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
solitons
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 locali ...
or
topological defects), can in fact be viewed as "elementary"
quantized particles with
electrons
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
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
British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories and Crown Dependencies.
* British national identity, the characteristics of British people and culture ...
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
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, 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
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English chemist and physicist who contributed to the study of electrochemistry and electromagnetism. His main discoveries include the principles underlying electromagnetic inducti ...
, 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:
:
The symmetry between
and
in these equations is striking. If one ignores the sources, or adds magnetic sources, the equations are invariant under
and
.
Why should there be such symmetry between
and
? 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
and magnetic charge
satisfy the quantization condition:
:
Note that from the above if just one monopole of some charge
exists anywhere, then all electric charges must be multiples of the unit
. 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 10
21. This led Dirac to state:
The magnetic monopole line of research took a step forward in 1974 when
Gerard 't Hooft
Gerardus "Gerard" 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating t ...
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
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:
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
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
case, but soon after a solution to it was obtained for the case of
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
Curtright–
Freund, 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
,
and
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
massive gravity
Massive is an adjective related to mass.
Massive may refer to:
Arts, entertainment, and media
* Massive (band), an Australian Hard Rock band
* ''Massive'', an album by The Supervillains released in 2008
* Massive Attack, a British musical group ...
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
:
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 ''L''''G'' and
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
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
. Since experimentally tested evidence bears no resemblance to the
String theory landscape
In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...
; 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
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{{DEFAULTSORT:Montonen-Olive duality
Gauge theories
Duality theories