c i)c i) i)) c i)c i))bi) c i) + c i) ) c i)/bi) )c i)c i) + ) c i)/bi) c i)c i)))), α 4 i) = Bi)Bi) c i)) Bi) + c )c i)/bi) c i)c i) i)) c i)c i)) Bi) c i) + c i) + ) c i)/bi) )c i)c i) ) c i)/bi) c i)c i)))).
42 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS Coefficients in Theorem 4.3. The coefficients b, p, q, r, u, appearing in Theorem 4.3 are given by b = dw i; s) + 4 dw i; s) w i; s)e w i;s), b = 8 dw i; s) w i; s)wj; s)e w i;s) e w j;s), b 3 = dw k; s) e w k;s), b 4 =3 dw k; s) e w k;s), b 5 = 4 dw k; s) w i; s)e w k;s) e w i;s), b 6 =4 dw k; s) w i; s)wj; s)e w j;s) e w i;s), b 7 = dw i; s) w i; s)e w i;s) e w k;s), b 8 = dw k; s) e w k;s) e w m;s), b 9 = dw k; s) w j; s)e w j;s) e w m;s), b = dw k; s) w i; s)e w i;s) e w m;s), b = wj; s)e w j;s) e w i;s), b =4w i; s)w j; s)e w j;s) e w i;s), b 3 = e w k;s) e w i;s), b 4 = 4w j; s)e w j;s) e w i;s), b 5 = 4w i; s)w j; s)e w j;s) e w i;s), b 6 =w i; s)e w i;s) e w k;s), b 7 =w i; s)e w i;s) e w k;s), b 8 =w i; s)w j; s)e w j;s) e w i;s), b 9 = w i; s)e w k;s) e w i;s), b = e w m;s) e w i;s), b =e w m;s) e w k;s), b = w i; s)e w m;s) e w i;s), b 3 = w i; s)e w m;s) e w i;s), b 4 = dw k; s) e w m;s) e w l;s),
43 34 LUIGI ACCARDI AND ANDREAS BOUKAS b 5 =w i; s)e w i;s) e w l;s), b 6 = e w m;s) e w l;s), b 7 =w i; s)e w i;s) e w l;s), b 8 = e w k;s) e w l;s), b 9 =e w n;s) e w l;s), p = dw i; s) e w i;s) e w j;s), p =e w i;s) e w j;s), p 3 = dw k; s) e w i;s) e w j;s), p 4 = e w i;s) e w j;s), p 5 = e w i;s) e w j;s), p 6 =e w i;s) e w j;s), q = dw j; s) e w j;s) + ), q =4 dw j; s) w i; s)e w i;s) e w j;s), q 3 = dw i; s) w i; s), q 4 =e w i;s) e w j;s), q 5 = w i; s) + w j; s))e w j;s) e w i;s), q 6 = dw k; s), q 7 =4 dw k; s) w i; s)e w i;s) e w j;s), q 8 = e w k;s) e w j;s), q 9 =e w j;s) e w i;s), q =w i; s)e w i;s) e w j;s), q = e w j;s) e w k;s), q =w i; s)e w j;s) e w i;s), q 3 =w i; s)e w j;s) e w i;s), q 4 = dw k; s) e w j;s) e w m;s),
44 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 35 q 5 = dw k; s) e w j;s) e w m;s), q 6 =e w j;s) e w k;s), q 7 =e w j;s) e w m;s), q 8 =e w j;s) e w k;s), q 9 =e w j;s), q = w i; s)e w j;s) e w i;s), q = w i; s)e w j;s) e w i;s), q = e w j;s) e w l;s), r =e w i;s), r = e w i;s) dw i; s) r 3 = dw k; s) dw i, k; s) r 4 = dw k, i; s) e w i;s),, wk, i; s)e w i;s) e w k;s) + dw i; s) e w i;s) e w k;s) e w i;s) e w k;s) dw k, i; s) e w i;s) e w k;s), r 5 = dw k; s) w k, i; s)wk, m; s)e w i;s) e w m;s) + dw i, k; s) w k, m; s)e w i;s) e w m;s) + dw k, m; s) w k, i; s)e w i;s) e w m;s) + dw k, i; s) w k, m; s)e w i;s) e w m;s), r 6 = dw k, m; s) w m, i; s)e w i;s) e w k;s), r 7 = dw k, m; s) w k, i; s)wm, l; s) + wk, l; s)wm, i; s)) e w i;s) e w l;s),
45 36 LUIGI ACCARDI AND ANDREAS BOUKAS u = j dwi, j; s) e w i;s) e w j;s) u = dw i; s) u = w i; s)e w i;s), + 4 dw i; s) w i; s), u 3 = w k, i; s) + w i, k; s))e w k;s), u 4 = dw k; s) w k, i; s) + wi, k; s))e w k;s) 4 dw i; s) w i, k; s)wi; s)e w i;s) e w k;s) 4 dw k; s) w k, i; s)wi; s)e w i;s) e w k;s) + dw i, k; s) w i; s)e w i;s) e w k;s) + dw k, i; s) w i; s)e w i;s) e w k;s), u 5 = dw k, i; s) w i; s)e w i;s), u 6 = dw k; s) e w k;s) e w j;s) w k, j; s) wj, i; s) + wi, j; s) ) dw j; s) w j, k; s) wj, i; s) + wi, j; s) ) e w k;s) e w j;s) + 4 dw j; s) w j, i; s)wj, k; s)wi; s)e w i;s) e w k;s) dw i, j; s) w j, k; s)wi; s)e w i;s) e w k;s) dw k, j; s) w j, i; s)wi; s)e w i;s) e w k;s) dw j, k; s) w j, i; s)wi; s)e w i;s) e w k;s) dw j, i; s) w j, k; s)wi; s)e w i;s) e w k;s) + dw j, k; s) w j, i; s)e w j;s) e w k;s) + dw j, k; s) w i, j; s)e w j;s) e w k;s) + dw k, j; s) w j, i; s)e w j;s) e w k;s) + dw k, j; s) w i, j; s)e w j;s) e w k;s),
46 DISENTANGLEMENT OF SCHRÖDINGER ALGEBRA EXPONENTIALS 37 u 7 = dw k, j; s) w j, i; s) + w i, j; s) ) e w j;s), u 8 = dw k; s) w k, j; s)wk, m; s) wj, i; s) + wi, j; s) ) e w j;s) e w m;s) + dw k, j; s) w k, i; s)wj, m; s)wi; s)e w i;s) e w m;s) dw k, j; s) w k, i; s)wj, m; s)e w k;s) e w m;s) dw k, j; s) w i, k; s)wj, m; s)e w k;s) e w m;s) + dw k, j; s) w k, m; s)wj, i; s)wi; s)e w i;s) e w m;s) dw m, j; s) w j, k; s)wk, i; s)e w k;s) e w m;s) dw m, j; s) w j, k; s)wi, k; s)e w k;s) e w m;s) dw k, m; s) w k, j; s)wj, i; s)e w j;s) e w m;s) dw k, m; s) w k, j; s)wi, j; s)e w j;s) e w m;s) dw k, j; s) w k, m; s)wj, i; s)e w j;s) e w m;s) dw k, j; s) w k, m; s)wi, j; s)e w j;s) e w m;s), u 9 = dw k, j; s) w k, m; s)wj, l; s)wm, i; s)e w l;s) e w m;s) + dw k, j; s) w k, m; s)wj, l; s)wi, m; s)e w l;s) e w m;s) + dw k, j; s) w k, l; s)wj, m; s) wm, i; s) + wi, m; s) ) e w l;s) e w m;s). References. Accardi, L. and Boukas, A.: On the characteristic function of random variables associated with Boson Lie algebras, Communications on Stochastic Analysis, 4 ), no. 4, Accardi, L. and Boukas, A.: Fourier transform of random variables associated with the multi-dimensional Heisenberg Lie algebra, Proc. Amer. Math. Soc. 43 5), Accardi, L., Boukas, A., and Lu,Y. G.: The vacuum distributions of the truncated Virasoro fiel are products of Gamma distributions, Open Systems & Information Dynamics, 4 7), no., pages). 4. Accardi, L., Ouerdiane H., and Rebei H. : On the quadratic Heisenberg group, Infinite Dimensional Anal. Quantum Probab. Related Topics, 3 ), no. 4, Berezin, F. A.: The Method of Second Quantization, Pure Appl. Phys. 4, Academic Press, New York, Chebotarev, A. M. and Tlyachev, T. V. : Normal forms, inner producst and Maslov indices of general multimode squeezings, Math. Notes 95 4), 7.
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