arXiv subjects
Publications and source records attributed to Francisco J. Turrubiates.
Wave function of a single linear graviton and its interpretation are proposed. The evolution equation for this function is given. A Hermitian operator with mutually commuting components canonically conjugated to the momentum operator of the linear graviton is found.
A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bialynicki-Birula scalar product and defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over $\mathbb{R}^3 \setminus \{(0,0,0)\}$. This observation enables us to reformulate quantum mechanics of a~single photon on $(\mathbb{R}^{3} \setminus \{(0,0,0)\}) \times \mathbb{C}^2$.
It is shown that the photon position operator $\hat{\vec{X}}$ with commuting components can be written in the momentum representation as $\hat{\vec{X}}=i \hat{\vec{D}}$, where $\hat{\vec{D}}$ is a flat connection in the tangent bundle $T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})$ over $\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}$ equipped with the Cartesian structure. Moreover, $\hat{\vec{D}}$ is such that the tangent $2$-planes orthogonal to the momentum are parallelly propagated with respect to $\hat{\vec{D}}$ and, also, $\hat{\vec{D}}$ is an anti-Hermitian operator with respect to the scalar product $\langle \mathbfΨ | \hat{H}^{-2s} |\mathbfΦ \rangle$. The eigenfunctions $\mathbfΨ_{\vec{X}} (\vec{x})$ of the position operator $\hat{\vec{X}}$ are found.
Classical model of light in helicity formalism is presented. Then quantum point of view at photons -- construction and interpretation of photon wave function is proposed. Quantum mechanics of photon is investigated. The Białynicki -- Birula scalar product $\langle {\bf Ψ}_{1}|{\bf Ψ}_{2}\rangle_{BB}$ and the generalized Hermitian conjugation $\widehatγ^{+\hspace{-0.45em}+}$ of linear operator $\widehatγ$ are discussed. Quantum description of light on a phase space is developed. A photon Wigner function is built.
Quantum cosmology of the Friedmann-Lemaître-Robertson-Walker model with cosmological constant in the Ho$\check{\rm r}$ava-Lifshitz type gravity is studied in the phase space by means of the Wigner function. The modification of the usual general relativity description by the Ho$\check{\rm r}$ava-Lifshitz type gravity induces a new scenario for the origin of the Universe with an embryonic era where the Universe can exist classically before the tunneling process takes place and which gives rise to the current evolution of the Universe. The Wigner functions corresponding to the Hartle-Hawking, Vilenkin and Linde boundary conditions are obtained by means of numerical calculations. In particular three cases were studied for the potential of the Wheeler-DeWitt equation: tunneling barrier with and without embryonic era and when the potential barrier is not present. The quantum behavior of these three cases are analyzed using the Wigner function for the three boundary conditions considered.
The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space $L^{2}(\mathbb{R}^{3})\otimes{\mathcal{H}}^{(s+1)}$ and functions on the phase space $\mathbb{R}^{3}\times\mathbb{R}^{3}\times \{0,...,s\} \times\{0,...,s\}$ is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin $\frac{1}{2}$ nonrelativistic particle as well as the magnetic resonance for a spin $\frac{1}{2}$ nonrelativistic uncharged particle are analysed.
An adaptation of the WKB method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase $σ(\vec{r})$ of a wave function $\exp \left(\frac{i}{\hbar} σ(\vec{r}) \right)$ and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi - classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl - Teller potential have been found.
The Robertson -- Schrödinger, Heisenberg -- Robertson and Trifonov uncertainty relations for arbitrary two functions $f_{1}$ and $f_{2}$ depending on the quantum phase and the number of photons respectively, are given. Intelligent states and states which minimize locally the product of uncertainties $(Δf_{1})^{2}\cdot (Δf_{2})^{2}$ or the sum $(Δf_{1})^{2}+(Δf_{2})^{2}$ are investigated for the cases $f_{1}=ϕ,\exp{(iϕ)}, \exp{(-iϕ)}, \cosϕ, \sinϕ$ and $f_{2}=n$.
Deformation quantization is applied to quantize gravitational systems coupled with matter. This quantization procedure is performed explicitly for quantum cosmology of these systems in a flat minisuper(phase)space. The procedure is employed in a quantum string minisuperspace corresponding to an axion-dilaton system in an isotropic FRW Universe. The Wheeler-DeWitt-Moyal equation is obtained and its corresponding Wigner function is given analytically in terms of Meijer's functions. Finally, this Wigner functions is used to extract physical information of the system.
The deformation quantization formalism is applied to the linearized gravitational field. Standard aspects of this formalism are worked out before the ground state Wigner functional is obtained. Finally, the propagator for the graviton is also discussed within the context of this formalism.
The Weyl-Wigner-Groenewold-Moyal formalism of deformation quantization is applied to cosmological models in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in a flat minisuperspace. The de Sitter cosmological model is worked out in detail and the computation of the Wigner functions for the Hartle-Hawking, Vilenkin and Linde wave functions are done numerically. The Wigner function is analytically calculated for the Kantowski-Sachs model in (non)commutative quantum cosmology and for string cosmology with dilaton exponential potential. Finally, baby universes solutions are described in this context and the Wigner function is obtained.
It is shown that the isomorphism between the generalized Moyal algebra and the matrix algebra follows in a natural manner from the generalized Weyl quantization rule and from the well known matrix representation of the destruction and creation operators.
An example of a sequence of the sl(N;C) chiral fields, for N$\geq 2$, tending to the complex heavenly metric (nonlinear graviton) of the type [4]x[-] when N --> infinity is given.