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Represent a 3 dimensional rotation with a 2 dimensional complex matrix

Posted 1 year ago

In this book "Space Groups and their Representations", I noticed the following method to represent a 3 dimensional rotation with a 2 dimensional complex matrix:

enter image description here

I try to reproduce this method to obtain the same 2 dimensional complex matrix, i.e., the following:

In[12]:= u={{Cos[1/2 \[Theta]] Exp[-I (\[CurlyPhi]+\[Psi])/2],  Sin[1/2 \[Theta]] Exp[I (\[CurlyPhi]-\[Psi])/2]},
{-Sin[1/2 \[Theta]] Exp[I (\[CurlyPhi]-\[Psi])/2], Cos[1/2 \[Theta]] Exp[-I (\[CurlyPhi]+\[Psi])/2]}}

Out[12]= {{E^(-(1/2) I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2], 
  E^(1/2 I (\[CurlyPhi] - \[Psi]))
    Sin[\[Theta]/2]}, {-E^(1/2 I (\[CurlyPhi] - \[Psi])) Sin[\[Theta]/
    2], E^(-(1/2) I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2]}}

On the other hand, I noticed the following description given in the book "Theory of spinors : an introduction", as shown below: enter image description here

So I tried the following method to establish the relationship between them, but failed to verify the equivalence between these two different forms:

In[133]:= u={{Cos[1/2 \[Theta]] Exp[-I (\[CurlyPhi]+\[Psi])/2], Sin[1/2 \[Theta]] Exp[I (\[CurlyPhi]-\[Psi])/2]},
{-Sin[1/2 \[Theta]] Exp[I (\[CurlyPhi]-\[Psi])/2], Cos[1/2 \[Theta]] Exp[-I (\[CurlyPhi]+\[Psi])/2]}}//FullSimplify

Out[133]= {{E^(-(1/2) I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2], 
  E^(1/2 I (\[CurlyPhi] - \[Psi]))
    Sin[\[Theta]/2]}, {-E^(1/2 I (\[CurlyPhi] - \[Psi])) Sin[\[Theta]/
    2], E^(-(1/2) I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2]}}

In[134]:= u1={{Exp[I \[CurlyPhi]/2],0},{0, Exp[-I \[CurlyPhi]/2]}};
u2={{Cos[\[Theta]/2], I Sin[\[Theta]/2]},{I Sin[\[Theta]/2], Cos[\[Theta]/2]}};
u3={{Exp[I \[Psi]/2],0},{0, Exp[-I \[Psi]/2]}};
u3 . u2 . u1//FullSimplify

Out[137]= {{E^(1/2 I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2], 
  I E^(-(1/2) I (\[CurlyPhi] - \[Psi])) Sin[\[Theta]/2]}, {I E^(
   1/2 I (\[CurlyPhi] - \[Psi])) Sin[\[Theta]/2], 
  E^(-(1/2) I (\[CurlyPhi] + \[Psi])) Cos[\[Theta]/2]}}

Any hints/comments/tips for this problem will be appreciated.

Regards, Zhao

POSTED BY: Hongyi Zhao
2 Replies
Posted 1 year ago

MatrixExp[ i/2 alpha ( sin theta' cos phi' sigma + sin theta' sin phi' sigma2 + cos theta' sigma3) ]

It seems that there is a typo: sigma should be written as sigma1 in your above description.

The nontrivial comparison is possible by comparing Eigensystem[] for both representations

Do you mean that in this way we can find out the correspondence and transformation between them? I still feel some difficulties. Can you give me a specific example to further explain?

Regards, Zhao

POSTED BY: Hongyi Zhao

The Euler angles representation is a rotation by phi in the 1,2-plane, a rotation by theta in the (new) 1,3-plane followed by a rotation by psi again in the new 1,2-plane. Its spin represenation by 2x2 matrices is therefore MatrixExp[psi/2 i sigma3] . MatrixExp[theta/2 i sigma 2] MatricExp[ phi/3 i sigma3]

The standard Lie-algebra representation by Eulers theorem is a rotation by alpha around an axis pointing to the point (theta', phi ') on the unit sphere

MatrixExp[ i/2 alpha ( sin theta' cos phi' sigma + sin theta' sin phi' sigma2 + cos theta' sigma3) ]

The sigma matrices are the PauliMatrix[ {1,2,3} ].

The nontrivial comparison is possible by comparing Eigensystem[] for both representations

Regards Roland

POSTED BY: Roland Franzius
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