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 σ₁,σ₂ and σ₃ 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) → SO⁺(3,1) depends on the isomorphism chosen between Minkowski space and the space of 2×2 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 σ₂ (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 σ₂ 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 ∈ C^n,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 „Zufall", 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 u_i = −i σ_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 < math >"?

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∧Ŝc²ħεИ_τlk 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)#A note on representations? 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 J_z = (j+1−a) δ_b,a, and the J± added and subtracted to have Jx and Jy, which are more familiar? E.g., J_x = (δ_b,a+1+δ_b+1,a) √(j+1)(a+b−1)−ab /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)#A note on representations 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 J_z, (1,0,0), will be instantly put off by the corresponding eigenvector of L_z, 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 21:16, 12 January 2015 (UTC)

OK I removed higher spin matrices, having moved them to Rotation group SO(3)#A note on representations 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)

"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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 21:16, 12 January 2015 (UTC)