calculate the trace of noncommutative or anticommutative objects, including products of Dirac and Pauli matrices - Maple Programming Help

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Physics[Trace] - calculate the trace of noncommutative or anticommutative objects, including products of Dirac and Pauli matrices

Calling Sequence

Trace(f)

Trace(f, onlymatrices = ..., ismatricialexpression = ...)

Parameters

f

-

any algebraic expression, relation, or set, list or rtable of them

onlymatrices

-

optional, the right hand side can be true (default value) or false, to consider as scalars everything but the algebraic representation for Dirac (Dgamma) and Pauli (Psigma) matrices as tensors of one index, or objects explicitly of type matrix or Matrix (Array)

ismatricialexpression

-

optional, the right hand side can be true, false or "unknown",

Description

• 

The Trace command represents and when possible computes the generalized trace of an object. The %Trace command is the inert form of Trace; that is, it represents the same mathematical operation, while displaying the operation unevaluated. To evaluate the operation, use the value command.

• 

When tracing an algebraic expression, Trace considers as scalars everything but the Physics algebraic representation for Dirac (Dgamma) and Pauli (Psigma) matrices as tensors of one index, or objects that are of type matrix or Matrix (Array). This default is convenient in formulations in particle physics. This behavior, however, can be changed so that everything but constants are considered traceable objects - for instance any quantum operator set with Setup - for this purpose either pass the optional argument onlymatrices = false or set this behavior to be the default using Setup with its keyword onlymatrices = false.

• 

When tracing an algebraic expression, if the expression is polynomial in Dirac matrices, Trace considers all the expression as matricial, so that every term not containing Dirac matrices is assumed to have as implicit factor the identity matrix. For example, in pμγμ+m, the term m is assumed to have the identity matrix as a factor.

• 

The result returned by Trace when onlymatrices = false is built as follows:

– 

If f is a constant, then return f.

– 

If f is a single matrix, then return the trace of f.

– 

If f is a (noncommutative) product, then

– 

If the operands are Dirac matrices or Pauli matrices, then use standard related formulas (see below).

– 

If all of the operands are anticommutative, then

– 

If the number of operands is even, then return 0.

– 

Otherwise, return the Trace after normalizing the product.

– 

If there are constants in the operands, then return the constants times the Trace of the rest.

– 

If f is a (commutative) product, then return the constants times the Trace of the rest.

– 

If f is a sum, not just of noncommutative objects, distribute Trace according to:TraceA+BTraceA+TraceB.

– 

For all other cases, return the unevaluated expression, Tracef.

• 

The Trace of a product of Dirac matrices is based on their anticommutation relation gammamugammanu+gammanugammamu=2g_[mu,nu], where g_ is the metric, and the formula is valid in 2, 3, and 4 dimensions. Thus, traces of products of an odd number of Dirac matrices are always equal to zero, while traces Traceγν1,γν2,...,γν2n of an even number ( 2n ) of them can be expressed as a sum of terms of the form 4g_[mu1,mu2]g_[mu3,mu4]...g_[mu2n1, mu2n], with the sign of each term being determined by whether the permutation of indices, from [nu1,nu2,...,nu2n] to [mu1,mu2,...,mu2n], is odd or even.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

First, set prefixes identifying anticommutative and noncommutative variables.

Setupanticommutativeprefix=Q,noncommutativeprefix=Z

anticommutativeprefix=Q,noncommutativeprefix=Z

(2)

Compute some traces of expressions involving constants and commutative, anticommutative, and noncommutative variables.

The Trace of the product of an even number of anticommutative elements is zero.

TraceQ1Q2,onlymatrices=false

0

(3)

TraceQ3Q1Q2

Q1Q2Q3

(4)

TraceaQ3bQ1cQ2

abcQ1Q2Q3

(5)

TracePiQ3gammaQ1IQ2

IπγQ1Q2Q3

(6)

TraceA+B+PiQ3gammaQ1IQ2

A+B+IπγQ1Q2Q3

(7)

Products of Dirac matrices and Pauli matrices:

Dgamma3Dgamma5Dgamma5

γ3γ52

(8)

Trace

0

(9)

Dgamma3Dgammaξ

γ3γξ

(10)

Trace

4g3,ξ

(11)

DgammaαDgammaνDgammaξ

γαγνγξ

(12)

Trace

0

(13)

DgammaαDgammaνDgammaξDgammaρ

γαγνγξγρ

(14)

Trace

4gα,νgρ,ξ+4gα,ρgν,ξ4gα,ξgν,ρ

(15)

Mixed products involving constants, variables, Dirac matrices, or Pauli matrices:

Psigma3

σ3

(16)

Library:-RewriteInMatrixForm

1001

(17)

Trace

0

(18)

5Psigma2Psigma2

5σ22

(19)

Library:-RewriteInMatrixForm

5`^`0II0,2

(20)

Library:-PerformMatrixOperations

5005

(21)

Trace

10

(22)

λPsigma2Psigma3

λσ2σ3

(23)

Trace

0

(24)

Setupdimension=3:

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

(25)

3Dgamma1Dgamma2Psigma3

3γ1γ2σ3

(26)

Trace

6I

(27)

Traces of algebraic expressions involving contracted products of tensors

Setupdimension=4,signature=`-`,op=p,k

* Partial match of 'op' against keyword 'quantumoperators'

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

_______________________________________________________

dimension=4,quantumoperators=k,p,signature=- - - +

(28)

Definep,k,quiet

k,p,γμ,σμ,μ,gμ,ν,εα,β,μ,ν

(29)

pμDgammaμ+mkνDgammaν+m

pμγμμ+mkνγνν+m

(30)

Trace

4m2+4pνkνν

(31)

pμDgammaμ+mkνDgammaν+mkρDgammaρ+mkσDgammaσ+m

pμγμμ+mkνγνν+mkργρρ+mkσγσσ+m

(32)

Trace

4m4+12m2kμkμμ+12m2pμkμμ4pκkλkκκkλλ+4pμkμμkνkνν+4pσkτkττkσσ

(33)

See Also

Dgamma, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[*], Physics[Library], Psigma, Setup