Dirac Equation in Cosmological Inertial Frame

Dirac equation is a one order-wave equation. Wave function uses as a probability amplitude in quantum mechanics. We make Dirac Equation from wave function, Type A in cosmological inertial frame.The Dirac equation satisfy Klein-Gordon equation in cosmological inertial frame.

Modeling of U-shaped composite detectors

We have investigated the basic operation of a composite detector comprising of elements arranged in the shape of an U-shaped rectangular well. Considering an isotropic scattering of gamma-rays and partial energy absorptions in up to four detector modules, expressions for the addback factor and the peak-to-total ratio have been obtained in terms of only one probability amplitude. We have compared the performance of two U-shaped detectors having different geometries and observed negligible gain in addback due to the longer arms. For completeness, comparisons have been made with composite detectors like the two element stacked detector and the two level pyramidal detector, both being embedded inside the U-shaped detector. Our pen-on-paper approach could be used to understand the operation of modern arrays having detector elements arranged in various sophisticated ways.

Quantum Propagators for Geodesic Congruences

Using the Raychaudhuri equation, we show that a quantum probability amplitude (Feynman propagator) can be univocally associated to any timelike or null affinely parametrized geodesic congruence.

Path integral implementation of relational quantum mechanics

Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (Yang in Sci Rep 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born’s Rule, Schrödinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to calculate measurement probability. Second, we provide a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also allows us to elegantly explain the double slit experiment, to describe the interaction history between the measured system and a series of measuring systems, and to calculate entanglement entropy based on path integral and influence functional. In return, the implementation brings back new insight to path integral itself by completing the explanation on why measurement probability can be calculated as modulus square of probability amplitude. Lastly, we clarify the connection between our reformulation and the quantum reference frame theory. A complete relational formulation of quantum mechanics needs to combine the present works with the quantum reference frame theory.

Path Integral Implementation of Relational Quantum Mechanics

Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In the recent works (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born's Rule, Schr\"{o}dinger Equations, and measurement theory. This paper gives a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also gives several important applications. For instance, the double slit experiment can be elegantly explained. A path integral representation of the reduced density matrix of the observed system can be derived. Such representation is shown valuable to describe the interaction history of the measured system and a series of measuring systems. More interestingly, it allows us to develop a method to calculate entanglement entropy based on path integral and influence functional. Criteria of entanglement is proposed based on the properties of influence functional, which may be used to determine entanglement due to interaction between a quantum system and a classical field.

Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links

This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected.

If the propagator of QED were reversed, the mathematics of Nature would be much simpler

In Quantum ElectroDynamics (QED) the propagator is a function that describes the probability amplitude of a particle going from point A to B. It summarizes the many paths of Feynman’s path integral approach. We propose a reverse propagator (R-propagator) that, prior to the particle’s emission, summarizes every possible path from B to A. Wave function collapse occurs at point A when the particle randomly chooses one and only one of many incident paths to follow backwards with a probability of one, so it inevitably strikes detector B. The propagator and R-propagator both calculate the same probability amplitude. The R-propagator has an advantage over the propagator because it solves a contradiction inside QED, namely QED says a particle must take EVERY path from A to B. With our model the particle only takes one path. The R-propagator had already taken every path into account. We propose that this tiny, infinitesimal change from propagator to R-propagator would vastly simplify the mathematics of Nature. Many experiments that currently describe the quantum world as weird, change their meaning and no longer say that. The quantum world looks and acts like the classical world of everyday experience.

Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function

In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.