Physics[Dgamma]  the Dirac gamma matrices

Calling Sequence


Dgamma[mu]


Parameters


mu



an integer between 1 and the dimension, or any algebraic expression generically representing it (when the dimension is equal to 4, mu = 5 is also allowed)





Description


•

The Dgamma[mu] command is used to represent the Dirac matrix, where ranges from 1 to the dimension d of spacetime; these are noncommutative objects satisfying


where the products in the above are noncommutative, constructed by using the `*` operator of the Physics package, and is the metric tensor. When the spacetime is Euclidean (signature = `+`), g_[mu, nu] is replaced by delta[mu, nu], the Kronecker symbol. The properties of the Dirac matrices are derived from the defining relation above, which is invariant under a unitary transformation, thus these matrices are determined up to a transformation of that kind, and conventions are necessary to construct their representations. The Physics package's commands know about the properties of the Dirac matrices, and in particular the Trace command computes traces of products of gammas taking these properties into account.

•

The most common representations for the Dirac matrices are the standard (also known as Dirac), the chiral (also known as Weyl), and the Majorana representations. The conventions for the standard representation are uniform in the literature: in a Minkowski (also called pseudoEuclidean) spacetime (signature = ``), the contravariant Dirac matrices are:

>

Dgamma[0] = Matrix(2, [[Id, 0], [0, Id]]);

 (1) 

where Id represents the 2 x 2 identity matrix,

>

Dgamma[j] = Matrix(2, [[0, sigma[j]], [sigma[j], 0]]);

 (2) 

The conventions for the chiral and Majorana representations are not uniform in the literature. The conventions adopted here are the same ones shown in Wikipedia, so that in the chiral representation, the Dgamma[j] are the same as in the standard representation, while Dgamma[0] changes to

>

Dgamma[0] = Matrix(2, [[0, Id], [Id, 0]]);

 (3) 

The convention implemented for the Majorana representation, that is, a representation where all the components of the Dirac matrices are imaginary, is

>

Dgamma[0] = Matrix(2, [[0, sigma[2]], [sigma[2], 0]]),
Dgamma[1] = Matrix(2, [[I*sigma[3], 0], [0, I*sigma[3]]]),
Dgamma[2] = Matrix(2, [[0, sigma[2]], [sigma[2], 0]]),
Dgamma[3] = Matrix(2, [[I*sigma[1], 0], [0, I*sigma[1]]]);

 (4) 
•

Note that the form of the Dirac matrices depends on the signature of spacetime: in a Euclidean spacetime (signature = `+`), for the standard, chiral, and Majorana representations, that form is obtained from the formulas above by performing a Wick rotation, equivalent to multiplying the Dgamma[j] by , while Dgamma[0] = Dgamma[d] remains unchanged, where is the dimension of spacetime.

•

In all of these representations, in a four dimensional spacetime, a matrix Dgamma[5] satisfying


where is the imaginary unit (to represent it with a lowercase , see interface). In a four dimensional Euclidean spacetime, Dgamma[5] is given by


where .

•

When the Physics package is loaded, no particular representation is enforced, and the symbol Dgamma[mu] just represents the corresponding Dirac matrix. To set a representation and concretely make Dgamma[mu] be the corresponding matrix, use the Setup command. For example, enter Physics[Setup](Dgammarepresentation = standard); at the Maple prompt.

•

The following are some representationfree frequently used identities for the Dirac matrices, valid provided the dimension, , is greater than 1, expressed by using the sum rule for repeated indices:

•

Some representationfree identities for the traces of products of Dirac matrices in four dimensions are:


where in these formulas, Trace is the Physics command to compute traces, g_ is the metric, epsilon is the LeviCivita totally antisymmetric symbol, and is the imaginary unit.



Examples


>


>


 (5) 
As is conventional in the Physics package, the value "0" for a spacetime index is mapped into the value , the dimension, so that in the input, you can use Dgamma[0] and it will be interpreted as
>


 (6) 
Note that even after the package is loaded, no representation for the Dirac matrices is set, so Dgamma[] is just a symbol representing these matrices, not even a matrix.
>


 (7) 
The Dirac matrices have representationfree properties; for example, for the trace of the product of two of them,
>


 (8) 
>


 (9) 
>


 (10) 
>


 (11) 
To set a representation, for example the chiral one (which is only valid in 4 dimensions), use the Setup command.
>


 (12) 
After setting the representation, the symbol Dgamma[mu] for mu ranging from 1 to the dimension of spacetime, is a matrix. You can compute using it as a symbol representing a matrix, or as a matrix itself (its components are computed as usual).
>


 (13) 
>


 (14) 
>


 (15) 
With the conventions used, among , only (consequently, when too) changes in form between the chiral and standard representations. Note below that Dgamma[0] and Dgamma[1] are symbols representing matrices, which also evaluate to matrices, and so can be referenced in the usual way.
>


 (16) 
>


 (17) 
>


 (18) 
>


 (19) 
Note that the representation also depends on the signature of spacetime; with the conventions used, among , only does not change when the signature of spacetime is changed.
>


 (20) 
>


 (21) 
>


 (22) 
The standard representation is also defined in lower dimensions. For example, if you change the dimension and signature again:
>


 (23) 
>


 (24) 
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 (25) 
>


 (26) 
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