Montonen–Olive Duality
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Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or
S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...
according to current terminology. It generalizes the electro-magnetic symmetry of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
by stating that magnetic monopoles, which are usually viewed as emergent
quasiparticles In condensed matter physics, a quasiparticle is a concept used to describe a collective behavior of a group of particles that can be treated as if they were a single particle. Formally, quasiparticles and collective excitations are closely relate ...
that are "composite" (i.e. they are
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or topological defects), can in fact be viewed as "elementary" quantized particles with
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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. It is named after Finnish physicist Claus Montonen and
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physicist David Olive after they proposed the idea in their academic paper '' Magnetic monopoles as gauge particles?'' where they state: S-duality is now a basic ingredient in topological quantum field theories and string theories, especially since the 1990s with the advent of the
second superstring revolution The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum ...
. This duality is now one of several in string theory, the
AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) that are used ...
which gives rise to the holographic principle, being viewed as amongst the most important. These dualities have played an important role in
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, from predicting fractional charges of the electron, to the discovery of the
magnetic monopole In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
.


Electric–magnetic duality

The idea of a close similarity between electricity and magnetism, going back to the time of
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as ''electrodynamics''. He is also the inventor of ...
and
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, was first made more precise with
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
's formulation of his famous equations for a unified theory of electric and magnetic fields: : \begin \nabla \cdot \mathbf &= \rho \quad & \nabla \times \mathbf + \dot &= 0 \\ \nabla \cdot \mathbf &= 0 \quad & \nabla \times \mathbf - \dot &= \mathbf. \end The symmetry between \mathbf and \mathbf in these equations is striking. If one ignores the sources, or adds magnetic sources, the equations are invariant under \mathbf \rightarrow \mathbf and \mathbf \rightarrow -\mathbf. Why should there be such symmetry between \mathbf and \mathbf? In 1931
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
was studying the quantum mechanics of an electric charge moving in a magnetic monopole field, and he found he could only consistently define the wavefunction if the electric charge e and magnetic charge q satisfy the quantization condition: : \begin eq=2\pi\hbar n \quad \quad & n = 0, \pm1, \pm2 ... \\ \end Note that from the above if just one monopole of some charge q exists anywhere, then all electric charges must be multiples of the unit 2\pi\hbar/q. This would "explain" why the magnitude of the electron charge and proton charge should be exactly equal and are the same no matter what electron or proton we are considering, a fact known to hold true to one part in 1021. This led Dirac to state: The magnetic monopole line of research took a step forward in 1974 when
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and Alexander Markovich Polyakov independently constructed monopoles not as quantized point particles, but as
soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s, in a \operatorname(2) Yang–Mills–Higgs system, previously magnetic monopoles had always included a point singularity. The subject was motivated by Nielsen–Olesen vortices. At weak coupling, the electrically and magnetically charged objects look very different: one an electron point particle that is weakly coupled and the other a monopole soliton that is strongly coupled. The magnetic fine structure constant is roughly the reciprocal of the usual one: \alpha_q^/4\pi\hbar=n^/4\alpha In 1977 Claus Montonen and David Olive conjectured that at strong coupling the situation would be reversed: the electrically charged objects would be strongly coupled and have non-singular cores, while the magnetically charged objects would become weakly coupled and point like. The strongly coupled theory would be equivalent to weakly coupled theory in which the basic quanta carried magnetic rather than electric charges. In subsequent work this conjecture was refined by Ed Witten and David Olive, they showed that in a supersymmetric extension of the Georgi–Glashow model, the N = 2 supersymmetric version (N is the number of conserved supersymmetries), there were no quantum corrections to the classical mass spectrum and the calculation of the exact masses could be obtained. The problem related to the monopole's unit spin remained for this N = 2 case, but soon after a solution to it was obtained for the case of N = 4 supersymmetry: Hugh Osborn 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.


Dual gravity

In 1979–1980, Montonen–Olive duality motivated developing mixed symmetric higher-spin Curtright field. For the spin-2 case, the gauge-transformation dynamics of Curtright field is dual to graviton in D>4 spacetime. Meanwhile, the spin-0 field, developed by CurtrightFreund, is dual to the Freund- Nambu field, that is coupled to the trace of its energy–momentum tensor. The massless linearized dual gravity was theoretically realized in 2000s for wide class of higher-spin gauge fields, especially that is related to \mathrm(8), E_7 and E_ supergravity. A massive spin-2 dual gravity, to lowest order, in ''D'' = 4 and ''N''-''D'' is recently introduced as a theory dual to the
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of Ogievetsky–Polubarinov theory. The dual field is coupled to the curl of the energy momentum tensor.


Mathematical formalism

In a four-dimensional Yang–Mills theory with ''N'' = 4 supersymmetry, which is the case where the Montonen–Olive duality applies, one obtains a physically equivalent theory if one replaces the gauge
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
''g'' by 1/''g''. This also involves an interchange of the electrically charged particles and
magnetic monopole In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
s. See also
Seiberg duality In quantum field theory, Seiberg duality, conjectured by Nathan Seiberg in 1994, is an S-duality relating two different super QCD, supersymmetric QCDs. The two theories are not identical, but they agree at low energies. More precisely under a re ...
. 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 : \tau = \frac+\frac. Since the theta-angle is periodic, there is a symmetry : \tau \mapsto \tau + 1. 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 : \tau \mapsto \frac while the gauge group ''G'' is simultaneously replaced by its Langlands dual group ''L''''G'' and n_G is an integer depending on the choice of gauge group. In the case the theta-angle is 0, this reduces to the simple form of Montonen–Olive duality stated above.


Philosophical implications

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. The philosophy of
reductionism Reductionism is any of several related philosophical ideas regarding the associations between phenomena which can be described in terms of simpler or more fundamental phenomena. It is also described as an intellectual and philosophical positi ...
states that if we understand the "fundamental" or "elementary" parts of a system we can then deduce all the properties of the system as a whole. 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. This seems to favour the view of
emergentism Emergentism is the philosophical theory that higher-level properties or phenomena emerge from more basic components, and that these emergent properties are not fully reducible to or predictable from those lower-level parts. A property of a sys ...
, as both the Noether charge (particle) and topological charge (soliton) have the same ontology. Several notable physicists underlined the implications of duality: 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 In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) that are used ...
, and such deep mathematical connections as
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. Since experimentally tested evidence bears no resemblance to the
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; where philosophically an
Anthropic principle In cosmology, the anthropic principle, also known as the observation selection effect, is the proposition that the range of possible observations that could be made about the universe is limited by the fact that observations are only possible in ...
is at its strongest a self-justification for any unprovable theory.


Notes


References


Further reading

Academic papers * * * * * * * * * * * * * * * * * * * * * * * * Books * {{DEFAULTSORT:Montonen-Olive duality Gauge theories Duality theories