Time Evolution of the Position Operators in a Bilayer Graphene

In this work, the researchers mainly focus on the trembling motion which is known as Zitterbewegung in a bilayer grapheme. This is effectively achieved by means of the long-wave approximation. That is, the Heisenberg representation is ultimately employed in order to derive the analytical expression concerning the expectation value related to the position operator along the longitudinal and transversal orientation, which describes the motion concerning the electronic wave packet inside the bilayer graphene. Parameters’ numbers are considered to explicate the packet of Gaussian wave, including the polarization of initial pseudo-spin as well as the wave number of the initial carrier number along with the localized wave packet’s width along the longitudinal as well as transversal orientation. Consequently, the researchers show that the obvious oscillation in position operator can be effectively controlled not only by what is known as the initial parameters concerning the wave packet. Rather, it can mainly be controlled by selecting the localized quantum state’s components. Furthermore, the interference’s analysis between the conduction as well as valence bands concerning quantum states is really emphasized as the ability of what can be described as the transient’s emergence, or in a sense, aperiodic temporal oscillations concerning the average value of position operator in the bilayer graphene.

From quantum field theory to quantum mechanics

The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momentum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the context of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theories, since it relies crucially on particle number conservation, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner position operator. We make some clarifications on the interpretation of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations. From a complex Klein–Gordon field, position and momentum operators can be constructed for both particles and antiparticles.

Accurate study from adaptive perturbation method

The adaptive perturbation method decomposes a Hamiltonian by the diagonal elements and nondiagonal elements of the Fock state. The diagonal elements of the Fock state are solvable but can contain the information about coupling constants. We study the harmonic oscillator with the interacting potential, [Formula: see text], where [Formula: see text] and [Formula: see text] are coupling constants, and [Formula: see text] is the position operator. In this study, each perturbed term has an exact solution. We demonstrate the accurate study of the spectrum and [Formula: see text] up to the next leading-order correction. In particular, we study a similar problem of Higgs field from the inverted mass term to demonstrate the possible nontrivial application of particle physics.

Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in non-relativistic quantum mechanics, can be expressed as a matrix element of a time evolution operator: i.e. GNR(x) = 〈x2|UNR(t)|x1〉 in terms of the orthonormal eigenkets |x〉 of the position operator. In quantum field theory, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator GR(x) as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in quantum field theory, as a matrix element 〈x2|UR(t)|x1〉 for a suitably defined time evolution operator and (non-orthonormal) kets |x〉 labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this quantum-gravity-corrected propagator can be expressed as a matrix element 〈x2|UQG(t)|x1〉. I describe these results and explore several consequences. It turns out that the evolution operator UQG(t) becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalized to any ultrastatic curved spacetime.

Asymptotic generalized extended uncertainty principle

We present a formalism which allows for the perturbative derivation of the Extended Uncertainty Principle (EUP) for arbitrary spatial curvature models and observers. Entering the realm of small position uncertainties, we derive a general asymptotic EUP. The leading 2nd order curvature induced correction is proportional to the Ricci scalar, while the 4th order correction features the 0th order Cartan invariant $$\Psi _2$$Ψ2 (a scalar quadratic in curvature tensors) and the curved space Laplacian of the Ricci scalar all of which are evaluated at the expectation value of the position operator i.e. the expected position when performing a measurement. This result is first verified for previously derived homogeneous space models and then applied to other non-trivial curvature related effects such as inhomogeneities, rotation and an anisotropic stress fluid leading to black hole “hair”. Our main achievement combines the method we introduce with the Generalized Uncertainty Principle (GUP) by virtue of deformed commutators to formulate a generic form of what we call the Asymptotic Generalized Extended Uncertainty Principle (AGEUP).

Proper relativistic position operators in 1+1 and 2+1 dimensions

We have revisited the Dirac theory in [Formula: see text] and [Formula: see text] dimensions by using the covariant representation of the parity-extended Poincaré group in their native dimensions. The parity operator plays a crucial role in deriving wave equations in both theories. We studied two position operators, a canonical one and a covariant one that becomes the particle position operator projected onto the particle subspace. In [Formula: see text] dimensions the particle position operator, not the canonical position operator, provides the conserved Lorentz generator. The mass moment defined by the canonical position operator needs an additional unphysical spin-like operator to become the conserved Lorentz generator in [Formula: see text] dimensions. In [Formula: see text] dimensions, the sum of the orbital angular momentum given by the canonical position operator and the spin angular momentum becomes a constant of motion. However, orbital and spin angular momentum do not conserve separately. On the other hand the orbital angular momentum given by the particle position operator and its corresponding spin angular momentum become a constant of motion separately.

Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics

First, we examine how spin is treated in special relativity and the necessity of introducing spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a spinning particle. Next, we discuss quantum electrodynamics and the Foldy–Wouthuysen transformation which we note is a position operator identical to the Pryce–Newton–Wigner position operator. The classical version of the operators are shown to be essential for the treatment of classical relativistic particles in general relativity, of special interest being the case of binary systems (black holes/neutron stars) which emit gravitational radiation.