User:Sparkyscience/Monopoles

File:Monopole_SU(2)_vortices.jpgThe simplest neutral quadruplet of chiral vortices. Electromagnetism with SU(2) symmetry allows for complex nonlinear phenomena such as topological vortices, known as monopoles.

Magnetic monopoles are emergent quasiparticles that exist if and only if the electromagnetic field has non-trivial topology. Magnetic monopoles conserve magnetism, in an analogous way that electrons and protons conserve electric charge. They can be thought of as a kind of domain wall or topological soliton that occurs when the symmetry of electric-magnetic duality is spontaneously broken. Condensation of magnetic monopoles induces the quantization of electric charge, in an exactly analogous way in the Higgs mechanism will quantize the mass of the electron. They can exhibit rich behaviour not encountered in conventional electromagnetism, and are far less constrained then the symmetry considerations required of a relativistically invariant electromagnetic field.

Electromagnetic particles (e.g. electons, protons etc) described in the standard Maxwell equations, or quantum electrodynamics (QED) form a $$\operatorname{U}(1)$$ symmetry group. When a particle is translated into any other position, orientation, speed etc. the value of the particle's charge does not vary and is conserved under what is called a $\operatorname{U}(1)$ gauge transformation, the magnetisation of the particle is not conserved. By definition, a particle that would conserve magnetic and electric charge, a monopole or dyon, would have to have higher symmetry, monopoles are not allowed to exist in $$\operatorname{U}(1)$$ in order for divergence-less solenoidal magnetic fields to exist (i.e. it is a requirement of Gauss's law which states that $∇⋅B = 0$). Only electric charge is conserved in $$\operatorname{U}(1)$$ symmetry. In order for magnetism to be conserved in particles we must use $$\operatorname{SU}(2)$$ or higher symmetry groups.

Aharonov–Bohm effect
It is tempting to believe that if the electromagnetic field has no net force there here should be no electromagnetic action. The outcome of the Aharonov–Bohm experiment is widely regarded as the strongest single piece of evidence that supports the view that electromagnetism is a gauge theory, and that the notion of local forces fails to describe all electromagnetic phenomena.

History of searches for the monopole
Pierre Curie suggested in 1894 that monopoles might exist. Henri Poincaré one of the founders of topology and who laid the groundwork for chaos theory, was aware that the lack symmetry in Maxwell's reformulated equations prevented monopoles, and in 1896 he investigated the motion of an electron in the presence of a monopole, spurred on by Kristian Birkeland's earlier report of anomalous motion of cathode rays in the presence of a magnetized needle. Joseph John Thomson in 1900 investigated the case of a single magnetic pole Paul Ulrich Villard known as the discover of gamma radiation believed he had discovered magnetic monopoles or has he called them magnetons, but the turned out to be the effect of cathode rays.

https://books.google.co.uk/books?id=e9wEntQmA0IC&pg=PA96&lpg=PA96&dq=heaviside+monopole&source=bl&ots=f2oWrxzTRu&sig=15j2R-6DfsjsmMUk7CSK19XGR1o&hl=en&sa=X&ved=0ahUKEwjwnJ6C5JrSAhXhAMAKHd5uCawQ6AEIIzAB#v=onepage&q=no%20magnetic%20charge&f=false

https://books.google.co.uk/books?id=zXm1Bso1VREC&printsec=frontcover#v=onepage&q=villard%201905&f=false

Quaternionic electromagnetism
19th century mathematicians were inspired by the mathematical beauty of complex numbers and complex analysis to seek new "generalized complex numbers" which would apply in a natural way to higher dimensions, and in particular, 3-dimensional space. In 1843 William Rowan Hamilton famously discovered quaternions while walking across Broom Bridge. Since complex numbers describe rotations in two dimensions and have two components, it might be thought that to describe three dimensions you need numbers which have three components, counterintuitively you need four dimensions to describe rotations in three dimension. A quaternion consists of an ordinary number, or scalar, combined with a vector, which has both magnitude and direction. The vector can be broken down into three components i, j and k that are perpendicular, or orthogonal, to each other such that there is no linear relationship between these components. A quaternion is generally represented in the form a + bi + cj + dk and the vector identities can be related via:

where:


 * $$\begin{alignat}{2}

ij & = k, & \qquad ji & = -k, \\ jk & = i, & kj & = -i, \\ ki & = j, & ik & = -j, \end{alignat}$$

These quantities satisfy all the usual laws of algebra we are familiar with from real and complex numbers bar one rule: the commutative law of multiplication ab = ba is violated by quaternions. The quaternions were the first time in history that a non-commutative product appeared in mathematics. This non-commutativity can be visualised by considering rotation of an object in 3D, if we rotate about the object 900 about the X axis and then 900 about the Z axis it does not end up in the same configuaration as rotation about Z axis then X axis. Order of the operation matters, just like solving a rubix cube.

Hamilton's quaternions were the first step towards what is now called a Clifford algebra, developed in 1876 by William Kingdon Clifford. In modern group theory, a group that preserves commutativity, like for example the complex numbers $$\mathbb{C}$$, is said to form an abelian group named after Niels Henrik Abel; the quaternion group, $$\mathbb{H}$$, is said to be a non-abelian. The complex numbers, $$\mathbb{C}$$, can be written by Euler's formula in the form $$e^{i\theta}=\cos\theta + i \sin\theta$$ and form a $$\operatorname{U}(1)$$ symmetry group. Geometrically, $$\operatorname{U}(1)$$ is the unit circle $$S^{1}$$. We can identify the plane $$\mathbb{R}^2$$ with the complex plane $$\mathbb{C}$$, letting $$z =x +iy \in \mathbb{C}$$ represent $$(x, y) \in \mathbb{R}^2$$. Multiplication of any complex number $$e^{i\theta}\in\operatorname{U}(1)$$ can be thought of as a rotation by an angle $$\theta$$ and is the same kind of symmetry as rotating a unit circle about the origin. For the quaternions, $$\mathbb{H}$$, it is often helpful two write them in the form of a $2 × 2$ complex matrix. The quaternion a + bi + cj + dk can be represented as:


 * $$\begin{bmatrix}a+bi & c+di \\ -c+di & a-bi \end{bmatrix}.$$



\mathbf{1}=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \mathbf{i}=\begin{bmatrix}i & 0 \\ 0 & -i \end{bmatrix} \mathbf{j}=\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix} \mathbf{k}=\begin{bmatrix}0 & i \\ -i & 0 \end{bmatrix} $$

Instead of the unit circle $$S^{1}$$ in $$\mathbb{R}^2$$, we need to consider the 3-sphere $$S^{3}$$ embedded in four dimentional $$\mathbb{R}^4$$.The 3-sphere $$S^{3}$$ is the set of points $$(x, y, z, t) \in \mathbb{R}^4$$ such that $$x^2 + y^2 + z^2 + t^2 = 1$$. It can be shown that the $$\operatorname{U}(1)$$ symmetry group is replaced by $$\operatorname{SU}(2)$$.

In 1858 Hamilton begun corresponding with Peter Guthrie Tait. Tait showed that quaternions had relevance to many problems in physics, and he used them to describe the theoretical properties of "magnetic particles". Tait's friend James Clerk Maxwell enigmatically called him the "Chief Musician upon Nabla" and wrote favourably about the benefits of using quaternion notation for science:

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.

What are called Maxwell's equations today are usually presented in a very different form to what was originally published in the first edition. Maxwell gave the main results in his Treatise on Electricity and Magnetism as twenty quaternion equations as well as cartesian equations. Maxwell later remarked to Tait that "The like of you may write everything and prove everything in pure 4nions, but in the transition period the bilingual method may help to introduce and explain the more perfect." Though Maxwell expressed a strong liking of the ideas behind quaternions, he was somewhat dismayed by their lack of linear symmetry and hence complex practicality. He was also troubled by the fact that they led to results which were not homogenous, and the fact that the square of the velocity vector made the kinetic energy negative.

Oliver Heaviside read Maxwell's Treatise in the 1870's where he came across the quaternion notation for the first time. He turned to the work of Tait to get a better understanding of the strange algebra, however he found that it was "without exception the hardest book to read I ever saw". Even after mastering the subject Heaviside was no fan of the algebra and was particularly troubled by the square of the vector being negative. He wrote

on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectoral aspects antiphysical and unnatural and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalars and vectors using a very simple vectorial algebra in my papers from 1883 onwards."

The work of William Thomson and Tait focused on potentials based on the principle of least action and Lagrangians; Heaviside by contrast focused on what he coined the "principle of activity" through the use of local forces. He regarded the notion of force rather then potential as the best guide to understand local transformations of energy and give the best insight into the real workings of dynamical systems. He saw the potentials as unphysical mathematical constructs that could have no observable effect in the context of electromagnetism. Heaviside developed the tools of vector analysis to help achieve this, and had great success in simplifying Maxwell's equations. Heaviside along with Josiah Willard Gibbs, John Henry Poynting, George Francis FitzGerald and Oliver Lodge significantly reformulated Maxwell's equations to make them more practical. They did this by ensuring that the potentials where symmetrical and scalar (and thus unobservable) to ensure the E and B fields were topologically homogenous.

It is interesting to note that Nikola Tesla made use of quaternion notation. It is broadly accepted that he used a very different means of electrical engineering then that present in today's conventional circuits. It has been argued convincingly that due to the reactive (capacitive, etc.) coupling in his oscillating-shuttle-circuits (OSC), they cannot be analysed using standard distributed circuit frameworks. The OSC is more compatible with the physics of nonlinear optics and as it takes advantage of the nonlinear features of the quaternion $$\operatorname{SU}(2)$$ symmetry.

It has sometimes been argued that quaternions present difficult issues of compatibility when describing space-time, though models have been proposed.

Dirac
The discovery of the spin of the electron by to two Dutch scientists, George Uhlenbeck and Samuel Goudsmit, forced physicists to search for mathematical tools to describe, within the framework of quantum mechanics, this new degree of freedom. In 1928 Paul Dirac argued that in order to do this he needed to find an equation for the electron in which the time-derivative appears in a first-order form $${\partial/\partial t}$$ (rather then second order form $$({\partial/\partial t})^2$$). His reason for this was to ensure that the probability of finding an electron in any given time would be always positive, and thus the probability can never be negative. In order to do this he was forced to add non-commuting quantities to the equations; it was these non-commutative quantities that turned out to describe the physical spin of a particle. In finding non-commuting spin quantities, Dirac had inadvertently rediscovered an instance of Clifford algebra. Dirac appears to have been completely unaware of the work of Clifford and Hamilton before him. Clifford and Hamilton had noticed that non-commutative algebras can be used to "take the square root" of Laplacians, where the dimension is 4 and the signature $$+---$$. Hamilton had shown that a square root of the ordinary 3-dimensional Laplacian can be obtained by using quaternions:


 * $$\left (\mathbf{i}{\partial\over\partial x}+\mathbf{j}{\partial\over\partial x}+\mathbf{k}{\partial\over\partial x}\right )^2=-\left ({\partial\over\partial x}\right )^2-\left ({\partial\over\partial x}\right )^2-\left ({\partial\over\partial x}\right )^2=-\nabla^2$$

Clifford generalised this idea for higher dimensions

The Dirac equation with no external field is:

Consider a global gauge transformation where $$\Gamma$$ is a hermitian matrix and $$\theta$$ a constant phase:

$$\psi\rightarrow e^{i\Gamma\theta}\psi$$

A necessary condition of the gauge invariance of the Dirac equation is that the factor does not depend on $$\mu$$, this is possible with two and only two matrices

Supersymmetry
Since the gauge field is Abelian, $∇⋅B = 0$, and isolated magnetic monopoles are necessarily singular in semilocal models. The only way to make the singularity disappear is by embedding the theory in a larger non-Abelian theory which provides a regular core, or by putting the singularity behind an event horizon.



For longer then a century it was believed that all physical theory could be derived from some unknown Theory of Everything. The Seiberg-Witten theory in 1994 changed that ideology profoundly. It is now believed that there is no unique fundamental theory, from which all the properties of the universe can be described based on reductive properties, but dualities that offer equivalent perspectives, each useful in different contexts. Donaldson theory



Spin ice








Magnetricity


https://www.newscientist.com/article/dn17983-magnetricity-observed-for-first-time/

Mathematics
Electromagnetism can be placed in the broader context of Yang-Mills theory.

A particle that conserves both magnetic and electric charge is known as a dyon and requires a greater symmetry group SU(2) to describe the preservation of magnetic symmetry. https://arxiv.org/abs/hep-th/9603086

Einstein insisted that all fundamental laws of nature could be understood in terms of geometry and symmetry. Before 1980 all states of matter could be classified by the principle of symmetry breaking. The quantum Hall state provided the first example of a quantum state that had no spontaneously broken symmetry. Its behaviour depends only on its topology and not its specific geometry. The quantum Hall effect earned Klaus von Klitzing the Nobel Prize in Physics for 1985. Though it was not understood at the time, the quantum Hall effect is an example of topological order, the subject of the 2016 Nobel Prize in Physics. Topological order violates the long-held belief that order in nature requires symmetry breaking. Fundamental aspects of nature can now also be understood in the context of topology, and physicists such as Edward Witten have developed topological field theory to develop this paradigm.

when $$m(\rho^2)$$ is constant in eq. (13.2) and (13.4) it has been shown that the presence of a monopole may be considered as a local torsion of an affine twisted space, the total curvature of which is ...

Indirect empirical evidence of monopoles has been suggested in the form of particles that move in a helical motion in ferromagnetic aerosols.

Instantons are topologically nontrivial solutions of Yang–Mills equations that absolutely minimize the energy functional within their topological type. The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the four-dimensional sphere, and turned out to be localized in space-time, prompting the names pseudoparticle and instanton.

Many methods developed in studying instantons have also been applied to monopoles. This is because magnetic monopoles arise as solutions of a dimensional reduction of the Yang–Mills equations.

In the context of (classical) field theory, the partial differential equations are the field equations of the theory and the trivial solutions are the vacuum solutions, i.e. the solutions that minimise the energy

an ordinary homotopy between two smooth functions is equivalent to the existence of a smooth homotopy.

A dipole such as an electron above the surface of a topological insulator induces an emergent quasi-particle image magnetic monopole, known as a dyon, which is a composite of electric and magnetic charges. This new particle obeys neither Bose nor Fermi statistics but behave like a so called anyon named as such because it is governed by with "any possible" statistics.

An electromagnetic field which contains a magnetic monopole is not Lorentz invariant.

Since the gauge field is Abelian, divB = 0, and isolated magnetic monopoles are necessarily singular in semilocal models. The only way to make the singularity disappear is by embedding the theory in a larger non-Abelian theory which provides a regular core, or by putting the singularity behind an event horizon.





Philosophy
Monopoles are everywhere - https://www.youtube.com/watch?v=seBwiL9InII&feature=youtu.be&t=48m14s

Ontology