't Hooft symbol

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.

ηaμν is the 't Hooft symbol:

\eta^a_{\mu\nu} = \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} .

In other words they are defined by

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

\eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} + \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4}
\bar \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} + \delta_{a \nu} \delta_{\mu 4}

The (anti)self-duality properties are

\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} \

Some other properties are

\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}
\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} \ ,
\eta_{a\mu\rho} \eta_{b\mu\sigma} = \delta_{ab} \delta_{\rho\sigma} + \epsilon_{abc} \eta_{c\rho\sigma} \ ,
\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} \ ,
\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} \ .

The same holds for \bar\eta except for

\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} \ .

and

\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} \ ,

Obviously \eta_{a\mu\nu} \bar\eta_{b\mu\nu} = 0 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

References

  1. ↑ 't Hooft, G. (1976). "Computation of the quantum effects due to a four-dimensional pseudoparticle". Physical Review D 14 (12): 3432. Bibcode:1976PhRvD..14.3432T. doi:10.1103/PhysRevD.14.3432.
  2. ↑ Belitsky, A. V.; Vandoren, S.; Nieuwenhuizen, P. V. (2000). "Yang-Mills and D-instantons". Classical and Quantum Gravity 17 (17): 3521. arXiv:hep-th/0004186. Bibcode:2000CQGra..17.3521B. doi:10.1088/0264-9381/17/17/305.


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