User:Sparkyscience/Monopoles

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^[a] 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.^[6]

Berry phase

Quaternionic electromagnetism

19th century mathematicians were inspired by the mathematical beauty of complex numbers and complex analysis^[b] to seek new "generalized complex numbers" which would apply in a natural way to higher dimensions, and in particular, 3-dimensional space.^[17] In 1843 William Rowan Hamilton famously discovered quaternions while walking across Broom Bridge.^[c] 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.^[20] 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:

${\displaystyle {\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}}$

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.^[21] The quaternions were the first time in history that a non-commutative product appeared in mathematics.^[22] 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.^[d][24] In modern group theory, a group that preserves commutativity, like for example the complex numbers ${\displaystyle \mathbb {C} }$, is said to form an abelian group named after Niels Henrik Abel; the quaternion group, ${\displaystyle \mathbb {H} }$, is said to be a non-abelian.^[e] The complex numbers, ${\displaystyle \mathbb {C} }$, can be written by Euler's formula in the form ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$ and form a ${\displaystyle \operatorname {U} (1)}$ symmetry group. Geometrically, ${\displaystyle \operatorname {U} (1)}$ is the unit circle ${\displaystyle S^{1}}$. We can identify the plane ${\displaystyle \mathbb {R} ^{2}}$ with the complex plane ${\displaystyle \mathbb {C} }$, letting ${\displaystyle z=x+iy\in \mathbb {C} }$ represent ${\displaystyle (x,y)\in \mathbb {R} ^{2}}$. Multiplication of any complex number ${\displaystyle e^{i\theta }\in \operatorname {U} (1)}$ can be thought of as a rotation by an angle ${\displaystyle \theta }$ and is the same kind of symmetry as rotating a unit circle about the origin. For the quaternions, ${\displaystyle \mathbb {H} }$, 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:

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

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

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

In 1858 Hamilton begun corresponding with Peter Guthrie Tait.^[26] Tait showed that quaternions had relevance to many problems in physics,^[20] and he used them to describe the theoretical properties of "magnetic particles".^[f] Tait's friend James Clerk Maxwell enigmatically called him the "Chief Musician upon Nabla"^[28] and wrote favourably about the benefits of using quaternion notation for 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^[29] as well as cartesian equations.^[20] 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."^[30] 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.^[31]

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

The work of William Thomson and Tait focused on potentials based on the principle of least action and Lagrangians;^[g] 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.^[33] 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)^[h] 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 ${\displaystyle \operatorname {SU} (2)}$ symmetry.^[34][4]

It has sometimes been argued that quaternions present difficult issues of compatibility when describing space-time,^[35] though models have been proposed.^[i]

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.^[37] 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 ${\displaystyle {\partial /\partial t}}$ (rather then second order form ${\displaystyle ({\partial /\partial t})^{2}}$). 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 ${\displaystyle +---}$. Hamilton had shown that a square root of the ordinary 3-dimensional Laplacian can be obtained by using quaternions:^[j][38]

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

Clifford generalised this idea for higher dimensions^[k]

The Dirac equation with no external field is:

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

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

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

Schwinger

Magnetic photon

Wu & Yang

Weinberg-Salam model

't Hooft–Polyakov monopole

Montonen–Olive conjecture

Nambu

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.^[39] Donaldson theory

Stanford

• Cabrera, Blas (1982). "First Results from a Superconductive Detector for Moving Magnetic Monopoles" (PDF). Physical Review Letters. 48 (20): 1378–1381. Bibcode:1982PhRvL..48.1378C. doi:10.1103/PhysRevLett.48.1378. ISSN 0031-9007.

Spin ice

• Castelnovo, C.; Moessner, R.; Sondhi, S. L. (2008). "Magnetic monopoles in spin ice" (PDF). Nature. 451 (7174): 42–45. arXiv:0710.5515. Bibcode:2008Natur.451...42C. doi:10.1038/nature06433. ISSN 0028-0836. PMID 18172493.

• Tchernyshyov, Oleg (2008). "Magnetism: Freedom for the poles" (PDF). Nature. 451 (7174): 22–23. Bibcode:2008Natur.451...22T. doi:10.1038/451022b. ISSN 0028-0836. PMID 18172484.

• Sondhi, Shivaji (2009). "Condensed-matter physics: Wien route to monopoles". Nature. 461 (7266): 888–889. Bibcode:2009Natur.461..888S. doi:10.1038/461888a. ISSN 0028-0836. PMID 19829360.

Magnetricity

• Bramwell, S. T.; Giblin, S. R.; Calder, S.; Aldus, R.; Prabhakaran, D.; Fennell, T. (2009). "Measurement of the charge and current of magnetic monopoles in spin ice" (PDF). Nature. 461 (7266): 956–959. arXiv:0907.0956v1. Bibcode:2009Natur.461..956B. doi:10.1038/nature08500. ISSN 0028-0836. PMID 19829376.

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

Ontology

• Castellani, Elena (2016). "Duality and 'particle' democracy" (PDF). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 59: 100–108. doi:10.1016/j.shpsb.2016.03.002. ISSN 1355-2198.

• Rehn, J.; Moessner, R. (2016). "Maxwell electromagnetism as an emergent phenomenon in condensed matter" (PDF). Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 374 (2075): 20160093. arXiv:1605.05874v1. Bibcode:2016RSPTA.37460093R. doi:10.1098/rsta.2016.0093. ISSN 1364-503X. PMID 27458263.{{cite journal}}: CS1 maint: article number as page number (link)

• Duff, M. J.; Khuri, Ramzi R.; Lu, J. X. (1995). "String solitons" (PDF). Physics Reports. 259 (4–5): 213–326. arXiv:hep-th/9412184v1. Bibcode:1995PhR...259..213D. doi:10.1016/0370-1573(95)00002-X. ISSN 0370-1573.