'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.

η^a_μν is the 't Hooft symbol:

\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}}.

In other words, they are defined by

( $a=1,2,3;~\mu ,\nu =1,2,3,4;~\epsilon _{{1234}}=+1$ )

\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]

References

↑ '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.
↑ 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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't Hooft symbol

See also

References