The Dirac Equation

Working with spinors

1. Chiral representation
   
2. Completeness relations
   
3. A practical representation
   
4. Products of spinors
   

   SIMPLE SPINOR PRODUCTS, ONCE MORE

   Let us start by calculating the following simple spinor products: When both helicities are identical, then only mass-terms are projected out, namely The signs here reflect the fact that particles 1 and 2 may be particles (+) or anti-particles (-). To allow for more compact notation, avoiding the signs, here the entities μ = ± m/η have been introduced.
   For different helicities, straightforward calculation yields and
   
   

   SNEAK PREVIEW: REPLACING SLASHS AND PROPAGATORS

   Before continuing in the construction of spinor products, it is worth remembering/anticipating that sometimes in calculations slashed four-momenta occur, i.e. terms of the form In order to deal with them in terms of spinor products, the "slash" needs to be replaced by some suitable projector. In order to deal with this, let us make a little detour and remember that by using projectors over positive and negative energy states a unity operator can be constructed, namely Here, it is understood that this equation holds true as long as the relation m=√p² is fulfilled, where p²<0 is included. In such a case, the mass clearly becomes complex, but even these cases do not harm. Inserting such a projector in front of the middle term of the above equation yields Here, the E.o.M. for the spinors have been used in going from the second to the third line. Anyways, the remarkable thing to note here is that this procedure allows to replace all slashes in a chain of Dirac matrices by spinors, thus cutting even long chains into simple pieces with known solutions (the Y-functions discussed above).
   
   

   GETTING MORE INVOLVED

   The next thing to be calculated is a spinor product with a slashed four-vector in between, namely Obviously, once the slashed four-momentum has been replaced with a dyadic product of spinors, this new object can be expressed by the Y-functions that have been calculated before. Here, it should be noted that the mass corresponding to Q is m=√ Q², and that it occurs with two signs, following the extended completeness relation from above.
   
   

   EVEN HARDER

   The last object in our menue is, at first glance, the most complex one, namely In order to calculate this, a little detour will be needed. This detour consists of two tricks:
   • The first one known as the "line-reversal" trick. It reads: where Γ is any string of &gamma,-matrices (including slashes), and Γ^R is the same string, but in reversed order.
     It is clear that such relations come in handy to relate different spinor products with each other and thus minimise the computational effort. However, before becoming too enthusiastic, let's prove this relation. To do so, consider and Let us now make a specific choice, relating the two Γs, namely With this choice the line-reversal trick is proven.
   • The second trick is a bit more involved. It is known as the Chisholm identity, which reads where spinor indices have been explicitly indicated. Clearly, this trick breaks open the "scalar" product of the spinors and transforms it into a dyadic product. To prove this, consider any string Γ of an odd number of γ matrices. It can be shown that there are two four vectors V and A such that The sum of Γ and its reverse string reads The Chisholm identity emerges when making specific choices. One of them, to prove the identity for one helicity combination, reads It should by now be easy to realise that this is a part of a spinor product as soon as a trace operation is applied.
   With help of this Chisholm identity, the Z functions can be rewritten as sums of products of Y-functions. This is exemplified below, for a simple, non-chiral coupling structure. These functions are part of a code package that will be used later when Feynman amplitudes will be calculated.