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## Simplification of products of Dirac matrices

Maple

The computation of traces of products of Dirac matrices was implemented years ago - see Physics,Trace .

The simplification of products of Dirac matrices, however, was not. Now it is, and illustrating this new feature is the matter of this post. To reproduce the results below please update the Physics library with the one distributed at the Maplesoft R&D Physics webpage.

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First of all, when loading Physics, a frequent question is about the signature, the default is (- - - +)

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 (1)

This is important because the convention for the Algebra of Dirac Matrices depends on the signature. With the signatures (- - - +) as well as (+ - - -), the sign of the timelike component is 1

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 (2)

With the signatures (+ + + -) as well as (- + + +), the sign of the timelike component is of course -1

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 (3)

The simplification of products of Dirac Matrices, illustrated below with the default signature, works fine with any of these signatures, and works without having to set a representation for the Dirac matrices -- all the results are representation-independent.

The examples below, however, also illustrate a new feature of Physics, for now implemented as a Library:-PerformMatrixOperations command (there is a related, also new, command, , to just present the underlying matrix operations, without performing them). To illustrate this other new functionality , set a representation for the Dirac matrices, say the standard one

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 (4)

The four Dirac matrices are

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 (5)

The definition of the Dirac matrices is implemented in Maple following the conventions of Landau books ([1] Quantum Electrodynamics, V4), and  does not depend on the signature, ie the form of these matrices is

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 (6)

With the default signature, the space part components of   change sign when compared with corresponding ones from  while the timelike component remains unchanged

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 (7)
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 (8)

For the default signature, the algebra of the Dirac Matrices, loaded by default when Physics is loaded, is (see page 80 of [1])

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 (9)

When the sign of the timelike component of the signature is -1, we have a -1 factor on the right-hand side of (9).

Note as well that in (9) the right-hand side has no matrix elements. This is standard in particle physics where the computations are performed algebraically, without performing the matrix operations. For the purpose of actually performing the underlying matrix operations, however, one may want to rewrite this algebra including a 4x4 identity matrix. For that purpose, see Algebra of Dirac Matrices with an identity matrix on the right-hand side. For the purpose of this illustration, below we proceed with the algebra as shown in (9), interpreting right-hand sides as if they involve an identity matrix.

Verify the algebra rule by performing all the involved matrix operations

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 (10)

Note that, regarding the spacetime indices, this is a 4x4 matrix, whose elements are in turn 4x4 matrices. Compute first the external 4x4 matrix related to  and

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 (11)

Perform now all the matrix operations involved in each of the elements of this 4x4 matrix

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 (12)

By eye everything checks OK.

Consider now the following five products of Dirac matrices

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 (13)
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 (14)
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 (15)
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 (16)
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New: the simplification of these products is now implemented

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 (18)

Verify this result performing the underlying matrix operations

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 (19)
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 (20)

The same with the other expressions

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 (21)
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 (22)
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 (23)
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 (24)

For e2

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 (25)
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 (26)
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 (27)
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 (28)

For e3 we have

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 (29)

Verify this result,

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 (30)

In this case, with three free spacetime indices lambda, nu, rho, the spacetime components form an array 4x4x4 of 64 components, each of which is a matrix equation

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 (31)

For instance, the first element is

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 (32)

and it checks OK:

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 (33)

How can you test the 64 components of T all at once?

1. Compute the matrices, without displaying the whole thing, take the elements of the array and remove the indices (ie take the right-hand side); call it M

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For instance,

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 (34)

Now verify all these matrix equations at once: take the elements of the arrays on each side of the equations and verify that the are the same: we expect for output just

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 (35)

The same for e4

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 (36)
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 (37)

Regarding the spacetime indices this is now an array 4x4x4x4

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 (38)

For instance the first of these 256 matrix equations

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 (39)

verifies OK:

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 (40)

Now all the 256 matrix equations verified at once as done for e3

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 (41)

Finally, although there is more work to be done here, let's define some tensors and contract their product with these expressions involving products of Dirac matrices.

For example,

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 (42)

Contract with e1 and e2 and simplify

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 (43)
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 (44)