QFT Index Notation: A Beginner's Guide

[dyn]

Hi. I'm just starting QFT for the first time. I've just finished a course in relativity but I'm confused about the index notation I've found in QFT. Here are 2 examples yⁱ = Σ M_ij x_j and y_j = δ_i^j yⁱ . These examples don't seem right after what I have learned in relativity unless the index notation changes in QFT ?

 

[haushofer]

The i-index on M should indeed be upstairs, but people are sometimes sloppy when considering flat space in Cartesian coordinates; there, covariant and contravariant components are numerically the same because the metric is given by the identity matrix. You'll encounter this often, but a good habit is to be precise when doing your own calculations. In older papers you can also encounter this notation for curved indices in general.

 

[vanhees71]

Where does this come from? I'd look for a more carefully written source on the subject. QFT is difficult enough to understand. You don't need unnecessary sloppyness in notation. Good starting points are

L. H. Ryder. Quantum Field Theory. Cambridge University Press, Cambridge, New York, Melbourne, 2 edition, 1996.
D. Bailin and A. Love. Introduction to Gauge Field Theory. Adam Hilger, Bristol and Boston, 1986.
Lowell S. Brown. Quantum Field Theory. Cambridge University Press, 1992.
M.E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley Publ. Comp., 1995.

The last one has to be read with great care, because it has pretty many typos (and sometimes even quite annoying inprecisions), but overall it's didactically well written.

The non-plus-ultra are of course the 3 volumes by Weinberg

S. Weinberg, Quantum Theory of Fields, Cambridge University Press, 3 Vols.

I'd, however, not take them as a first text.