Does anyone know of websites or texts that have an abundance of examples of computing time-ordered products of fields using Wick's Theorem for both bosons and fermions?

I'm not just talking about the simple product of 4 fields, something such as

$T(\phi^4(x) \phi^4(y))$

$T(:\phi^4(x): :\phi^4(y):)$


These are actually the only examples I have and they're found in "Problem Book in Quantum Field Theory" by Voja Radovanovic.


1 Answer 1


Computing time-ordered products via Wick's theorem is fairly straightforward, schematically,

$$\mathcal{T} \left\{ \phi_1 \phi_2 \dots \phi_n\right\} = \; : \sum \mathrm{all \;possible \; contractions}:$$

where colons denote normal-ordering, and for simplicity we have chosen a real scalar field, and the notation $\phi_n$ denotes a field evaluated at the point $x_n$. A contraction of a pair of scalar fields at different points is the propagator evaluated at the difference of the respective space-time points, i.e. the Feynman propagator $\Delta_{ij} \equiv \Delta(x_i-x_j)$. Consider as a first example the product,

$$\mathcal{T} \left\{ \phi_1 \phi_2 \phi_3 \phi_4 \right\} = \; :\phi_1 \phi_2 \phi_3 \phi_4 : + \Delta_{12} :\phi_3\phi_4: + \Delta_{34}:\phi_1 \phi_2:$$ $$+ \Delta_{13}:\phi_2 \phi_4: + \Delta_{24}:\phi_1 \phi_3: + \Delta_{14}:\phi_2 \phi_3: + \Delta_{23}:\phi_1 \phi_4:$$

If we consider, for example, a Dirac field then we must take into account that they anti-commute when expanding the time-ordered product. For every swap, we introduce a minus sign. In addition, the propagator is only constructed from the contraction of a Dirac conjugate field with another field; any other type of contraction vanishes. In your example,

$$\mathcal{T} \left\{ \bar{\psi}_x \psi_x \bar{\psi}_y \psi_y \right\} = \; :\bar{\psi}_x \psi_x \bar{\psi}_y \psi_y: + \Delta_{xx}:\bar{\psi}_y \psi_y:$$ $$+ \Delta_{yy}:\bar{\psi}_x \psi_x: + \Delta_{xy}:\psi_x \bar{\psi}_y: + \Delta_{yx} :\bar{\psi}_x \psi_y:$$

Notice in this case there are no overall minus signs because there were an even number of required operator swaps. For additional resources on Wick's theorem, I recommend:

  1. Tong's lecture notes, they provide a proof of Wick's theorem, followed by a few examples, as well as applications of the theorem in computing scattering amplitudes. It is also shown which terms in the expansion correspond to certain processes, and how one selects which yield contributions to a particular Feynman diagram.
  2. Srednicki's Quantum Field Theory which includes a treatment of Wick's theorem, with examples, as well as problems to work through.
  3. Tutorial on Wick's Theorem which features a long computation using Wick's theorem, in complete explicit detail, and shows how to handle certain other insertions in time-ordered products, such as gamma matrices.

Unfortunately, you will not be able to find dozens of examples in a single resource because the computations are quite mechanical and tedious. In addition, the approach to scattering amplitudes involving Wick's theorem is mostly obsolete, and the preferred approach uses Feynman rules, and Feynman diagrams, as well as methods built upon them.

  • $\begingroup$ Thanks for the references. They don't really approach the direct calculations the way I'm looking for though (I've seen the first two before). With the answer you gave before the links, as I said, the products of four fields (whether fermions or bosons) is no difficulty. The problem I really have it when you start to have 3/4 different fields were are some are already normal ordered and you have to calculate the number of permutation etc. $\endgroup$
    – Phibert
    Apr 16, 2014 at 13:29
  • $\begingroup$ If you're having trouble with a time-ordered product, as a last resort, you can always do it explicitly without using Wick's theorem. $\endgroup$
    – JamalS
    Apr 16, 2014 at 13:32
  • $\begingroup$ That's true; it's more of a case of practising it as I think it's one of those of the more you try, the quicker you become at doing it. $\endgroup$
    – Phibert
    Apr 16, 2014 at 13:38
  • $\begingroup$ Shouldn't the time ordered products here have further terms where all the fields are contracted? As they're written now the vacuum matrix elements would just evaluate to zero because all terms have a normal-ordered product of fields. $\endgroup$ May 25, 2021 at 10:03

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