't Hooft symbol

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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.

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.

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]


  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/0004186free to read. Bibcode:2000CQGra..17.3521B. doi:10.1088/0264-9381/17/17/305.