169 research outputs found
Recent Results Regarding Affine Quantum Gravity
Recent progress in the quantization of nonrenormalizable scalar fields has
found that a suitable non-classical modification of the ground state wave
function leads to a result that eliminates term-by-term divergences that arise
in a conventional perturbation analysis. After a brief review of both the
scalar field story and the affine quantum gravity program, examination of the
procedures used in the latter surprisingly shows an analogous formulation which
already implies that affine quantum gravity is not plagued by divergences that
arise in a standard perturbation study. Additionally, guided by the projection
operator method to deal with quantum constraints, trial reproducing kernels are
introduced that satisfy the diffeomorphism constraints. Furthermore, it is
argued that the trial reproducing kernels for the diffeomorphism constraints
may also satisfy the Hamiltonian constraint as well.Comment: 32 pages, new features in this alternative approach to quantize
gravity, minor typos plus an improved argument in Sec. 9 suggested by Karel
Kucha
Reduction of Lie-Jordan Banach algebras and quantum states
A theory of reduction of Lie-Jordan Banach algebras with respect to either a
Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared
with the standard reduction of C*-algebras of observables of a quantum system
in the presence of quantum constraints. It is shown that the later corresponds
to the particular instance of the reduction of Lie-Jordan Banach algebras with
respect to a Lie-Jordan subalgebra as described in this paper. The space of
states of the reduced Lie-Jordan Banach algebras is described in terms of
equivalence classes of extensions to the full algebra and their GNS
representations are characterized in the same way. A few simple examples are
discussed that illustrates some of the main results
Quantum control without access to the controlling interaction
In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum
system and its controller. We show under which conditions measurements, state
preparations, and unitary implementations on the system can be performed by
quantum operations on the controller only.
It turns out that a measurement of the observable A and an implementation of
the one-parameter group exp(iAr) can be performed by almost the same sequence
of control operations. Furthermore measurement procedures for A+B, for (AB+BA),
and for i[A,B] can be constructed from measurements of A and B. This shows that
the algebraic structure of the set of observables can be explained by the Lie
group structure of the unitary evolutions on the joint Hilbert space of the
measuring device and the measured system.
A spin chain model with nearest neighborhood coupling shows that the border
line between controller and system can be shifted consistently.Comment: 10 pages, Revte
Reduction Procedures in Classical and Quantum Mechanics
We present, in a pedagogical style, many instances of reduction procedures
appearing in a variety of physical situations, both classical and quantum. We
concentrate on the essential aspects of any reduction procedure, both in the
algebraic and geometrical setting, elucidating the analogies and the
differences between the classical and the quantum situations.Comment: AMS-LaTeX, 35 pages. Expanded version of the Invited review talk
delivered by G. Marmo at XXIst International Workshop On Differential
Geometric Methods In Theoretical Mechanics, Madrid (Spain), 2006. To appear
in Int. J. Geom. Methods in Modern Physic
Representable states on quasi-local quasi *-algebras
Continuing a previous analysis originally motivated by physics, we consider
representable states on quasi-local quasi *-algebras, starting with examining
the possibility for a {\em compatible} family of {\em local} states to give
rise to a {\em global} state. Some properties of {\em local modifications} of
representable states and some aspects of their asymptotic behavior are also
considered.Comment: In press in Jpurnal of Mathematical Physic
The Possibility of Reconciling Quantum Mechanics with Classical Probability Theory
We describe a scheme for constructing quantum mechanics in which a quantum
system is considered as a collection of open classical subsystems. This allows
using the formal classical logic and classical probability theory in quantum
mechanics. Our approach nevertheless allows completely reproducing the standard
mathematical formalism of quantum mechanics and identifying its applicability
limits. We especially attend to the quantum state reduction problem.Comment: Latex, 14 pages, 1 figur
The Geometric Phase and Ray Space Isometries
We study the behaviour of the geometric phase under isometries of the ray
space. This leads to a better understanding of a theorem first proved by
Wigner: isometries of the ray space can always be realised as projections of
unitary or anti-unitary transformations on the Hilbert space. We suggest that
the construction involved in Wigner's proof is best viewed as an use of the
Pancharatnam connection to ``lift'' a ray space isometry to the Hilbert space.Comment: 17 pages, Latex file, no figures, To appear in Pramana J. Phy
Semiclassical properties and chaos degree for the quantum baker's map
We study the chaotic behaviour and the quantum-classical correspondence for
the baker's map. Correspondence between quantum and classical expectation
values is investigated and it is numerically shown that it is lost at the
logarithmic timescale. The quantum chaos degree is computed and it is
demonstrated that it describes the chaotic features of the model. The
correspondence between classical and quantum chaos degrees is considered.Comment: 30 pages, 4 figures, accepted for publication in J. Math. Phy
Towards a definition of quantum integrability
We briefly review the most relevant aspects of complete integrability for
classical systems and identify those aspects which should be present in a
definition of quantum integrability.
We show that a naive extension of classical concepts to the quantum framework
would not work because all infinite dimensional Hilbert spaces are unitarily
isomorphic and, as a consequence, it would not be easy to define degrees of
freedom. We argue that a geometrical formulation of quantum mechanics might
provide a way out.Comment: 37 pages, AmsLatex, 1 figur
On non-completely positive quantum dynamical maps on spin chains
The new arguments based on Majorana fermions indicating that non-completely
positive maps can describe open quantum evolution are presented.Comment: published; small change
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