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 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 was studying the quantum mechanics of an electric charge moving in a magnetic monopole field, 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 1021. This led Dirac to state: The magnetic monopole line of research took a step forward in 1974 when Gerard 't Hooft and Alexander Markovich Polyakov independently constructed monopoles not as quantized point particles, but as solitons, 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 nonsingular 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, 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 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 constantg by 1/g. This also involves an interchange of the electrically charged particles and magnetic monopoles. See also Seiberg duality. In fact, there exists a larger SL symmetry where both g as well as theta-angle are transformed non-trivially. The gauge coupling and theta-angle can be combined together to form one complex coupling Since the theta-angle is periodic, there is a symmetry The quantum mechanical theory with gauge groupG is also invariant under the symmetry while the gauge group G is simultaneously replaced by its Langlands dual groupLG 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 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, as both the Noether charge and topological charge have the same ontology. Several notable physicists underlined the implications of duality:
