An introduction to quantum stochastic calculus

Transcription

1 An introduction to quantum stochastic calculus Robin L Hudson Loughborough University July 21, 214 (Institute) July 21, / 31

2 What is Quantum Probability? Quantum probability is the generalisation of the classical theory of probability made necessary by the noncommutative multiplication of quantum observables, which are usually represented by self-adjoint operators in a Hilbert space. (Institute) July 21, / 31

3 What is Quantum Probability? Quantum probability is the generalisation of the classical theory of probability made necessary by the noncommutative multiplication of quantum observables, which are usually represented by self-adjoint operators in a Hilbert space. For example the momentum and position observables p and q for a one-dimensional particle satisfy the Heisenberg commutation relation where h is Planck s constant, [p, q] = h i 2π h = m 2 kg/s, which is quite small in everyday units. It is more convenient to take h = 4π so that [p, q] = 2i but to remeber that these are not everyday units. (Institute) July 21, / 31

4 Some things don t work in quantum probability. For example if you try to de ne a joint probability distribution ρ p,q for such a canonical pair (p, q) by Z R 2 e i(xu+yv ) ρ p,q (u, v) dudv = you nd a nice joint Gaussian distribution De i(xp+yq)e D E =: ψ, e i(xp+yq) ψ, ψ ρ p,q (u, v) = (2π) 1 e 1 2 (u 2 +v 2 ) when ψ is the ground state of the oscillator 1 2 it s the rst excited state you nd p 2 + q 2, but when ρ p,q (u, v) = (2π) 1 u 2 + v 2 1 e 1 2 (u 2 +v 2 ) which may look OK till you try to calculate the probability that (p, q) lies inside the unit circle. (Institute) July 21, / 31

5 Some things don t work in quantum probability. For example if you try to de ne a joint probability distribution ρ p,q for such a canonical pair (p, q) by Z R 2 e i(xu+yv ) ρ p,q (u, v) dudv = you nd a nice joint Gaussian distribution De i(xp+yq)e D E =: ψ, e i(xp+yq) ψ, ψ ρ p,q (u, v) = (2π) 1 e 1 2 (u 2 +v 2 ) when ψ is the ground state of the oscillator 1 2 it s the rst excited state you nd p 2 + q 2, but when ρ p,q (u, v) = (2π) 1 u 2 + v 2 1 e 1 2 (u 2 +v 2 ) which may look OK till you try to calculate the probability that (p, q) lies inside the unit circle. Another thing that doesn t always work is the notion of conditional expectation and the associated probabilistic concept of martingale. (Institute) July 21, / 31

6 Notice that if (p 1, q 1 ), (p 2, q 2 ),... is a sequence of canonical pairs satisfying the canonical commutation relations [p j, q k ] = 2iδ j,k, [p j, p k ] = [q j, q k ] =, then. for N = 1.2,... p1 + p p p N, q 1 + q q p N N N = 2i. Assuming that the input canonical pairs are "independent, identically distributed and of zero means and nite variance", this new canonical pair converges in distribution as N! to a Gaussian limit, in the manner of the central limit theorem. (Institute) July 21, / 31

7 Notice that if (p 1, q 1 ), (p 2, q 2 ),... is a sequence of canonical pairs satisfying the canonical commutation relations [p j, q k ] = 2iδ j,k, [p j, p k ] = [q j, q k ] =, then. for N = 1.2,... p1 + p p p N, q 1 + q q p N N N = 2i. Assuming that the input canonical pairs are "independent, identically distributed and of zero means and nite variance", this new canonical pair converges in distribution as N! to a Gaussian limit, in the manner of the central limit theorem. But what can these words mean and what is the meaning of convergence in distribution when there is no joint probability distribution? (Institute) July 21, / 31

8 The answer lies in the Stone-von-Neumann uniqueness theorem, according to which there is, up to unitary equivalence, exactly one irreducible representation of the canonical commutation relations for nitely many degrees of freedom. In the one degree of freedom case, denoting this representation by (p, q ) and the carrier Hilbert space by H, a consequence is that, for an arbitrary canonical pair (p, q) in a Hilbert space H and an arbitrary state in H, there exists a unique unit vector Ψ (p,q) 2 H H which is invariant under the ip, the conjugate-unitary map from H H to itself for which each ψ φ 7! φ ψ, such that E D DΨ (p,q), e i(xp +yq 9 ) Ψ (p,q) = e i(xp+yq)e. Ψ (p,q) is called the distribution vector of (p, q) (in this state). Two canonical pairs are identically distributed if they have the same distribution vector. A sequence of canonical pairs converges in distribution if the sequence of distribution vectors converges in the usual Hilbert space sense. (Institute) July 21, / 31

9 To de ne independence we rst de ne the joint distribution vector Ψ (p,q),(p,q ) 2 H H H H of two mutually commuting canonical pairs using the two-dimensional Stone-von-Neumann theorem analogously. Then (p, q) and (p, q ) are independent if Ψ (p,q),(p,q ) = Ψ (p,q) Ψ (p,q ). A canonical pair (p, q) is of zero mean and variance σ 2 1 if hpi = hqi = and the covariance matrix is of form p 2 hpqi hqpi q 2 σ 2 i = i σ 2 Finally we say that such a canonical pair (p, q) is Gaussian : If σ 2 = 1, if Ψ (p,q) = ψ ψ where ψ is the harmonic oscillator ground state. If σ 2 > 1, if Ψ (p,q) = 1/2 e 2β(2n+1)! e β(2n+1) ψ n ψ n where n= n= ψ n is the n-th excited oscillator state and the reciprocal temperature is given by coth β 4 = σ2. (Institute) July 21, / 31

10 Notes (1) Provided all second moments are nite, an arbitrarily distributed canonical pair (p, q) can be reduced to one of zero mean and variance σ 2 by an inhomogeneous linear canonical transformation (p, q) 7! (αp + βq + a, γq + δq + b), where αδ βγ = 1. (2) For sequence of iid canonical pairs (p 1, q 1 ), (p 2, q 2 ),...of zero means and nite variance σ 2 the sequence p 1, p 2,... of mutually commuting observables is iid of zero means and nite variance σ 2 in the classical sense. In particular, by the classical central limit theorem, as N!, (N!) 2 1 (p 1 + p p N ) converges in distribution to the standard Gaussian limit distribution N, σ 2. Likewise for q 1, q 2,...The limit state for the sequence of pairs is consistent with this. (3) By Donsker s invariance principle, the process P N (t) de ned by P N (t) = N 1 2 p 1 + p p [Nt] + (Nt [Nt]) p [Nt]+1 must converge to a Brownian motion P of variance σ 2. Similarly Q N (t) = N 1 2 q 1 + q q [Nt] + (Nt [Nt]) q [Nt]+1 converges to a Brownian motion Q, also of variance σ 2. (Institute) July 21, / 31

11 Quantum planar Brownian motion. Since [P N (s), Q N (t)] = N 1 h p 1 + p p [Ns], q 1 + q q [Nt] i +N 1 h (Ns [Ns]) p [Ns]+1, (Nt [Nt]) q [Nt]+1 i [N (s ^ t)] = 2i N! N! 2is ^ t 2i (Ns [Ns]) (Nt [Nt]) δ N [Ns]+1,[Nt]+1 we expect the limit Brownian motions P and Q to satisfy the commutation relation [P (s), Q (t)] = 2is ^ t. (Institute) July 21, / 31

12 Quantum planar Brownian motion. Since [P N (s), Q N (t)] = N 1 h p 1 + p p [Ns], q 1 + q q [Nt] i +N 1 h (Ns [Ns]) p [Ns]+1, (Nt [Nt]) q [Nt]+1 i [N (s ^ t)] = 2i N! N! 2is ^ t 2i (Ns [Ns]) (Nt [Nt]) δ N [Ns]+1,[Nt]+1 we expect the limit Brownian motions P and Q to satisfy the commutation relation [P (s), Q (t)] = 2is ^ t. How can we construct two such Brownian motions? We again distinguish the cases σ = 1 and σ > 1. (Institute) July 21, / 31

13 The Fock space F (H) over a Hilbert space H is usually de ned by physicists as the Hilbert space in nite direct sum F (H) = CH (H H) sym (H H H) sym of symmetric parts of the n-fold tensor product of H with itself. A more useful de nition for our purposes is that F (H) is a Hilbert space generated by a family (e (f )) f 2H of socalled exponential vectors, satisfying he (f ), e (g)i F(H) = exp hf, gi H. The connection between the two de nitions is made by realising the exponential vectors in the rst de nition as e (f ) = 1f f p f f f f p. 2! 3! One reason why this second de nition is useful is that it makes clear the exponential property of Fock spaces, that there is a natural isomorphism allowing us to identify the Fock space over a direct sum with the tensor product of the Fock spaces over the summands; F (H 1 H 2 ) = F (H 1 ) F (H 2 ), with e (f 1 f 2 ) = e (f 1 ) e (f 2 ). (Institute) July 21, / 31

14 To construct two Brownian motions P and Q satisfying [P (s), Q (t)] = 2i (1) of unit variance σ 2 = 1 in the Fock space F L 2 (R + ) over L 2 (R + ), rst de ne the mutually adjoint creation and annihilation processes A = A (t) t and A = (A (t)) t by their actions on exponential vectors A (t) e (f ) = d D E dz e f + zχ [,t[, A (t) e (f ) = χ [,t[, f e (f ). These satisfy the commutation relations h i h i A (s), A (t) = [A (s), A (t)] =, A (s), A (t) = s ^ t, (2) in the sense that for example, for arbitrary exponential vectors, D E A (s) e (f ), A (t) e (g) ha (s) e (f ), A (t) e (g)i = s ^ t he (f ), e (g)i De ne P and Q by P = i A A, Q = A + A Then (2) implies (1) and commutativity of the process P (likewise Q). (Institute) July 21, / 31

15 Moreover in the vacuum state e () they are both Brownian motions of unit variance in the sense of the following Theorem: Denote by (Ω, F, W) the standard Wiener probability space and by X = (X (t)) t the standard realisation on it of unit variance Brownian motion, so that for ω 2 Ω, X (t) (ω) = ω (t) and F is the σ- eld generated by the X (t). Then there exists a unique Hilbert space isomorphism D P (resp. D Q ) from the Fock space F L 2 (R + ) onto L 2 (Ω, F, W) with the properties D P e () (ω) (resp. D Q e () (ω) ) = 1 for all ω 2 Ω, D P P (t) DP 1 Q Q (t) D Q =1 = mult X (t) for all t 2 R + where mult X (t) denotes the operator of multiplication by X (t). But because P and Q don t commute with eachother they cannot be simultaneously diagonalized; D P 6= D Q. Note: Although they don t commute, they have a property amounting to stochastic Dindependence, namely E factorization D of E joint D characteristic E functions, e i(σ j x j P (s j )+Σ k y k Q (t k )) = e i Σ j x j P (s j ) e i Σ k y k Q (t k ), so we regard them as jointly a quantum planar Brownian motion. (Institute) July 21, / 31

16 Now assume σ 2 > 1. Then we can write σ 2 = α 2 + β 2 where α 2 β 2 = 1. In the tensor product F F of the Fock space F = F L 2 (R + ) with its dual Hilbert space, equipped with the unit vector e () e (), de ne P σ (t) = αp (t) Ī + βi P (t), Q σ (t) = αq (t) Ī + βi Q (t), where for example P (t) is the dual operator to P (t), P (t) ψ = P (t) ψ. Then, in the product state e () e (), αp Ī is a Brownian motion of variance α 2 and βi P is an independent Brownian motion of variance β 2 which commutes with it. Hence the sum αp Ī + βi P is itself a Brownian motion of variance α 2 + β 2 = σ 2. Similarly αq Ī + βi Q is a Brownian motion of variance σ 2. On the other hand [P σ (s), Q σ (t)] = α 2 [P (s), Q (t)] Ī + β 2 I [ P (s), Q (t)] = α 2 β 2 ( 2is ^ t) = 2is ^ t since [ P (s), Q (t)] = [P (s), Q (t)] = 2is ^ t = 2is ^ t. The (P σ, Q σ ) with σ > 1 is technically easier to work with than (P, Q) because e () e () is cyclic and separating for fp σ, Q σ g. (Institute) July 21, / 31

17 Quantum stochastic calculus. Brownian motion X is non-di erentiable. Intuitively this is because (for s > t) X (s) X (t) is N (, s t), so it is of order of magnitude p s t. So when you form the di erence quotient X (s) X (t) s t there can be no sensible limit as s! t. However the two components P and Q of quantum planar Brownian motion show some hints of di erentiability. For example Q = A + A and so, for exponential vectors e (f ) and e (g), he (f ), Q (t) e (g)i = ha (t) e (f ), e (g)i + he (f ), A (t) e (g)i D E D = χ, f [,t[ + χ [,t[, ge he (f ), e (g)i = ( f (s) + g (s)) he (f ), e (g)i which suggests that, at least for well-behaved f and g, dq (t) e (f ), e (g) = ( f (t) + g (t)) he (f ), e (g)i. dt (Institute) July 21, / 31

18 dq (t) Unfortunately this does not de ne an operator dt. But it does suggest a theory of operator-valued quantum stochastic integrals, of the form M (t) = F (s) da (s) + G (s) da (s) + H (s) dt (s) for which the " rst fundamental formula" he (f ), M (t) e (g)i = he (f ), (F (s) f (s) + G (s) g (s) + H (s)) e (g)i ds holds, provided that da (s) may be commuted through F (s) to reach the left of the inner product. Assuming that di erentials like da "point into the future" this requires that the integrand processes F = (F (t)) t are adapted, meaning that each operator F (t) is of form in so far as the Fock space F (t) = F t I t, F L 2 (R + ) = F L 2 ([, t[) F L 2 ([t, [) (Institute) July 21, / 31

19 Everyone agrees how to de ne the stochastic integral R t F (s) dl (s), L 2 A, A, T of an elementary adapted process F (t) = χ [a,b[ (t) F (a) = χ [a,b[ (t) F a I a ; it is R t F (s) dl (s) = F (a) (L (t ^ b) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only nitely many di erent values. (Institute) July 21, / 31

20 Everyone agrees how to de ne the stochastic integral R t F (s) dl (s), L 2 A, A, T of an elementary adapted process F (t) = χ [a,b[ (t) F (a) = χ [a,b[ (t) F a I a ; it is R t F (s) dl (s) = F (a) (L (t ^ b) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only nitely many di erent values. Note: The products of unbounded operators occurring here can be de ned rigorously on the exponential domain as tensor product operators. (Institute) July 21, / 31

21 Everyone agrees how to de ne the stochastic integral R t F (s) dl (s), L 2 A, A, T of an elementary adapted process F (t) = χ [a,b[ (t) F (a) = χ [a,b[ (t) F a I a ; it is R t F (s) dl (s) = F (a) (L (t ^ b) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only nitely many di erent values. Note: The products of unbounded operators occurring here can be de ned rigorously on the exponential domain as tensor product operators. The rst fundamental formula is easily proved for such integrands: (Institute) July 21, / 31

22 Everyone agrees how to de ne the stochastic integral R t F (s) dl (s), L 2 A, A, T of an elementary adapted process F (t) = χ [a,b[ (t) F (a) = χ [a,b[ (t) F a I a ; it is R t F (s) dl (s) = F (a) (L (t ^ b) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only nitely many di erent values. Note: The products of unbounded operators occurring here can be de ned rigorously on the exponential domain as tensor product operators. The rst fundamental formula is easily proved for such integrands: Theorem: For simple adapted processes F, G and H * + e (f ), F (s) da (s) + G (s) da (s) + H (s) dt (s) e (g) = he (f ), (F (s) f (s) + G (s) g (s) + H (s)) e (g)i ds. (Institute) July 21, / 31

23 The second fundamental formula is the heart of quantum stochastic calculus. It is a rule for expressing the product of two stochastic integrals as a sum of iterated stochastic integrals. The prototype is the rule for Brownian motion X : Z 2 Z (X (t)) 2 = dx (s) = 2 X (s) dx (s) + t s<t s<t Z Z = dx (s 1 ) dx (s 2 ) + dx (s 1 ) dx (s 2 ) s 1 <s 2 <t s 1 <s 2 <t Z + dt (s) (3) s<t where the unexpected additional "Itô correction" t = R dt s<t (s) is rather mysterious in classical Itô calculus. In the quantum generalization it can be seen as a consequence of the basic commutation relations between the annihilation and creation processes. We write (3) in di erential form as d X 2 = 2XdX + (dx ) 2 where (dx ) 2 = dt. To avoid multiplying unbounded operators, the second fundamntal formula places the two stochastic integrals on the two sides of the Hilbert space inner product between exponential vectors. (Institute) July 21, / 31

24 Thus let M j (t) = F j (s) da (s) + G j (s) da (s) + H j (s) dt (s), j = 1, 2 be two stochastic integrals where the F j, G j and H j are (for the moment at least) simple processes, which we assume have adjoint processes de ned on the exponential vectors. = Theorem : hm 1 (t) e (f ), M 2 (t) e (g)i fh(f 1 (s) ḡ (s) + G 1 (s) f (s) + H 1 (s)) e (f ), M 2 (s) e (g)i + hm 1 (s) e (f ), (F 2 (s) f (s) + G 2 (s) g (s) + H (s)) e (g)i + hf 1 (s) e (f ), F 2 (s) e (g)ig ds. Proof. By manipulation of the commutation relations between the creation and annihilation processes. (Institute) July 21, / 31

25 In di erential form the second fundamental formula becomes d MM = (dm) M + MdM + dmdm where M = M 1, M = M 2 and for dm = FdA + GdA + HdT and dm = F da + G da + H dt, dmdm is evaluated form the quantum Itô product table da da dt da, da dt dt or in the (dp, dq, dt ) basis, dp dq dt dp dt idt dq idt dt dt. Note: The latter table contains the classical rule (dx ) 2 = dt twice over, with X = P and X = Q. But it is manifestly noncommutative. (Institute) July 21, / 31

26 The real importance of the second fundamental formula is that it provides estimates which allow us to extend stochastic integration beyond simple processes as integrands. For example, putting g = f and M 1 = M 2 = M and putting G = H =, we get, for M (t) = R t F (s) da (s), = km (t) e (f )k 2 n 2 Re hf (s) f (s) e (f ), M (s) e (f )i + kf (s) e (f )k 2o ds. so d dt km (t) e (f )k2 = 2 Re hf (t) f (t) e (f ), M (t) e (f )i + kf (t) e (f )k 2. Using the Hilbert space inequality 2 Re hψ 1, ψ 2 i kψ 1 k 2 + kψ 2 k 2 we get d dt km (t) e (f )k2 jf (t)j 2 km (t) e (f )k kf (t) e (f )k 2. Multiplying by the integrating factor e F (s) da 2 (s) e (f ) 2e R t jf (s)j2 ds we obtain the estimate R t jf (s)j2 ds kf (s) e (f )k 2 ds. (Institute) July 21, / 31

27 This and similar (but simpler) inequalities for the creation and time integrals allow us to extend quantum stochastic integration from simple adapted integrands to adapted integrands F which are locally square-integrable in the sense that all seminorms of the form r kf k t,f = kf (s) e (f )k 2 ds are nite. The two fundamental formulas and the derived estimates hold for the extended integral. Compare this with classical Itô calculus where the extension is based on R t the isometry E F (s) dx (s) 2 = R t hjf E (s)j 2i ds. In the non-fock case σ 2 > 1 the extension uses two isometry relations F (s) da 2 σ (s) e () e () = α 2 F (s) e () e () 2 ds, 2 F (s) da σ (s) e () e () = β 2 F (s) e () e () 2 ds where σ 2 = α 2 + β 2, α 2 β 2 = 1. Thus the non-fock theory is simpler. (Institute) July 21, / 31

28 The gauge process. Stochastic integrals against the creation and annihilation processes are always martingales; 2 3 Z s E 4 FdA + GdA jn s 5 = FdA + GdA for s < t, where N s is the von Neumann algebra A (r), A (r) jr s Is the converse true? Can every martingale for the ltration of the quantum planar Brownian motion be represented as a stochastic integral? Answer: No in the Fock case σ 2 = 1, Yes in the non-fock case σ 2 > 1. In the Fock case we de ne the gauge process Λ = (Λ (t)) t by Λ (t) e (f ) = d dz e (ez χ [o,t[ f ). z= The gauge process, also called the number process, is sometimes attributed by physicists to V P Belavkin but in fact it originates with myself and Parthasarathy [1]. (Institute) July 21, / 31

29 Its matrix elements between exponential vectors are given by he (f ), Λ (t) e (g)i = f (s) g (s) ds he (f ), e (g)i. We compare this with corresponding formulas for e (f ), A (t) e (g) and he (f ), A (t) e (g)i which are f (s) ds he (f ), e (g)i, g (s) ds he (f ), e (g)i. And, just as A = R da and A = R da are martingales, so too is Λ, essentially because for s < t, if f and g vanish on [s, [ he (f ), Λ (t) e (g)i = he (f ), Λ (s) e (g)i because f (r) g (r) dr = Z s f But Λ cannot be expressed as a stochastic integral against da and da. (Institute) July 21, / 31

30 Instead we incorporate Λ as a new fundamental martingale in the Fock case. The rst fundamental formula becomes * = t Z e (f ), E (s) dλ (s) + F (s) da (s) + G (s) da (s) + H (s) dt (s) e (g) he (f ), (E (s) f (s) g (s) + F (s) f (s) + G (s) g (s) + H (s)) e (g)i ds. The quantum Itô product table becomes dλ da da dt dλ dλ da da da da dt dt In this way the classical Poisson process of rate l can be incorporated into quantum stochastic calculus as Π l = Λ + p la + p la + lt, for which (dπ l ) 2 = dπ l. But in the non-fock case when σ > 1 there is no gauge process. (Institute) July 21, / 31.

31 Quantum stochastic di erential equations for unitary processes. In physical applications of quantum stochastic calculus the physics usually takes place in a separate "initial" Hilbert space H which is coupled to the "noise" space carrying the calculus as a Hilbert space tensor product. Consider the quantum stochastic di erential equation (id d) U = U E dλ + F da + G da + H dt where E, F, G and H are bounded operators on H. Writing E dλ + F da + G da + H dt = 4 E j dλ j j=1 we always interpret such sde s as the equivalent integral equation U (t) = I + U (s) 4 j=1 E j dλ j (s)., U () = I (Institute) July 21, / 31

32 We may try to solve this iteratively, starting with U (t) I and U N (t) = I + U N 1 (s) 4 j=1 E j dλ j (s). Thus U 1 (t) = I + + U N (t) = I + Z 4 j=1 4 j 1,j 2 =1 N M =1 E j dλ j (s), U 2 (t) = I + 4 j=1 E j dλ j (s) Z E j1 E j2 dλ j1 (s 1 ) dλ j2 (s 2 ),..., <s 1 <s 2 <t 4 j 1,j 2,...,j M =1 <s 1 <s 2 <<s M <t E j1 E j2...e jm dλ j1 (s 1 ) dλ j2 (s 2 )...dλ jm (s M ) (Institute) July 21, / 31

33 Provided that the coe cient operators E j are bounded you can then use estimates for iterated stochastic integrals derived from the fundamental estimates to show that the limit U (t) = I + Z N =1 4 j 1,j 2,...,j N =1 <s 1 <s 2 <<s N <t E j1 E j2...e jn dλ j1 (s 1 ) dλ j2 (s 2 )...dλ jn (s N ) is a well de ned operator on the span of vectors of form ψ e (f ), and that it is a solution of the original sde and that the solution is unique. If we start with the corresponding "left-driven" sde (id d) V = EV dλ + FV da + GV da + HV dt, V () = I we obtain similarly the solution V (t) = I + Z N =1 4 j 1,j 2,...,j N =1 <s 1 <s 2 <<s N <t E jn E jn 1...E j1 dλ j1 (s 1 ) dλ j2 (s 2 )...dλ jn (s N ). (Institute) July 21, / 31

34 When is the solution U of (id d) U = U E dλ + F da + G da + H dt, U () = I unitary-valued? The adjoint process U satis es (id d) U = E U dλ + G U da + F U da + H U dt, U () By applying the Leibniz-Itô formula d (MM ) = (dm) M + MdM + dmdm to the products UU and U U using the quantum Itô product rule, we nd that the conditions E + E + EE =, F + G + EG =, G + F + GE =, H + H + GG = are su cient for coisometry (UU = I ) while the conditions E + E + E E =, G + F + E F =, F + G + F E =, H + H + F F = are necessary for isometry (U U = I ). (Institute) July 21, / 31

35 But by using backward-adapted quantum stochastic calculus, which is completely equivalent to forward adapted for iterated integrals such as occur in the solutions above, you can show that both conditions are both necessary and su cient [2]. Both are equivalent to the condition (E, F, G, H) = S 1, L, SL, ih 1 2 L L, S unitary, h self-adjoint which I understand is known nowadays in the physics communiity as the (S, L, h) formalism. (Institute) July 21, / 31

36 (Institute) July 21, / 31

37 Quantum stochastic di erential equations for ows. (Institute) July 21, / 31

38 References. R L Hudson and K R Parthasarathy, Quantum Ito s formula and stochastic evolutions, Communications in Mathematical Physics 93, (1984). R L Hudson, Forward and backward adapted quantum stochastic calculus and double product integrals. Submitted to Russian Journal of Mathematical Physics, V P Belavkin Memorial Volume. (Institute) July 21, / 31