Talk:Pauli matrices

Untitled 2003
Somebody should redirect Pauli Gate to go to this page, I have no experience doing this, and it wasn't as easy as #REDIRECT Pauli Gate so I didn't do it.


 * I did this. Rattatosk (talk) 00:11, 27 April 2009 (UTC)

I removed this:
 * , and the Pauli matrices generate the corresponding Lie group SU(2)

I don't see in what sense four matrices can "generate" an uncountable group, especially if they aren't even elements of that group. AxelBoldt 00:34 Apr 29, 2003 (UTC)


 * By exponentiation. -- CYD

Looking at the replacement


 * so the Pauli matrices are a representation of the generators of the corresponding Lie group SU(2).

I think I see the source of my confusion. We are not talking about generators in the sense of group theory, but rather "infinitesimal generators" of a Lie group, i.e. the elements of its Lie algebra. This should be clarified somewhere. So what we are really saying is that &sigma;1,&sigma;2 and &sigma;3 form an R-basis of the Lie algebra su(2) of all Hermitian 2x2 matrices with trace 0, is that correct?

Also, the above link to group representation is misleading, since we are really representing a Lie algebra, not a group. I'll try to weave that into the article. AxelBoldt 20:02 Apr 29, 2003 (UTC)


 * That sounds right to me. -- CYD

Sign of second Pauli matrix
I think the sign of $$\sigma_2$$ has been inadvertently flipped. Indeed, I don't know what books y'all are looking at, but it at least some textbooks this guy does appear with the other sign.

Why is the other sign preferable? Self-consistency, but more for aesthetics than anything else. The problem is that with the present sign, multiplying by $$t>0$$ and exponentiating gives clockwise rotation, whereas $$\sigma_1, \sigma_3$$ give counterclockwise rotation. That's a bit awkward and is a minor annoyance in related articles like Lorentz group. OK, this might be my most pedantic quibble yet, but if anyone agrees the sign needs fixing, please do it (don't forget to check the commutators, which you'll probably also need to modify).---CH (talk) 16:42, 13 July 2005 (UTC)


 * No, the sign given in the article is correct:
 * $$\sigma_2 = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}$$
 * Please don't change it. This sign gives rise to a counterclockwise rotation, just as do $$\sigma_1, \sigma_3$$. Check your math. (The Lorentz group article needs to be changed as well). -- Fropuff 17:48, 13 July 2005 (UTC)


 * Actually, let me qualify my previous response. The spinor map SL(2,C) &rarr; SO+(3,1) depends on the isomorphism chosen between Minkowski space and the space of 2&times;2 Hermitian matrices. As long as this isomorphism is chosen to be
 * $$\{t,x,y,z\} \leftrightarrow t\sigma_0 + x\sigma_1 + y\sigma_2 + z\sigma_3$$
 * it doesn't matter which sign is chosen for &sigma;2 (choosing a different sign means choosing a different spinor map). With this choice of isomorphism the image under the spinor map of the exponential of a Pauli matrix always represents a counterclockwise rotation about the corresponding axis.


 * However, I again request that no one change the sign of &sigma;2 as this sign is conventional in physics papers and textbooks worldwide. -- Fropuff 05:02, 16 July 2005 (UTC)

Here you are:


\sigma_x \times \sigma_y = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}

\times \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}

=

i \sigma_z = i\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} $$



\sigma_x \times \sigma_z = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \times \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} =i\sigma_y $$



\sigma_y \times \sigma_x =

\begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}

\times \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}

=

\begin{pmatrix} -i&0\\ 0&i \end{pmatrix} =-i \sigma_z $$



\sigma_y \times \sigma_z =

\begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}

\times \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} =

\begin{pmatrix} 0&i\\ i&0 \end{pmatrix} =i\sigma_x $$



\sigma_z \times \sigma_x = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}

\times \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} =

\begin{pmatrix} 0&1\\ -1&0 \end{pmatrix} =-i\sigma_y $$



\sigma_z \times \sigma_y = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}

\times \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} = \begin{pmatrix} 0&-i\\ -i&0 \end{pmatrix} =-i\sigma_x $$

At this point, I feel, it may be useful to emphasize the anticommutator relations of the $$\sigma$$ matrices as their defining equations. This clarifies their relationship to the invariant metric tensor defining SO(3) and their role in the corresponding Clifford algebra. This algebraic definition allows for a manifold of alternative $$\sigma$$ representations. Please, have a look at the $$\gamma$$ matrices for an analogy.

You may, of course, insist on $$[\sigma_x,\sigma_y] = 2i \sigma_z$$ etc. to keep conventions of chirality in 3-dimensional space.

Real algebra
Direct calculation shows that the Lie algebra su(2) is the 3 dimensional real algebra spanned by the set {i σj}.

What is meant by a real algebra here? Surely the elements of the set {i σj} are complex.Wiki me (talk) 22:30, 27 February 2008 (UTC)


 * The coefficients are only allowed to be real. Compare with the statement "the complex numbers are the real algebra spanned by the set {1, i}." -- Fropuff (talk) 06:38, 28 February 2008 (UTC)

Lie algebras in lowercase
I think it may help eliminate confusion to use the normal convention of denoting Lie groups with uppercase letters and their corresponding Lie algebras with lowercase letters. I changed some instances that I noticed in the article. Thanks. Idempotent (talk) 12:02, 1 August 2008 (UTC)

Section on measurement?
With regard to Quantum mechanics, would a section on probability of measurement of the electron's spin not be good/informative? —Preceding unsigned comment added by 92.236.96.97 (talk) 12:25, 2 September 2008 (UTC)

Problems printing Pauli page?
I've tried printing the article as it stands, using four different printers, all of which print other Wikipedia articles OK, but for the Pauli Matrices article I find the Commutation relations (near top of 2nd page, printing as normal A4 in portrait orientation) don't come out, neither do the contents of the "Proof of (1)" box (lower on 2nd page), nor do parts of "Proof of (2)" box; and a single line for p = span{isigma1,isigma2}. Unless others find the printing is AOK, it would be nice if someone could amend this please (I'd rather not mess with it myself). Thanks PaulGEllis (talk) 20:14, 7 September 2008 (UTC)

Pauli vector.
As a new reader (despite already knowing clifford algebra) I found the commutator section exceptionally unclear. The Pauli vector was defined, but only by context could one see the mechanism that it provided to relate a vector to a "Pauli vector".

Additionally the statement "(as long as the vectors a and b commute with the pauli matrixes)" was confusing since one doesn't ever directly multiple these R^3 vectors with these 2x2 matrixes.

I've attempted to clarify this, adding in a bit of the reverse engineering context that was required to understand the text. In doing so I've split the Pauli vector definition out of the commutator section.

As somebody who doesn't have any text that covers this material I can't comment on how well used the Pauli vector concept is. If one's aim is to learn how to use the matrix algebra (ie: for things like rotations that aren't even covered in this article), I'd be inclined to define a vector in terms of coordinates directly:


 * $$a \equiv \sum_i a_i \sigma_i$$

and omit (or defer to an afternote) the Pauli vector entirely.

Peeter.joot (talk) 05:28, 6 December 2008 (UTC)

Pauli algebra.
Isn't the Pauli algebra just the good ol' real algebra of 2 by 2 complex matrices? It seems worth to mention it, along with the much more exotic reference to the real Clifford algebra 3,0. 147.122.52.70 (talk) 11:39, 20 April 2009 (UTC)
 * Yes, the edit should be made. Furthermore, early research in relativity used this algebra as biquaternions but Pauli's expositions turned the terminology. One of our challenges in WP is merging the physics and mathematical cultures that claim the same namespaces. The editor that can make the change will require special sensitivity.Rgdboer (talk) 21:06, 1 September 2009 (UTC)

Quantum Information and Generalised Pauli matrices: this article looks very old-fashioned
This article is far from being complete. The Pauli matrices play a big role in Quantum Information wich should be highlighted. This is a big mistake, because Quantum Information is one of the most clearest ways to understand Quantum Mechanics.

This article should have separated sections for the following three topics: 1) Connection of the Pauli matrices with quantum error correcting codes. 2) Information about the generalised Pauli group: pauli matrices can be defined for any finite group (abelian or not). 3) The stabiliser formalism and the Gottesman-Knill theorem! Relation to Clifford operations! — Preceding unsigned comment added by Garrapito (talk • contribs) 02:22, 18 June 2011 (UTC)


 * Don't demand that something you consider significant be done by others; DO IT YOURSELF! --Netheril96 (talk) 04:43, 18 June 2011 (UTC)


 * I can do it, but I do not have much free time for it before my summer holidays. Since its going to take some time before them, I just metioned that this things are missing and that the article would need some re-structuring. I would help anyone who wants to work on this :) — Preceding unsigned comment added by Garrapito (talk • contribs) 13:54, 19 June 2011 (UTC)


 * Please sign your comments using four tildes ( ~ ), now SineBot did it for you, but sometimes it gets confused and is unable to do so.


 * There already is a Physics section in this article speaking about quantum mechanics and quantum information, maybe that section could serve as a start for anything you feel is missing in the article. --Kri (talk) 12:42, 22 June 2011 (UTC)

Eigenvectors and ~values
Here it should be mentioned that this is the quantum-mechanics of the simple alternative (eigenvalues +1,-1), i.e. the lowest-dimensional non-trivial quantum-mechanics (in Hilbert-space C2). This was used by Carl-Friedrich von Weizsäcker for his Ur-theory - Urs are the basic two elementary particles in this theory, corresponding to the two inequivalent representations mentioned here. Mathematically - thanks for mentioning the Clifford-algebra here. The Pauli-matrices generate the real, associative Clifford-algebra over an Euclidean R3 (defined by a positive-definite real bilinear-form). There is an alternative on R3 with respect to an indefinite non-degenerate bilinear-form of signature (++-), with a two-dimensional representation by complex 2x2 matrices. These are given by an alternative to the Pauli-matrices, changing some signs. Another mathematical remark: With respect to the canonical bilinear-form trace(AB)-trace(A)trace(B)  for matrices A,B &isin; Cn,n (i.e. square matrices), the 4-dimensional real vector-space, spanned by the 2x2 identity-matrix and the three Pauli-matrices is a real Minkowski-space with signature (+---). Sofar there is no physical meaning behind this, it is just &bdquo;Zufall&quot;, like the other one, namely that the only unit-spheres Sn that are Lie-groups are those for n=1 and n=3. Taking the above alternative to the Pauli-matrices, this signature on the four dimensional vector-space becomes (++--). — Preceding unsigned comment added by 130.133.155.70 (talk) 13:51, 24 September 2012 (UTC)

Thus Fropuff's isomorphism above not only is one of 4-dimensional real vector-spaces, but also one of Minkowski-spaces. There is another remark above: Certainly there is an equality of this Clifford-algebra to the general linear complex algebra of endomorphisms of C2. The proof even is easy, it suffices to show, that the matrices you get by matrix multiplication are linearly independent. But - this remark also is misleading, since both associative algebras have n-dimensional generalizations, and these are not isomorphic for higher dimensions, for instance in the case of Dirac-matrices. Relativistic physics neads Dirac matrices, which with respect to the above bilinear-form of matrices are a Minkowski-space as well. The same holds for the Duffin-Kemmer-Petiau matrices. So referring to this isomorphism makes sense only for the 2-dimensional case, corresponding to the fact, that simple Lie (and Jordan) algebras of lower dimensions collapse to only a few isomorphism classes. Let me add, that the Clifford-algebras are universal envelops of a class of Jordan-algebras, defined by the underlying non-degenerate symmetric bilinear-forms in the same way, as the Heisenberg Lie-algebras are defined in terms of symplectic forms. Thus Bose-Einstein and Fermi-Dirac creation and annihilation operators are traced back to the two types of non-degenerate bilinear forms, the symmetric and the skew ones (and therefore there is no third type of statistics).

Relationship of spinors to points on the Riemann sphere and physical interpretation
As the Pauli matrices were developed in the study of spin 1/2 particles, I think the article should have a physics bias. I'd propose that the explanation of their use in physics be moved to be more prominent, and the interpretation of the two-component vectors/spinors be explained. If I understand rightly, each spinor corresponds to a point on the sphere, and is the state of the system with a definite spin in that direction. The components map to the Riemann sphere by dividing one component by the other. Explaining this in the article would explain how to find eigenvectors of linear combinations of Pauli matrices, which is important as these are the observable states for the observables these combinations represent. At the moment, the article only explains the eigenvectors of the Pauli matrices themselves. Count Truthstein (talk) 17:11, 29 November 2012 (UTC)
 * This doesn't quite work - the eigenvectors of $$\vec{n} \cdot \vec{\sigma} = \begin{pmatrix}

z  &x - iy\\ x+iy&-z \end{pmatrix}$$ are $$\begin{pmatrix}{x - iy}\\{1 - z}\end{pmatrix}$$ and $$\begin{pmatrix}{-x + iy}\\{1 + z}\end{pmatrix}$$, which look like projective coordinates except for the sign of y. This could be related to the discussion above. Count Truthstein (talk) 22:57, 12 December 2012 (UTC)

Second Pauli matrix (continued from above)
I think the problem comes from the fact that there are multiple choices for generators of the Lie algebra su(2). Looking at Special unitary group: n = 2, we see that the algebra is generated by u1, u2 and u3 with [u1,u2] = u3 and cyclic permutations of the indices. The most obvious relation to the Pauli matrices (from the definitions of the matrices in this article, and using their commutation relations) would be to have ui = &minus;i &sigma;i. However, as is apparent at the other article, u1 = i σ1,u2 = −i σ2 and u3 = i σ3 works as well, with an unexpected minus sign on the second matrix (the minus sign could of course be on any of the matrices). Count Truthstein (talk) 17:03, 9 March 2013 (UTC)

"Indent breaking "?
An IP made this edit, and I reverted since there didn't appear to be any problem at all before the indents were removed. it looks odd to have some formulae not indented and pressed against the screen, and the rest indented. Can anyone confirm any technical problems of this nature? Thanks, M&and;Ŝc2ħεИτlk 07:01, 4 May 2013 (UTC)

Lean length
Shouldn't the lead section be shorter? --Mortense (talk) 11:32, 27 December 2014 (UTC)

Higher spins
How about moving the higher spin matrices, without the ħ, of course, to Rotation group SO(3)? Higher spins hardly belong here. Concerning the arbitrary j case commented out, how about indexing a from 1 to 2j+1, as the novice would expect, in which case  Jz = (j+1−a)  δb,a, and the J± added and subtracted to have Jx and Jy, which are more familiar? E.g., Jx = (δb,a+1+δb+1,a)        /2, which agrees with the examples. I don't presume to foist work orders on the commentor, though... Cuzkatzimhut (talk) 17:03, 9 January 2015 (UTC)
 * Just my thought. We could still leave a residue here. In turn, Rotation group SO(3) can, in expanded form, rely a bit on representation theory of the Lorentz group where most everything is spelled out in painful detail, some of it directly applicable to $SO(3)$. I see a collection of articles that could work extremely well together: Rotation matrix, Rodrigues' rotation formula, Rotation group SO(3), SU(2), Pauli matrix, Lorentz group, representation theory of the Lorentz group, perhaps including Euler angles, Axis-angle representation and even Möbius group (missed something?). YohanN7 (talk) 18:29, 9 January 2015 (UTC)


 * Sounds good. Residue summaries and ample linking could result in a reasonable nexus... Some uniformity of notation might well be desirable. Cuzkatzimhut (talk) 19:12, 9 January 2015 (UTC) I realized that this last statement is obscure. I prepared the ground in this article, Pauli matrix, and i suspect everything here is consistent. However, the spin 1 (triplet rep) one has here does not automatically convert to the real antisymmetric L's of the rotation group SO(3) by dividing by ħi. It turns to an apparently (but not really) complex (and antihermitean) one, which satisfies the same algebra as the L's, so a similarity equivalent of the L's --- such is the nature of the QM choice representation. (A physicist used to the spin +1 eigenvector of Jz, (1,0,0), will be instantly put off by the corresponding eigenvector of Lz, namely (1, i, 0)/√2.)
 * Now, the easiest thing to do is to reinsert the i in the commutation relations (just) in that section there, and use the physics one here, without the ħ, and footnote the fact that Just the Ls come out different--not the ts, and provide them (the spin 1 rep here). I'd give it some time to be finessed. Cuzkatzimhut (talk) 20:34, 9 January 2015 (UTC)

The higher-spin matrix elements (including the hbar/2) could also be added to spin operator. I agree the higher spin matrices should be moved out of the article since they are not "Pauli matrices". Correct me if wrong, but Pauli matrices refers to the spin half case only. M&and;Ŝc2ħεИτlk 17:40, 12 January 2015 (UTC)


 * Agreed! you want to do it? The only purpose of the section is to remind the reader Pauli matrices are the simplest sibling of all spin matrices, and also parent, since they are all recoverable systematically out of tensor products of Paulis... As YohanN7 suggested, a bare skeleton trace residue (maybe the general j formula?) may be left here, to incite the reader to go to the spin operators... Cuzkatzimhut (talk) 19:55, 12 January 2015 (UTC)


 * Yes, I'll come back to this a later though. I'm neutral on what is left in this article, feel free to keep in the general formula, but IMO clearing out all the higher-spin equations and leaving links pointing to general results in other articles would be enough. M&and;Ŝc2ħεИτlk 21:16, 12 January 2015 (UTC)


 * OK I removed higher spin matrices, having moved them to Rotation group SO(3) without the ħ,  but leaving some background on how the fundamental one, Pauli, can lead to the rest---there is a nice formula for that, but too technical for here... In any case, the older (deleted) version, with the  ħs, is what you could easily lift and inject in the spin operators. Enjoy... Cuzkatzimhut (talk) 01:16, 13 January 2015 (UTC)


 * Done. M&and;Ŝc2ħεИτlk 19:18, 13 January 2015 (UTC)

It makes NO SENSE not to have the higher spin matrices described. Even if you are so pedantic to not consider them pauli matrices (in contradiction to many standard texts), they are direct generalizations. For example, if I will google spin matrices, it leads me to this page.

In addition, I can not seem to find ANY link on this page that will take me to the place describing the higher spin matrices. This is absolutely ridiculous. 136.152.38.211 (talk) 23:19, 13 April 2015 (UTC)


 * Pedantic or not, Pauli matrices are 2×2 spin matrices, and all higher reps are distinctly not Pauli matrices. No reputable text considers higher SU(2) representation as Pauli matrices, that, as explained here, have distinctive properties in their universal Lie algebra. Moreover, it is stated quite clearly, I think, in section 3.1, that all higher spins are constructible from tensor products of Paulis, and a distinct link sends you to where you should find them all! Try reading more carefully and reign in the emotive venting: A smoothie on Telegraph avenue might well help? Cuzkatzimhut (talk) 00:21, 14 April 2015 (UTC)


 * To the IP... Provide one or a list of reliable references then, let's see how many there are. Also, the article does link to spin operator, for example in section Pauli matrices, which is probably why google will point to here. M&and;Ŝc2ħεИτlk 08:37, 14 April 2015 (UTC)

"General expression" using Kronecker delta in lead?
What is the point of having the formula



\sigma_j = \begin{pmatrix} \delta_{j3}               &  \delta_{j1} - i\delta_{j2}\\ \delta_{j1} + i\delta_{j2} & -\delta_{j3} \end{pmatrix}. $$

in the lead? Who uses or remembers it? It seems like it should be deleted, but if others have a good reason it could be kept. M&and;Ŝc2ħεИτlk 17:47, 12 January 2015 (UTC)
 * I agree. Hermiticity and tracelessness is manifest for all three by just viewing one matrix (what a bargain), but so what? YohanN7 (talk) 17:55, 12 January 2015 (UTC)

I beg to differ... the first thing that one does is to run to the compact expression and dot to a 3-vector, as detailed in the Pauli-vector section later.. a bit of duplication might not hurt anyone... Will address other discussion later, but L&L is online out there... Cuzkatzimhut (talk) 18:05, 12 January 2015 (UTC)


 * If people want to keep it, it could be moved somewhere in the main text - but where? Maybe in the very first section after determinants and traces are mentioned? It's trivial to check this for each matrix anyway. M&and;Ŝc2ħεИτlk 18:08, 12 January 2015 (UTC)


 * Sure, I do want to keep it. Maybe where you said, or right above the definition of the Pauli vector? You might try whichever you like, but I do think that some repetition cannot hurt, obvious or not... After all, the entire article is obvious for someone in the know, but one should not give novices the excuse to decouple and go elsewhere for insight... Cuzkatzimhut (talk) 19:51, 12 January 2015 (UTC)


 * If it is useful then by all means keep it, repetition is not the problem, I just thought it may not be useful. An analogous case Yohan and me stumbled on was a closed formula for Levi-Civita symbol, which can be written down, but who uses or remembers it when you can just permute indices? For now I'll just move it down to the first section mentioned. M&and;Ŝc2ħεИτlk 21:16, 12 January 2015 (UTC)

"Inappropriate and promotional" ??
by editor Cuzkatzimhut deleting my edit showing the relation between Pauli vectors and the geometric product with the comment "Inappropriate and promotional. At best a footnote in 3.1 or 2.2.)"

What is it in the edit that is promotional? Why is is it inappropriate? How does a "promotional and inappropriate" edit suddenly become an OK edit as long as its a footnote?

I have reverted pending an explanation. Selfstudier (talk) 17:19, 26 October 2017 (UTC)


 * The idea is to first discuss these peremptory insertions, and then make them. In any case, I object to the insert on several grounds, clearly stated in the reverts. Firstly, that section establishes the relations to the dot and cross product, to be used right away, below. It has no place for asides ("footnote") or diversions of the type "while we are on cross products, here is what some believe is a better way to play with them". This is a brief reminder to mainstream students of how to handle the symbols at hand, not an invitation to leave the page. I could see a footnote of this type in section 3.1, or 2.2 where it might not break the continuity. So I consider the attempt to steal eyeballs from here to promote alternate notations inappropriate. The reader going down the article is checking most steps for validity, or familiarity, and throwing a stumbling log at her/him is just not cool. I will let another editor remove the inappropriate paragraph, since we appear to have a self-entitled edit war brewing here. Cuzkatzimhut (talk) 18:43, 26 October 2017 (UTC)


 * Prior discussion is merited if the matter is controversial, I did not expect a simple factual insertion to lead to controversy. The edit does not even mention the cross product (it mentions the dot and the geometric product). The following section is about trace so there is no interruption in the flow that I can see. I don't agree that this page is solely for mainstream students (of physics?) (I am not a mainstream student yet here I am on the page). If your objection is really about the wikified link to the geometric algebra page and although I do not really accept that a wikilink equates to stealing eyeballs nor that it amounts to an invitation to leave the page, I am quite happy to dewikify it and have done so. Selfstudier (talk) 21:31, 26 October 2017 (UTC)
 * Quaternions and versors, later down, as indicated, are also factual, but resolutely ancillary angles to it, and that was my point in recommending other venues for the parenthetical remark / footnote. Yes, I do mean section 1.6: you fail to see the flow?  Indeed, the mainstream of physics, computing, and chemistry students come here. In my view, this is the last chance for students to understand su(2) concretely, which is otherwise mired in ritualized formal abstraction. There is nothing wrong with communicating to everybody, but there has to be some standardization of language and technique. This is not a rambling open forum. Cuzkatzimhut (talk) 22:26, 26 October 2017 (UTC)
 * The su2 (/so3)business is a bit awkward to deal with absent a sensible treatment of spinors (in a Pauli matrix context). In that sense, I agree that the section on quaternions, while relevant, does just appear to be sitting there in a sort of unrelated way(the link to "versor" is not useful as that is merely the historical usage of that term by Hamilton, it is not the geometric algebra usage if that's what you are thinking, personally I would remove that link). In any case, this point directly relates to my edit, if you follow that line of algebra, it a) is helping to show why Pauli "vectors" work as opposed to the non explanatory footnote above about being a "formal device" and b) it (and the quaternion stuff, suitably reworked and put in the proper sequence) leads quite naturally via the isomorphisms into a discussion of the spin operators (and thereby su2, although I can understand it if you would prefer not to proceeed in that manner). In any case I am not trying to communicate with everybody, I am simply adding another equally valid factorization of products which I would have thought a point of interest not something to be disparaged (as for whether it is "elegant" I will leave that for others to judge) Selfstudier (talk) 14:37, 27 October 2017 (UTC)

Not even wrong
In my opinion, the equation $$\frac{1}{2} \mathrm{tr} [(\vec{a} \cdot \vec{\sigma}) \vec{\sigma}] = \vec{a}$$ is not even wrong but much worse. On the right side of the equal sign there is a vector and on the left side there is a scalar (i.e. a trace value). (unsigned by User:217.95.166.32)
 * New comments go to the bottom of the list. Please move yours there, or, better delete it. You are misreading the notation. The trace on the left acts on a vector of matrices, and traces each matrix to yield a vector with scalar entries. Please do due diligence in reading to appreciate the notation well specified. Cuzkatzimhut (talk) 21:23, 25 October 2018 (UTC)
 * (1) Please do due diligence in writing articles and using formalisms in a way to make them readable. (2) Please adjust WP user interface to put the burden of policy conforming editing on the machine instead of the user as every forum is able to do. (3) Please appreciate the statement by Wittgenstein that if one uses a language where things CAN be said/written clearly then one also should SAY/WRITE things clearly and not try to abuse the notation. (4) If insisting on such a particular (ab)use of trace, please amend WP article on trace to reflect this. (5) Probably there will be some way to turn $$(\vec{a} \cdot \vec{\sigma}) \vec{\sigma}$$ into a linear tensor-type thing where then certain dimensions can be traced out that you get this result as expected. Probably there is a WP policy template which explains that this is not the style to write in an encyclopedia. — Preceding unsigned comment added by 217.95.169.2 (talk) 22:38, 14 November 2018 (UTC)

We might have cognitive dissonance here. This is not a forum. If you have an improvement to readability, you may propose it on this page. The notation employed, and explained clearly for the Pauli vector, is universal in the physics community, and often detailed in most textbooks. The quantity in the parenthesis is a matrix. The quantity in the square bracket is a vector with matrix components. The trace of the square bracket is therefore a vector with scalar components, which comports with the right-hand side. The page has 102 watchers/editors, and 25 of them have engaged recently, as you presumably checked from the page info. If you could convince some of them of your improvement, well.... Cuzkatzimhut (talk) 22:55, 14 November 2018 (UTC)
 * We do have cognitive dissonance here. While agreeing that the Pauli vector is universal I dispute the universality and readability of the particular trace notation in discussion here. We seem to agree that the quantity in the parentheses is a traceless Hermitian 2x2 matrix. The quantity $$\vec{\sigma}$$ is (un)defined via notions $$\vec{x}$$ not further clarified in the article and which presumably shall denote some unexplained spin-observables in some unmentioned space, for which there are at least three plausible guesses which come to my mind. If the article is so detailed as even to explain the Kronecker delta or the imaginary unit then the article should at least also address this notation. The footnote nb2 is even worse. It seems garnished with categorical blurb which can be spelled out in more detail if one wants to and/or is able to. A mapping has a domain and a range, it is mathematical nonsense to talk of a mapping from one basis to another basis, since a basis is an ordered n-tuple of vectors in the finite case or a family of vectors in the general case. The trace is an operation which can be defined and is defined in WP on a square matrix in the finite case or on a trace-class linear operator in the general case, none of which is the case here. Traces, contractions, partial traces and operator invariants may be defined in all kinds of different manners in more complex ways of which one particular variant should apply here - but neither this article nor the trace articles in WP outline this. Defining a trace on a vector with matrix components or on tensors can be done but it can be done in different ways and therefore needs more notational clarity. An encyclopedia is - in my perception - an arena for those who want to learn and not a playground for those who believe they understood. I am not in the business of convincing anybody except my own mind, sorry, nobody is perfect. I am also not in the business of being convinced by numbers of people, and again I apologize for that as well if it is perceived as a deficit. I am aware of quite a number of people who struggle with quite a number of articles in WP/physics and WP/mathematics for which I have sympathy, particularly if things can be explained and are explained much clearer in many other places. I agree, however, that the various imprecisions in WP are very helpful in so far as they prompt the curious mind to find the mistakes and fix them for themselves. — Preceding unsigned comment added by 217.95.169.103 (talk) 14:00, 15 November 2018 (UTC)


 * This comment is nonsense. The notation is both clear, and standard: the trace of a vector of matrices is a vector. The LHS and the RHS of the equation are vectors. — Preceding unsigned comment added by 155.41.43.158 (talk) 20:21, 3 September 2019 (UTC)


 * Cool. "Vector of matrices". A vector is an abstract entity. There may be models of vector spaces with entries from a set, but in this case these entries are elements of a commutative field and not matrices. Matrices usually do not form fields and are not commutative. Traces are operations on matrices or linear functions (and that's also how WP, correctly, introduces them). They can be extended to partial traces or contraction operators on more complicated linear objects, such as tensor products. These are more abstract things, which is why we use a different name for them. Ok, I agree that all of this can be repaired and rephrased so that it finally is correct. Things should be made as easy as possible - but not easier. — Preceding unsigned comment added by 87.163.197.249 (talk) 17:51, 7 December 2019 (UTC)

Ground rules: NO extraneous arcane vanity cites. Proceed to propose concrete phrase substitutes here, not in the article, instead of tendentious philosophical peroration.Cuzkatzimhut (talk) 19:13, 7 December 2019 (UTC)
 * Don't worry. I am not attempting to argue with you if you are not interested in the arguments. While science is the art of finding ever better models of our world by listening to arguments which might challenge ones own position, dogmatism is the character trait of insisting on ground rules and discarding criticism based on prior belief systems. While everybody is free, of course, to choose their own perspective, nobody can gain insight from the dogmatism of ground rules. — Preceding unsigned comment added by 87.163.197.249 (talk) 20:42, 7 December 2019 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 03:22, 25 April 2020 (UTC)
 * Wolfgang Pauli.jpg

(2,0), (1,1), or (0,2)?
Are Pauli matrices (2,0), (1,1), or (0,2)? Just granpa (talk) 17:33, 2 July 2020 (UTC)

Problem with visualization of the Pauli Matrices on Chrome and Edge
When I open the page on Chrome or Edge, the third Pauli matrix reads [0 i; i 0], instead of [0 -i;i 0]. My version of Google Chrome is 87.0.4280.88 (Official Build) (64-bit) and Microsoft Edge is 87.0.664.66 (Official build) (64-bit). When I open the page on Firefox it looks fine.

Mimigdal (talk) 09:38, 22 December 2020 (UTC)
 * Solutions for chrome. Also wikiwand.

Cuzkatzimhut (talk) 12:16, 22 December 2020 (UTC)

Pauli matrices
Hermitian and unitary implies involutory. The identity matrix and its opposite are both hermitian unitary and then involutory, but not Pauli. It seems that null trace should be included in the definition of a Pauli matrix. And strip out the involutory condition BatracioVerde (talk) 12:21, 12 July 2024 (UTC)


 * Appended tracelessness. Involution is harmless, and might help the less sophisticated reader. Cuzkatzimhut (talk) 16:28, 12 July 2024 (UTC)