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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 γμ matrices, where μ ranges from 1 to the dimension d of spacetime; these are noncommutative objects satisfying

γμγν+γνγμ=2gμ,ν

  

where the products in the above are noncommutative, constructed by using the `*` operator of the Physics package, and gμ,ν is the metric tensor. The properties of the Dirac matrices are derived from this defining relation (anticommutator algebra) above. The Simplify command simplifies products of Dirac matrices and the Trace command computes traces of these products, taking this defining relation into account.

• 

This defining anticommutator algebra satisfied by the Dirac matrices 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 most common representations are the standard (also known as Dirac), the chiral (also known as Weyl or spinor), and the Majorana representations. The form of the contravariant Dirac matrices in a Minkowski spacetime in each of these three representations, shown below, does not change with the signature, that could be (- - - +) (default when you load Physics), (+ - - -), (+ + + -) or (- + + +). To query about the signature, enter Setup(signature), to change the value of the signature see Setup. When using the signatures (+ + + -) or (- + + +), the right-hand side of the anticommutator satisfied by the Dirac matrices, shown in the first paragraph, has an additional factor -1.

• 

Since Maple 2019, when the Physics package is loaded, the standard representation for Dirac's matrices is automatically set. 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 and in [1], [2] and [3], 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 nonzero components of the Dirac matrices are imaginary, is

γ0=0σ2σ20,γ1=σ300σ3,γ2=0σ2σ20,γ3=σ100σ1

  

where I is the imaginary unit (to represent it with a lowercase i, see interface).

• 

In all of these three representations, the Dirac matrices satisfy γμ=γμ, they are anti-Hermitian and, in a four dimensional Minkowski spacetime, an Hermitian matrix γ5=γ5 satisfying

γμγ5+γ5γμ=0,γ5γ5=1

  

is given by:

γ5=γ0γ1γ2γ3

  

Note the minus sign on the right-hand side of this definition, according to [1], [2] and [3], but not Wikipedia and not uniform in the literature. This definition of γ5 can also be written as

γ5=4!εα,β,μ,νγμγνγαγβ

  

from where, being a pseudo-scalar, there is no distinction between a covariant or contravariant character for its index, so γ5=γ5.

• 

The form of the Dirac matrices implemented in the case of an Euclidean spacetime, for the standard, chiral, and Majorana representations, is obtained from the formulas above for the contravariant Dirac matrices by performing a Wick rotation, equivalent to multiplying the γk by , while γ0 remains unchanged, and γ5 is given by

γ5=γ1γ2γ3γ0

  

These Euclidean Dirac matrices are all Hermitian, γμ=γμ, including γ5, and they all satisfy the same defining equations and anticommutation algebra rules stated in the previous paragraphs for a Minkowski spacetime.

• 

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)

Represent the imaginary unit with the lowercase i to distinguish it clearly from the number 1.

interfaceimaginaryunit=i

I

(2)

The Dgamma command is implemented as a tensor, which means you can compute with Dgamma[mu] entering tensorial expressions and have all the properties of tensors taken into account (see Physics,Tensors). For instance, the covariant and components are given by

Dgamma[]

Dgammaμ=Dgamma1Dgamma2Dgamma3Dgamma4

(3)

Dgamma`~`

Dgamma~mu=Dgamma~1Dgamma~2Dgamma~3Dgamma~4

(4)

Note that (since Maple 2019) when Physics is loaded the standard representation for the Dirac matrices is automatically set. For the default signature, (- - - +), the traditional standard matrix representation is that of the contravariant components

Library:-RewriteInMatrixFormDgamma`~`

Dgamma~mu=0001001001001000000I00I00I00I00000100001100001001000010000100001

(5)

To change that representation to the chiral or Majorana representations see Setup. As is conventional in the Physics package, you can get the definition of a tensor indexing it with the keyword definition

Dgammadefinition

%AntiCommutatorDgamma~mu,Dgamma~nu=2g_~mu,~nu

(6)

The value 0 of a spacetime index of a tensor is always mapped into the value of the position of the time-like component (the different sign in the signature)

Setupsignature

signature=- - - +

(7)

Library:-PositionOfTimelikeComponent

4

(8)

So with the current signature you can use Dgamma[0] to represent Dgamma[4]

Dgamma0

γ4

(9)

You can access the current matrix representation of each component of γμ in several ways, the simplest being

Dgamma0,matrix

Dgamma4=1000010000100001

(10)

Dgamma`~1`,matrix

Dgamma~1=0001001001001000

(11)

The γ5 matrix in the standard representation

Dgamma5,matrix

Dgamma5=0010000110000100

(12)

Dgamma5,definition

γ5=γ55,γ55=−ⅈγ00γ11γ22γ33,γ55γ55=1,%AntiCommutatorDgamma~mu,Dgamma~5=0,%AntiCommutatorDgamma~mu,Dgamma~nu=2gμ,νμ,ν

(13)

This definition includes several equations, two of which have AntiCommutators on the left-hand sides. A quick way of verifying tensorial equations including their matricial form is to use valueTensorArray%,performmatrixoperations. One can apply this command selectively, for example for the first three of these defining equations, then to only the fourth one

valueTensorArray1..3,performmatrixoperations

0010000110000100=0010000110000100,0010000110000100=0010000110000100,1000010000100001=1

(14)

valueTensorArray4,performmatrixoperations

0000000000000000=00000000000000000=00000000000000000=00000000000000000=0

(15)

You can compute with the tensor components and later represent them in matrix form, or perform the corresponding matrix operations

Dgamma1Dgamma2+Dgamma0

γ1γ2+γ4

(16)

Library:-RewriteInMatrixForm

`.`0001001001001000,000I00I00I00I000+1000010000100001

(17)

Library:-PerformMatrixOperations

1I00001+I00001I00001+I

(18)

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

DgammaμDgammaν

γμγν

(19)

Trace

4gμ,ν

(20)

Dgamma1Dgamma2+Dgamma2Dgamma1

γ1γ2+γ2γ1

(21)

Trace

0

(22)

Consider the following five products of Dirac matrices and their simplification using Simplify

e0Dgammaμ2

e0γμγμμ

(23)

Simplifye0

4

(24)

e1DgammaμDgamma`~nu`Dgammaμ

e1γμγννγμμ

(25)

Simplifye1

2γνν

(26)

e2DgammaμDgamma`~lambda`Dgamma`~nu`Dgammaμ

e2γμγλλγννγμμ

(27)

Simplifye2

4gλ,νλ,ν

(28)

e3DgammaμDgamma`~lambda`Dgamma`~nu`Dgamma`~rho`Dgammaμ

e3γμγλλγννγρργμμ

(29)

Simplifye3

2γρργννγλλ

(30)

e4DgammaμDgamma`~lambda`Dgamma`~nu`Dgamma`~rho`Dgamma`~sigma`Dgammaμ

e4γμγλλγννγρργσσγμμ

(31)

Simplifye4

2γσσγλλγννγρρ+2γρργννγλλγσσ

(32)

Verify the simplification of e1.

e1=

γμγννγμμ=2γνν

(33)

SumOverRepeatedIndices

γ1γννγ11+γ2γννγ22+γ3γννγ33+γ4γννγ44=2γνν

(34)

Rewrite this equation as an array with 4 tensorial equations as components (for each of the values o the contravariant spacetime index ν)

TTensorArray

`*`Dgamma1,`^`Dgamma~1,2+`*`Dgamma2,Dgamma~1,Dgamma~2+`*`Dgamma3,Dgamma~1,Dgamma~3+`*`Dgamma4,Dgamma~1,Dgamma~4=2Dgamma~1`*`Dgamma1,Dgamma~2,Dgamma~1+`*`Dgamma2,`^`Dgamma~2,2+`*`Dgamma3,Dgamma~2,Dgamma~3+`*`Dgamma4,Dgamma~2,Dgamma~4=2Dgamma~2`*`Dgamma1,Dgamma~3,Dgamma~1+`*`Dgamma2,Dgamma~3,Dgamma~2+`*`Dgamma3,`^`Dgamma~3,2+`*`Dgamma4,Dgamma~3,Dgamma~4=2Dgamma~3`*`Dgamma1,Dgamma~4,Dgamma~1+`*`Dgamma2,Dgamma~4,Dgamma~2+`*`Dgamma3,Dgamma~4,Dgamma~3+`*`Dgamma4,`^`Dgamma~4,2=2Dgamma~4

(35)

Perform all the matrix operations in each of the components of this array.

Library:-PerformMatrixOperationsT

0002002002002000=00020020020020000002I002I002I002I000=0002I002I002I002I0000020000220000200=00200002200002002000020000200002=2000020000200002

(36)

With the conventions used, among γμ, only γ0 (consequently, when d=4,γ5 too) changes in form between the chiral and standard representations.

The standard representation is also defined in lower dimensions. For example, if you change the dimension to 3 and check the signature

Setupdimension=3,signature

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

dimension=3,signature=- - +

(37)

Dgamma`~1`,matrix

Dgamma~1=0II0

(38)

Setupdimension=2

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

dimension=2

(39)

Dgamma`~1`,matrix

Dgamma~1=0110

(40)

Reset the dimension to 4 and check the metric

Setupdimension=4

The dimension and signature of the tensor space are set to 4,- - - +

dimension=4

(41)

g_[]

g_μ,ν=1000010000100001

(42)

The form of the four contravariant Dirac matrices in each of the three representations, all the matrices are anti-Hermitian but for Dgamma[5] which is Hermitian

SetupDgamma=standard,quiet

Dgammarepresentation=standard

(43)

TensorArray%Dgamma`~mu`=Dgamma`~mu`,performmatrixoperations

%Dgamma~1=0001001001001000%Dgamma~2=000I00I00I00I000%Dgamma~3=0010000110000100%Dgamma~4=1000010000100001

(44)

SetupDgamma=chiral,quiet

Dgammarepresentation=chiral

(45)

TensorArray%Dgamma`~mu`=Dgamma`~mu`,performmatrixoperations

%Dgamma~1=0001001001001000%Dgamma~2=000I00I00I00I000%Dgamma~3=0010000110000100%Dgamma~4=0010000110000100

(46)

The definition of Dgamma[5] is the independent of the representation

Dgamma`~5`,definition

γ55=−ⅈγ00γ11γ22γ33,γ55γ55=1,%AntiCommutatorDgamma~mu,Dgamma~5=0,%AntiCommutatorDgamma~mu,Dgamma~nu=2gμ,νμ,ν

(47)

The matrix form of the first of these equations

TensorArray1,performmatrixoperations

1000010000100001=1000010000100001

(48)

In the Majorana representation, all the components of the Dirac matrices are imaginary

SetupDgamma=Majorana,quiet

Dgammarepresentation=Majorana

(49)

TensorArray%Dgamma`~mu`=Dgamma`~mu`,performmatrixoperations

%Dgamma~1=I0000I0000I0000I%Dgamma~2=000I00I00I00I000%Dgamma~3=0I00I000000I00I0%Dgamma~4=000I00I00I00I000

(50)

The Euclidean form of the four contravariant Dirac matrices in each of the three representations: all of them are Hermitian

Setupmetric=Euclidean,Dgamma=standard,quiet

Dgammarepresentation=standard,metric=1,1=1,2,2=1,3,3=1,4,4=1

(51)

g_[]

g_μ,ν=1000010000100001

(52)

TensorArray%Dgammaμ=Dgammaμ,performmatrixoperations

%Dgamma1=000I00I00I00I000%Dgamma2=0001001001001000%Dgamma3=00I0000II0000I00%Dgamma4=1000010000100001

(53)

SetupDgamma=chiral,quiet