ExpandScalarProduct
ExpandScalarProduct[expr]
expands scalar products of sums of momenta in expr
.
ExpandScalarProduct
does not use Expand
on expr
.
See also
Overview, Calc, MomentumExpand, MomentumCombine, FCVariable
Examples
SP[p1 + p2 + p3, p4 + p5 + p6]
% // ExpandScalarProduct
(p1+p2+p3)⋅(p4+p5+p6)
p1⋅p4+p1⋅p5+p1⋅p6+p2⋅p4+p2⋅p5+p2⋅p6+p3⋅p4+p3⋅p5+p3⋅p6
SP[p, p - q]
ExpandScalarProduct[%]
p⋅(p−q)
p2−p⋅q
FV[p - q, \[Mu]]
ExpandScalarProduct[%]
(p−q)μ
pμ−qμ
SPD[p - q, q - r]
ExpandScalarProduct[%]
(p−q)⋅(q−r)
p⋅q−p⋅r+q⋅r−q2
Using the option Momentum
one can limit the expansion to particular momenta
SP[p1 + p2 + p3, p4 + p5 + p6]
ExpandScalarProduct[%, Momentum -> {p1}]
(p1+p2+p3)⋅(p4+p5+p6)
p1⋅(p4+p5+p6)+(p2+p3)⋅(p4+p5+p6)
By default ExpandScalarProduct
does not apply linearity to Levi-Civita tensors
LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6]
ExpandScalarProduct[%]
ϵˉμp1+p2p3+p4p5+p6
ϵˉμp1+p2p3+p4p5+p6
Using the option EpsEvaluate
takes care of that
LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6]
ExpandScalarProduct[%, EpsEvaluate -> True]
ϵˉμp1+p2p3+p4p5+p6
ϵˉμp1p3p5+ϵˉμp1p3p6+ϵˉμp1p4p5+ϵˉμp1p4p6+ϵˉμp2p3p5+ϵˉμp2p3p6+ϵˉμp2p4p5+ϵˉμp2p4p6
One can use the options EpsEvaluate
and Momentum
together
LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6]
ExpandScalarProduct[%, EpsEvaluate -> True, Momentum -> {p1}]
ϵˉμp1+p2p3+p4p5+p6
ϵˉμp1p3+p4p5+p6+ϵˉμp2p3+p4p5+p6
Of course, the function is also applicable to Cartesian quantities
CSP[p1 + p2, p3 + p4]
ExpandScalarProduct[%]
(p1+p2)⋅(p3+p4)
p1⋅p3+p1⋅p4+p2⋅p3+p2⋅p4
CLC[][p1 + p2, p3 + p4, p5 + p6]
ExpandScalarProduct[%, EpsEvaluate -> True]
ϵˉp1+p2p3+p4p5+p6
ϵˉp1p3p5+ϵˉp1p3p6+ϵˉp1p4p5+ϵˉp1p4p6+ϵˉp2p3p5+ϵˉp2p3p6+ϵˉp2p4p5+ϵˉp2p4p6
Sometimes one would like to have external momenta multiplied by symbolic parameters in the propagators. In this case one should first declare the corresponding variables to be of FCVariable
type
DataType[a, FCVariable] = True;
DataType[b, FCVariable] = True;
ExpandScalarProduct[SP[P, Q] /. P -> a P1 + b P2]
StandardForm[%]
a(P1⋅Q)+b(P2⋅Q)
(*a Pair[Momentum[P1], Momentum[Q]] + b Pair[Momentum[P2], Momentum[Q]]*)