How to write a space-like four-vector in FeynCalc? #222

alessiotbceleste · 2023-06-09T21:05:20Z

alessiotbceleste
Jun 9, 2023

Hi, please ...

How to write a space-like quadrivector in FeynCalc in Mathematica?

vsht · 2023-06-10T13:20:13Z

vsht
Jun 10, 2023
Maintainer

Hi,

sorry, but I don't really understand what you are trying to achieve?

0 replies

alessiotbceleste · 2023-06-10T14:01:46Z

alessiotbceleste
Jun 10, 2023
Author

for example, in FeynCalc the command TC[p] is the temporal component of a 4-vector, pt = (p0, 0, 0, 0). Is there a similar one for the spatial component? ps = (0, p1, p2, p3)

0 replies

vsht · 2023-06-10T16:06:46Z

vsht
Jun 10, 2023
Maintainer

You mean CV[p,i]?

https://feyncalc.github.io/FeynCalcBookDev/CV.html

0 replies

alessiotbceleste · 2023-06-10T20:05:10Z

alessiotbceleste
Jun 10, 2023
Author

hi thanks for the tips, I had thought of using this command but it would have 3 components, I was thinking a four-vector with 4 components, where the temporal part is equal to zero.
Would this be possible?

0 replies

vsht · 2023-06-11T12:33:26Z

vsht
Jun 11, 2023
Maintainer

This is why I asked what are you trying to achieve in the first place. For a Lorentz covariant calculation it should be sufficient to satisfy
all such constraints by just assigning dedicated values to the corresponding via scalar products.

How do you expect your vector to behave differently from a usual FV-vector?

0 replies

alessiotbceleste · 2023-06-11T15:07:11Z

alessiotbceleste
Jun 11, 2023
Author

the calculation I'm trying to implement is the trace below, where vector b is like space

bt = TemporalPair[TemporalMomentum[b], ExplicitLorentzIndex[0]];

b_alpha1 = Contract[FV[b, \alpha_1] - bt];
b_alpha2 = Contract[FV[b, \alpha_2]] - bt];

LINE = GS[p_2] . b_alpha1 . GA[\beta_1] . FV[p_1 - p_3, \rho_1] . LC[\alpha_1, \beta_1, \rho_1, \lambda_1] . MT[\lambda_1, \mu] . GS[p_1 - p_3] .
b_alpha2 . GA[\beta_2] . FV[p_1 - p_3, \rho_2] . LC[\alpha_2, \beta_2, \rho_2, \lambda_2] . MT[\lambda_2, \nu] . GS[p_1] . GA[\nu] .
GS[p_1 - p_3] . GA[\mu]

Contract[Tr[LINE]]

0 replies

vsht · 2023-06-11T15:19:52Z

vsht
Jun 11, 2023
Maintainer

Try to paste your code back into a Mathematica notebook: it doesn't get interpreted correctly. Please use Copy as -> Input text.

3 replies

alessiotbceleste Jun 11, 2023
Author

bt=TemporalPair[TemporalMomentum[b],ExplicitLorentzIndex[0]];

balpha1=Contract[FV[b,Subscript[[Alpha], 1]]-bt];

balpha2=Contract[FV[b,Subscript[[Alpha], 2]]-bt];

LINE=prop[Subscript[p, 2],-m].(-balpha1.GA[Subscript[[Beta], 1]].FV[Subscript[p, 1]-Subscript[p, 3],Subscript[[Rho], 1]].LC[Subscript[[Alpha], 1],Subscript[[Beta], 1],Subscript[[Rho], 1],Subscript[[Lambda], 1]].MT[Subscript[[Lambda], 1],[Mu]]).prop[Subscript[p, 1]-Subscript[p, 3],m].(-balpha2.GA[Subscript[[Beta], 2]].FV[Subscript[p, 1]-Subscript[p, 3],Subscript[[Rho], 2]].LC[Subscript[[Alpha], 2],Subscript[[Beta], 2],Subscript[[Rho], 2],Subscript[[Lambda], 2]].MT[Subscript[[Lambda], 2],[Nu]]).prop[Subscript[p, 1],m].GA[[Nu]].prop[Subscript[p, 1]-Subscript[p, 3],m].GA[[Mu]];

M = Contract[Tr[LINE]];

vsht Jun 11, 2023
Maintainer

Thanks. Does this help?

FCClearScalarProducts[]
Pair[Momentum[b], Momentum[x_]] := FCI[TC[b] TC[x]]
M = Contract[DiracTrace[LINE]]

M // DiracSimplify // Simplify

-16*prop[Subscript[p, 1], m]*prop[Subscript[p, 2], -m]*prop[Subscript[p, 1] - Subscript[p, 3], m]^2*TC[b]*
 (-(FV[b, Subscript[\[Alpha], 1]]*(SP[Subscript[p, 1], Subscript[p, 1]] - 2*SP[Subscript[p, 1], Subscript[p, 3]] + SP[Subscript[p, 3], Subscript[p, 3]])) - 
  FV[b, Subscript[\[Alpha], 2]]*(SP[Subscript[p, 1], Subscript[p, 1]] - 2*SP[Subscript[p, 1], Subscript[p, 3]] + SP[Subscript[p, 3], Subscript[p, 3]]) + 
  TC[b]*(SP[Subscript[p, 1], Subscript[p, 1]] + MT[Subscript[\[Alpha], 1], Subscript[\[Alpha], 2]]*SP[Subscript[p, 1], Subscript[p, 1]] - 2*SP[Subscript[p, 1], Subscript[p, 3]] - 
    2*MT[Subscript[\[Alpha], 1], Subscript[\[Alpha], 2]]*SP[Subscript[p, 1], Subscript[p, 3]] + SP[Subscript[p, 3], Subscript[p, 3]] + MT[Subscript[\[Alpha], 1], Subscript[\[Alpha], 2]]*SP[Subscript[p, 3], Subscript[p, 3]] + 
    FV[Subscript[p, 1], Subscript[\[Alpha], 2]]*TC[Subscript[p, 1]] - FV[Subscript[p, 3], Subscript[\[Alpha], 2]]*TC[Subscript[p, 1]] - TC[Subscript[p, 1]]^2 + 
    FV[Subscript[p, 1], Subscript[\[Alpha], 1]]*(-FV[Subscript[p, 1], Subscript[\[Alpha], 2]] + FV[Subscript[p, 3], Subscript[\[Alpha], 2]] + TC[Subscript[p, 1]] - TC[Subscript[p, 3]]) - 
    FV[Subscript[p, 1], Subscript[\[Alpha], 2]]*TC[Subscript[p, 3]] + FV[Subscript[p, 3], Subscript[\[Alpha], 2]]*TC[Subscript[p, 3]] + 2*TC[Subscript[p, 1]]*TC[Subscript[p, 3]] - TC[Subscript[p, 3]]^2 + 
    FV[Subscript[p, 3], Subscript[\[Alpha], 1]]*(FV[Subscript[p, 1], Subscript[\[Alpha], 2]] - FV[Subscript[p, 3], Subscript[\[Alpha], 2]] - TC[Subscript[p, 1]] + TC[Subscript[p, 3]])))

alessiotbceleste Jun 12, 2023
Author

Thank you very much, although I didn't understand how these command lines work, I'll test it here.
I forgot to inform you that the command prop[p_,m_] = GS[p]+m

vsht · 2023-06-12T14:39:24Z

vsht
Jun 12, 2023
Maintainer

The command just defines that every scalar product of the form b.x should be replaced with b^0 x^0, which effectively implements the constraint that the spatial part of b is zero. Am 12. Juni 2023 02:01:43 UTC schrieb alessiotbceleste ***@***.***>:

…

Thank you very much, although I didn't understand how these command lines work, I'll test it here. I forgot to inform you that the command prop[p_,m_] = GS[p]+m -- Reply to this email directly or view it on GitHub: #222 (reply in thread) You are receiving this because you commented. Message ID: ***@***.***>

0 replies

alessiotbceleste · 2023-06-20T03:10:08Z

alessiotbceleste
Jun 20, 2023
Author

Hi, first thank you so much for the comments, but what I intended was that the temporal part of b is zero and the spatial one is not zero. Follow my code again ...

<<FeynCalc`

m=0;
prop[p_,m_]:=GS[p]+m

bt=TemporalPair[TemporalMomentum[b],ExplicitLorentzIndex[0]];

balpha1=Contract[FV[b,Subscript[[Alpha], 1]]-bt];
balpha2=Contract[FV[b,Subscript[[Alpha], 2]]-bt];
balpha3=Contract[FV[b,Subscript[[Alpha], 3]]-bt];
balpha4=Contract[FV[b,Subscript[[Alpha], 4]]-bt];

LINE=prop[Subscript[p, 2],-m].(-balpha1.GA[Subscript[[Beta], 1]].FV[Subscript[p, 1]-Subscript[p, 3],Subscript[[Rho], 1]].LC[Subscript[[Alpha], 1],Subscript[[Beta], 1],Subscript[[Rho], 1],Subscript[[Lambda], 1]].MT[Subscript[[Lambda], 1],[Mu]]).prop[Subscript[p, 1]-Subscript[p, 3],m].(-balpha2.GA[Subscript[[Beta], 2]].FV[Subscript[p, 1]-Subscript[p, 3],Subscript[[Rho], 2]].LC[Subscript[[Alpha], 2],Subscript[[Beta], 2],Subscript[[Rho], 2],Subscript[[Lambda], 2]].MT[Subscript[[Lambda], 2],[Nu]]).prop[Subscript[p, 1],m].GA[[Nu]].prop[Subscript[p, 1]-Subscript[p, 3],m].GA[[Mu]];

M=Contract[Tr[LINE]];

3 replies

vsht Jun 20, 2023
Maintainer

Ok, then you can just write

FCClearScalarProducts[]
TC[b] = 0;
Pair[Momentum[b], Momentum[x_]] := FCI[CSP[b, x]];
M = DiracSimplify[DiracTrace[LINE]]

alessiotbceleste Jun 20, 2023
Author

Can I use LINE the way I wrote it myself?

vsht Jun 21, 2023
Maintainer

In FeynCalc 10 Tr will not invoke DiracTrace anymore. It was a rather ugly hack that involve redefining system functions of Mathematica (Tr) so for the next release we want to get away from that.

alessiotbceleste · 2024-04-07T15:15:30Z

alessiotbceleste
Apr 7, 2024
Author

Hello! Could you help me with this code, because Mathematica doesn't perform the contraction?

FCClearScalarProducts[]
TC[b] = 0;
Pair[Momentum[b], Momentum[x_]] := FCI[CSP[b, x]];
bt = TemporalPair[TemporalMomentum[b], ExplicitLorentzIndex[0]];
bmu = FV[b, [Mu]] - bt;
bnu = FV[b, [Nu]] - bt;

line1 = GS[Subscript[p, 1]].(-FV[Subscript[p, 1] + Subscript[p, 2], [Beta]].bmu.MT[[Mu], [Beta]] - SP[bmu, Subscript[p, 1] + Subscript[p, 2]].GA[[Mu]]).GS[Subscript[p, 2]].GA[[Nu]];

trace1 = DiracTrace[line1, DiracTraceEvaluate -> True];

Message produced by Mathematica

Contract: Error! Contract has encountered a fatal problem and must abort the computation. The problem reads: Something went wrong during prepareProductContractions.

1 reply

vsht Apr 9, 2024
Maintainer

Hi,

the expression SP[bmu, Subscript[p, 1] + Subscript[p, 2]] makes no sense to me. The same goes for the definitions of bmu and bnu for that matter.

Still, the evaluation works for me, even though the result is not meaningful:

FCClearScalarProducts[]
TC[b] = 0;
Pair[Momentum[b], Momentum[x_]] := FCI[CSP[b, x]];
bt = TemporalPair[TemporalMomentum[b], ExplicitLorentzIndex[0]];
bmu = FV[b, \[Mu]] - bt;
bnu = FV[b, \[Nu]] - bt;
line1 = GS[
    Subscript[p, 
     1]] . (-FV[Subscript[p, 1] + Subscript[p, 2], \[Beta]] . bmu . 
       MT[\[Mu], \[Beta]] - 
     SP[bmu, Subscript[p, 1] + Subscript[p, 2]] GA[\[Mu]]) . 
   GS[Subscript[p, 2]] . GA[\[Nu]];
trace1 = DiracTrace[line1, DiracTraceEvaluate -> True];

trace1 // StandardForm
(*
-4 (Pair[LorentzIndex[[Nu]], Momentum[Subscript[p, 2]]] Pair[
Momentum[
CartesianPair[CartesianMomentum[b],
CartesianMomentum[Subscript[p, 1]]]],
Momentum[Subscript[p, 1]]] +
Pair[LorentzIndex[[Nu]], Momentum[Subscript[p, 2]]] Pair[
Momentum[
CartesianPair[CartesianMomentum[b],
CartesianMomentum[Subscript[p, 1]]]],
Momentum[Subscript[p, 2]]] +
Pair[LorentzIndex[[Nu]], Momentum[Subscript[p, 1]]] Pair[
Momentum[
CartesianPair[CartesianMomentum[b],
CartesianMomentum[Subscript[p, 2]]]],
Momentum[Subscript[p, 1]]] +
Pair[LorentzIndex[[Nu]], Momentum[Subscript[p, 1]]] Pair[
Momentum[
CartesianPair[CartesianMomentum[b],
CartesianMomentum[Subscript[p, 2]]]],
Momentum[Subscript[p, 2]]] -
Pair[Momentum[Pair[LorentzIndex[[Nu]], Momentum[b]]],
Momentum[Subscript[p, 1]]] Pair[Momentum[Subscript[p, 1]],
Momentum[Subscript[p, 2]]] -
Pair[Momentum[Pair[LorentzIndex[[Nu]], Momentum[b]]],
Momentum[Subscript[p, 2]]] Pair[Momentum[Subscript[p, 1]],
Momentum[Subscript[p, 2]]])
*)

alessiotbceleste · 2024-04-14T14:53:34Z

alessiotbceleste
Apr 14, 2024
Author

Hello, thanks for the help. I can't get a correct result for the dash calculation, here's the code used

FCClearScalarProducts[]
TC[b]=0;
Pair[Momentum[b],Momentum[x_]]:=FCI[CSP[b,x]];

bt=TemporalPair[TemporalMomentum[b],ExplicitLorentzIndex[0]];
bmu=FV[b,[Mu]]-bt;

line=GS[p].(GS[q].GA[5].bmu-SP[bmu,q].GA[5].GA[[Mu]]).GS[k].GA[[Nu]];
trace=DiracTrace[line,DiracTraceEvaluate->True]

0 replies

vsht · 2024-04-15T15:01:29Z

vsht
Apr 15, 2024
Maintainer

Hi,

what is bmu supposed to mean? We do agree that writing $(p^\mu - p^0)$ inside a Lorentz covariant expression looks very weird, right?

Then, I also don't understand what are you trying to express with SP[bmu, q]? Could you please write the corresponding expressions in TeX?

1 reply

alessiotbceleste Apr 16, 2024
Author

Hi, answering your questions:

What should bmu mean?
Answer: I want b to be a space-like four vector.
What are you trying to express with SP[bmu, q]?
Answer: It would be the scalar product with the four-vector b with a four-momentum q.

$b \cdot q$

vsht · 2024-04-16T13:35:30Z

vsht
Apr 16, 2024
Maintainer

Hi, then you can just subtract a time-like vector n^mu from the full vector. In Lorentz covariant expressions the fact that n^mu = (1,0,0,0) will be relevant only for n^2 = 1. Am 16.04.24 um 13:44 schrieb alessiotbceleste:

…

Hi, answering your questions: 1. What should bmu mean? Answer: I want b to be a space-like four vector. 2. What are you trying to express with SP[bmu, q]? Answer: It would be the scalar product with the four-vector b with a four-momentum q. $b \cdot q$ — Reply to this email directly, view it on GitHub <#222 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ABXSD5CDZ7TOUQVCMXFKLFLY5UFJDAVCNFSM6AAAAAAZBDUWBCVHI2DSMVQWIX3LMV43SRDJONRXK43TNFXW4Q3PNVWWK3TUHM4TCMRZGIZDM>. You are receiving this because you commented.Message ID: ***@***.***>

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