Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPoisson bracket operator
65
0
0.0
(
0
)
Authors
T. Koide
Ask ChatGPT about the research
No Arabic abstract
We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this operator, we derive the quantum canonical equation which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, the quantum canonical equation is applicable to c-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to non-standard system where the Heisenberg is not applicable. As an example, we consider the application to the system where a c-number and a q-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
rate research
Read More
Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.
We apply the projection operator method (POM) to $phi^4$ theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term, in sharp contrast with other well-known results obtained using the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force field terms, these equations are similar to the corresponding equations obtained using IFM with the linear harmonic approximation, which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation in which the fluctuation force field is given in terms of the operators of the hard modes. These operators are then replaced with c-numbers using a certain procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation force fields introduced in this paper and those introduced in IFM. The arbitrariness of the definition of the fluctuation force field in IFM is also discussed.
Time operator is studied on the basis of field quantization, where the difficulty stemming from Paulis theorem is circumvented by borrowing ideas from the covariant quantization of the bosonic string, i.e., one can remove the negative energy states by imposing Virasoro constraints. Applying the index theorem, one can show that in a different subspace of a Fock space, there is a different self-adjoint time operator. However, the self-adjoint time operator in the maximal subspace of the Fock space can also represent the self-adjoint time operator in the other subspaces, such that it can be taken as the single, universal time operator. Furthermore, a new insight on Paulis theorem is presented.
We generalize our previous work on the phase stability and hydrodynamic of polar liquid crystals possessing local uniaxial $C_{infty v}$-symmetry to biaxial systems exhibiting local $C_{2v}$-symmetry. Our work is motivated by the recently discovered examples of thermotropic biaxial nematic liquid crystals comprising bent-core mesogens, whose molecular structure is characterized by a non-polar body axis $({bf{n}})$ as well as a polar axis $({bf{p}})$ along the bisector of the bent mesogenic core which is coincident with a large, transverse dipole moment. The free energy for this system differs from that of biaxial nematic liquid crystals in that it contains terms violating the ${bf{p}}to -{bf{p}}$ symmetry. We show that, in spite of a general splay instability associated with these parity-odd terms, a uniform polarized biaxial state can be stable in a range of parameters. We then derive the hydrodynamic equations of the system, via the Poisson-bracket formalism, in the polarized state and comment on the structure of the corresponding linear hydrodynamic modes. In our Poisson-bracket derivation, we also compute the flow-alignment parameters along the three symmetry axes in terms of microscopic parameters associated with the molecular geometry of the constituent biaxial mesogens.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
