Setting a value to a scalar product using Mathematica List

[YangPhy]

Hi all,

Today I was trying to set a value to a scalar product using MMA List but it did now work well. Let me show what was happening:

As we know, in FeynCalc, SPD[p] stands for the scalar product SPD[p,p], so it is understandable that SPD[p,p]=0 if we set SPD[p]=0. However, if one wants to set the values of several scalar products together, using {SPD[p1],SPD[p2],SPD[p3]}={0,0,0}, there would be SPD[p1]=0 but SPD[p1,p1]=p1^2--they are not equal to each other anymore!

Does anyone else also see the same thing?

Attached is an example in .nb file.

ScalarProduct.zip

Best,
Yang Ma

[vsht]

Unfortunately, it is not possible to set FeynCalc scalar products via direct list assignments.

The code

{SPD[p1]}={0}

actually assigns 0 only to SPD[p1], but not to the more fundamental Pair[Momentum[p1,D],Momentum[p1,D]],
so it doesn't work as expected neither for SPD[p1] nor for SPD[p1,p1].

The reason for this is a technical limitation within Mathematica: Scalar product assignments in FeynCalc are implemented
via UpValues using TagSetDelayed. However, the assignments cannot be arbitrarily complicated, as that would slow down Mathematica's
pattern matcher too much. One can have e.g.

SP/:
	Set[SP[a_, b_] , c_]:=
		(ScalarProduct[a,b,Dimension->4,SetDimensions->{4}]=c)

but a definition such as

SP /: Set[{a__SP}, {b__}] := MapThread[Set, {{a}, {b}}]

is already too complex and is not allowed.

Of course, one could redefine Set itself, as in

Unprotect[Set];
Set[{a__SP}, {b__}] := MapThread[Set, {{a}, {b}}];
Protect[Set];

which would make

{SP[p1], SP[p2]} = {x2, y2}

work as expected. However, messing with the definitions of protected Mathematica symbols is almost always
considered a very poor programming practice and could potentially introduce numerous unwanted side effects.
So I certainly don't want to implement something like that in FeynCalc.

tl;dr Please do not use direct list assignments as in your example. This won't work.
A possible workaround for your problem would be to define scalar products via the following construction

MapThread[Set, {{SPD[p1], SPD[p2], SPD[p3]}, {0, 0, 0}}]

This is perfectly fine and should always work as expected.

I'll consider adding something like FCSetScalarProducts[{SP[..],SPD[...],...},{val1,val2,...}] to the dev version
that implements the above idea.

[YangPhy]

Thanks for the reply, and I do look forward to the implementation.

[vsht]

You can try out FCSetScalarProducts in the dev version.

[YangPhy]

Sure, thanks!