en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-19T16:16:10Z

Fix CS1 deprecated date parameter errors

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The ''''t Hooft η symbol''' is a symbol which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the [[Kronecker delta]] and the [[Levi-Civita symbol]]. It was introduced by [[Gerard 't Hooft]]. It is used in the construction of the [[BPST instanton]].
They are defined by

(<math> a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_{1 2 3 4}=+1</math>)

:<math> \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} + \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4} </math>

:<math> \bar \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} + \delta_{a \nu} \delta_{\mu 4} </math>

The (anti)self-duality properties are
:<math>
\eta_{a\mu\nu} = \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} \eta_{a\rho\sigma} \ ,
\qquad
\bar\eta_{a\mu\nu} = - \frac{1}{2} \epsilon_{\mu\nu\rho\sigma}
\bar\eta_{a\rho\sigma} \
</math>

Some other properties are

:<math>
\epsilon_{abc} \eta_{b\mu\nu} \eta_{c\rho\sigma}
= \delta_{\mu\rho} \eta_{a\nu\sigma}
+ \delta_{\nu\sigma} \eta_{a\mu\rho}
- \delta_{\mu\sigma} \eta_{a\nu\rho}
- \delta_{\nu\rho} \eta_{a\mu\sigma}
</math>
:<math>
\eta_{a\mu\nu} \eta_{a\rho\sigma}
= \delta_{\mu\rho} \delta_{\nu\sigma}
- \delta_{\mu\sigma} \delta_{\nu\rho}
+ \epsilon_{\mu\nu\rho\sigma} \ ,
</math>
:<math>
\eta_{a\mu\rho} \eta_{b\mu\sigma}
= \delta_{ab} \delta_{\rho\sigma} + \epsilon_{abc} \eta_{c\rho\sigma} \ ,
</math>
:<math>
\epsilon_{\mu\nu\rho\theta} \eta_{a\sigma\theta}
= \delta_{\sigma\mu} \eta_{a\nu\rho}
+ \delta_{\sigma\rho} \eta_{a\mu\nu}
- \delta_{\sigma\nu} \eta_{a\mu\rho} \ ,
</math>
:<math>
\eta_{a\mu\nu} \eta_{a\mu\nu} = 12 \ ,\quad
\eta_{a\mu\nu} \eta_{b\mu\nu} = 4 \delta_{ab} \ ,\quad
\eta_{a\mu\rho} \eta_{a\mu\sigma} = 3 \delta_{\rho\sigma} \ .
</math>

The same holds for <math>\bar\eta</math> except for

:<math>
\bar\eta_{a\mu\nu} \bar\eta_{a\rho\sigma}
= \delta_{\mu\rho} \delta_{\nu\sigma}
- \delta_{\mu\sigma} \delta_{\nu\rho}
- \epsilon_{\mu\nu\rho\sigma} \ .
</math>

Obviously <math>\eta_{a\mu\nu} \bar\eta_{b\mu\nu} = 0</math> due to different
duality properties.

Many properties of these are tabulated in the appendix of 't Hooft's paper<ref>{{cite doi|10.1103/PhysRevD.14.3432}}</ref> and also in the article by Belitsky et al.<ref>{{cite doi|10.1088/0264-9381/17/17/305}}</ref>

==See also==
*[[Instanton]]

==References==
{{reflist}}

[[Category:Gauge theories]]

{{Quantum-stub}}

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en>Monkbot: Fix CS1 deprecated date parameter errors