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Open Dynamical Systems for Beginners: Algebraic Foundations

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Open Dynamical Systems for Beginners: Algebraic Foundations 1 ∗ 2 Open dynamical systems for beginners: algebraic foundations Rolando Rebolledo B. Facultades de Ingenier´ıay Matem´aticas Pontificia Universidad Cat´olicade Chile [email protected] ∗ 2 Introduction Open system theory is deeply connected with stochastic analysis foundation in both, commutative and non-commutative versions. Biologist von Bertalanffy pointed out in 1950 the importance of defining living matter as an open-dynamical-system. In parallel, physicists working on radiation theory introduced quantum dynamical semigroups to model the interaction between a system and its reservoir. And the open-system point of view invaded numerous fields like Finance, supported by stochastic differential equations, as well as Markov processes and many other probabilistic branches. These lectures provide a panorama of the algebraic setting which allows to synthesize commu- tative and non-commutative open-system dynamics via semigroup theory. I do not suppose any previous knowledge of Quantum Mechanics and will not develop applications to Physics which have been the object of a number of research papers and books, some of them quoted in the references. Our goal is simply to explain the passage from the customary classical Markov Theory to the non commutative one. I have in mind that the reader has a basic knowledge of classical Stochastic Analysis. Markov Theory is especially well adapted to deal with classical open systems, which are described by stochastic differential equations. Stochastic differential equations, Markov pro- cesses, Markov semigroups are all connected with mathematical descriptions of open systems in classical Physics. So do quantum stochastic differential equations, quantum flows and quantum Markov semigroups, which are mathematical descriptions of open quantum systems. In both cases one looks for a memoryless approach to the dynamics of a composed system. However, although their similarities classical and quantum Markov semigroups have a deep difference: observables and dynamics have to suitably describe the Uncertainty Principle in the quantum case. This forces us to deal with non-commutative stochastic analysis and non-commutative geometry, both deeply founded in operator algebras. Contents 1 From Newton to Langevin 5 1.1 The classical flow . .5 1.2 The algebraic flow . .6 1.3 Semigroups . .7 1.4 Opening the main system . .7 1.5 Introducing probabilities . .9 2 An algebraic view on Probability 11 2.1 The basic closed quantum dynamics . 11 2.2 Algebraic probability spaces . 12 3 Completely positive and completely bounded maps 17 3.1 From transition kernels to completely positive maps . 17 3.2 Completely bounded maps . 22 3.3 Dilations of CP and CB maps . 23 4 Quantum Markov Semigroups and Flows 29 4.1 Semigroups . 30 4.2 Representation of the generator . 30 4.3 Conclusions and outlook. An invitation to further reading . 37 3 4 CONTENTS Chapter 1 From Newton to Langevin We denote by Σ ⊆ R3 × R3 the state space of a single particle mechanical system, that is, each element x = (q; p) = (q1; q2; q3; p1; p2; p3) 2 Σ corresponds to the pair of position and momentum of a particle, which is supposed to have mass m. In Newtonian Mechanics, the dynamics is entirely characterised by the Hamilton operator which, in the homogeneous case, can be written as 1 H(x) = jpj2 + V (q); (x 2 Σ); (1.0.1) 2m where |·| denotes here the euclidian norm in R3. This allows to write the initial value problem which characterises the evolution of states as 8 q0 = @H (x); <> i @pi 0 @H pi = − @q (x); 1 ≤ i ≤ 3; (1.0.2) > i : x(0) = x0: If I denotes the 3 × 3 identity matrix call 0 I J = : (1.0.3) −I 0 Then (1.0.2) becomes 0 x = JrH(x); x(0) = x0; (1.0.4) where r is the customary notation for the gradient of a function. 1.1 The classical flow Let denote t 7! θt(x0) the solution of (1.0.2), so that θ0 = id and θt(θs(x0)) = θt+s(x0). When t varies on R, the family (θt)t2R is a group of transformations of Σ, known as the (classical) flow of solutions. The orbit of an element x0 2 Σ is θ(x) = (θt(x0))t2R . We will construct a different representation of our system, which will prepare notations for the sequel. Call Ω = D([0; 1[; Σ) the space of functions ! = (!(t); t ≥ 0) defined in [0; 1[ 5 6 CHAPTER 1. FROM NEWTON TO LANGEVIN with values in Σ, which have left-hand limits (!(t−) = lims!t; s<t !(s)) and are right-continuous (!(t+) = lims!t; s>t !(s) = !(t)) on each t ≥ 0 (with the convention !(0−) = !(0)). Define Xt(!) = !(t) for all t ≥ 0. So that for each trajectory ! 2 Ω, and any time t ≥ 0, Xt(!) is the state of the system at time t when it follows the trajectory !. We can write 3 Xt(!) = (Qt(!);Pt(!)), where Qt;Pt :Ω ! R represent respectively, the position and momentum applications. These are particular examples of stochastic processes. Thus, (1.0.2) may be written dXt(!) = JrH(Xt(!))dt; X0(!) = x0: There is no great change in this writing of the equations of motion, however let us agree that such expression is a short way of writing an integral equation that is Z t Xt(!) = x0 + JrH(Xs(!))ds: 0 Solutions of the above equation are obviously continuous and differentiable. Moreover they preserve the total energy of the system: H(Xt(!)) = H(x0); (1.1.1) for all t ≥ 0. 1.2 The algebraic flow The set of bounded continuous functions (one may consider also bounded measurable functions) from Σ to C (respectively, from Ω to C), constitute an algebra A, (resp. B). These algebras are endowed with an involution operation ∗ which associates to each function f its conjugate f¯. We define the algebraic flow associated to (1.0.2) by the map jt : A ! B given by jt(f)(!) = f(Xt(!)); (t 2 R;! 2 Ω; f 2 A): (1.2.1) Notice that jt(f) satisfies a differential equation as well, since by the chain rule for any differ- entiable function f: djt(f) = hrf(Xt); dXti = hrf(Xt);JrH(Xt)idt = fH; fg (Xt)dt; where fu; vg = hru; Jrvi = P3 @u @v − @u @v for two differentiable functions u; v :Σ ! i=1 @qi @pi @pi @qi C is the so-called Poisson-bracket. So that, the equation of the flow, for all differentiable function f, can be written as djt(f) = jt(fH; fg)dt; j0 = id: (1.2.2) 1.3. SEMIGROUPS 7 1.3 Semigroups For each x 2 Σ, the orbit θ(x) is -as we explained- the solution of (1.0.2) with starting point x. Call δθ(x) the Dirac measure supported by fθ(x)g, so that δθ(x)(A) = 1 if and only if θ(x) 2 A. On the algebra A introduced before, we can define Z Z Ttf(x) = jt(f)(!)δθ(x)(d!) = f(Xt(!))δθ(x)(d!); (1.3.1) Ω Ω for all t 2 R, f 2 A and x 2 Σ. Notice that for each s; t 2 R, Tt(Ts(f)) = Tt+s(f), T0(f) = f, so that (Tt)t2R is a group. This property is due to the fact that the system is conservative so that for a given t ≥ 0, T−t is the inverse of Tt, which is characteristic of reversibility. In which follows we will introduce dissipation in our model so that the energy will not be conserved and the system will become irreversible. Thus, instead of taking time running over all the real numbers, we will consider t 2 R+. In this + A case (Tt)t2R is no more a group, but a semigroup of maps acting on . This semigroup admits a generator L defined as 1 lim (Ttf − f) ; t!0 t for all f for which this limit exists (in the pointwise sense, for instance). That is, Z t Ttf(x) = f(x) + L(Tsf(x))ds: 0 A way to formally recover the above expression is Z dTt djt Lf(x) = f(x)jt=0 = f(Xt)jt=0δθ(x)(d!); dt Ω dt that is, Lf(x) = fH; fg (x); (1.3.2) for all x 2 Σ whenever we take as domain of the generator the set D(L) of all C1-functions with bounded derivatives. 1.4 Opening the main system Our basic space of trajectories Ω allows discontinuities. Thus, we may modify our simple model by introducing kicks. Assume for instance that at a given time t0 the particle collides with another object which introduces an instantaneous modification (force) on the momentum. Mathematically that variation on the momentum is given by a jump at time t0, that is ∆Pt0 (!) = Pt0 (!)−Pt0−(!). From the physical point of view, we have changed the system: we no more have a single particle but a two-particle system. In the new system the jump in the momentum of the first particle is (−1)× the jump in the momentum of the second particle via the law of conservation of the momentum. Suppose that the magnitude of the jump in the momentum of the colliding particle (the instantaneous force) is c > 0, and call ξ(!) its sign, that is ξ(!) = 1 if the main particle is pushed forward, ξ(!) = −1 if it is pushed backwards. We then have ∆Pt0 (!) = ξ(!)c = c∆Vt0 (!); 8 CHAPTER 1. FROM NEWTON TO LANGEVIN where Vt(!) = ξ(!)1[t0;1[(t) and 1[t0;1[ is the characteristic function of [t0; 1[ (or the Heaviside function at t0).
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