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The Pauli spin matrices $$ \sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}), \qquad\qquad \sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 \end{smallmatrix}), \qquad\qquad \sigma_3 ~=~ (\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}),$$

are mathematically symmetric in the sense that (like $i$ and $-i$) they can be universally exchanged with each other in several ways without altering any mathematical result. However, the visual forms for these three matrices are unexpectedly diverse (e.g., only $\sigma_2$ uses $i$ and $-i$). My understanding of physics history is that Pauli (and also Dirac) developed his matrices by trial and error, rather than by applying any specific theory.

Does a deeper theoretical explanation exist for why these very different visual representations of spin are nonetheless interchangeable in multiple ways?

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up vote 5 down vote accepted

As Qmechanic states: Most calculations in modern physics do not actually depend on the explicit realization of the Pauli matrices. At the end the physical quantities depend on the bilinear functions like the vector and axial currents.

However, it's entirely possible to transform the standard Pauli matrix representation into a spatially symmetric and entirely real valued representation that produces the exact same physics but is much easier to interpret as the complex asymmetric representation.

This representation uses 4x4 real valued matrices instead of 2x2 complex ones and the slightly larger group structure of $SO(4)\cong Spin(3)\otimes Spin(3)$ allows us to make the representation symmetric in the x, y and z-coordinates.

The 4 complex components of the bispinor field become 8 real values and the spatial symmetry of the representation becomes apparent if we visualize the linear relations between these 8 parameters for each of the matrices used, as shown in the images below. The red numbers represent the 4 right chiral parameters and the black numbers represent the 4 left chiral parameters.

enter image description here

enter image description here

For the detailed meaning of this all you can have a look here:

The real symmetric representation of the Dirac equation

Short overviews are given here: Physics-quest site and here: blog post

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Hans, wow, thanks!! I've been going over this very problem recently and remain absolutely fascinated by such notational asymmetries, which happen for example when the dimensionality of the representation space drops to low to capture the symmetries of the original problem. That situation is a given for attempting to represent the spin axes while using the 2D tool we call matrices, but the nature of the mapping continues to elude me (not a big surprise, that, but I do like to keep trying!). I look forward to looking at your answer in detail over the next couple of days (not tonight probably). – Terry Bollinger Sep 24 '12 at 1:23
I went ahead and flagged this as the answer -- sorry @Qmechanic, your answer was helpful, but Hans has nailed exactly the down-from-higher-dimensions nature of my main question. I even recognize some bits and pieces with which I was dabbling myself, quite unsuccessfully I should note. Again, Hans, thanks for the extra effort on this one. To me what you just did is where a lot of physics courses on this should start, so that that math stays connected to the underlying physics and doesn't just drift off into unexplained coincidences where cryptic notations all decide to behave nicely. Cool! – Terry Bollinger Sep 24 '12 at 1:33
One more: Is this published? I certainly haven't found anything like it, though I've not done a sufficiently deep dive to be sure. If it's original work by you, I really think you should submit this somewhere, assuming you have not already. – Terry Bollinger Sep 24 '12 at 1:39

One answer is that most calculations in modern physics do not actually depend on the explicit realization of the Pauli matrices $\sigma_a$, $a=1,2,3$, but rather on the relations

$$ \sigma_{a}^{\dagger}~=~ \sigma_{a},\qquad\qquad {\rm tr}(\sigma_{a})~=~0,\qquad\qquad a=1,2,3, $$ $$ \sigma_a \sigma_b ~=~ \delta_{ab} {\bf 1}_{2\times 2} + i\sum_{c=1}^3\varepsilon_{abc} \sigma_c, \qquad\qquad a,b=1,2,3, $$

which treat the three Pauli matrices $\sigma_a$ on a manifestly equal footing. Here ${\bf 1}_{2\times 2}$ is a $2\times 2$ unit matrix; $\delta_{ab}$ is the Kronecker delta; and $\varepsilon_{abc}$ is the Levi-Civita symbol.

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Thank you; this is quite helpful. I will examine your answer closely to make sure I understand it fully. The Levi-Civita $\epsilon_{abc}$ reminds me of a negated version of the clockwise and counterclockwise face products of the ijk unit-axes octahedral subspace of H. Quaternions do seem to capture the underlying symmetries much more directly. – Terry Bollinger Feb 29 '12 at 2:06

You might also look at the geometric algebra presentation of the Pauli spin matrices given in the paper "Imaginary Numbers are not Real-the Geometric Algebra of Spacetime" by Gull, Lasenby, and Doran (pg. 8).

(this should be a comment to the answer, but I don't have any rep points)

You can also follow up on some other good links in Norm Cimon's comment to MathOverflow Q22247.

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Tom, thanks, that's a very interesting looking paper that I'm definitely going to read in detail. The authors seem to be addressing some areas that I find highly interesting even if it's not an exact match. – Terry Bollinger Mar 23 '12 at 20:34
A possible equivalent is the Oersted Medal Lecture by Hestenes – Helder Velez Mar 12 '15 at 12:54

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