[note 2] It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite" (i.e. they are solitons or topological defects), can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality.
It was later proven to hold true when dealing with a N = 4 supersymmetric Yang–Mills theory[citation needed].
This duality is now one of several in string theory, the AdS/CFT correspondence which gives rise to the holographic principle,[note 3] being viewed as amongst the most important.
These dualities have played an important role in condensed matter physics, from predicting fractional charges of the electron, to the discovery of the magnetic monopole.
The idea of a close similarity between electricity and magnetism, going back to the time of André-Marie Ampère and Michael Faraday, was first made more precise with James Clerk Maxwell's formulation of his famous equations for a unified theory of electric and magnetic fields: The symmetry between
[5] This led Dirac to state: The interest of the theory of magnetic poles is that it forms a natural generalization of the usual electrodynamics and it leads to the quantization of electricity.
[...] The quantization of electricity is one of the most fundamental and striking features of atomic physics, and there seems to be no explanation for it apart from the theory of poles.
This provides some grounds for believing in the existence of these poles.The magnetic monopole line of research took a step forward in 1974 when Gerard 't Hooft[6] and Alexander Markovich Polyakov[7] independently constructed monopoles not as quantized point particles, but as solitons, in a
In subsequent work this conjecture was refined by Ed Witten and David Olive,[10] they showed that in a supersymmetric extension of the Georgi–Glashow model, the
supersymmetry: Hugh Osborn[11] was able to show that when spontaneous symmetry breaking is imposed in the N = 4 supersymmetric gauge theory, the spins of the topological monopole states are identical to those of the massive gauge particles.
In 1979–1980, Montonen–Olive duality motivated developing mixed symmetric higher-spin Curtright field.
[12] For the spin-2 case, the gauge-transformation dynamics of Curtright field is dual to graviton in D>4 spacetime.
The massless linearized dual gravity was theoretically realized in 2000s for wide class of higher-spin gauge fields, especially that is related to
In fact, there exists a larger SL(2,Z) symmetry where both g as well as theta-angle are transformed non-trivially.
The gauge coupling and theta-angle can be combined to form one complex coupling Since the theta-angle is periodic, there is a symmetry The quantum mechanical theory with gauge group G (but not the classical theory, except in the case when the G is abelian) is also invariant under the symmetry while the gauge group G is simultaneously replaced by its Langlands dual group LG and
The Montonen–Olive duality throws into question the idea that we can obtain a full theory of physics by reducing things into their "fundamental" parts.
Duality says that there is no physically measurable property that can deduce what is fundamental and what is not, the notion of what is elementary and what is composite is merely relative, acting as a kind of gauge symmetry.
Several notable physicists underlined the implications of duality: Under a duality map, often an elementary particle in one string theory gets mapped to a composite particle in a dual string theory and vice versa.
Personally, I would bet that this kind of anti-reductionist behaviour is true in any consistent synthesis of quantum mechanics and gravity.The first conclusion is that Dirac’s explanation of charge quantisation is triumphantly vindicated.
The magnetic monopole here has been treated as bona fide particle even though it arose as a soliton, namely as a solution to the classical equations of motion.
It therefore appears to have a different status from the “Planckian particles” considered hitherto and discussed at the beginning of the lecture.
These arose as quantum excitations of the original fields of the initial formulation of the theory, products of the quantisation procedures applied to these dynamical variables (fields).However, this argument bears little consequence to the reality of string theory as a whole, and perhaps a better perspective might quest for the implications of the AdS/CFT correspondence, and such deep mathematical connections as Monstrous moonshine.