arXiv subjects
Publications and source records attributed to G. Marmo.
In this paper we present a theory of reduction of quantum systems in the presence of symmetries and constraints. The language used is that of Lie--Jordan Banach algebras, which are discussed in some detail together with spectrum properties and the space of states. The reduced Lie--Jordan Banach algebra is characterized together with the Dirac states on the physical algebra of observables.
The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.
We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter dependent family - via the Poincaré Normal Form approach. We discuss this at first formally, and later pay attention to the convergence of the linearizing procedure. We also discuss some generalizations of our main result
In this paper we develop a picture of Quantum Mechanics based on the description of physical observables in terms of expectation value functions, generalizing thus the so called Ehrenfest theorems for quantum dynamics. Our basic technical ingredient is the set of tools which has been developed in the last years for the geometrical formulation of Quantum Mechanics. In the new picture, we analyze the problem of the dynamical equations, the uncertainty relations and interference and illustrate the construction with the simple case of a two-level system.
Quantum dynamics of the density operator in the framework of a single probability vector is analyzed. In this framework quantum states define a proper convex quantum subset in an appropriate simplex. It is showed that the corresponding dynamical map preserving a quantum subset needs not be stochastic contrary to the classical evolution which preserves the entire simplex. Therefore, violation of stochasticity witnesses quantumness of evolution.
Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.
Relevant algebraic structures for the description of Quantum Mechanics in the Heisenberg picture are replaced by tensorfields on the space of states. This replacement introduces a differential geometric point of view which allows for a covariant formulation of quantum mechanics under the full diffeomorphism group.
The quantizer-dequantizer method allows to construct associative products on any measure space. Here we consider an inverse problem: given an associative product is it possible to realize it within the quantizer-dequantizer framework? The answer is positive in finite dimensions and we give a few examples in infinite dimensions.
The spin-statistics connection, quantum gravity and other physical considerations suggest that classical space-time topology is not an immutable attribute and can change in quantum physics. The implementation of topology change using quantum principles has been studied for over two decades by a few of us. There has been a recent revival of interest in some of our work, dating back to as early as 1995. The present paper is meant as a resource article to our major relevant papers. It contains summaries of the contents of the cited papers and the corresponding links wherever available.
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture,which has evolved in the $C^\star-$algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept. The equivalence between these two pictures amounts essentially, to the Gelfand-Naimark-Segal construction. In this construction, the abstract $% C^\star-$algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation one defines may be reducible or irreducible, but in either case it allows to identify an unitary group associated with the $C^\star-$algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group, it follows that also quantum tomograms are strictly related with appropriate functions (positive-type)on the group. In this paper we present, by means of very simple examples, the tomographic description emerging from the set of ideas connected with the $C^\star-$algebra picture of quantum mechanics. In particular, the tomographic probability distributions are introduced for finite and compact groups and an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state is formulated.
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical structure like symplectic, Poisson, etc. Such geometrical structure provides a generalized toroidal bundle on the carrier space of the system. Non--canonical diffeomorphisms of such structure generate alternative Hamiltonian structures for complete integrable Hamiltonian systems. The energy-period theorem provides the first non--trivial obstruction for the equivalence of integrable systems.
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.
After a general description of the tomographic picture for classical systems, a tomographic description of free classical scalar fields is proposed both in a finite cavity and the continuum. The tomographic description is constructed in analogy with the classical tomographic picture of an ensemble of harmonic oscillators. The tomograms of a number of relevant states such as the canonical distribution, the classical counterpart of quantum coherent states and a new family of so called Gauss--Laguerre states, are discussed. Finally the Liouville equation for field states is described in the tomographic picture offering an alternative description of the dynamics of the system that can be extended naturally to other fields.
In the case of two qubits, standard entanglement monotones like the linear entropy fail to provide an efficient quantum estimation in the regime of weak entanglement. In this paper, a more efficient entanglement estimation, by means of a novel class of entanglement monotones, is proposed. Following an approach based on the geometric formulation of quantum mechanics, these entanglement monotones are defined by inner products on invariant tensor fields on bipartite qubit orbits of the group SU(2)xSU(2).
The experimental check of two--mode Robertson uncertainty relations and inequalities for highest quadrature moments is suggested by using homodyne photon detection. The relation between optical tomograms and symplectic tomograms is used to connect the tomographic dispersion matrix and the quadrature components dispersion matrix of the two--mode field states.
We experimentally verify uncertainty relations for mixed states in the tomographic representation by measuring the radiation field tomograms, i.e. homodyne distributions. Thermal states of single-mode radiation field are discussed in details as paradigm of mixed quantum state. By considering the connection between generalised uncertainty relations and optical tomograms is seen that the purity of the states can be retrieved by statistical analysis of the homodyne data. The purity parameter assumes a relevant role in quantum information where the effective fidelities of protocols depend critically on the purity of the information carrier states. In this contest the homodyne detector becomes an easy to handle purity-meter for the state on-line with a running quantum information protocol.
Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincarè group algebra acts on it with a Drinfel'd-twisted coproduct. But the latter is not appropriate for more complicated spacetimes such as those containing the Friedman-Sorkin (topological) geons. They have rich diffeomorphism groups and in particular mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group $S_N$. We generalise the Drinfel'd twist to (essentially) generic groups including to finite and discrete ones and use it to modify the commutative spacetime algebras of geons as well to noncommutative algebras. The latter support twisted actions of diffeos of geon spacetimes and associated twisted statistics. The notion of covariant fields for geons is formulated and their twisted versions are constructed from their untwisted versions. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli principle, seem to be the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.