Abstract
We examine the discrete symmetries $\hat C$ (charge conjugation), $\hat P$ (parity) and $\hat T$ (time reversal) in higher-dimensional field theories. In the first chapter, we set up the notation by working out the Lorentz transformation, and by deriving a relation between the spinor representation of the generators of Lorentz group and the Clifford algebra. Finally, we introduce the spinor representation which is an useful method to produce a specific representation of $\gamma$-matrices in any dimension. In the second chapter, we first consider the equivalence of different representations of the Clifford algebra in single-time dimension of signature (1,$d_-$). Then, using this representation of Clifford algebra, we discuss whether a representation of Lorentz group is reducible or not. Two conditions, Chiral condition and Majorana condition, are introduced to reduce the Lorentz representations. In the third chapter, following the same procedure as in Chapter 2, we consider the arbitrary ($d_+,d_-$) dimensions. First, using the spinor representation method, one can produce a special representation of the Clifford algebra. The properties of $\gamma$-matrices are list in Table 3.1 and Table 3.2. Then we discuss whether this representation is irreducible or not. The Chiral as well as the Majorana Conditions in arbitrary dimensions are discussed and the result is listed in Table 3.3. In chapter 4, we start to discuss the discrete symmetries using the special spinor representation. Demanding that the Dirac equation is invariant under these discrete transformations, we can write down all the operators associated with the discrete symmetries. The results are listed in Table 4.1. In chapter 5, we will check these operators associated with $\hat C$,$\hat P$,$\hat T$ deduced in the chapter 4 in the field theory. Writing explicity down the spinor-field (which is the solutions of the Dirac equation) and how the discrete symmetries act on the spinor-field to check if these operators correct. In chater 6, we find it is possible to write down a Majorana mass term in arbitrary dimensions. We exam the parity and time-reversal properties of the Majorana mass term in arbitrary dimensions.