The t Hooft symbol is a collection of numbers which allows one to express the generators of the

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Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the

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and the

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. It was introduced by

Gerard 't Hooft Gerardus (Gerard) 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating the ...

. It is used in the construction of the

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. η^''a''_μν is the 't Hooft symbol: :

\eta^a_ = \begin \epsilon^ & \mu,\nu=1,2,3 \\ -\delta^ & \mu=4 \\ \delta^ & \nu=4 \\ 0 & \mu=\nu=4 \end .

In other words, they are defined by (

a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_=+1

) :

\eta_ = \epsilon_ + \delta_ \delta_ - \delta_ \delta_

\bar \eta_ = \epsilon_ - \delta_ \delta_ + \delta_ \delta_

where the latter are the anti-self-dual 't Hooft symbols. More explicitly, these symbols are :

\eta_ =  
\begin
    0 & 0 & 0 & 1  \\
    0 & 0 & 1 & 0   \\
    0 & -1 & 0 & 0  \\
    -1 & 0 & 0 & 0 
  \end,
\quad
\eta_ =  
\begin
    0 & 0 & -1 & 0  \\
    0 & 0 & 0 & 1   \\
    1 & 0 & 0 & 0  \\
    0 & -1 & 0 & 0 
  \end,
\quad 
\eta_ =  
\begin
    0 & 1 & 0 & 0  \\
    -1 & 0 & 0 & 0   \\
    0 & 0 & 0 & 1  \\
    0 & 0 & -1 & 0 
  \end,

and :

\bar_ =  
\begin
    0 & 0 & 0 & -1  \\
    0 & 0 & 1 & 0   \\
    0 & -1 & 0 & 0  \\
    1 & 0 & 0 & 0 
  \end,
\quad
\bar_ =  
\begin
    0 & 0 & -1 & 0  \\
    0 & 0 & 0 & -1   \\
    1 & 0 & 0 & 0  \\
    0 & 1 & 0 & 0 
  \end,
\quad 
\bar_ =  
\begin
    0 & 1 & 0 & 0  \\
    -1 & 0 & 0 & 0   \\
    0 & 0 & 0 & -1  \\
    0 & 0 & 1 & 0 
  \end.

Properties

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

\eta_ = \frac \epsilon_ \eta_ \ ,
\qquad
\bar\eta_ = - \frac \epsilon_
\bar\eta_ \

Some other properties are :

\epsilon_ \eta_ \eta_
= \delta_ \eta_
+ \delta_ \eta_
- \delta_ \eta_
- \delta_ \eta_

\eta_ \eta_
= \delta_ \delta_
- \delta_ \delta_
+ \epsilon_ \ ,

\eta_ \eta_
= \delta_ \delta_ + \epsilon_ \eta_ \ ,

\epsilon_ \eta_
= \delta_ \eta_
+ \delta_ \eta_
- \delta_ \eta_ \ ,

\eta_ \eta_ = 12 \ ,\quad
\eta_ \eta_ = 4 \delta_ \ ,\quad
\eta_ \eta_ = 3 \delta_ \ .

The same holds for

\bar\eta

except for :

\bar\eta_ \bar\eta_
= \delta_ \delta_
- \delta_ \delta_
- \epsilon_ \ .

and :

\epsilon_ \bar\eta_
= -\delta_ \bar\eta_
- \delta_ \bar\eta_
+ \delta_ \bar\eta_ \ ,

Obviously

\eta_ \bar\eta_ = 0

due to different duality properties. Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.

References

Gauge theories Mathematical symbols {{Quantum-stub

Properties

See also

References