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't Hooft symbol

Mathematical symbol used in algebras From Wikipedia, the free encyclopedia

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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

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is the 't Hooft symbol: Where and are instances of the Kronecker delta, and is the Levi–Civita symbol.

In other words, they are defined by

()

where the latter are the anti-self-dual 't Hooft symbols.

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Matrix form

In matrix form, the 't Hooft symbols are and their anti-self-duals are the following:

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Properties

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They satisfy the self-duality and the anti-self-duality properties:

Some other properties are

The same holds for except for

and

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]

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See also

References

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