File r37/lisp/csl/html/r37_0411.html artifact f5b2024172 part of check-in 5f584e9b52

<A NAME=G>

<TITLE>G</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>G</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
<em>g</em> is an n-ary operator used to denote a product of gamma matrices 
contracted with Lorentz four-vectors, in high-energy physics. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>g</em>(&lt;identifier&gt;,&lt;vector-expr&gt; 
{,&lt;vector-expr&gt;}*) 
<P>
<P>
<P>
&lt;identifier&gt; is a scalar identifier representing a fermion line 
identifier, &lt;vector-expr&gt; can be any valid vector expression, 
representing a vector or a gamma matrix. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
vector aa,bb,cc; 

vector a; 

g(line1,aa,bb); 

  AA.BB 


g(line2,aa,a); 

  0 


g(id,aa,bb,cc); 

  0 


g(li1,aa,bb) + k; 

  AA.BB + K 


let aa.bb = m*k; 

g(ln1,aa)*g(ln1,bb); 

  K*M 


g(ln1,aa)*g(ln2,bb); 

  0

</TT></PRE><P>The vector <em>A</em> is reserved in arguments of <em>g</em> to de
note the 
special gamma matrix gamma_5. It must be declared to 
be a vector before you use it. 
<P>
<P>
Gamma matrix expressions are associated with fermion lines in a Feynman 
diagram. If more than one line occurs in an expression, the gamma 
matrices involved are separate (operating in independent spin space), as 
shown in the last two example lines above. A product of gamma matrices 
associated with a single line can be entered either as a single <em>g</em> 
command with several vector arguments, or as products of separate <em>g</em> 
commands each with a single argument. 
<P>
<P>
While the product of vectors is not defined, the product, sum and 
difference of several gamma expressions are defined, as is the product of 
a gamma expression with a scalar. If an expression involving gamma 
matrices includes a scalar, the scalar is treated as if it were the 
product of itself with a unit 4 x 4 matrix. 
<P>
<P>
Dirac expressions are evaluated by computing the trace of the expression 
using the commutation algebra of gamma matrices. The algorithms used are 
described in articles by J. S. R. Chisholm in &lt;Il Nuovo Cimento X,&gt; Vol. 
30, p. 426, 1963, and J. Kahane, &lt;Journal of Mathematical Physics&gt;, 
Vol. 9, p. 1732, 1968. The trace is then divided by 4 to distinguish 
between the trace of a scalar and the trace of an expression that is the 
product of a scalar with a unit 4 x 4 matrix. 
<P>
<P>
Trace calculations may be prevented over any line identifier by declaring it 
to be 
<A HREF=r37_0415.html>nospur</A>. If it is later desired to evaluate these trace
s, 
the declaration can be undone with the 
<A HREF=r37_0417.html>spur</A> declaration. 
<P>
<P>
The notation of Bjorken and Drell, &lt;Relativistic Quantum Mechanics,&gt; 
1964, is assumed in all operations involving gamma matrices. For an 
example of the use of <em>g</em> in a calculation, see the &lt;REDUCE 
User's Manual&gt;. 
<P>
<P>
<P>