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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 gammamu matrix, where mu ranges from 1 to the dimension d of spacetime; these are noncommutative objects satisfying

γμγν+γνγμ=2g_μ,ν

  

where the products in the above are noncommutative, constructed by using the `*` operator of the Physics package, and g_mu,nu is the metric tensor. 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  spacetime, with signature - - - +, the contravariant Dirac matrices are:

γ0=σ000σ0, γk=0σkσk0

  

where σ0 is the 2 x 2 identity matrix and σk, with 1 k 3 are the three Pauli matrices. All of σ0 and σk are represented in the Physics package with Psigma and, as is the case for all spacetime tensors, the value 0 of a spacetime index can also be represented by the number d, the spacetime dimension.

  

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 γk are the same as in the standard representation, while γ0 changes to

γ0=0σ0σ00

  

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

γ0=0σ2σ20,γ1=Iσ300Iσ3,γ2=0σ2σ20,γ3=Iσ100Iσ1

• 

Note that the form of the Dirac matrices depends on the signature of spacetime: in a Euclidean spacetime, for the standard, chiral, and Majorana representations, that form is obtained from the formulas above by performing a Wick rotation, equivalent to multiplying the γk by I, while γ0 remains unchanged.

• 

In all of these representations, in a four dimensional spacetime, a matrix γ5 satisfying

γμγ5+γ5γμ=0,γ5γ5=1

  

is given in the case of a Minkowski spacetime by:

γ5=Iγ0γ1γ2γ3

  

where I is the imaginary unit (to represent it with a lowercase i, see interface). In a four dimensional Euclidean spacetime, γ5 is given by

γ5=γ1γ2γ3γ0

• 

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

• 

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

γμγμ=d

γμγνγμ=2dγν

γμγαγβγμ=4g_α,β+d4γαγβ

γμγαγβγργμ=2γργβγα+4dγαγβγρ

  

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

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

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

Dgamma0

γ4

(2)

Note that even after the package is loaded, no representation for the Dirac matrices is set, so Dgamma[mu] is just a symbol representing these matrices, not even a matrix. So this input is just echoed

Dgamma0matrix

γ4matrix

(3)

The Dirac matrices have representation-free properties; for example, for the trace of the product of two of them,

DgammaμDgammaν

γμγν

(4)

Trace

4gμ,ν

(5)

Dgamma1Dgamma2+Dgamma2Dgamma1

γ1γ2+γ2γ1

(6)

Trace

0

(7)

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

SetupDgammarepresentation=chiral:

Setting lowercaselatin letters to represent spinor indices

Defined Dirac gamma matrices (Dgamma) in chiral representation,γ1,γ2,γ3,γ4

__________________________________________________

(8)

After setting the representation, the symbol Dgamma[mu] for mu ranging from 1 to the dimension of spacetime, can be manipulated as a tensor with two spinor indices, or as a matrix by using the corresponding matrix keyword

Dgamma0

γ4

(9)

Dgamma0matrix

γ4a,b=0010000110000100

(10)

Dgamma01,3

1

(11)

With the conventions used, among gammamu, only gamma0 (consequently, when d=4,gamma5 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.

SetupDgammarepresentation=standard:

Defined Dirac gamma matrices (Dgamma) in standard representation,γ1,γ2,γ3,γ4

__________________________________________________

(12)

Dgamma0+Dgamma1

γ4+γ1

(13)

As with any tensor of the Physics package, to see its components, in this case the matrix for of gammamu, you can index gammamu with the keyword matrix or just with no indices, both show the matrix form:

Dgamma0[]

γ4a,b=1000010000−10000−1

(14)

Dgamma5matrix

γ5a,b=0010000110000100

(15)

Dgamma13,2

1

(16)

In a Minkowski spacetime with signature - - - +, the matrix representation of the spacial part of the covariant gammamu and contravariant `^`gamma,mu changes sign, while gamma5 and `^`gamma,5 are equal

Dgamma2[]

γ2a,b=000I00−I00−I00I000

(17)

Dgamma`~2`[]

γ22a,b=000−I00I00I00−I000

(18)

Dgamma`~5`[]

γ55a,b=0010000110000100

(19)

Note that the representation also depends on the signature (Euclidean or Minkowski) of spacetime; with the conventions used, among gammamu, only gamma0 does not change when the signature of spacetime is changed.

Setupsignature=`+`

Changing the signature of the tensor spacetime to: + + + +

signature=+ + + +

(20)

Dgamma`~0`[]

γ44a,b=1000010000−10000−1

(21)

Dgamma`~1`[]

γ11a,b=000−I00−I00I00I000

(22)

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

Setupdimension=3,`-`

The dimension and signature of the tensor space are set to: [3, - - +]

dimension=3,signature=- - +

(23)

Dgamma`~1`[]

γ11a,b=0II0

(24)

Setupdimension=2

The dimension and signature of the tensor space are set to: [2, - +]

dimension=2

(25)

Dgamma`~1`[]

γ11a,b=01−10

(26)

See Also

Physics, Physics conventions, Physics examples, Physics/`*`, Setup, Trace


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