In this chapter we fix our terminology and notation, mostly related to (real and complex) linear algebra. We will consider only algebraic properties. Infinite-dimensional vector spaces will not be equipped with any topology.

Let us stress that using precise terminology and notation concerning linear algebra is very useful in describing various aspects of quantization and quantum fields. Even though the material of this chapter is elementary, the terminology and notation introduced in this chapter will play an important role throughout our work. In particular we should draw the reader's attention to the notion of the complex conjugate space (Subsect. 1.2.3), and of the holomorphic and antiholomorphic subspaces (Subsect. 1.3.6).

Throughout the book K will denote either the field ℝ or ℂ, all vector spaces being either real or complex, unless specified otherwise.

Elementary linear algebra

The material of this section is well known and elementary. Among other things, we discuss four basic kinds of structures, which will serve as the starting point for quantization:

1. (1) Symplectic spaces – classical phase spaces of neutral bosons,

2. (2) Euclidean spaces – classical phase spaces of neutral fermions,

3. (3) Charged symplectic spaces – classical phase spaces of charged bosons,

4. (4) Unitary spaces – classical phase spaces of charged fermions.