- Bogomol'nyi-Prasad-Sommerfield bound
The Bogomol'nyi-Prasad-Sommerfeld bound is a series of inequalities for solutions of
partial differential equation s depending on thehomotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied (called "saturated"), one can come up with a simpler set of partial differential equations to solve. Solutions saturating the bound are called BPS states and play an important role instring theory . [citation|title=Exploring the vicinity of the Bogomol'nyi-Prasad-Sommerfield bound |author=CAG Almeida, D Bazeia, L Losano|publisher= Journal of Physics A: Mathematical and General|year=2001 |url= http://www.iop.org/EJ/article/0305-4470/34/16/302/a11602.pdf]Examples:
*Seeinstanton .
*"Incomplete:" Yang-Mills-Higgs partial differential equations.The energy at a given time "t" is given by
:E=int d^3x, left [ frac{1}{2}overrightarrow{Dvarphi}^T cdot overrightarrow{Dvarphi} +frac{1}{2}pi^T pi + V(varphi) + frac{1}{2g^2}operatorname{Tr}left [vec{E}cdotvec{E}+vec{B}cdotvec{B} ight] ight]
where "D" is the
covariant derivative and "V" is the potential. If we assume that "V" is nonnegative and is zero only for the Higgs vacuum and that the Higgs field is in theadjoint representation , then:E geq int d^3x, left [ frac{1}{2}operatorname{Tr}left [overrightarrow{Dvarphi} cdot overrightarrow{Dvarphi} ight] + frac{1}{2g^2}operatorname{Tr}left [vec{B}cdotvec{B} ight] ight]
::geq int d^3x, operatorname{Tr}left [ frac{1}{2}left(overrightarrow{Dvarphi}mpfrac{1}{g}vec{B} ight)^2 pmfrac{1}{g}overrightarrow{Dvarphi}cdot vec{B} ight]
::geq pm frac{1}{g}int d^3x, operatorname{Tr}left [overrightarrow{Dvarphi}cdot vec{B} ight]
::pmfrac{1}{g}int_{S^2 mathrm{boundary operatorname{Tr}left [varphi vec{B}cdot dvec{S} ight] .
Therefore,
:Egeq left|int_{S^2} operatorname{Tr}left [varphi vec{B}cdot dvec{S} ight] ight |.
This quantity is the absolute value of the
magnetic flux .upersymmetry
In supersymmetry, the BPS bound is saturated when half (or a quarter or an eighth) of the SUSY generators are unbroken. This happens when the mass is equal to the central extension, which is typically a
topological charge .In fact, most bosonic BPS bounds actually come from the bosonic sector of a supersymmetric theory and this explains their origin.
References
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