12 research outputs found
Towards the deformation quantization of linearized gravity
We present a first attempt to apply the approach of deformation quantization
to linearized Einstein's equations. We use the analogy with Maxwell equations
to derive the field equations of linearized gravity from a modified Maxwell
Lagrangian which allows the construction of a Hamiltonian in the standard way.
The deformation quantization procedure for free fields is applied to this
Hamiltonian. As a result we obtain the complete set of quantum states and its
discrete spectrum.Comment: 13 pages, no figures **preliminary entry **
Twist Deformations of the Supersymmetric Quantum Mechanics
The N-extended Supersymmetric Quantum Mechanics is deformed via an abelian
twist which preserves the super-Hopf algebra structure of its Universal
Enveloping Superalgebra. Two constructions are possible. For even N one can
identify the 1D N-extended superalgebra with the fermionic Heisenberg algebra.
Alternatively, supersymmetry generators can be realized as operators belonging
to the Universal Enveloping Superalgebra of one bosonic and several fermionic
oscillators. The deformed system is described in terms of twisted operators
satisfying twist-deformed (anti)commutators. The main differences between an
abelian twist defined in terms of fermionic operators and an abelian twist
defined in terms of bosonic operators are discussed.Comment: 18 pages; two references adde
Star products made (somewhat) easier
We develop an approach to the deformation quantization on the real plane with
an arbitrary Poisson structure which based on Weyl symmetrically ordered
operator products. By using a polydifferential representation for deformed
coordinates we are able to formulate a simple and effective
iterative procedure which allowed us to calculate the fourth order star product
(and may be extended to the fifth order at the expense of tedious but otherwise
straightforward calculations). Modulo some cohomology issues which we do not
consider here, the method gives an explicit and physics-friendly description of
the star products.Comment: 20 pages, v2, v3: comments and references adde
Wigner's quantum phase space current in weakly anharmonic weakly excited two-state systems
This is an open access article distributed under the terms of the Creative Commons Attribution License CC BY 4.0 (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.There are no phase-space trajectories for anharmonic quantum systems, but Wigner’s phase-space representation of quantum mechanics features Wigner current J . This current reveals fine details of quantum dynamics – finer than is ordinarily thought accessible according to quantum folklore invoking Heisenberg’s uncertainty principle. Here, we focus on the simplest, most intuitive, and analytically accessible aspects of J . We investigate features of J for bound states of time-reversible, weakly-anharmonic one-dimensional quantum-mechanical systems which are weakly-excited. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd potentials. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline patterns can be characterised by J ’s discrete stagnation points, how these arise and how a quantum system’s dynamics is constrained by the stagnation points’ topological charge conservation. We additionally show that quantum dynamics in phase space, in the case of vanishing Planck constant ̄ h or vanishing anharmonicity, does not pointwise converge to classical dynamics.Peer reviewe