Mathematical symbol used in algebras
The 't Hooft symbol is a collection of numbers 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.
Definition[edit]
is the 't Hooft symbol:
![{\displaystyle \eta _{\mu \nu }^{a}={\begin{cases}\epsilon ^{a\mu \nu }&\mu ,\nu =1,2,3\\-\delta ^{a\nu }&\mu =4\\\delta ^{a\mu }&\nu =4\\0&\mu =\nu =4\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/007efdc73995cbb61c51c39c57df4103196bbde8)
Where
and
are instances of the Kronecker delta, and
is the Levi-Civita symbol.
In other words, they are defined by
(
)
![{\displaystyle \eta _{a\mu \nu }=\epsilon _{a\mu \nu 4}+\delta _{a\mu }\delta _{\nu 4}-\delta _{a\nu }\delta _{\mu 4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c145f25ec41b8be6732b406e1b37a9d57f51d1b2)
![{\displaystyle {\bar {\eta }}_{a\mu \nu }=\epsilon _{a\mu \nu 4}-\delta _{a\mu }\delta _{\nu 4}+\delta _{a\nu }\delta _{\mu 4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fafce9da0d9c1da1e4e9f0dd3017bf360626e249)
where the latter are the anti-self-dual 't Hooft symbols.
Matrix Form[edit]
In matrix form, the 't Hooft symbols are
![{\displaystyle \eta _{1\mu \nu }={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{bmatrix}},\quad \eta _{2\mu \nu }={\begin{bmatrix}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{bmatrix}},\quad \eta _{3\mu \nu }={\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2981943424baec23bd65044b3a04fb6b67688cc6)
and their anti-self-duals are the following:
![{\displaystyle {\bar {\eta }}_{1\mu \nu }={\begin{bmatrix}0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}},\quad {\bar {\eta }}_{2\mu \nu }={\begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{bmatrix}},\quad {\bar {\eta }}_{3\mu \nu }={\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7f129d4bafb01e49d7a73612df94eb70f6dc11)
Properties[edit]
They satisfy the self-duality and the anti-self-duality properties:
![{\displaystyle \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 }\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/547ac99fa0dc7c9185e7ef3907156c8fed6d3edb)
Some other properties are
![{\displaystyle \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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2f11677be947d0f30bbfbd2cbcaad938d2ce3f)
![{\displaystyle \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 }\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b1407e7a76d8c66d8329d73e97282d1109179d)
![{\displaystyle \eta _{a\mu \rho }\eta _{b\mu \sigma }=\delta _{ab}\delta _{\rho \sigma }+\epsilon _{abc}\eta _{c\rho \sigma }\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f842d696a89f2bc085f793197b22cb2c0e89d9e)
![{\displaystyle \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 }\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116b917d7e8c5421a0e8f93fc15d3d509c0e69c3)
![{\displaystyle \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 }\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eed202323b31047829b9c95d78ed1776150b7e8a)
The same holds for
except for
![{\displaystyle {\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 }\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ee09237791b4fba1b7c5e4cda8f8cb3a0d5af3)
and
![{\displaystyle \epsilon _{\mu \nu \rho \theta }{\bar {\eta }}_{a\sigma \theta }=-\delta _{\sigma \mu }{\bar {\eta }}_{a\nu \rho }-\delta _{\sigma \rho }{\bar {\eta }}_{a\mu \nu }+\delta _{\sigma \nu }{\bar {\eta }}_{a\mu \rho }\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c83bad9566cffed82b39da5fef82361db5af449e)
Obviously
due to different duality properties.
Many properties of these are tabulated in the appendix of 't Hooft's paper[1] and also in the article by Belitsky et al.[2]
See also[edit]
References[edit]