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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 eo
represents 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 p
may be represented by (p.u)
. If
contraction over components of vectors is required, then the declaration index
must be
used. Thus
index u;
declares u
as 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 u
and v
is
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 V1
through Vn
. However, these variables remain vectors in the
system.
Syntax:
g(id:identifier[,exprn:vector_expression]) :gamma_matrix_expression.
g
is 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 g
is a line identification identifier (not a number) used to distinguish
different lines.
Thus
g(l1,p) * g(l2,q)
denotes the product of \(\gamma \).p
associated with a fermion line identified as l1
, and \(\gamma \).q
associated with another line identified as l2
and 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 a
is reserved in arguments of G to 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 u
flagged as an index if
contraction over u
is 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 eps
has 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 i
and j
flagged as
indices, and so on.
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