Davies E. B., Quantum theory of open systems. Academic Press, London 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... is prepared, the positions x of the particles are distributed according to the quantum equilibrium measure IP with density ae = j j ( normalized) 18] Nor are they given by a positive operator valued measure (POV) which has been proposed as a generalized quantum observable, see [16] In particular, the continuity equation for the probability shows that the probability flux (j t j ; j t j ) is conserved, since j t j = j . Hence, given t , the solutions x(t; x 0 ) of equation (8) are random trajectories, where the randomness comes from the IP ....

E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, New York, San Francisco, 1976).

....needed) A second motivation for our approach is that the reformulation (1.10) becomes impossible for open quantum systems, since the eigenvalues j would then be time dependent. Open quantum systems are important in many elds of applications (quantum di usion, coupling to a heat bath, cf. 14] [15], 16] and references therein) and they are modelled by augmenting the right hand side of (1.1) by interaction terms of Lindblad form (cf. 17] 18] In our subsequent analysis we shall not include such (bounded) Lindblad operators, but they would not pose any additional analytical problems. ....

E. B. Davies, Quantum Theory of Open Systems (Academic Press, 1976).

.... The proper framework of such sort of di#usive models is that of open quantum systems, i.e. an ensemble of electrons interacting with a heat bath (an infinite set of harmonic oscillators in thermodynamic equilibrium) that can exchange matter (conserved particles) with their environment (see [4] [7], 11] 6] The quantum Wigner Fokker Planck equation reads x )W #[V ]W # # W 2#div # (#W ) 2 div x (# # W ) D qq # x W , 1) where W is the (quasi) probability distribution function, D pp , D pq , D qq , m and # are physical constants and #[V ]W is the (quadratic) nonlinear ....

Davies, E. B. "Quantum Theory of Open Systems," Academic Press, New York, 1976.

....and Control (PhysCon 2003) August 20 22, St. Petersburg, Russia. Electronic Mail: maxim ece.northwestern.edu 1 Introduction and background In quantum information theory [1] all admissible devices are described mathematically by means of the so called quantum operations (or quantum channels) [2, 3]. Given a complex Hilbert space H, the # algebra of all bounded operators on H. In this paper we will work primarily with finite dimensional Hilbert spaces, so that includes all linear operators on H. Given Hilbert spaces 2 , a quantum channel T is a completely positive ....

E.B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.

.... positive maps, quantum operations, quantum channels, noncommutative RadonNikodym theorem Electronic Mail: maxim ece.northwestern.edu 1 Introduction In the mathematical framework of quantum information theory [18] all admissible devices are modelled by the so called quantum operations [9, 20] that is, completely positive linear contractions on the algebra of observables of the physical system under consideration. Thus it is of paramount importance to have at one s disposal a good analysis toolkit for completely positive (CP) maps. There are many useful structure theorems for CP ....

....This is accomplished by means of theorems of the Radon Nikodym type, as in the case of, e.g. partial ordering of positive measures or positive linear functionals. There are a number of Radon Nikodym theorems for CP maps (see, e.g. the work of Arveson [1] Belavkin and Staszewski [2] Davies [9], Holevo [14] Ozawa [24] and Parthasarathy [26] that di er widely in scope and in generality. Thus, the results of Davies, Ozawa, and Holevo have to do with RadonNikodym derivatives of CP instruments [24] with respect to scalar measures. On the other hand, ideas common to the Arveson and ....

E.B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).

....F x F y = xy F x . Example 2.2.9 (irreversible quantum dynamics) In Example 2.2.5, we have considered the case of unitarily implemented channels. Such channels arise whenever we talk about 18 reversible quantum dynamics. A general theory of irreversible quantum dynaimcs proceeds as follows [31]. The system, initially in some state 2 B(H ) is brought into contact with another system, the reservoir, initially in some xed state R 2 B(K ) where K is the Hilbert space of the reservoir. The combined system reservoir entity is assumed to be closed. Then the two are caused to ....

....and Kraus Up to this point, our treatment of channels has been largely axiomatic. However, we can adopt the pragmatic point of view and demand that only those transformations that can be built up from certain basic blocks can serve as channels. We take our cue from quantum theory of open systems [31] and say that any physically acceptable channel can be realized as a sequence of the following steps: a) adjunction of an auxiliary system (called the ancilla in the terminology of Helstrom [58] in some xed initial state, b) unitarily implemented evolution of the enlarged system, and (c) ....

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E.B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).

....1 Introduction In this paper we analyze a class of dissipative quantum systems, modeling the motion of a particle ensemble, say electrons, interacting with a heat bath of oscillators. The resulting irreversible dynamics for the electrons, is a typical example of a so called open quantum system [Da], i.e. a system in which the interaction with the environment is taken into account. The evolution equation, sometimes called Master equation, for the density (matrix) operator R(t) of the particles reads dt R = H; R] A(R) 1.1) R(t = 0) R 0 ; 1.2) where H is the self adjoint ....

....the positivity, hermiticity and the normalization (unit trace) of the density operator R. However, for unbounded operators L, which is the case in our work, the so called Lindblad condition is necessary but not sucient to guarantee the conservation of these properties (see, e.g. AlLe] CF] [Da] and the references therein) For a most recent review article on this subject see [Al] Using the Wigner transform [Wi] dissipative quantum models can be equivalently represented in phase space, resulting in a kinetic transport equation with interaction terms for the quasiprobability ....

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E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).

....section is devoted to a brief review of the basic ideas and di erent approximations involved from ab initio in the derivation of master equations for open quantum systems coupled to in nite reservoirs. Excellent books on the subject are available where the reader can nd more details (see e.g. [7, 1, 3, 32]) Even though, for the sake of clarity we feel necessary to state here our basic approximations and assumptions on times scales, to precise for which physical systems the kinetic models of the next section could be relevant. 2.1. Starting from Quantum Statistical Mechanics. Within the framework ....

E.B. Davies, Quantum Theory of Open Systems, Academic Press (1976).

....and Cipriani [Cip] have extended the abstract theory to a faithful normal state OE 0 in the context of standard forms. The need to construct Markovian semigroups on von Neumann algebras, which are symmetric with respect to a non tracial state, is clear for various applications to open systems [Dav], quantum statistical mechanics [BR] and quantum probability theory [Acc, AFL, Part] Although on an abstract level we have quite well developed theory as mentioned above, the progress in concrete application is very slow. We would like to mention a few recent works in this direction. The ....

E.B. Davies, Quantum theory of open systems, Academic Press, London-New York-San Francisco (1976).

....Introduction We investigate the problem of the localization of a free quantum particle moving in a one dimensional box with periodic boundary conditions, adopting the point of view that observables are represented as appropriate normalised positive operator measures. For that approach, see, e.g. [1, 2, 3, 4]) Therefore, if one chooses the one dimensional torus T as the con guration space of the system, then a localization observable E is a map that de nes for any (Borel) subset X T a bounded operator E(X) such that, if is the (vector) state of the system, the number h jE(X) i is the probability ....

Davies, E.B.: Quantum Theory of Opens Systems, Academic Press, London, 1976.

....tr = X i2I j i ; T i j; T 2 T (H) where f i ; i 2 Ig is an ONB of H, then kaek tr = Z R n j (x)j 2 dx: 1.4) It is well known that ae governs the classical Liouville equation i ae = H; ae] ae(0) ae 0 (1. 5) on which the quantum theory of open system is based (see [A L] and [Da] ) The solution of (1.5) can be expressed by so called quantum dynamical semigroup S(t) ae(t) S(t)ae 0 = e GammaitH ae 0 e itH (1.6) which is a unitary group on T (H) There is also another link between quantum theory of the open systems and the theory of statistical mechanics. Let us ....

E.B. Davies, Quantum Theory of Open Systems, Acad. Press, London, New York, 1976.

....in terms of measurements that return two outcomes, whereas the Kolmogorov distance K can be defined for an arbitrary number of outcomes. Let us show that, if the inequality K 4ffl holds for any binary outcome measurement, the same inequality holds for an arbitrary measurement. It is shown in [6,14] that the most general measurement on the n photons that is allowed by quantum mechanics can be described by operators M (B) 1 ; M (B) k given by equations M (B) j = P j U where U is an isometry from the space of the n photons to some other Hilbert space H and the operators P j are ....

...., both together, determine an overall measurement M 1 on the overall system. The classical outcome of this overall measurement is denoted y = c; c) This measurement is determined by an isometry U 1 and projection operators P 1;y that define an orthogonal measurement on the image of U 1 [6,14]. We have that M 1;y = P 1;y U 1 is the collapse operator associated with the outcome y. We have Pr(X = y j Phi = OE) kM 1;y jOEik 2 and Pr(T 1 (Y ) ok j Phi = OE) kM 1;ok jOEik 2 , where M 1;ok = P y ; T 1 (y) ok M 1;y . We obtain: j Pr(T 1 (Y ) ok j Phi = OE 0 ) Gamma Pr(T 1 ....

E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976),

....linear maps T : A#A such that T (AB) T (A)T (B) and T (A # ) T (A) # . For a pure quantum system the symmetries are precisely the unitarily implemented maps, i.e. the maps of the form T (A) UAU # ,withU a unitary element of A. To readers familiar with Wigner s Theorem (e.g. Corollary 3.3. in [22]) another class of maps is conspicuously absent here, namely positive maps of the form T (A) #A # # # with # anti unitary.Itiswellknown that due to the positivity of energy a time reversal symmetry can only be implemented by such an anti unitary transformation. But since such symmetries are not ....

....we get a composite output algebra C(X) # ,whereX is the set of classical outcomes of a measurement, and B describes the output systems, which can be of a di#erent type in general form the input systems with observable algebra A. The term instrument for such devices was coined by Davies [22]. As in the case of observables, it is convenient to expand in the basis e x of the classical algebra. Thus T : C(X) # #Acan be considered as a collection of maps T x : B#A, such that T (f# B) # x f(x)T x (B) The conditions for T x are T x : B#A completely positive, and # x ....

E.B. Davies, Quantum theory of open systems, (Academic Press, 1976)

....notation and terminology: When H is a Hilbert space, we denote by B(H) the space of bounded linear operators on H. A channel converting quantum systems with Hilbert space H in into systems with Hilbert space H out is a linear operator T : B(H out ) B(H in ) which is completely positive [Da, Pa] and normalized as T(1I) 1I. A (discrete) observable F on H over an output parameter space X is a collection of positive op3 erators F x 2 B(H) such that P x F x = 1I. A density operator on H is a positive operator with trace 1. The basic probabilistic interpretation of these objects is fixed ....

....a factor ( 2 C . But then every vector is an eigenvector of the linear operator K ff , which is only possible, if K ff is a multiple of the identity. A collection of completely positive maps adding up to a normalized one should be understood as an instrument in the terminology of Davies [Da], i.e. a device which produces classical measurement results k , such that the probability for obtaining this result and a response to a subsequent measurement F on an input state ae is tr(aeT k (F ) The channel P k T k then describes the overall state change, when the measuring results are ....

E.B. Davies, Quantum theory of open systems, Academic Press, 1976

.... = we have for the emergence of an effective dynamics, to mention only the Boltzmann and Vlasov equation, hydrodynamics [21] homogenization i= n periodic and random environments [3, 8] interface and vortex dynamics in Ginzburg Landau theories [11] quantum systems weakly coupled to = a heat bath [6], and a quantum particle in the semiclassical limit = ite Ha,Ro,Sprl. Their common thread is a separation of space time scales together with some sort of local stationarity in such a way that the slow= ly varying dynamical variables are governed by an effective dynamics. Howeve= r, the detailed ....

E.B. Davies, "Quantum Theory of Open Systems". Academic Press, London, 1976.

....semigroup on H. 1 Applying our result we are able to construct translation invariant Markovian semigroups for quantum spin systems with finite range interactions. The efforts in construction and study of quantum Markov semigroups are as old as the equilibrium description of quantum systems [Dav]. The study of non commutative Dirichlet forms was pioneered by Albeverio and Hegh Krohn [AH K] Sauvageot [Sau 1,2,3] and extensively developed by Davies and Lindsay [DL] and Guido, Isola and Scarlatt [GIS] All these authors considered Markovianity of forms and semigroups only with respect to a ....

....[Cip 1,2] have extended the theory to a faithful normal state OE 0 in the context of standard forms. The need to construct continuous Markovian semigroups on von Neumann algebras, which are symmetric with respect to a non tracial state, is clear for various applications to open quantum systems [Dav], quantum statistical mechanics [BR] and quantum probability [Acc,Par] Although on an abstract level we have quite well developed theory as stated above, the progress in concrete application is very slow. We would like to mention few recent works in this direction. The completely positive ....

E. B. Davies, Quantum theory of open systems, Academic Press, London-New York-San Francisco, (1976).

.... Hartree approximation interacting dissipatively with an idealized heat bath consisting of an ensemble of harmonic oscillators [7] This system generally fits the framework of open quantummechanical systems, present in a wide range of situations in quantum statistical mechanics [21] [15], where the particle background interaction is important. Dissipative phenomena play a relevant role in microelectronics, essentially through the modeling of quantum transport of charge carriers in quantum semiconductor devices ( 23] 35] Some other significant fields of application are quantum ....

Davies, E. B. Quantum Theory of Open Systems. Academic Press, New York, 1976.

....and information theory, quantum tomography, quantum optics, and quantum measurement theory. Also many conceptual problems, like the problem of joint measurability of noncommutative quantities, or the problem of classical limit of quantum mechanics have greatly advanced by this tool. The monographs [1, 2, 3, 4, 5, 6, 7] exhibit various aspects of these developments. Any positive trace one operator T (a state) defines a phase space observable Q T according to the rule Q T (E) 1 2# # E e i(qP pQ) T e i(qP pQ) dq dp, where E is a Borel subset of the (two dimensional) phase space. It is well known that all ....

E.B. Davies, Quantum Theory of Open Systems, Academic Press, New York, 1976.

....2. POSITIVE AND COMPLETELY POSITIVE MAPS We start by recalling and establishing some definitions and facts related to trace ideals and to the concept of positive and completely positive maps ( for more details on trace ideals see e.g. 31, 32] and on positive and completely positive maps e.g. [33, 34, 5, 6, 35, 26]) Let H be any complex, separable Hilbert space with scalar product denoted by h ; i. By B(H) we denote the C algebra of all bounded operators on H equipped with the norm jj jj. By I2 B(H) we denote the identity map on H. For any A 2 B(H) we set jAj = A A) 1=2 2 B(H) where ....

....map in J p (1 p 1) is continuous, i.e. it satisfies jj jj p 1. Here we denote by jj jj p the norm of any continuous linear map in J p , such that jj jj p = sup jjAjj p 1 jj (A)jj p : 4) In particular jj jj 1 = sup A2C1;1 Tr( A) 5) if is completely positive on J 1 (see e.g. [6, 22]) Proof. We adapt a standard proof (see e.g. 26] p. 19) used in the context of positive maps on non unital C algebras. We first claim that it suffices to prove boundedness on C p . Indeed, this follows from the decomposition (2) and the related bounds. Assume that is not bounded on C p ....

E.B. Davies, Quantum Theory of Open Systems, Academic Press, New York, 1976.

....changes state in some stochastic manner. For physical reasons, these changes are subject to certain constraints. A careful mathematical analysis shows that the only possible transitions are described by quantum instruments satisfying the further condition of complete positivity (Kraus, 1983, Davies, 1976). Let B(H) be the space of bounded linear operators on the Hilbert space H. If H has dimension d then we may think of elements of B (H) as complex d Theta d matrices. We shall consider the space F of linear maps of B(H) into itself. An element Phi of F is positive if Phi[SA (H) ae SA (H) ....

....to the set of self adjoint operators on H. The prototypical examples of quantum transformation models are of the form (S(H) OProM(X ; H) together with actions of G satisfying (4.7) and ae 7 U g aeU g . The main result on the structure of equivariant measurements is the following. Theorem 3 (Davies, 1976; Holevo, 1982) Let G be a Lie group with a measure which is both left and right invariant (so that G is unimodular) Let g 7 U g be a continuous projective unitary action of G on a Hilbert space H. Let H be a closed subgroup of G and put X = G=H. Then G acts on X on the left in the usual ....

Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London 1976.

....can be described as a nite weighted mean of properties of nitely many elementary events. Probabilities are introduced in a generality supporting so called e ects, a sort of fuzzy events (related to POV measures that play a signi cant role in measurement theory; see Busch et al. 8, 9] Davies [14], Peres [62] The weak law of large numbers provides the relation to the frequency interpretation of probability. As a special case of the de nition, one gets without any e ort the well known squared probability amplitude formula for transition probabilities. States are de ned as partial mappings ....

E.B. Davies, Quantum theory of open systems, Academic Press, London 1976.

.... algebra B(H) of all bounded operators on the Hilbert space H, and L(B(H) is the lattice of projections, then there does not exist a conditional expectation from B(H) onto a subalgebra B 0 , if B 0 is generated by the spectral projectors of a selfadjoint operator A having non discrete spectrum ([Davies 1976] Chapter 4.3) Translated into the language of quantum physics, where L(B(H) describes the simplest case of a quantum event structure and A represents an observable physical quantity with non discrete spectra, the non existence of conditional expectation means that it is far from clear, what ....

: E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976

.... we have for the emergence of an effective dynamics, to mention only the Boltzmann and Vlasov equation, hydrodynamics [21] homogenization in periodic and random environments [3, 8] interface and vortex dynamics in GinzburgLandau theories [11] quantum systems weakly coupled to a heat bath [6], and a quantum particle in the semiclassical limit [10, 18, 22] Their common thread is a separation of space time scales together with some sort of local stationarity in such a way that the slowly varying dynamical variables are governed by an effective dynamics. However, the detailed mechanisms ....

E.B. Davies, "Quantum Theory of Open Systems". Academic Press, London, 1976.

.... 4 this if. then inference is describable as the dual of the Mittelstaedt s if. then conditional. Proposition 1 shows that this is not the case if L is a conditional expectation. However, as mentioned, for an arbitrary M 0 a conditional expectation onto M 0 may not exist (see e.g. ref. [13]) It is therefore natural to ask whether (3) breaks down also in the case where the conditional probabilities are given by (LB) with L as generalization of conditional expectation. However, even if L is not a conditional expectation, a minimal requirement of L is that 0 L be a state, i.e. a ....

E.B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976)

....and fermions. Non commutative probability theory has grown into a substantial branch of analysis with a number of physical applications. The mathematical theory is reviewed and developed in [Me85] and [Me86] while other sorts of physical applications, besides those discussed here, are treated in [Da76] and [HuPa] for example. It is a pleasure to thank Leonard Gross for discussing his results and conjectures with us, and for encouraging us to take up the latter. We are indebted to Keith Ball for his collaboration on the subject of convexity inequalities that led to Theorem 1 [BCL] which is one ....

Davies, E.B.: Quantum Theory of Open Systems, Academic Press, New York, 1976.

....a place where its problems are less damaging to our understanding. Since interaction with a macroscopic device is involved, it must be treated independent of foundational problems as a question in the thermodynamics of open systems. This has been done repeatedly in the literature; e.g. Davies [9], Busch et al. 6] Zurek [65] Joos Zeh [26] Ghirardi, Rimini Weber [17] The dynamics of states is deterministic, and the weak equations turn out to be elements of physical reality in the sense of Einstein, Podolsky Rosen [14] But there is no need to introduce a classical framework with ....

E.B. Davies, Quantum theory of open systems, Academic Press, London 1976.

....works are noted, but may be ignored by the reader. Sect. 1 formulates notions and results. Sect. 2 develops a new criterion of linearizability for continuous tensor products of probability spaces. Sect. 3 relates unitary Brownian motions to the theory of continuous quantum measurements (see Davies [11]) Sect. 4 establishes local finiteness of the corresponding quantum stochastic process, which implies linearizability. It is interesting to observe quantum probability helping to classical probability. Naturally, driving forces behind the matter should be more intelligible for readers acquainted ....

....interesting to observe quantum probability helping to classical probability. Naturally, driving forces behind the matter should be more intelligible for readers acquainted with continuous tensor products of Hilbert spaces and probability spaces [2, 13, 3, 4, 34] continuous quantum measurements [11, 22, 7, 1], the relation between the former and the latter [6] and quantum stochastic calculus [25, 26, 15] Brownian motions in a linear topological space are evidently Gaussian, if the space possesses sufficiently many continuous linear functions. However, L p for 0 p 1 is an example of a separable ....

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E.B. Davies, Quantum theory of open systems, Academic Press, London, 1976.

....B n 6= P n i O( i ) Thus for nonrepeatable measurements it is not possible to cast the information about the entire statistics into a bilinear form involving a single self adjoint operator. POV s have been proposed as a means of providing a generalized description for fuzzy measurements [6]. Note, however, that POV s arise naturally from a measurement analysis in Bohmian mechanics, in which there is no intrinsic fuzziness. 6 The Phase Problem in Quantum Optics 6.1 A brief history of the phase operator For the following discussion it will be sufficient to focus on a single mode ....

E. B. Davies, Quantum Theory of Open Systems, (Academic Press, London---New York---San Francisco, 1976).

.... regard we must emphasize that the following question arises for quantum orthodoxy, but does not arise for Bohmian mechanics: For precisely which theory is the quantum formalism an idealization (For further elaboration on this point, as well as for a discussion of how generalized observables (Davies, 1976) naturally arise in Bohmian mechanics, see Durr et al. 1996 and Daumer et al. 1996. 7 We should probably distinguish two senses of primitive : i) the strongly primitive variables, which describe what the theory is fundamentally about , and ii) the weakly primitive variables, the basic variables ....

Davies, E.B.: 1976, Quantum Theory of Open Systems, Academic Press, London.

.... algebraic statement whose clarity is compelling, and the subsequent veri cation of its violation through experiments by Aspect [1] Finally, it seems that measurement problems can be adequately analysed by generalized observables de ned as positive operator valued (POV) measures; see, e.g. Davies [13], Busch et al. 10] The gradual and approximate state reduction can be explained thermodynamically through dissipative interaction by the coupling to a macroscopic apparatus or heat bath; see, e.g. Zurek [40] Joos Zeh [18] Ghirardi, Rimini Weber [15] That obtaining information about a ....

....quantum logic calculus. But what sense should it make to talk about the probability of a statement that is not even logically well formed It seems to me that the only formula for probabilities veri ed (abundantly) by experiments is the formula (see, e.g. equation (1. 8) in Chapter 3 of Davies [13]) P (B 2 Ej ) tr B(E) 6) for the probability that, in a mixed state with the density matrix , the generalized observable (POV measure) B has a value from a set E. For ordinary observables B and singleton sets E = fbg, the e ect B(E) reduces to a projector b . Equation (6) contains a ....

E.B. Davies, Quantum Theory of Open Systems, Academic Press, London 1976.

....let j be the corresponding eigenvectors. We will colloquially refer to A as an atom or a small system. Let B be an infinite heat bath. In this paper B will be an infinite free Bose gas at inverse temperature fi = 1=kT without Bose Einstein condensate. The system B is described (see e.g. BR] [D], D1] by a triple fHB ; Omega B ; HB g, where HB is a Hilbert space, HB is a self adjoint operator on HB and Omega B is a unit vector in HB . There is a representation of CCR on HB , WB (f) exp(i B (f) f; f= p 2 L 2 (R 3 ) 1.1) where B (f) are field operators, satisfying ....

....The extensively studied spin boson Hamiltonian also has the form (1.8) When the heat bath is at positive temperature, the Hilbert space of the joint system is H = HA Omega HB and the Hamiltonian is formally given by H = HA Omega I Q Omega B (f) I Omega HB = HA H I HB ; 1. 9) see [D], D1] PU] H] BR] In Section 3 we will prove that if f= p ; f 2 L 2 (R 3 ) then H is essentially self adjoint on HA Omega D(H bos ) Omega D(H bos ) However, H I is not a relatively bounded perturbation of H 0 . Note that at zero temperature (fi = 1) the operator H decouples ....

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Davies E. B., Quantum Theory of Open Systems. Academic Press, London, 1976.

....while paying little attention to the proposals of the other groups. For an analysis of time measurements within the framework of Bohmian mechanics, see Daumer, Durr, Goldstein and Zangh i 1994 and the contribution of Leavens to this volume. Because of such difficulties, it has been proposed (Davies 1976) that we should go beyond operators as observables, to generalized observables, described by mathematical objects even more abstract than operators. The basis of this generalization lies in the observation that, by the spectral theorem, the concept of self adjoint operator is completely ....

Davies, E.B. (1976), Quantum Theory of Open Systems. London, Academic Press.

....set of all perceptions p, has a fundamental measure (S) for each subset S of M . Quantum Consciousness Connection: The measure (S) for each set S of conscious perceptions is given by the expectation value of a corresponding awareness operator A(S) a positive operator valued (POV) measure [26], in the state oe of the quantum world: S) hA(S)i j oe[A(S) 1) Here a perception p is the entirety of a single conscious experience, all that one is consciously aware of or consciously experiencing at one moment, the total raw feel that one has at one time, or [27] a phenomenal ....

E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).

....in a natural way to that of positive operator valued (POV) measure. POV s have been employed to construct generalized quantum observables for which the usual framework of self adjoint operators has been unsuccessful, mainly in the fields of quantum optics and the theory of open quantum systems [18]. Moreover, POV s emerge naturally from an analysis of quantum experiments in Bohmian mechanics [19] A POV measure is a set function M , which maps measurable subsets Delta ae IR to bounded positive operators M ( Delta) such that for any quantum state ae Delta 7 ae M ( Delta) tr ae M ....

E.B. Davies, Quantum Theory of Open Systems (Academic Press, 1976).

....of wave functions satisfying the CPC. The expression for the probability that the escape time is within an arbitrary set Delta IP(T e 2 Delta) Z ( Delta) Z Delta Z(t)dt ) 15) leads us now to the map Delta 7 Z ( Delta) which is a positive operator valued measure (POV) [13]. A POV is a generalization of a projection valued measure (PV) in the sense that the operators (Z ( Delta) need not be projections. 5 As in the spectral theorem, we may associate with a POV a self adjoint operator, namely its first moment. This association is, however, many to one, and the ....

D. Davies, Quantum Theory of Open Systems, Academic Press, London-New York-San Franzisco (1976).

....has a unitary dilation acting on a larger Hilbert space. The theory was subsequently extended to non contractive operators by Davis [11] at the cost of introducing indefinite inner product spaces. Unitary dilations have previously found physical applications in the quantum theory of open systems [12], and have also been employed by one of us in an inverse scattering construction of point like interactions in quantum mechanics [13, 14] Put concisely, starting with a non unitary evolution X, we pass to a unitary dilation of X, mapping between enlarged inner product spaces whose inner product ....

E.B. Davies, Quantum Theory of Open Systems, (Academic Press, London, 1976)

.... is prepared, the positions x of the particles are distributed according to the quantum equilibrium measure IP with density ae = j j 2 ( normalized) 18] 2 Nor are they given by a positive operator valued measure (POV) which has been proposed as a generalized quantum observable, see [16] In particular, the continuity equation for the probability shows that the probability flux (j t j 2 ; j t j 2 v t ) is conserved, since j t j 2 v t = j t . Hence, given t , the solutions x(t; x 0 ) of equation (8) are random trajectories, where the randomness comes from ....

E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, New York, San Francisco, 1976).

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Davies E. B., Quantum theory of open systems. Academic Press, London 1976.

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Davies, E.B: Quantum Theory of Open Systems, London, Academic Press (1976).

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E. B. Davies. Quantum Theory of Open Systems. Academic Press, London, 1976.

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E. B. Davies. Quantum Theory of Open Systems. Academic Press, London, 1976.

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E.B. Davies, Quantum theory of open systems, Academic Press, London 1976.

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E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).

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E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976)

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E.B. Davies. Quantum Theory of Open Systems. Academic Press, 1976.

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E.B. Davies, Quantum theory of open systems, (Academic Press, 1976)

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E. B. Davies, Quantum Theory of Open Systems, Academic Press, London 1976

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E.B. Davies, Quantum Theory of Open Systems (Academic Press, 1976).

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E. B. Davies, Quantum Theory of Open Systems, London 1976

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Davies, E.B., Quantum Theory of Open Systems, (Academic Press, 1976).

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