scholarly journals A Feynman–Kac formula for magnetic monopoles

2021 ◽  
pp. 2150015
Author(s):  
J. Dimock
Keyword(s):  
Quantum Mechanics ◽  
Charged Particle ◽  
Line Bundle ◽  
Eigenfunction Expansion ◽  
Magnetic Monopole ◽  
Magnetic Monopoles ◽  
Wave Functions ◽  
Stochastic Integrals ◽  
Complex Line ◽  
Imaginary Time

We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

Scattering on Magnetic Monopoles

1980 ◽  
Vol 35 (12) ◽  
pp. 1276-1284
Author(s):  
Herbert-Rainer Petry
Keyword(s):  
Line Bundle ◽  
Cross Section ◽  
Scattering Theory ◽  
Charged Particles ◽  
Scattering Matrix ◽  
Magnetic Monopoles ◽  
Wave Functions ◽  
Complex Line ◽  
Complex Line Bundle ◽  
Frame Work

Abstract The time-dependent scattering theory of charged particles on magnetic monopoles is investigated within a mathematical frame-work, which duely pays attention to the fact that the wave-functions of the scattered particles are sections in a non-trivial complex line-bundle. It is found that Möller operators have to be defined in a way which takes into account the peculiar long-range behaviour of the monopole field. Formulas for the scattering matrix and the differential cross-section are derived, and, as a by-product, a momentum space picture for particles, which are described by sections in the underlying complex line-bundle, is presented.

scholarly journals All roads lead to the Dirac quantization condition

2019 ◽  
Vol 6 (1) ◽  
pp. 102-127
Author(s):  
Pedro Aguilar
Keyword(s):  
Quantum Mechanics ◽  
Charged Particle ◽  
Magnetic Monopole ◽  
Short Review ◽  
Magnetic Monopoles ◽  
Quantization Condition ◽  
Formal Description ◽  
Mathematical Structures ◽  
Dirac Quantization ◽  
The Status

The existence of magnetic monopoles is a sufficient argument to explain the quantization of electric charge, an argument that was presented by Dirac. Regardless of the status of any search for magnetic monopoles, the formal description of the quantum mechanics of a charged particle in the field of a magnetic monopole is very rich and has increased our understanding of the mathematical structures underlying this description, as well as of its physical implications. In this short review, we present four different arguments all leading to the Dirac quantization condition, emphasizing their geometrical and topological aspects.

scholarly journals Witten effect, anomaly inflow, and charge teleportation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hajime Fukuda ◽  
Kazuya Yonekura
Keyword(s):  
Quantum Mechanics ◽  
Magnetic Monopole ◽  
Charged Particles ◽  
Magnetic Monopoles ◽  
Three Dimensions ◽  
Total Charge ◽  
Four Dimensions ◽  
Linking Number ◽  
M Theory ◽  
Chern Simons

Abstract We study a phenomenon that electric charges are “teleported” between two spatially separated objects without exchanging charged particles at all. For example, this phenomenon happens between a magnetic monopole and an axion string in four dimensions, two vortices in three dimensions, and two M5-branes in M-theory in which M2-charges are teleported. This is realized by anomaly inflow into these objects in the presence of cubic Chern-Simons terms. In particular, the Witten effect on magnetic monopoles can be understood as a general consequence of anomaly inflow, which implies that some anomalous quantum mechanics must live on them. Charge violation occurs in the anomalous theories living on these objects, but it happens in such a way that the total charge is conserved between the two spatially separated objects. We derive a formula for the amount of the charge which is teleported between the two objects in terms of the linking number of their world volumes in spacetime.

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

A Student's Guide to the Schrödinger Equation

2020 ◽  
Author(s):  
Daniel A. Fleisch
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Plain Language ◽  
Science And Engineering ◽  
Popular Approach ◽  
Matlab Code ◽  
Mathematical Techniques

Quantum mechanics is a hugely important topic in science and engineering, but many students struggle to understand the abstract mathematical techniques used to solve the Schrödinger equation and to analyze the resulting wave functions. Retaining the popular approach used in Fleisch's other Student's Guides, this friendly resource uses plain language to provide detailed explanations of the fundamental concepts and mathematical techniques underlying the Schrödinger equation in quantum mechanics. It addresses in a clear and intuitive way the problems students find most troublesome. Each chapter includes several homework problems with fully worked solutions. A companion website hosts additional resources, including a helpful glossary, Matlab code for creating key simulations, revision quizzes and a series of videos in which the author explains the most important concepts from each section of the book.

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