Define the matrices
where
are the Pauli Matrices, I is the Identity Matrix, , 2, 3, and
is the matrix Direct Product. Explicitly,
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
These matrices satisfy the anticommutation identities
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(10) |
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(11) |
where is the Kronecker Delta, the commutation identity
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(12) |
and are cyclic under permutations of indices
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(13) |
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(14) |
A total of 16 Dirac matrices can be defined via
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(15) |
for , 1, 2, 3 and where
. These matrices satisfy
- 1.
, where
is the Determinant,
- 2.
,
- 3.
, making them Hermitian, and therefore unitary,
- 4.
, except
,
- 5. Any two
multiplied together yield a Dirac matrix to within a multiplicative factor of or ,
- 6. The
are linearly independent,
- 7. The
form a complete set, i.e., any constant matrix may be written as
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(16) |
where the are real or complex and are given by
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(17) |
(Arfken 1985).
Dirac's original matrices were written and were defined by
for , 2, 3, giving
The additional matrix
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(24) |
is sometimes defined. Other sets of Dirac matrices are sometimes defined as
and
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(28) |
for , 2, 3 (Arfken 1985) and
for , 2, 3 (Goldstein 1980).
Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with the
other eight. Let
, then
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(31) |
In addition
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(32) |
The products of and satisfy
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(33) |
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(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.
© 1996-9 Eric W. Weisstein
1999-05-24