4
Supersymmetry Demystified
dizzying profusion of indices that sometimes do not even seem be used consistently,
as in χ
a
, χ
a
, χ
˙a
, χ
˙a
, (σ
2
)
ab
, (σ
2
)
˙a
˙
b
,¯σ
μ
a ˙a
, and so on.
There is, of course, a very good reason for introducing all this notation: It helps
immensely in constructing invariant quantities and making expressions as compact
as possible. Unfortunately, this also makes it difficult for newcomers to the field
because more effort must be put into making sense of the notation than into learning
supersymmetry itself! It is possible to keep the notation very simple by not using
those strange dotted indices and by not making any distinction between upper and
lower indices, but the price to pay is that expressions are more lengthy, and some
calculations become quite awkward. In itself, this is not such a big deal. But the
goal of this book is not only to help you learn the rudiments of supersymmetry; it is
also to provide you with the background required to “graduate” to more advanced
references on the subject. And this necessitates familiarizing you with the notation
of dotted and undotted spinors. For this reason, a compromise has been made. At
first, the simpler and less compact notation will be used to ensure that the fancy
indices do not get in the way of learning the basic concepts. After a few chapters of
practice, however, the more advanced notation will be introduced with an emphasis
on the rationale behind it.
A second difficulty in learning supersymmetry stems from the language used to
describe the fermionic fields of the theory. Supersymmetry requires the use of Weyl
or Majorana spinors, whereas most of us became familiar only with Dirac spinors
in our introductory quantum field theory classes. The next two chapters therefore
will be entirely devoted to introducing these types of spinors and explaining their
relation to Dirac spinors.
A third difficulty is that there are actually two methods for constructing super-
symmetric theories. As you might expect, one method, the so-called superspace or
superfield formalism, is more powerful and compact but also much more abstract
and difficult for a beginner than the second, down- and dirty-method. Alas, most
references on supersymmetry use right away or very early on the superfield ap-
proach. In this book we will start with the less efficient but easier to learn approach
and will only later introduce superfields and superspace.
Supersymmetry is a huge and formidably complex topic. In writing a (finite)
book on the subject, one is faced with a gut-wrenching dilemma. Should one try to
touch on as many aspects of the theory as possible, albeit in a necessarily superfi-
cial manner? Or should one restrict oneself to a very small (almost infinitesimal)
subset of the theory with the goal of making the presentation, and especially the
mathematical derivations, as thorough and detailed as possible?
Well, for the present book, the decision was not difficult because the goal of
the Demystified series is to provide readers with self-teaching guides. This implies
that the accent must be put on covering in depth the more basic concepts, even
if it means paying the price of sacrificing several advanced topics. For example,