Covariant quantum Markovian evolutions

Quantum Markovian master equations with generally unbounded generators, having physically relevant symmetries, such as Weyl, Galilean or boost covariance, are characterized. It is proven in particular that a fully Galilean covariant zero spin Markovian evolution reduces to the free motion perturbed by a covariant stochastic process with independent stationary increments in the classical phase space. A general form of the boost covariant Markovian master equation is discussed and a formal dilation to the Langevin equation driven by quantum Boson noises is described.

[1]  Bart Demoen,et al.  Completely positive quasi-free maps of the CCR-algebra , 1979 .

[2]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[3]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[4]  A. Chebotarev.  Necessary and sufficient conditions for conservativeness of dynamical semigroups , 1991 .

[5]  Edward Davies.  Quantum dynamical semigroups and the neutron diffusion equation , 1977 .

[6]  V. Bargmann,et al.  On Unitary ray representations of continuous groups , 1954 .

[7]  Unstable particles and the Poincaré semigroup in quantum field theory , 1986 .

[8]  H. Araki.  Factorizable Representation of Current Algebra : Non commutative extension of the Levy-Kinchin formula and cohomology of a solvable group with values in a Hilbert space , 1969 .

[9]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[10]  Bart Demoen,et al.  Completely positive maps on the CCR-algebra , 1977 .

[11]  Collett,et al.  Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. , 1985, Physical review. A, General physics.

[12]  A. Frigerio.  Quantum poisson processes: Physical motivations and applications , 1988 .

[13]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[14]  Quantum mechanics with only positive-time evolution for an isolated system , 1978 .

[15]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[16]  A. Holevo.  On conservativity of covariant dynamical semigroups , 1993 .

[17]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[18]  Hans Maassen,et al.  The essentially commutative dilations of dynamical semigroups onMn , 1987 .

[19]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .