“Dirac Matrices”的概念、定义、习题解题方法及翻译-数学辞典

单词

Dirac Matrices

释义

Dirac Matrices

Define the matrices

			(1)
			(2)

where

are the Pauli Matrices, I is the Identity Matrix,

, 2, 3, and

is the matrix Direct Product. Explicitly,

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

These matrices satisfy the anticommutation identities

(10)

(11)

where

is the Kronecker Delta, the commutation identity

(12)

and are cyclic under permutations of indices

(13)

(14)

A total of 16 Dirac matrices can be defined via

(15)

for

, 1, 2, 3 and where

. These matrices satisfy

, where

is the Determinant,

, making them Hermitian, and therefore unitary,

, except

5. Any two

multiplied together yield a Dirac matrix to within a multiplicative factor of

6. The

are linearly independent,

7. The

form a complete set, i.e., any

constant matrix may be written as

(16)

where the

are real or complex and are given by

(17)

(Arfken 1985).

Dirac's original matrices were written and were defined by

			(18)
			(19)

for

, 2, 3, giving

			(20)
			(21)
			(22)
			(23)

The additional matrix

(24)

is sometimes defined. Other sets of Dirac matrices are sometimes defined as

			(25)
			(26)
			(27)

and

(28)

for

, 2, 3 (Arfken 1985) and

			(29)
			(30)

for

, 2, 3 (Goldstein 1980).

Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with theother eight. Let , then

(31)

In addition

(32)

The products of

and

satisfy

(33)

(34)

The 16 Dirac matrices form six anticommuting sets of five matrices each:

: 1. , , , , ,
: 2. , , , , ,
: 3. , , , , ,
: 4. , , , , ,
: 5. , , , , ,
: 6. , , , , .

See also Pauli Matrices

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211-213, 1985.

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。