本文主要的目的是：檢驗在高維度場論中的不連續對稱，如電荷共軛、宇稱、時間反演。
第一章，藉由羅倫茲轉換定義本文所用的符號，並導出羅倫茲群生成元的旋量表示 Clifford 代數之間的關係。在本章最後，引進 Clifford 代數的旋量表示，這個表示法可以在任意維度下製造出一組特別的狄拉克矩陣的表示。

第二章，首先考慮一個時間維度下，不同的 Clifford 代數的表示之間彼此的同等性。接著，利用這個同等性質討論羅倫茲群的可約性。考慮手徵條件和 Majorana 條件以約化羅倫茲表示。

第三章，遵循和第二章相同的步驟，考慮任意時空維度的情況。首先，由旋量表示法，我們可以寫下 Clifford 代數的一組特定的表示。然後檢驗狄拉克矩陣的性質並列於表3.1及表3.2。接著，討論羅倫茲群的可約性。討論手徵條件和 Majorana 條件並將結果列於表3.3。

第四章，藉由旋量表示法開始討論不連續對稱。狄拉克方程不變的要求，使我們得以寫下所有和不連續對稱相關的算符。這些結果列於表4.1。

第五章，以場論的方式檢驗在第四章中得到的這些算符是否正確。明確的寫下旋量場的形式（也就是狄拉克方程的解），將不連續對稱相關的算符作用在旋量場上，檢驗這些算符是否正確。

在第六章中，我們發現在任意維度中都可能寫下一個 Majorana 質量項。檢驗任意維度下，這個質量項在宇稱、時間反演轉換下的性質。

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.

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