'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:
In other words they are defined by
The (anti)self-duality properties are
Some other properties are
The same holds for except for
Obviously due to different duality properties.
- '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 andD-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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