That is if we transform them to a basis that say \sigma_x is diagonal can we still use the commutation relationships that are valid in \sigma_z diagonal basis?
Of course. As long as you transform a set M_1,....,M_p of nxn matrices with a single invertible nxn matrix T by the formula
M_i ---> T M_i T^-1 (sometimes called a similarity transformation or a basis transformation)
every relation among the M_1,...,M_p that can be expressed by matrix multiplication, addition, and subtraction (all commutation relations in particular) hold in the same form also for the transformed matrices. This would, of course, not hold if you would consider 'nonlinear transformations' such as M_i ---> M_i*M_i.
The chapter on matrices in any textbook on linear algebra explains the background for this. Good luck.
Of course. As long as you transform a set M_1,....,M_p of nxn matrices with a single invertible nxn matrix T by the formula
M_i ---> T M_i T^-1 (sometimes called a similarity transformation or a basis transformation)
every relation among the M_1,...,M_p that can be expressed by matrix multiplication, addition, and subtraction (all commutation relations in particular) hold in the same form also for the transformed matrices. This would, of course, not hold if you would consider 'nonlinear transformations' such as M_i ---> M_i*M_i.
The chapter on matrices in any textbook on linear algebra explains the background for this. Good luck.
Exactly Ulrich,
Masoud, may I ask what prompted you to ask such a simple question?
The algebra of Pauli matrices, or Pauli algebra, is actually defined by the commutation relations. Then it can be represented by matrices, and this representation is not unique. Any transformation that preserve the commutation relations gives an equivalent representation. That is the case of a unitary transformation. The Pauli algebra is analogous of the algebra of Dirac matrices, or gamma matrices.
Claude, the algebra of Pauli matrices is not only defined by the commutation relations but also by rules for products of Pauli matrices ( as a linear combination of Pauli matrices and the unit matrix, i.e. as an associative algebra with unit). See the Wikipedia article on Pauli matrices. Correspondingly also the Dirac algebra is defined as an associative algebra with unit. With this understanding, all irreducible representations of the Pauli algebra are by 2x2 matrices and all irreducible representations of the Dirac algebra are by 4x4 matrices. If one would consider only the commutation relations (i.e.define the Pauli algebra as a Lie algebra) one gets irreducible representations as nxn matrices for any integer n>1.
Of course matricies, in general, do not comute and there is a qustion of Pauli's understanding of some transformations because of his advice agains publishing research which was awarded a Noble prise. Look into the Noble prise wining research.
Thanks for the answers. I was answering a question from Iranian PhD exam. The question was something like what is the representation of sigma_x and sigma_z in the basis that sigma_y is diagonal. From four provided answer only one was obeying the algebra, so I went for it. But when I decided to go through complete calculation I ended up with other results. That made me confused because I couldn't believe that a question in such exam can be wrong, but it was. After calculating such simple thing again and again I became exhausted enough to reach wrong and paradoxical results, then I asked myself is Pauli algebra really a Lie algebra or I am totally wrong? So the question was prompted by my exhaustion.
Masoud: Of course, not the Pauli matrices themselves form a Lie-algebra but the products of the Pauli matrices with the imaginary unit i. Is this the point that disturbed you?
Under what kind of transformation ? If it is similarity transformation, then yes it is invariant and so all Lie-Algebra, any non-singular square matrix "S" of the same dimension as that of the matrices involved (say A_i) in the algebra can keep it invariant if we transform like this :
$ A^{\prime} = S^{-1} A_i S $, then the primed matrices will satisfy the same algebra as the unprimed A's once did.
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