文档介绍：MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 65
11. Free field theories in higher dimensions
. Minkowski and Euclidean we pass from quantum mechanics to quantum field
theory in dimensions d ≥ 1. As we explained above, we have two main settings.
1. Minkowski space. Fields are functions on a spacetime VM , which is a real inner product space of
signature (1,d − 1). This is where physical processes actually “take place”. The symmetry group of V ,
G = SO(1,d − 1), is called the Lorenz group; it is the group of transformation of spacetime in special
relativity. Therefore, field theories in Minkowski space which are in an appropriate sense “compatible”
with the action of G are called relativistic.
Recall some standard facts and definitions. The light conein V is the cone described by the equation
|v|2 =0, where |v|2 := (v, v). Vectors belonging to the light cone are called lightlike. The light cone
divides the space V into spacelike vectors |v|2 < 0 (the outside of the cone), and timelike vectors
|v|2 > 0 (inside the cone). We will choose one of the ponents of the interior of the cone and
call it positive; it will be denoted by V+. The opposite (negative) component is denoted by V−.The
group of g ∈ SO(V )=SO (1,d − 1) which preserve V+ is denoted by SO+(1,d − 1); it is the connected
component of the identity of the grou