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

SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...

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

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 ...

and the

Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...

. 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 th ...

. It is used in the construction of the

BPST instanton In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills th ...

. 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