Instanton: Difference between revisions
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If we insist that the solutions to the [[Yang-Mills]] equations have finite [[energy]], then the curvature of the solution at infinity (taken as a [[limit]]) has to be zero. This means that the [[Chern-Simons]] invariant can be defined at the 3-space boundary. |
If we insist that the solutions to the [[Yang-Mills]] equations have finite [[energy]], then the curvature of the solution at infinity (taken as a [[limit]]) has to be zero. This means that the [[Chern-Simons]] invariant can be defined at the 3-space boundary. |
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This is equivalent, via [[Stoke's theorem]], to taking the integral <math>\int_{\mathbb{R}^4}Tr[\bold{F}\wedge\bold{F}]</math>. This is a homotopy invariant and it tells us which [[homotopy class]] the instanton belongs to. The [[Yang-Mills]] energy is given by <math>\frac{1}{2}\int_{\mathbb{R}^4}Tr[*\bold{F}\wedge \bold{F}]</math> where * is the [[Hodge dual]]. |
This is equivalent, via [[Stoke's theorem]], to taking the integral |
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:<math>\int_{\mathbb{R}^4}Tr[\bold{F}\wedge\bold{F}]</math>. |
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This is a homotopy invariant and it tells us which [[homotopy class]] the instanton belongs to. The [[Yang-Mills]] energy is given by <math>\frac{1}{2}\int_{\mathbb{R}^4}Tr[*\bold{F}\wedge \bold{F}]</math> where * is the [[Hodge dual]]. |
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Since the integral of a nonnegative [[integrand]] is always nonnegative, <math>0\leq\frac{1}{2}\int_{\mathbb{R}^4}Tr[(*\bold{F}+e^{-i\theta}\bold{F})\wedge(\bold{F}+e^{i\theta}*\bold{F})] |
Since the integral of a nonnegative [[integrand]] is always nonnegative, <math>0\leq\frac{1}{2}\int_{\mathbb{R}^4}Tr[(*\bold{F}+e^{-i\theta}\bold{F})\wedge(\bold{F}+e^{i\theta}*\bold{F})] |
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Revision as of 05:29, 10 October 2003
In mathematical physics, the concept of instanton is more complicated in Minkowski space: in this article, we will focus on instantons in 4D Euclidean space.
If we insist that the solutions to the Yang-Mills equations have finite energy, then the curvature of the solution at infinity (taken as a limit) has to be zero. This means that the Chern-Simons invariant can be defined at the 3-space boundary.
This is equivalent, via Stoke's theorem, to taking the integral
- .
![{\displaystyle \int _{\mathbb {R} ^{4}}Tr[{\mathbf {F}}\wedge {\mathbf {F}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efdbef784057ec889803b691e87e6f2a19dfcd9b)
This is a homotopy invariant and it tells us which homotopy class the instanton belongs to. The Yang-Mills energy is given by where * is the Hodge dual.
![{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}Tr[*{\mathbf {F}}\wedge {\mathbf {F}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8b98172d6a61b176cc133f6508115bbb4738e8)
Since the integral of a nonnegative integrand is always nonnegative, for all real θ. So, this means If this bound is saturated, then the solution is a BPS state. For such states, either *F=F or *F=-F depending on the sign of the homotopy invariant.
![{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}Tr[(*{\mathbf {F}}+e^{-i\theta }{\mathbf {F}})\wedge ({\mathbf {F}}+e^{i\theta }*{\mathbf {F}})]=\int _{\mathbb {R} ^{4}}Tr[*{\mathbf {F}}\wedge {\mathbf {F}}+2\cos \theta {\mathbf {F}}\wedge {\mathbf {F}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40849b8d1d2869e3ceb212051b89bc1c5131c221)
![{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}Tr[*{\mathbf {F}}\wedge {\mathbf {F}}]\geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}Tr[{\mathbf {F}}\wedge {\mathbf {F}}]\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d71e74ed87aa5a3754f0b9a8233e50c4373d84)