Bogomol\'nyi-Prasad-Sommerfield bound

The Bogomol'nyi-Prasad-Sommerfeld bound is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical This set of inequalities is very useful for solving soliton equations. Also see base concepts Topology, Differential equations Quantum theory & Condensed matter physics. 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 in string theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings ^[1]

Examples:

See instanton. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics.
Incomplete: Yang-Mills-Higgs partial differential equations.

The energy at a given time t is given by

$E=\int d^3x\, \left[ \frac{1}{2}\overrightarrow{D\varphi}^T \cdot \overrightarrow{D\varphi} +\frac{1}{2}\pi^T \pi + V(\varphi) + \frac{1}{2g^2}\operatorname{Tr}\left[\vec{E}\cdot\vec{E}+\vec{B}\cdot\vec{B}\right]\right]$

where D is the covariant derivative and V is the potential. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. If we assume that V is nonnegative and is zero only for the Higgs vacuum and that the Higgs field is in the adjoint representation, then

$E \geq \int d^3x\, \left[ \frac{1}{2}\operatorname{Tr}\left[\overrightarrow{D\varphi} \cdot \overrightarrow{D\varphi}\right] + \frac{1}{2g^2}\operatorname{Tr}\left[\vec{B}\cdot\vec{B}\right] \right]$

$\geq \int d^3x\, \operatorname{Tr}\left[ \frac{1}{2}\left(\overrightarrow{D\varphi}\mp\frac{1}{g}\vec{B}\right)^2 \pm\frac{1}{g}\overrightarrow{D\varphi}\cdot \vec{B}\right]$

$\geq \pm \frac{1}{g}\int d^3x\, \operatorname{Tr}\left[\overrightarrow{D\varphi}\cdot \vec{B}\right]$

$= \pm\frac{1}{g}\int_{S^2\ \mathrm{boundary}} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right].$

Therefore,

$E\geq \left\|\int_{S^2} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right]\right \|.$

This quantity is the absolute value of the magnetic flux. In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic

Supersymmetry

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 Physics, a topological quantum number is any quantity in a physical theory that takes on only one of a discrete set of values due to topological considerations

In fact, most bosonic BPS bounds actually come from the bosonic sector of a supersymmetric theory and this explains their origin.

References

^ CAG Almeida, D Bazeia, L Losano (2001), Exploring the vicinity of the Bogomol'nyi-Prasad-Sommerfield bound, Journal of Physics A: Mathematical and General, <http://www.iop.org/EJ/article/0305-4470/34/16/302/a11602.pdf>

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