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Kindly ignore if some +- signs are placed wrongly in the equations. Thank you.

Rotation in three dimensions can be represented using pauli matrices [itex]\sigma^{i}[/itex], by writing coordinates as

[itex]X= x_{i}\sigma^{i}[/itex], and applying the transform [itex]X'= AXA^{-1}[/itex]. Here [itex]A= I + n_{i}\sigma^{i}d\theta/2[/itex].

The pauli matrices are closely related to two-dimensional representations of [itex]SO(3)[/itex] and [itex]SU(2)[/itex] groups (SO(3) and SU(2) groups are isomorphic). And they are used in representing rotation. Necessary, and sufficient, condition for the rotation to be represented is that the matrices satisfy- [itex]\sigma^{i}\sigma^{j}+\sigma^{j}\sigma^{i}= 2\delta^{ij}[/itex] and [itex]\sigma^{i}\sigma^{j} - \sigma^{j}\sigma^{i}= 2i\epsilon^{ijk}\sigma^{k}[/itex].

But it can be easily verified that only two dimensional representations of SO(3) satisfy this property. My question is if there is a more fundamental reason for this, rather than just a mathematical coincidence. Or is there a more general result which relates all representations of SO(3) to rotation in three dimensions (like the one two-dimensional representations have, that is more than just having the same commutation relation for the generator). Also can the three dimensional representation of SO3 be written in terms of usual rotation matrices (Like [itex]X'= RX[/itex]) ?

Rotation in three dimensions can be represented using pauli matrices [itex]\sigma^{i}[/itex], by writing coordinates as

[itex]X= x_{i}\sigma^{i}[/itex], and applying the transform [itex]X'= AXA^{-1}[/itex]. Here [itex]A= I + n_{i}\sigma^{i}d\theta/2[/itex].

The pauli matrices are closely related to two-dimensional representations of [itex]SO(3)[/itex] and [itex]SU(2)[/itex] groups (SO(3) and SU(2) groups are isomorphic). And they are used in representing rotation. Necessary, and sufficient, condition for the rotation to be represented is that the matrices satisfy- [itex]\sigma^{i}\sigma^{j}+\sigma^{j}\sigma^{i}= 2\delta^{ij}[/itex] and [itex]\sigma^{i}\sigma^{j} - \sigma^{j}\sigma^{i}= 2i\epsilon^{ijk}\sigma^{k}[/itex].

But it can be easily verified that only two dimensional representations of SO(3) satisfy this property. My question is if there is a more fundamental reason for this, rather than just a mathematical coincidence. Or is there a more general result which relates all representations of SO(3) to rotation in three dimensions (like the one two-dimensional representations have, that is more than just having the same commutation relation for the generator). Also can the three dimensional representation of SO3 be written in terms of usual rotation matrices (Like [itex]X'= RX[/itex]) ?

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