the Pauli's 2 x 2 sigma matrices - Maple Programming Help

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Physics[Psigma] - the Pauli's 2 x 2 sigma matrices

Calling Sequence

Psigma[n]

Parameters

n

-

an integer between 0 and 4, or an algebraic expression representing it, identifying a Pauli matrix

Description

• 

The Psigma[n] command, where n ranges from 1 to 3, represents the three Pauli matrices, displayed on the screen as σn; these are the Hermitian and unitary matrices

σ1=0110,σ2=0−II0,σ3=100−1

  

where I is the imaginary unit (to represent it with a lowercase i, see interface,imaginaryunit). Psigma[...] can be also be indexed with the letters x, y and z, with the standard correspondence σx=σ1 σy=σ2 and σz=σ3, and also with `+` and `-` (include the ``), representing the ladder operators

σ+σ1+Iσ2=0200

σ-σ1Iσ2=0020

  

To see the matrix form of any of these formulas use Physics:-Library:-RewriteInMatrixForm.

• 

The Pauli matrices satisfy the commutation relations

σa,σb=2Iεa,b,cσc

σa,σb+=2δa,b

  

where δa,b and εa,b,c are respectively the KroneckerDelta and LeviCivita symbols, and a,b,c range from 1 to 3. The Pauli matrices satisfy Detσa=−1,Traceσa=0, and σa2=1 (the 2 x 2 identity matrix), where Det represents the determinant, and Trace represents/computes the trace.

• 

When Physics is loaded, the three Pauli matrices Psigma[a], with a ranging from 1 to 3, together with Psigma[4] representing the 2 x 2 identity, are the components of a 4-vector in spacetime, Psigma[mu], with mu ranging from 1 to 4 and displayed as σμ. As with all spacetime tensors, you can use the value 0 of an index to refer to the position of the timelike component (by default equal to 4). The defining algebra for the 3D Pauli matrices σa shown above is extended to the 4D σμ as follows

σμ,σν=2Iε4,μ,να4,μ,νασα

σμ,σν+=2σμδν4ν4+2σνδμ4μ42gμ,ν

• 

Note: The default metric is of Minkowski type with signature (---+), so the space components of the contravariant σμ change in sign with respect to the definition of the Pauli matrices shown above.

• 

You can use Setup to change the metric or set a different signature, for example with the timelike component in position 1, as in (+---) or (-+++). That, however, makes the values 1, 2 and 3 of the index μ respectively refer to the identity matrix and the Pauli matrices 1, 2. To work with (+---) or (-+++) and avoid the inconvenience of having Psigma[1] referring to the identity matrix instead of σx, set spaceindices or su2indices, for example via Setupsu2indices=lowercaselatin_ah, and use Define with its redo option to redefine the type of tensor of Psigma, for example entering Defineredo,Psigmaa. In that way Psigma[a] becomes a 3D tensor, free of the issue mentioned about the numerical value of the index 1 in Psigma[mu].

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Psigma1

σ1

(2)

You can see the matrix contents of σ1 in different ways, for example:

Psigma1,matrix

Psigma1=0110

(3)

or for generic expressions involving tensors that represent matrices use

Library:-RewriteInMatrixForm

0110

(4)

Psigma1Psigma2+Psigma2Psigma1

σ1σ2+σ2σ1

(5)

Library:-RewriteInMatrixForm

`.`0110,0II0+`.`0II0,0110

(6)

Besides Library:-RewriteInMatrixForm, which shows the matrix contents of a tensorial expression, to perform those matrix operations you can use

Library:-PerformMatrixOperations

0000

(7)

Among the basic properties of Pauli matrices, there are

Psigma1Psigma1

σ12

(8)

Trace

2

(9)

Psigma1Psigma2

σ1σ2

(10)

Trace

0

(11)

To see the algebra satisfied by the Pauli matrices at any moment use Library:-DefaultAlgebraRules

Library:-DefaultAlgebraRulesPsigma

%CommutatorPsigmaμ,Psigmaν=2Iε4,μ,να4,μ,νασα,%AntiCommutatorPsigmaμ,Psigmaν=2σμg_ν4ν4+2σνg_μ4μ42gμ,ν

(12)

These equations can be verified in different ways. For example, construct an array with their components, then use its simplifier option to evaluate the commutators and anticommutators

TensorArray

%CommutatorPsigma1,Psigma1=0%CommutatorPsigma1,Psigma2=2IPsigma3%CommutatorPsigma1,Psigma3=2IPsigma2%CommutatorPsigma1,Psigma4=0%CommutatorPsigma2,Psigma1=2IPsigma3%CommutatorPsigma2,Psigma2=0%CommutatorPsigma2,Psigma3=2IPsigma1%CommutatorPsigma2,Psigma4=0%CommutatorPsigma3,Psigma1=2IPsigma2%CommutatorPsigma3,Psigma2=2IPsigma1%CommutatorPsigma3,Psigma3=0%CommutatorPsigma3,Psigma4=0%CommutatorPsigma4,Psigma1=0%CommutatorPsigma4,Psigma2=0%CommutatorPsigma4,Psigma3=0%CommutatorPsigma4,Psigma4=0,%AntiCommutatorPsigma1,Psigma1=2%AntiCommutatorPsigma1,Psigma2=0%AntiCommutatorPsigma1,Psigma3=0%AntiCommutatorPsigma1,Psigma4=2Psigma1%AntiCommutatorPsigma2,Psigma1=0%AntiCommutatorPsigma2,Psigma2=2%AntiCommutatorPsigma2,Psigma3=0%AntiCommutatorPsigma2,Psigma4=2Psigma2%AntiCommutatorPsigma3,Psigma1=0%AntiCommutatorPsigma3,Psigma2=0%AntiCommutatorPsigma3,Psigma3=2%AntiCommutatorPsigma3,Psigma4=2Psigma3%AntiCommutatorPsigma4,Psigma1=2Psigma1%AntiCommutatorPsigma4,Psigma2=2Psigma2%AntiCommutatorPsigma4,Psigma3=2Psigma3%AntiCommutatorPsigma4,Psigma4=4Psigma42

(13)

Note that in the lines above the matricial operations are performed abstractly, with the 2x2 matrices 0 and 1 (identity) omitted. To represent the algebra of the Pauli matrices with those two matrices not omitted, see the approach used in the MaplePrimes post Algebra of the Dirac matrices with an identity matrix on the right-hand side.

TensorArray,simplifier=value

0=02IPsigma3=2IPsigma32IPsigma2=2IPsigma20=02IPsigma3=2IPsigma30=02IPsigma1=2IPsigma10=02IPsigma2=2IPsigma22IPsigma1=2IPsigma10=00=00=00=00=00=0,2=20=00=02Psigma1=2Psigma10=02=20=02Psigma2=2Psigma20=00=02=22Psigma3=2Psigma32Psigma1=2Psigma12Psigma2=2Psigma22Psigma3=2Psigma34Psigma42=4Psigma42

(14)

Alternatively, for instance, rewrite in matrix form the equations before computing the commutators, then activate the inert commutators using value

Library:-RewriteInMatrixForm1

%Commutator0110,0110=0%Commutator0110,0II0=2I1001%Commutator0110,1001=2I0II0%Commutator0110,1001=0%Commutator0II0,0110=2I1001%Commutator0II0,0II0=0%Commutator0II0,1001=2I0110%Commutator0II0,1001=0%Commutator1001,0110=2I0II0%Commutator1001,0II0=2I0110%Commutator1001,1001=0%Commutator1001,1001=0%Commutator1001,0110=0%Commutator1001,0II0=0%Commutator1001,1001=0%Commutator1001,1001=0

(15)

mapexpand,value

0000=02I002I=2I002I0220=02200000=02I002I=2I002I0000=002I2I0=02I2I00000=00220=022002I2I0=02I2I00000=00000=00000=00000=00000=00000=0

(16)

A notational issue, correct but that could be seen as an inconvenience, happens when you set the signature with the timelike component in position 1, as in (+---) or (-+++), in that Psigma[1] points to Psigma[0], the identity 2x2 matrix instead of to Psigma[x]

Setupsignature=`+---`

signature=+ - - -

(17)

Psigma1=Psigma0

σ1=σ1

(18)

Psigma1,matrix

Psigma1=1001

(19)

You can still refer to σx indexing with the letter x

Psigmax=Library:-RewriteInMatrixFormPsigmax

Psigma2=0110

(20)

To avoid this potential inconvenience you could set Psigma to be a 3D tensor. One way of doing that is to set spaceindices and redefine Psigma using Define with its redo option (necessary)

Setupspaceindices=lowercaselatin_is

spaceindices=lowercaselatin_is

(21)

Note that the redefinition requires passing Psigma indexed with a space index

Defineredo,Psigmaj

Defined Pauli sigma matrices (Psigma): σ1, σ2, σ3

__________________________________________________

Defined objects with tensor properties

γμ,σj,μ,gμ,ν,γi,j,εα,β,μ,ν

(22)

Now we have σ1 = σx

Psigma1,matrix

Psigma1=0110

(23)

Note that in this case the algebra is expressed in terms of the 3D metric, gamma_, displayed as γi,j.

Library:-DefaultAlgebraRulesPsigma

%CommutatorPsigmai,Psigmaj=2Iεi,jki,jkσk,%AntiCommutatorPsigmai,Psigmaj=2γi,j

(24)

TensorArray

%CommutatorPsigma1,Psigma1=0%CommutatorPsigma1,Psigma2=2IPsigma3%CommutatorPsigma1,Psigma3=2IPsigma2%CommutatorPsigma2,Psigma1=2IPsigma3%CommutatorPsigma2,Psigma2=0%CommutatorPsigma2,Psigma3=2IPsigma1%CommutatorPsigma3,Psigma1=2IPsigma2%CommutatorPsigma3,Psigma2=2IPsigma1%CommutatorPsigma3,Psigma3=0,%AntiCommutatorPsigma1,Psigma1=2%AntiCommutatorPsigma1,Psigma2=0%AntiCommutatorPsigma1,Psigma3=0%AntiCommutatorPsigma2,Psigma1=0%AntiCommutatorPsigma2,Psigma2=2%AntiCommutatorPsigma2,Psigma3=0%AntiCommutatorPsigma3,Psigma1=0%AntiCommutatorPsigma3,Psigma2=0%AntiCommutatorPsigma3,Psigma3=2

(25)

Alternatively, to entirely detach the definition of the Pauli matrices from the details of the spacetime or space metric and signatures you can set Psigma as a tensor of a generic SU(2) space setting su2indices and redefining Psigma in the same way

Setupsu2indices=lowercaselatin_ah

su2indices=lowercaselatin_ah

(26)

Note that the redefinition requires passing Psigma indexed with a su2 index

Defineredo,Psigmaa

Defined objects with tensor properties

γμ,σa,μ,gμ,ν,γi,j,εα,β,μ,ν

(27)

Now, again, we have σ1 = σx

Psigma1,matrix

Psigma1=0110

(28)

This time the algebra is expressed using δa,b, the KroneckerDelta, used to represent the metric in the SU(2) space, and the components of these tensorial equations are the same as those computed for Psigma as a 3D space tensor lines above

Library:-DefaultAlgebraRulesPsigma

%CommutatorPsigmaa,Psigmab=2Iεa,b,cσc,%AntiCommutatorPsigmaa,Psigmab=2δa,b

(29)

TensorArray

%CommutatorPsigma1,Psigma1=0%CommutatorPsigma1,Psigma2=2IPsigma3%CommutatorPsigma1,Psigma3=2IPsigma2%CommutatorPsigma2,Psigma1=2IPsigma3%CommutatorPsigma2,Psigma2=0%CommutatorPsigma2,Psigma3=2IPsigma1%CommutatorPsigma3,Psigma1=2IPsigma2%CommutatorPsigma3,Psigma2=2IPsigma1%CommutatorPsigma3,Psigma3=0,%AntiCommutatorPsigma1,Psigma1=2%AntiCommutatorPsigma1,Psigma2=0%AntiCommutatorPsigma1,Psigma3=0%AntiCommutatorPsigma2,Psigma1=0%AntiCommutatorPsigma2,Psigma2=2%AntiCommutatorPsigma2,Psigma3=0%AntiCommutatorPsigma3,Psigma1=0%AntiCommutatorPsigma3,Psigma2=0%AntiCommutatorPsigma3,Psigma3=2

(30)

To activate the inert commutators and anticommutators use value

value

0=02IPsigma3=2IPsigma32IPsigma2=2IPsigma22IPsigma3=2IPsigma30=02IPsigma1=2IPsigma12IPsigma2=2IPsigma22IPsigma1=2IPsigma10=0,2=20=00=00=02=20=00=00=02=2

(31)

See Also

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

Compatibility

• 

The Physics[Psigma] command was updated in Maple 2020.