7 - Anti-symmetric calculus

Summary

In almost every respect there exists a strong analogy between symmetric and anti-symmetric tensors, between bosons and fermions. It is often convenient to stress this analogy in terminology and notation.

Symmetric tensors over a vector space can be treated as polynomial functions on its dual. Such functions can be multiplied, differentiated and integrated, and we can change their variables.

There exists a similar language in the case of anti-symmetric tensors. It has been developed mostly by Berezin, hence it is sometimes called the Berezin calculus. It is often used by physicists, because it allows them to treat bosons and fermions within the same formalism.

Anti-symmetric calculus has a great appeal – it often allows us to express the analogy between the bosonic and fermionic cases in an elegant way. On the other hand, readers who see it for the first time can find it quite confusing and strange. Therefore, we devote this chapter to a presentation of elements of anti-symmetric calculus.

Note that the main goal of this chapter is to present a certain intriguing notation. Essentially no new concepts of independent importance are introduced here. Therefore, a reader in a hurry can probably skip this chapter on the first reading.

This chapter can be viewed as a continuation of Chap. 3, and especially of Sect. 3.6. In particular, we will use the anti-symmetric multiplication, differentiation and the Hodge star introduced already in Chap. 3.