Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics
64
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard free case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr{o}dinger problem, the free noise can also be extended to any infinitely divisible probability law, as covered by the L{e}vy-Khintchine formula. Since the relativistic Hamiltonians $| abla |$ and $sqrt {-triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (DAlembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr{o}dinger evolution is analyzed in detail. The relativistic covariance of related wave
rate research
Read More
The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard free case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr{o}dinger problem, the free noise can also be extended to any infinitely divisible probability law, as covered by the L{e}vy-Khintchine formula. Since the relativistic Hamiltonians $| abla |$
This article is the second in a series of two presenting the Scale Relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. Here, we show Schroedingers equation to be a reformulation of Newtons fundamental relation of dynamics as generalized to non-differentiable geometries in the first paper cite{paper1}. It motivates an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the Scale Relativistic approach and, at the same time, it allows to identify macroscopic chaotic systems considered at time scales exceeding their horizon of predictability as candidates in which to search for quantum-like structuring or behavior.
This article is a pedagogical introduction to relativistic quantum mechanics of the free Majorana particle. This relatively simple theory differs from the well-known quantum mechanics of the Dirac particle in several important aspects. We present its three equivalent formulations. Next, so called axial momentum observable is introduced, and general solution of the Dirac equation is discussed in terms of eigenfunctions of that operator. Pertinent irreducible representations of the Poincare group are discussed. Finally, we show that in the case of massless Majorana particle the quantum mechanics can be reformulated as a spinorial gauge theory.
We consider the quantum simulation of relativistic quantum mechanics, as described by the Dirac equation and classical potentials, in trapped-ion systems. We concentrate on three problems of growing complexity. First, we study the bidimensional relativistic scattering of single Dirac particles by a linear potential. Furthermore, we explore the case of a Dirac particle in a magnetic field and its topological properties. Finally, we analyze the problem of two Dirac particles that are coupled by a controllable and confining potential. The latter interaction may be useful to study important phenomena as the confinement and asymptotic freedom of quarks.
This brief article gives an overview of quantum mechanics as a {em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
