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We begin by introducing three new operators required in these calculations.
Syntax:
(exprn1:vector_expression) . (exprn2:vector_expression):algebraic.
The binary .operator, which is normally used to denote the addition of an element to the front of a list, can also be used in algebraic mode to denote the scalar product of two Lorentz four-vectors. For this to happen, the second argument must be recognizable as a vector expression at the time of evaluation. With this meaning, this operator is often referred to as the dot operator. In the present system, the index handling routines all assume that Lorentz four-vectors are used, but these routines could be rewritten to handle other cases.
Components of vectors can be represented by including representations of unit vectors in the system. Thus if eorepresents the unit vector (1,0,0,0), (p.eo)represents the zeroth component of the four-vector P. Our metric and notation follows Bjorken and Drell [JDB65]. Similarly, an arbitrary component pmay be represented by (p.u). If contraction over components of vectors is required, then the declaration indexmust be used. Thus
index u;
declares uas an index, and the simplification of
p.u * q.u
would result in
P.Q
The metric tensor \(g^{\mu \nu }\) may be represented by (u.v). If contraction over uand vis required, then they should be declared as indices.
Errors occur if indices are not properly matched in expressions.
If a user later wishes to remove the index property from specific vectors, he can do it with the declaration remind. Thus remind v1,…,vn;removes the index flags from the variables V1through Vn. However, these variables remain vectors in the system.
Syntax:
g(id:identifier[,exprn:vector_expression]) :gamma_matrix_expression.
gis an n-ary operator used to denote a product of \(\gamma \) matrices contracted with Lorentz four-vectors. Gamma matrices are associated with fermion lines in a Feynman diagram. If more than one such line occurs, then a different set of \(\gamma \) matrices (operating in independent spin spaces) is required to represent each line. To facilitate this, the first argument of gis a line identification identifier (not a number) used to distinguish different lines.
Thus
g(l1,p) * g(l2,q)
denotes the product of \(\gamma \cdot p\) associated with a fermion line identified as l1, and \(\gamma \cdot q\) associated with another line identified as l2and where \(p\) and \(q\) are Lorentz four-vectors. A product of \(\gamma \) matrices associated with the same line may be written in a contracted form.
Thus
g(l1,p1,p2,...,p3) = g(l1,p1)*g(l1,p2)*...*g(l1,p3) .
The vector ais reserved in arguments of gto denote the special \(\gamma \) matrix \(\gamma ^{5}\). Thus
g(l,a) \( = \gamma ^{5}\) associated with the line l g(l,p,a) \( = \gamma \cdot p \times \gamma ^{5}\) associated with the line l.
\(\gamma ^{\mu }\) (associated with the line l) may be written as g(l,u), with uflagged as an index if contraction over uis required.
The notation of Bjorken and Drell is assumed in all operations involving \(\gamma \) matrices.
eps(exprn1:vector_expression,...,exprn4:vector_exp) :vector_exp.
The operator epshas four arguments, and is used only to denote the completely antisymmetric tensor of order 4 and its contraction with Lorentz four-vectors. Thus
A contraction of the form \(\epsilon _{i j \mu \nu }p_{\mu }q_{\nu }\) may be written as eps(i,j,p,q), with iand jflagged as indices, and so on.
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