Physics[Psigma] - the Pauli's 2 x 2 sigma matrices
an integer between 0 and 4, or an algebraic expression representing it, identifying a Pauli matrix
The Psigma[n] command represents the three Pauli matrices; that is, the set of Hermitian and unitary matrices:
where I is the imaginary unit (to represent it with a lowercase i, see interface,imaginaryunit). Together with Psigma4=Psigma0, representing the 2 x 2 identity matrix, the Pauli matrices form an orthogonal basis. The Psigman matrices are displayed as σn.
When multiplied by the imaginary unit, these matrices are a realization of the Lie algebra of the SU(2) group, which is isomorphic to the Lie algebra of SO(3). So, the I⁢σi are also a matrix realization of infinitesimal rotations in 3D space, hence serving as representation for the 3D angular momentum operator in Physics.
The Pauli matrices satisfy the commutation relations σi,σj=2⁢I⁢εijk⁢σk, where εijk is the Levi-Civita symbol, and i,j,k range from 1 to 3. The σi also satisfy the anticommutation relations σi,σj=2⁢δij, where δij is the Kronecker delta. Those two relations can be written as σi⁢σj=I⁢εijk⁢σk+δij.
For i from 1 to 3, the Pauli matrices satisfy Det⁡σi=−1,Trace⁡σi=0, and σi2=1 (the 2 x 2 identity matrix), where Det represents the determinant, and Trace represents/computes the trace. In the context of the Physics package (see conventions), you can also use the index 0, as in Psigma0, and it will be automatically mapped into Psigma4.
conventions, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[*], Physics[Library], Trace
The Physics[Psigma] command was updated in Maple 2019.
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