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A few days ago I started to use the Mathematica package FeynCalc and one thing confuses me: Assume we have a four-vector $p_\mu$ and we contract it with the epsilon tensor. FeynCalc produces $\varepsilon^{\mu\nu\rho\sigma} p_\sigma = \varepsilon^{\mu\nu\rho p}$...what does the momentum as an index mean?

Furthermore, the FeynCalc documentation says "Eps[a, b, c, d] is the head of the totally antisymmetric $\varepsilon$ (Levi-Civita) tensor. The a,b,... may have head LorentzIndex, Momentum or Integer." What is an epsilon tensor with a momentum index? I never saw this before.

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    $\begingroup$ Would Mathematica be a better home for this question? $\endgroup$ – Qmechanic Apr 12 '14 at 18:59
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    $\begingroup$ I believe its just defined that way. The momentum as an index is defined to be an index contraction of a momentum with the Levi-Civita. $\endgroup$ – JeffDror Apr 12 '14 at 21:29
  • $\begingroup$ I agree with @JeffDror. This is just a weird notation. $\endgroup$ – Melquíades Apr 29 '14 at 20:57
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Personally I have only seen this notation used in FORM (another computer algebra package used in high energy physics). To my knowledge, or at least what the FORM docs tell me, this was introduced by its predecessor "SCHOONSCHIP".

As for what it means, JeffDror's comment is correct - "momentum as an index is defined to be an index contraction of a momentum with the Levi-Civita". In fact, in this notation, this extends to all tensors contracted with vectors.In this section of the FORM manual you can find the explanation in the paragraph starting "Formula w5 illustrates". (Note FORM has a more generic concept than tensor, which here is called a function).

As for why this notation exists, I'd suggest it's easier to read Eps[mu,nu,si,p] on a terminal!

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  • $\begingroup$ It is there to minimize problems with dummy summation indices. $\endgroup$ – Rolf Mertig Jun 12 '14 at 20:00
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here epsilon tensor is the same one that you know-the anti-symmetric Levi-civita tensor. Momentum as an index of it means that one of its index is contracted with a momentum four vector.

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