Talk:Pauli matrices

4241:

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.

2526:

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 R (defined by a positive-definite real bilinear-form). There is an alternative on R 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 (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 S 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 (++--).

3916:

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)

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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.

4071:(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 2550:

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.

2562:

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.

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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.

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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.

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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

3915:

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

3900:

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,

3763:

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

2525:

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 C). This was used by Carl-Friedrich von Weizsäcker for his Ur-theory - Urs are the basic two elementary particles

2397:

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

2272:

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

4240:

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

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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.

2561:

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

3881:

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

3415:

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

2552:

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

4337:

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

3862:

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

3334:

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...

3271:

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

2549:

There is another remark above: Certainly there is an equality of this Clifford-algebra to the general linear complex algebra of endomorphisms of C. 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

2241:

I've tried printing the article as it stands, using four different printers, all of which print other Knowledge 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

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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

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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

3390:

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.

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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 :)

4050:

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

3189:. 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 4464:

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

3294:

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.

4417:

When I open the page on Chrome or Edge, the third Pauli matrix reads , instead of . 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.

2959:

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,

2261:

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".

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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.

3703:

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

2195:

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.

871:(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. 861: 4133:

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.

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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.

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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

3723:

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.

4131: 1995:{\displaystyle \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}} 4158:

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

2268:

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.

1804:{\displaystyle \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}} 1437:{\displaystyle \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}} 1070:{\displaystyle \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}}} 1619:{\displaystyle \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}} 1252:{\displaystyle \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}} 2806: 2741: 4524: 4044: 2868:

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

3833:

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.)"

2327: 602:. 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).--- 2265:

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.

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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.

4166:

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....

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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)

732: 592: 3239:. 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. 4514: 4338:

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.

3786:

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

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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).

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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. --

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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?

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Ground rules: NO extraneous arcane vanity cites. Proceed to propose concrete phrase substitutes here, not in the article, instead of tendentious philosophical peroration.

4238: 2580: 2071: 2051: 2027: 448:, 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 σ 4376: 4266:

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.

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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.

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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.

3790:, 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. 753:) → 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 521:

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

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matrices as their defining equations. This clarifies their relationship to the invariant metric tensor defining SO(3) and their role in the corresponding

4554: 3128: 3096: 288: 3218:, 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. 4539: 4509: 183: 3633:{\displaystyle \sigma _{j}={\begin{pmatrix}\delta _{j3}&\delta _{j1}-i\delta _{j2}\\\delta _{j1}+i\delta _{j2}&-\delta _{j3}\end{pmatrix}}.} 4519: 620: 264: 136: 4549: 3161:

Sounds good. Residue summaries and ample linking could result in a reasonable nexus... Some uniformity of notation might well be desirable.

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The coefficients are only allowed to be real. Compare with the statement "the complex numbers are the real algebra spanned by the set {1,

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I don't see in what sense four matrices can "generate" an uncountable group, especially if they aren't even elements of that group.

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With regard to Quantum mechanics, would a section on probability of measurement of the electron's spin not be good/informative?

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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.

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works as well, with an unexpected minus sign on the second matrix (the minus sign could of course be on any of the matrices).

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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

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

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Thus Fropuff's isomorphism above not only is one of 4-dimensional real vector-spaces, but also one of Minkowski-spaces.

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is misleading, since we are really representing a Lie algebra, not a group. I'll try to weave that into the article.

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

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Quaternions and versors, later down, as indicated, are also factual, but resolutely ancillary angles to it, and

3420:! Try reading more carefully and reign in the emotive venting: A smoothie on Telegraph avenue might well help? 2537: 49: 4349: 4307: 4252: 3411:

Pedantic or not, Pauli matrices are 2×2 spin matrices, and all higher reps are distinctly not Pauli matrices.

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The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:

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I think the problem comes from the fact that there are multiple choices for generators of the Lie algebra

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in the commutation relations (just) in that section there, and use the physics one here, without the

2674:{\displaystyle {\vec {n}}\cdot {\vec {\sigma }}={\begin{pmatrix}z&x-iy\\x+iy&-z\end{pmatrix}}} 44:. Please limit discussion to improvement of this article. You may wish to ask factual questions about 4341: 4299: 4269: 4244: 4185: 4137: 3331: 3182: 3112: 3092: 3076:/2, which agrees with the examples. I don't presume to foist work orders on the commentor, though... 3021: 2529: 2449: 2431: 2402: 2247: 2243: 2218: 2201: 468: 432: 379: 63: 2457: 2410: 2079: 356: 2157: 263:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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Appended tracelessness. Involution is harmless, and might help the less sophisticated reader.

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Please help fix the broken anchors. You can remove this template after fixing the problems. |

2368:

Yes, the edit should be made. Furthermore, early research in relativity used this algebra as

464:-basis of the Lie algebra su(2) of all Hermitian 2x2 matrices with trace 0, is that correct? 4399: 3917: 3883: 3840: 2808:, which look like projective coordinates except for the sign of y. This could be related to 2335: 2056: 2036: 2030: 2012: 332: 3104: 3099:

where most everything is spelled out in painful detail, some of it directly applicable to

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Don't demand that something you consider significant be done by others; DO IT YOURSELF! --

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Quantum Information and Generalised Pauli matrices: this article looks very old-fashioned

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I realized that this last statement is obscure. I prepared the ground in this article,

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A Commons file used on this page or its Wikidata item has been nominated for deletion

3441: 3236: 3124: 3046:, and the J± added and subtracted to have Jx and Jy, which are more familiar? E.g., 2998: 2506: 735: 599: 524: 3791: 3725: 3705: 3689: 3645: 3449: 3357: 3296: 3240: 3174: 3144: 3132: 3120: 2961: 2557:

Relationship of spinors to points on the Riemann sphere and physical interpretation

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rotation. That's a bit awkward and is a minor annoyance in related articles like

3676:

I agree. Hermiticity and tracelessness is manifest for all three by just viewing

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formula?) may be left here, to incite the reader to go to the spin operators...

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The higher-spin matrix elements (including the hbar/2) could also be added to

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as this sign is conventional in physics papers and textbooks worldwide. --

679:{\displaystyle \sigma _{2}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}} 3103:. I see a collection of articles that could work extremely well together: 3196:, (1,0,0), will be instantly put off by the corresponding eigenvector of 2502: 480: 420: 4489: 4474: 4454: 4429: 4407: 4388: 4353: 4331: 4311: 4281: 4256: 4175: 4149: 4066: 3925: 3910: 3891: 3872: 3848: 3816: 3773: 3750: 3717: 3697: 3670: 3474: 3429: 3403: 3382: 3344: 3321: 3285: 3265: 3227: 3170: 3152: 3085: 3006: 2940: 2821: 2571: 2541: 2510: 2465: 2435: 2418: 2382: 2362: 2343: 2251: 2230: 2205: 2184: 2165: 884: 742: 609: 387: 340:

This article links to one or more target anchors that no longer exist.

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did it for you, but sometimes it gets confused and is unable to do so.

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Problem with visualization of the Pauli Matrices on Chrome and Edge

749:

Actually, let me qualify my previous response. The spinor map SL(2,

444:

I think I see the source of my confusion. We are not talking about

4126:{\displaystyle ({\vec {a}}\cdot {\vec {\sigma }}){\vec {\sigma }}} 3116: 2033:. This algebraic definition allows for a manifold of alternative 2156:

algebra here? Surely the elements of the set {i σj} are complex.

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Just my thought. We could still leave a residue here. In turn,

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the Gottesman-Knill theorem! Relation to Clifford operations!

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

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etc. to keep conventions of chirality in 3-dimensional space.

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from 1 to 2j+1, as the novice would expect, in which case

3024:? Higher spins hardly belong here. Concerning the arbitrary 2801:{\displaystyle {\begin{pmatrix}{-x+iy}\\{1+z}\end{pmatrix}}} 4439: 2736:{\displaystyle {\begin{pmatrix}{x-iy}\\{1-z}\end{pmatrix}}} 2005:

At this point, I feel, it may be useful to emphasize the

876:

However, I again request that no one change the sign of σ

3330:

OK I removed higher spin matrices, having moved them to

3016:

How about moving the higher spin matrices, without the

2480:

Please sign your comments using four tildes (~~~~), now

3831: 3353: 2956: 2952: 4039:{\displaystyle {\frac {1}{2}}\mathrm {tr} ={\vec {a}}} 3515: 3416:

distinct link sends you to where you should find them

2755: 2693: 2619: 1936: 1894: 1852: 1748: 1709: 1667: 1569: 1527: 1485: 1381: 1342: 1300: 1199: 1157: 1118: 1033: 972: 933: 642: 4525:

Knowledge level-5 vital articles in Physical sciences

4217: 4188: 4077: 3951: 3496: 2749: 2687: 2583: 2282: 2082: 2059: 2039: 2015: 1820: 1635: 1453: 1268: 1086: 901: 761: 700: 623: 560: 528: 497: 398:, and the Pauli matrices generate the corresponding 259:, a collaborative effort to improve the coverage of 154:, a collaborative effort to improve the coverage of 3483:"General expression" using Kronecker delta in lead? 3448:, which is probably why google will point to here. 4232: 4203: 4125: 4038: 3632: 2800: 2735: 2673: 2321: 2133: 2065: 2045: 2021: 1994: 1803: 1618: 1436: 1251: 1069: 855: 726: 690:Please don't change it. This sign gives rise to a 678: 586: 540: 510: 2322:{\displaystyle a\equiv \sum _{i}a_{i}\sigma _{i}} 3210:Now, the easiest thing to do is to reinsert the 2886:. However, as is apparent at the other article, 2809: 4375:Participate in the deletion discussion at the 3332:Rotation group SO(3)#A note on representations 3093:Rotation group SO(3)#A note on representations 3022:Rotation group SO(3)#A note on representations 2577:This doesn't quite work - the eigenvectors of 615:No, the sign given in the article is correct: 4515:Knowledge vital articles in Physical sciences 8: 3764:to decouple and go elsewhere for insight... 2053:representations. Please, have a look at the 867:it doesn't matter which sign is chosen for σ 786: 762: 4545:B-Class physics articles of High-importance 4530:B-Class vital articles in Physical sciences 4398:Are Pauli matrices (2,0), (1,1), or (0,2)? 4339: 4297: 4267: 4242: 4135: 3129:representation theory of the Lorentz group 3097:representation theory of the Lorentz group 2840:, we see that the algebra is generated by 2828:Second Pauli matrix (continued from above) 205: 100: 4219: 4218: 4216: 4190: 4189: 4187: 4112: 4111: 4097: 4096: 4082: 4081: 4076: 4025: 4024: 4007: 4006: 3992: 3991: 3977: 3976: 3962: 3952: 3950: 3858:discuss these peremptory insertions, and 3610: 3592: 3573: 3556: 3537: 3522: 3510: 3501: 3495: 3487:What is the point of having the formula 2997:Shouldn't the lead section be shorter? -- 2779: 2758: 2750: 2748: 2714: 2696: 2688: 2686: 2614: 2600: 2599: 2585: 2584: 2582: 2313: 2303: 2293: 2281: 2125: 2103: 2090: 2081: 2058: 2038: 2014: 1986: 1931: 1889: 1847: 1838: 1825: 1819: 1795: 1743: 1704: 1662: 1653: 1640: 1634: 1610: 1564: 1522: 1480: 1471: 1458: 1452: 1428: 1376: 1337: 1295: 1286: 1273: 1267: 1243: 1194: 1152: 1113: 1104: 1091: 1085: 1028: 1016: 967: 928: 919: 906: 900: 847: 831: 815: 799: 760: 738:article needs to be changed as well). -- 718: 705: 699: 637: 628: 622: 578: 565: 559: 527: 502: 496: 3839:I have reverted pending an explanation. 3028:case commented out, how about indexing 727:{\displaystyle \sigma _{1},\sigma _{3}} 587:{\displaystyle \sigma _{1},\sigma _{3}} 446:generators in the sense of group theory 435:of the generators of the corresponding 207: 102: 61: 3095:can, in expanded form, rely a bit on 7: 3681: 347:] The anchor (#via Laurent series) 253:This article is within the scope of 148:This article is within the scope of 91:It is of interest to the following 3966: 3963: 3827:"Inappropriate and promotional" ?? 14: 4555:Mid-priority mathematics articles 273:Knowledge:WikiProject Mathematics 4540:High-importance physics articles 4510:Knowledge level-5 vital articles 4204:{\displaystyle {\vec {\sigma }}} 3682: 3446:Pauli matrices#Quantum mechanics 2946:"Indent breaking < math : --> 318: 276:Template:WikiProject Mathematics 240: 230: 209: 135: 125: 104: 71: 62: 19: 293:This article has been rated as 188:This article has been rated as 4520:B-Class level-5 vital articles 4224: 4195: 4117: 4108: 4102: 4087: 4078: 4030: 4018: 4012: 4003: 3997: 3982: 3973: 3970: 2605: 2590: 2542:13:51, 24 September 2012 (UTC) 2134:{\displaystyle =2i\sigma _{z}} 2109: 2083: 2076:You may, of course, insist on 789: 1: 4455:12:16, 22 December 2020 (UTC) 4430:09:38, 22 December 2020 (UTC) 4282:20:21, 3 September 2019 (UTC) 4257:14:00, 15 November 2018 (UTC) 4176:22:55, 14 November 2018 (UTC) 4150:22:38, 14 November 2018 (UTC) 4057:the notation well specified. 3007:11:32, 27 December 2014 (UTC) 2822:22:57, 12 December 2012 (UTC) 2572:17:11, 29 November 2012 (UTC) 2383:21:06, 1 September 2009 (UTC) 2252:20:14, 7 September 2008 (UTC) 2237:Problems printing Pauli page? 2231:12:25, 2 September 2008 (UTC) 2185:06:38, 28 February 2008 (UTC) 2166:22:30, 27 February 2008 (UTC) 267:and see a list of open tasks. 168:Knowledge:WikiProject Physics 162:and see a list of open tasks. 32:for general discussion about 4550:B-Class mathematics articles 4354:20:42, 7 December 2019 (UTC) 4332:19:13, 7 December 2019 (UTC) 4312:17:51, 7 December 2019 (UTC) 4067:21:23, 25 October 2018 (UTC) 3945:In my opinion, the equation 3926:14:37, 27 October 2017 (UTC) 3911:22:26, 26 October 2017 (UTC) 3892:21:31, 26 October 2017 (UTC) 3873:18:43, 26 October 2017 (UTC) 3849:17:19, 26 October 2017 (UTC) 3817:21:16, 12 January 2015 (UTC) 3774:19:51, 12 January 2015 (UTC) 3751:18:08, 12 January 2015 (UTC) 3718:18:05, 12 January 2015 (UTC) 3698:17:55, 12 January 2015 (UTC) 3671:17:47, 12 January 2015 (UTC) 3383:19:18, 13 January 2015 (UTC) 3345:01:16, 13 January 2015 (UTC) 3322:21:16, 12 January 2015 (UTC) 3286:19:55, 12 January 2015 (UTC) 3266:17:40, 12 January 2015 (UTC) 2838:Special unitary group: n = 2 2344:05:28, 6 December 2008 (UTC) 479:That sounds right to me. -- 431:so the Pauli matrices are a 171:Template:WikiProject Physics 4211:is (un)defined via notions 3228:20:34, 9 January 2015 (UTC) 3171:19:12, 9 January 2015 (UTC) 3153:18:29, 9 January 2015 (UTC) 3109:Rodrigues' rotation formula 3086:17:03, 9 January 2015 (UTC) 511:{\displaystyle \sigma _{2}} 487:Sign of second Pauli matrix 426:Looking at the replacement 4571: 4389:03:22, 25 April 2020 (UTC) 4233:{\displaystyle {\vec {x}}} 3475:08:37, 14 April 2015 (UTC) 3430:00:21, 14 April 2015 (UTC) 3404:23:19, 13 April 2015 (UTC) 2363:11:39, 20 April 2009 (UTC) 2206:12:02, 1 August 2008 (UTC) 388:00:11, 27 April 2009 (UTC) 194:project's importance scale 4490:16:28, 12 July 2024 (UTC) 4475:12:21, 12 July 2024 (UTC) 3444:, for example in section 3137:Axis-angle representation 2941:17:03, 9 March 2013 (UTC) 2511:12:42, 22 June 2011 (UTC) 2466:13:54, 19 June 2011 (UTC) 2436:04:43, 18 June 2011 (UTC) 2419:02:22, 18 June 2011 (UTC) 2273:of coordinates directly: 2191:Lie algebras in lowercase 2073:matrices for an analogy. 885:05:02, 16 July 2005 (UTC) 743:17:48, 13 July 2005 (UTC) 610:16:42, 13 July 2005 (UTC) 550:and exponentiating gives 475:20:02 Apr 29, 2003 (UTC) 411:00:34 Apr 29, 2003 (UTC) 292: 225: 187: 120: 99: 4535:B-Class physics articles 4408:17:33, 2 July 2020 (UTC) 2521:Eigenvectors and ~values 734:. Check your math. (The 467:Also, the above link to 299:project's priority scale 4394:(2,0), (1,1), or (0,2)? 3708:is online out there... 2988:07:01, 4 May 2013 (UTC) 2211:Section on measurement? 2066:{\displaystyle \gamma } 2046:{\displaystyle \sigma } 2022:{\displaystyle \sigma } 256:WikiProject Mathematics 4505:B-Class vital articles 4234: 4205: 4127: 4040: 3680:matrix (what a bargain 3634: 2802: 2737: 2675: 2323: 2135: 2067: 2047: 2023: 1996: 1805: 1620: 1438: 1253: 1071: 857: 728: 680: 588: 543: 512: 4235: 4206: 4128: 4054:reading to appreciate 4041: 3635: 3143:(missed something?). 2803: 2738: 2676: 2324: 2136: 2068: 2048: 2024: 1997: 1806: 1621: 1439: 1254: 1072: 858: 729: 694:rotation, just as do 681: 589: 544: 541:{\displaystyle t: --> 513: 78:level-5 vital article 4436:Solutions for chrome 4215: 4186: 4075: 3949: 3494: 3183:rotation group SO(3) 3131:, perhaps including 3113:Rotation group SO(3) 2810:the discussion above 2747: 2685: 2581: 2280: 2080: 2057: 2037: 2013: 1818: 1633: 1451: 1266: 1084: 899: 759: 698: 621: 558: 526: 495: 491:I think the sign of 469:group representation 279:mathematics articles 36:. Any such comments 3272:(maybe the general 2497:There already is a 2152:What is meant by a 374:so I didn't do it. 151:WikiProject Physics 4381:Community Tech bot 4370:Wolfgang Pauli.jpg 4230: 4201: 4123: 4036: 3788:Levi-Civita symbol 3630: 3621: 3179:real antisymmetric 2798: 2792: 2733: 2727: 2671: 2665: 2319: 2298: 2131: 2063: 2043: 2019: 1992: 1967: 1922: 1880: 1801: 1776: 1734: 1695: 1616: 1594: 1555: 1513: 1434: 1409: 1367: 1328: 1249: 1227: 1185: 1143: 1067: 1061: 1000: 958: 853: 724: 676: 670: 584: 554:rotation, whereas 538: 508: 248:Mathematics portal 87:content assessment 31: 4356: 4344:comment added by 4314: 4302:comment added by 4284: 4272:comment added by 4259: 4247:comment added by 4227: 4198: 4152: 4140:comment added by 4120: 4105: 4090: 4033: 4015: 4000: 3985: 3960: 2608: 2593: 2532:comment added by 2469: 2452:comment added by 2422: 2405:comment added by 2289: 2233: 2221:comment added by 2009:relations of the 363: 362: 335:in most browsers. 313: 312: 309: 308: 305: 304: 204: 203: 200: 199: 56: 55: 27: 4562: 4239: 4237: 4236: 4231: 4229: 4228: 4220: 4210: 4208: 4207: 4202: 4200: 4199: 4191: 4132: 4130: 4129: 4124: 4122: 4121: 4113: 4107: 4106: 4098: 4092: 4091: 4083: 4045: 4043: 4042: 4037: 4035: 4034: 4026: 4017: 4016: 4008: 4002: 4001: 3993: 3987: 3986: 3978: 3969: 3961: 3953: 3688:), but so what? 3687: 3686: 3685: 3639: 3637: 3636: 3631: 3626: 3625: 3618: 3617: 3600: 3599: 3581: 3580: 3564: 3563: 3545: 3544: 3530: 3529: 3506: 3505: 3185:by dividing by 3102: 3075: 3074: 3020:, of course, to 2951:An IP made this 2933:Count Truthstein 2836:(2). Looking at 2814:Count Truthstein 2807: 2805: 2804: 2799: 2797: 2796: 2789: 2774: 2742: 2740: 2739: 2734: 2732: 2731: 2724: 2709: 2680: 2678: 2677: 2672: 2670: 2669: 2610: 2609: 2601: 2595: 2594: 2586: 2564:Count Truthstein 2544: 2468: 2446: 2421: 2399: 2328: 2326: 2325: 2320: 2318: 2317: 2308: 2307: 2297: 2216: 2140: 2138: 2137: 2132: 2130: 2129: 2108: 2107: 2095: 2094: 2072: 2070: 2069: 2064: 2052: 2050: 2049: 2044: 2031:Clifford algebra 2028: 2026: 2025: 2020: 2001: 1999: 1998: 1993: 1991: 1990: 1972: 1971: 1927: 1926: 1885: 1884: 1843: 1842: 1830: 1829: 1810: 1808: 1807: 1802: 1800: 1799: 1781: 1780: 1739: 1738: 1700: 1699: 1658: 1657: 1645: 1644: 1625: 1623: 1622: 1617: 1615: 1614: 1599: 1598: 1560: 1559: 1518: 1517: 1476: 1475: 1463: 1462: 1443: 1441: 1440: 1435: 1433: 1432: 1414: 1413: 1372: 1371: 1333: 1332: 1291: 1290: 1278: 1277: 1258: 1256: 1255: 1250: 1248: 1247: 1232: 1231: 1190: 1189: 1148: 1147: 1109: 1108: 1096: 1095: 1076: 1074: 1073: 1068: 1066: 1065: 1021: 1020: 1005: 1004: 963: 962: 924: 923: 911: 910: 862: 860: 859: 854: 852: 851: 836: 835: 820: 819: 804: 803: 733: 731: 730: 725: 723: 722: 710: 709: 692:counterclockwise 685: 683: 682: 677: 675: 674: 633: 632: 596:counterclockwise 593: 591: 590: 585: 583: 582: 570: 569: 549: 546: 545: 539: 517: 515: 514: 509: 507: 506: 393:I removed this: 357:Reporting errors 349:has been deleted 322: 321: 315: 281: 280: 277: 274: 271: 250: 245: 244: 234: 227: 226: 221: 213: 206: 176: 175: 174:physics articles 172: 169: 166: 145: 140: 139: 129: 122: 121: 116: 108: 101: 84: 75: 74: 67: 66: 58: 23: 22: 16: 4570: 4569: 4565: 4564: 4563: 4561: 4560: 4559: 4495: 4494: 4462: 4415: 4396: 4377:nomination page 4363: 4213: 4212: 4184: 4183: 4073: 4072: 3947: 3946: 3943: 3854:The idea is to 3829: 3813: 3747: 3683: 3667: 3620: 3619: 3606: 3601: 3588: 3569: 3566: 3565: 3552: 3533: 3531: 3518: 3511: 3497: 3492: 3491: 3485: 3471: 3379: 3318: 3262: 3201: 3194: 3105:Rotation matrix 3100: 3072:(j+1)(a+b−1)−ab 3070: 3068: 3065: 3058: 3051: 3044: 3037: 3014: 2995: 2983: 2949: 2930: 2922: 2915: 2907: 2900: 2892: 2885: 2876: 2867: 2860: 2853: 2846: 2830: 2791: 2790: 2776: 2775: 2751: 2745: 2744: 2726: 2725: 2711: 2710: 2689: 2683: 2682: 2664: 2663: 2655: 2640: 2639: 2625: 2615: 2579: 2578: 2559: 2527: 2523: 2447: 2400: 2391: 2351: 2309: 2299: 2278: 2277: 2259: 2239: 2213: 2193: 2147: 2121: 2099: 2086: 2078: 2077: 2055: 2054: 2035: 2034: 2011: 2010: 1982: 1966: 1965: 1960: 1951: 1950: 1942: 1932: 1921: 1920: 1915: 1909: 1908: 1900: 1890: 1879: 1878: 1870: 1864: 1863: 1858: 1848: 1834: 1821: 1816: 1815: 1791: 1775: 1774: 1769: 1760: 1759: 1754: 1744: 1733: 1732: 1727: 1721: 1720: 1715: 1705: 1694: 1693: 1685: 1679: 1678: 1673: 1663: 1649: 1636: 1631: 1630: 1606: 1593: 1592: 1587: 1581: 1580: 1575: 1565: 1554: 1553: 1545: 1539: 1538: 1533: 1523: 1512: 1511: 1506: 1500: 1499: 1491: 1481: 1467: 1454: 1449: 1448: 1424: 1408: 1407: 1402: 1396: 1395: 1390: 1377: 1366: 1365: 1360: 1354: 1353: 1348: 1338: 1327: 1326: 1321: 1315: 1314: 1306: 1296: 1282: 1269: 1264: 1263: 1239: 1226: 1225: 1220: 1214: 1213: 1205: 1195: 1184: 1183: 1175: 1169: 1168: 1163: 1153: 1142: 1141: 1136: 1130: 1129: 1124: 1114: 1100: 1087: 1082: 1081: 1060: 1059: 1051: 1045: 1044: 1039: 1029: 1012: 999: 998: 993: 987: 986: 978: 968: 957: 956: 951: 945: 944: 939: 929: 915: 902: 897: 896: 893: 879: 870: 843: 827: 811: 795: 757: 756: 714: 701: 696: 695: 669: 668: 663: 657: 656: 648: 638: 624: 619: 618: 574: 561: 556: 555: 523: 522: 498: 493: 492: 489: 459: 455: 451: 368: 359: 338: 337: 336: 319: 278: 275: 272: 269: 268: 246: 239: 219: 190:High-importance 173: 170: 167: 164: 163: 141: 134: 115:High‑importance 114: 85:on Knowledge's 82: 72: 20: 12: 11: 5: 4568: 4566: 4558: 4557: 4552: 4547: 4542: 4537: 4532: 4527: 4522: 4517: 4512: 4507: 4497: 4496: 4493: 4492: 4461: 4460:Pauli matrices 4458: 4444: 4443: 4414: 4411: 4395: 4392: 4373: 4372: 4362: 4359: 4358: 4357: 4346:87.163.197.249 4320: 4319: 4318: 4317: 4316: 4315: 4304:87.163.197.249 4288: 4287: 4286: 4285: 4261: 4260: 4249:217.95.169.103 4226: 4223: 4197: 4194: 4156: 4155: 4154: 4153: 4119: 4116: 4110: 4104: 4101: 4095: 4089: 4086: 4080: 4032: 4029: 4023: 4020: 4014: 4011: 4005: 3999: 3996: 3990: 3984: 3981: 3975: 3972: 3968: 3965: 3959: 3956: 3942: 3941:Not even wrong 3939: 3937: 3935: 3934: 3933: 3932: 3931: 3930: 3929: 3928: 3876: 3875: 3828: 3825: 3824: 3823: 3822: 3821: 3820: 3819: 3811: 3779: 3778: 3777: 3776: 3754: 3753: 3745: 3701: 3700: 3665: 3643: 3641: 3640: 3629: 3624: 3616: 3613: 3609: 3605: 3602: 3598: 3595: 3591: 3587: 3584: 3579: 3576: 3572: 3568: 3567: 3562: 3559: 3555: 3551: 3548: 3543: 3540: 3536: 3532: 3528: 3525: 3521: 3517: 3516: 3514: 3509: 3504: 3500: 3484: 3481: 3480: 3479: 3478: 3477: 3469: 3435: 3434: 3433: 3432: 3396:136.152.38.211 3388: 3387: 3386: 3385: 3377: 3348: 3347: 3327: 3326: 3325: 3324: 3316: 3289: 3288: 3260: 3233: 3232: 3231: 3230: 3208: 3199: 3192: 3156: 3155: 3063: 3056: 3049: 3042: 3035: 3013: 3010: 2994: 2991: 2981: 2948: 2944: 2928: 2920: 2913: 2905: 2898: 2890: 2881: 2872: 2865: 2858: 2851: 2844: 2829: 2826: 2825: 2824: 2795: 2788: 2785: 2782: 2778: 2777: 2773: 2770: 2767: 2764: 2761: 2757: 2756: 2754: 2730: 2723: 2720: 2717: 2713: 2712: 2708: 2705: 2702: 2699: 2695: 2694: 2692: 2668: 2662: 2659: 2656: 2654: 2651: 2648: 2645: 2642: 2641: 2638: 2635: 2632: 2629: 2626: 2624: 2621: 2620: 2618: 2613: 2607: 2604: 2598: 2592: 2589: 2558: 2555: 2551: 2548: 2534:130.133.155.70 2522: 2519: 2518: 2517: 2516: 2515: 2514: 2513: 2490: 2489: 2488: 2487: 2486: 2485: 2473: 2472: 2471: 2470: 2439: 2438: 2390: 2387: 2386: 2385: 2350: 2349:Pauli algebra. 2347: 2330: 2329: 2316: 2312: 2306: 2302: 2296: 2292: 2288: 2285: 2258: 2255: 2238: 2235: 2212: 2209: 2192: 2189: 2188: 2187: 2146: 2143: 2128: 2124: 2120: 2117: 2114: 2111: 2106: 2102: 2098: 2093: 2089: 2085: 2062: 2042: 2018: 2007:anticommutator 2003: 2002: 1989: 1985: 1981: 1978: 1975: 1970: 1964: 1961: 1959: 1956: 1953: 1952: 1949: 1946: 1943: 1941: 1938: 1937: 1935: 1930: 1925: 1919: 1916: 1914: 1911: 1910: 1907: 1904: 1901: 1899: 1896: 1895: 1893: 1888: 1883: 1877: 1874: 1871: 1869: 1866: 1865: 1862: 1859: 1857: 1854: 1853: 1851: 1846: 1841: 1837: 1833: 1828: 1824: 1812: 1811: 1798: 1794: 1790: 1787: 1784: 1779: 1773: 1770: 1768: 1765: 1762: 1761: 1758: 1755: 1753: 1750: 1749: 1747: 1742: 1737: 1731: 1728: 1726: 1723: 1722: 1719: 1716: 1714: 1711: 1710: 1708: 1703: 1698: 1692: 1689: 1686: 1684: 1681: 1680: 1677: 1674: 1672: 1669: 1668: 1666: 1661: 1656: 1652: 1648: 1643: 1639: 1627: 1626: 1613: 1609: 1605: 1602: 1597: 1591: 1588: 1586: 1583: 1582: 1579: 1576: 1574: 1571: 1570: 1568: 1563: 1558: 1552: 1549: 1546: 1544: 1541: 1540: 1537: 1534: 1532: 1529: 1528: 1526: 1521: 1516: 1510: 1507: 1505: 1502: 1501: 1498: 1495: 1492: 1490: 1487: 1486: 1484: 1479: 1474: 1470: 1466: 1461: 1457: 1445: 1444: 1431: 1427: 1423: 1420: 1417: 1412: 1406: 1403: 1401: 1398: 1397: 1394: 1391: 1389: 1386: 1383: 1382: 1380: 1375: 1370: 1364: 1361: 1359: 1356: 1355: 1352: 1349: 1347: 1344: 1343: 1341: 1336: 1331: 1325: 1322: 1320: 1317: 1316: 1313: 1310: 1307: 1305: 1302: 1301: 1299: 1294: 1289: 1285: 1281: 1276: 1272: 1260: 1259: 1246: 1242: 1238: 1235: 1230: 1224: 1221: 1219: 1216: 1215: 1212: 1209: 1206: 1204: 1201: 1200: 1198: 1193: 1188: 1182: 1179: 1176: 1174: 1171: 1170: 1167: 1164: 1162: 1159: 1158: 1156: 1151: 1146: 1140: 1137: 1135: 1132: 1131: 1128: 1125: 1123: 1120: 1119: 1117: 1112: 1107: 1103: 1099: 1094: 1090: 1078: 1077: 1064: 1058: 1055: 1052: 1050: 1047: 1046: 1043: 1040: 1038: 1035: 1034: 1032: 1027: 1024: 1019: 1015: 1011: 1008: 1003: 997: 994: 992: 989: 988: 985: 982: 979: 977: 974: 973: 971: 966: 961: 955: 952: 950: 947: 946: 943: 940: 938: 935: 934: 932: 927: 922: 918: 914: 909: 905: 892: 889: 888: 887: 877: 873: 872: 868: 865: 864: 863: 850: 846: 842: 839: 834: 830: 826: 823: 818: 814: 810: 807: 802: 798: 794: 791: 788: 785: 782: 779: 776: 773: 770: 767: 764: 746: 745: 721: 717: 713: 708: 704: 688: 687: 686: 673: 667: 664: 662: 659: 658: 655: 652: 649: 647: 644: 643: 641: 636: 631: 627: 581: 577: 573: 568: 564: 537: 534: 531: 505: 501: 488: 485: 484: 483: 457: 453: 449: 442: 441: 433:representation 424: 423: 417:exponentiation 405: 404: 391: 390: 367: 364: 361: 360: 354: 353: 352: 345: 333:case-sensitive 327: 326: 325: 323: 311: 310: 307: 306: 303: 302: 291: 285: 284: 282: 265:the discussion 252: 251: 235: 223: 222: 214: 202: 201: 198: 197: 186: 180: 179: 177: 160:the discussion 147: 146: 143:Physics portal 130: 118: 117: 109: 97: 96: 90: 68: 54: 53: 50:Reference desk 46:Pauli matrices 38:may be removed 34:Pauli matrices 24: 13: 10: 9: 6: 4: 3: 2: 4567: 4556: 4553: 4551: 4548: 4546: 4543: 4541: 4538: 4536: 4533: 4531: 4528: 4526: 4523: 4521: 4518: 4516: 4513: 4511: 4508: 4506: 4503: 4502: 4500: 4491: 4487: 4483: 4479: 4478: 4477: 4476: 4472: 4468: 4467:BatracioVerde 4459: 4457: 4456: 4452: 4448: 4441: 4437: 4434: 4433: 4432: 4431: 4427: 4423: 4419: 4412: 4410: 4409: 4405: 4401: 4393: 4391: 4390: 4386: 4382: 4378: 4371: 4368: 4367: 4366: 4360: 4355: 4351: 4347: 4343: 4336: 4335: 4334: 4333: 4329: 4325: 4313: 4309: 4305: 4301: 4294: 4293: 4292: 4291: 4290: 4289: 4283: 4279: 4275: 4274:155.41.43.158 4271: 4265: 4264: 4263: 4262: 4258: 4254: 4250: 4246: 4221: 4192: 4180: 4179: 4178: 4177: 4173: 4169: 4164: 4163: 4151: 4147: 4143: 4139: 4114: 4099: 4093: 4084: 4070: 4069: 4068: 4064: 4060: 4056: 4055: 4049: 4048: 4047: 4027: 4021: 4009: 3994: 3988: 3979: 3957: 3954: 3940: 3938: 3927: 3923: 3919: 3914: 3913: 3912: 3908: 3904: 3899: 3895: 3894: 3893: 3889: 3885: 3880: 3879: 3878: 3877: 3874: 3870: 3866: 3861: 3857: 3853: 3852: 3851: 3850: 3846: 3842: 3837: 3834: 3832: 3826: 3818: 3815: 3814: 3807: 3805: 3802: 3798: 3797: 3796: 3789: 3785: 3784: 3783: 3782: 3781: 3780: 3775: 3771: 3767: 3762: 3758: 3757: 3756: 3755: 3752: 3749: 3748: 3741: 3739: 3736: 3732: 3731: 3730: 3722: 3721: 3720: 3719: 3715: 3711: 3707: 3699: 3695: 3691: 3679: 3675: 3674: 3673: 3672: 3669: 3668: 3661: 3659: 3656: 3652: 3651: 3650: 3627: 3622: 3614: 3611: 3607: 3603: 3596: 3593: 3589: 3585: 3582: 3577: 3574: 3570: 3560: 3557: 3553: 3549: 3546: 3541: 3538: 3534: 3526: 3523: 3519: 3512: 3507: 3502: 3498: 3490: 3489: 3488: 3482: 3476: 3473: 3472: 3465: 3463: 3460: 3456: 3455: 3454: 3447: 3443: 3442:spin operator 3439: 3438: 3437: 3436: 3431: 3427: 3423: 3419: 3414: 3410: 3409: 3408: 3407: 3406: 3405: 3401: 3397: 3392: 3384: 3381: 3380: 3373: 3371: 3368: 3364: 3363: 3362: 3355: 3352: 3351: 3350: 3349: 3346: 3342: 3338: 3333: 3329: 3328: 3323: 3320: 3319: 3312: 3310: 3307: 3303: 3302: 3301: 3293: 3292: 3291: 3290: 3287: 3283: 3279: 3275: 3270: 3269: 3268: 3267: 3264: 3263: 3256: 3254: 3251: 3247: 3246: 3245: 3238: 3237:spin operator 3229: 3225: 3221: 3217: 3213: 3209: 3206: 3203:, namely (1, 3202: 3195: 3188: 3184: 3180: 3176: 3172: 3168: 3164: 3160: 3159: 3158: 3157: 3154: 3150: 3146: 3142: 3138: 3134: 3130: 3126: 3125:Lorentz group 3122: 3118: 3114: 3110: 3106: 3098: 3094: 3090: 3089: 3088: 3087: 3083: 3079: 3073: 3066: 3059: 3052: 3045: 3038: 3031: 3027: 3023: 3019: 3011: 3009: 3008: 3004: 3000: 2992: 2990: 2989: 2987:"?": --> 2985: 2984: 2977: 2975: 2972: 2968: 2967: 2966: 2958: 2954: 2945: 2943: 2942: 2938: 2934: 2926: 2919: 2911: 2904: 2896: 2889: 2884: 2880: 2875: 2871: 2864: 2857: 2850: 2843: 2839: 2835: 2827: 2823: 2819: 2815: 2811: 2793: 2786: 2783: 2780: 2771: 2768: 2765: 2762: 2759: 2752: 2728: 2721: 2718: 2715: 2706: 2703: 2700: 2697: 2690: 2666: 2660: 2657: 2652: 2649: 2646: 2643: 2636: 2633: 2630: 2627: 2622: 2616: 2611: 2602: 2596: 2587: 2576: 2575: 2574: 2573: 2569: 2565: 2556: 2554: 2545: 2543: 2539: 2535: 2531: 2520: 2512: 2508: 2504: 2500: 2496: 2495: 2494: 2493: 2492: 2491: 2483: 2479: 2478: 2477: 2476: 2475: 2474: 2467: 2463: 2459: 2455: 2451: 2443: 2442: 2441: 2440: 2437: 2433: 2429: 2425: 2424: 2423: 2420: 2416: 2412: 2408: 2404: 2395: 2388: 2384: 2380: 2376: 2371: 2370:biquaternions 2367: 2366: 2365: 2364: 2360: 2356: 2355:147.122.52.70 2348: 2346: 2345: 2341: 2337: 2333: 2314: 2310: 2304: 2300: 2294: 2290: 2286: 2283: 2276: 2275: 2274: 2270: 2266: 2263: 2257:Pauli vector. 2256: 2254: 2253: 2249: 2245: 2236: 2234: 2232: 2228: 2224: 2220: 2210: 2208: 2207: 2203: 2199: 2190: 2186: 2182: 2178: 2174: 2170: 2169: 2168: 2167: 2163: 2159: 2155: 2150: 2144: 2142: 2126: 2122: 2118: 2115: 2112: 2104: 2100: 2096: 2091: 2087: 2074: 2060: 2040: 2032: 2016: 2008: 1987: 1983: 1979: 1976: 1973: 1968: 1962: 1957: 1954: 1947: 1944: 1939: 1933: 1928: 1923: 1917: 1912: 1905: 1902: 1897: 1891: 1886: 1881: 1875: 1872: 1867: 1860: 1855: 1849: 1844: 1839: 1835: 1831: 1826: 1822: 1814: 1813: 1796: 1792: 1788: 1785: 1782: 1777: 1771: 1766: 1763: 1756: 1751: 1745: 1740: 1735: 1729: 1724: 1717: 1712: 1706: 1701: 1696: 1690: 1687: 1682: 1675: 1670: 1664: 1659: 1654: 1650: 1646: 1641: 1637: 1629: 1628: 1611: 1607: 1603: 1600: 1595: 1589: 1584: 1577: 1572: 1566: 1561: 1556: 1550: 1547: 1542: 1535: 1530: 1524: 1519: 1514: 1508: 1503: 1496: 1493: 1488: 1482: 1477: 1472: 1468: 1464: 1459: 1455: 1447: 1446: 1429: 1425: 1421: 1418: 1415: 1410: 1404: 1399: 1392: 1387: 1384: 1378: 1373: 1368: 1362: 1357: 1350: 1345: 1339: 1334: 1329: 1323: 1318: 1311: 1308: 1303: 1297: 1292: 1287: 1283: 1279: 1274: 1270: 1262: 1261: 1244: 1240: 1236: 1233: 1228: 1222: 1217: 1210: 1207: 1202: 1196: 1191: 1186: 1180: 1177: 1172: 1165: 1160: 1154: 1149: 1144: 1138: 1133: 1126: 1121: 1115: 1110: 1105: 1101: 1097: 1092: 1088: 1080: 1079: 1062: 1056: 1053: 1048: 1041: 1036: 1030: 1025: 1022: 1017: 1013: 1009: 1006: 1001: 995: 990: 983: 980: 975: 969: 964: 959: 953: 948: 941: 936: 930: 925: 920: 916: 912: 907: 903: 895: 894: 891:Here you are: 890: 886: 883: 875: 874: 866: 848: 844: 840: 837: 832: 828: 824: 821: 816: 812: 808: 805: 800: 796: 792: 783: 780: 777: 774: 771: 768: 765: 755: 754: 752: 748: 747: 744: 741: 737: 736:Lorentz group 719: 715: 711: 706: 702: 693: 689: 671: 665: 660: 653: 650: 645: 639: 634: 629: 625: 617: 616: 614: 613: 612: 611: 608: 605: 601: 600:Lorentz group 597: 579: 575: 571: 566: 562: 553: 535: 532: 529: 519: 503: 499: 486: 482: 478: 477: 476: 474: 470: 465: 463: 447: 440: 438: 434: 429: 428: 427: 422: 418: 414: 413: 412: 410: 403: 401: 396: 395: 394: 389: 385: 381: 377: 376: 375: 373: 366:Untitled 2003 365: 358: 350: 346: 343: 342: 341: 334: 330: 324: 317: 316: 300: 296: 290: 287: 286: 283: 266: 262: 258: 257: 249: 243: 238: 236: 233: 229: 228: 224: 218: 215: 212: 208: 195: 191: 185: 182: 181: 178: 161: 157: 153: 152: 144: 138: 133: 131: 128: 124: 123: 119: 113: 110: 107: 103: 98: 94: 88: 80: 79: 69: 65: 60: 59: 51: 47: 43: 39: 35: 30: 26:This page is 25: 18: 17: 4482:Cuzkatzimhut 4463: 4447:Cuzkatzimhut 4445: 4420: 4416: 4397: 4374: 4364: 4340:— Preceding 4324:Cuzkatzimhut 4321: 4298:— Preceding 4268:— Preceding 4243:— Preceding 4168:Cuzkatzimhut 4161: 4160: 4157: 4142:217.95.169.2 4136:— Preceding 4059:Cuzkatzimhut 4053: 4052: 3944: 3936: 3903:Cuzkatzimhut 3897: 3865:Cuzkatzimhut 3859: 3855: 3838: 3835: 3830: 3809: 3803: 3800: 3794: 3792: 3766:Cuzkatzimhut 3760: 3743: 3737: 3734: 3728: 3726: 3710:Cuzkatzimhut 3702: 3677: 3663: 3657: 3654: 3648: 3646: 3642: 3486: 3467: 3461: 3458: 3452: 3450: 3422:Cuzkatzimhut 3417: 3412: 3393: 3389: 3375: 3369: 3366: 3360: 3358: 3337:Cuzkatzimhut 3314: 3308: 3305: 3299: 3297: 3278:Cuzkatzimhut 3273: 3258: 3252: 3249: 3243: 3241: 3234: 3220:Cuzkatzimhut 3215: 3211: 3204: 3197: 3190: 3186: 3178: 3175:Pauli matrix 3163:Cuzkatzimhut 3141:Möbius group 3133:Euler angles 3121:Pauli matrix 3078:Cuzkatzimhut 3071: 3061: 3054: 3047: 3040: 3033: 3029: 3025: 3017: 3015: 3012:Higher spins 2996: 2979: 2973: 2970: 2964: 2962: 2950: 2924: 2917: 2909: 2902: 2894: 2887: 2882: 2878: 2873: 2869: 2862: 2855: 2848: 2841: 2833: 2831: 2560: 2546: 2528:— Preceding 2524: 2448:— Preceding 2401:— Preceding 2396: 2392: 2373:sensitivity. 2352: 2334: 2331: 2271: 2267: 2264: 2260: 2240: 2223:92.236.96.97 2214: 2194: 2172: 2153: 2151: 2148: 2145:Real algebra 2075: 2004: 750: 691: 595: 551: 520: 490: 466: 461: 443: 430: 425: 406: 397: 392: 378:I did this. 369: 339: 331:Anchors are 328: 295:Mid-priority 294: 254: 220:Mid‑priority 189: 149: 93:WikiProjects 76: 4400:Just granpa 3918:Selfstudier 3884:Selfstudier 3841:Selfstudier 3181:L's of the 2993:Lean length 2336:Peeter.joot 2217:—Preceding 270:Mathematics 261:mathematics 217:Mathematics 29:not a forum 4499:Categories 3041:(j+1−a) δ 2428:Netheril96 2244:PaulGEllis 2198:Idempotent 548:0}" /: --> 372:Pauli Gate 42:refactored 4162:universal 3207:, 0)/√2.) 3139:and even 2454:Garrapito 2407:Garrapito 552:clockwise 473:AxelBoldt 437:Lie group 409:AxelBoldt 400:Lie group 380:Rattatosk 81:is rated 4440:wikiwand 4438:. Also 4422:Mimigdal 4342:unsigned 4300:unsigned 4270:unsigned 4245:unsigned 4138:unsigned 3759:Sure, I 2999:Mortense 2957:reverted 2955:, and I 2861:with = 2530:unsigned 2462:contribs 2450:unsigned 2415:contribs 2403:unsigned 2219:unsigned 525:0}": --> 460:form an 3706:L&L 3690:YohanN7 3145:YohanN7 2499:Physics 2482:SineBot 2375:Rgdboer 2177:Fropuff 2175:}." -- 2158:Wiki me 882:Fropuff 740:Fropuff 297:on the 192:on the 165:Physics 156:Physics 112:Physics 83:B-class 48:at the 607:(talk) 439:SU(2). 89:scale. 3856:first 3117:SU(2) 3101:SO(3) 3064:b+1,a 3057:b,a+1 2877:= −i 594:give 533:: --> 456:and σ 419:. -- 402:SU(2) 70:This 4486:talk 4471:talk 4451:talk 4426:talk 4404:talk 4385:talk 4350:talk 4328:talk 4308:talk 4278:talk 4253:talk 4172:talk 4146:talk 4063:talk 3922:talk 3907:talk 3898:that 3888:talk 3869:talk 3860:then 3845:talk 3770:talk 3714:talk 3694:talk 3426:talk 3400:talk 3354:Done 3341:talk 3282:talk 3224:talk 3167:talk 3149:talk 3082:talk 3003:talk 2953:edit 2937:talk 2916:and 2854:and 2818:talk 2743:and 2681:are 2568:talk 2538:talk 2507:talk 2458:talk 2432:talk 2411:talk 2379:talk 2359:talk 2340:talk 2248:talk 2227:talk 2202:talk 2181:talk 2162:talk 2154:real 384:talk 329:Tip: 184:High 4379:. — 3812:τlk 3746:τlk 3678:one 3666:τlk 3470:τlk 3418:all 3378:τlk 3317:τlk 3261:τlk 3053:= ( 3043:b,a 2982:τlk 2908:= − 2503:Kri 604:CH 481:CYD 421:CYD 415:By 289:Mid 40:or 4501:: 4488:) 4473:) 4453:) 4428:) 4406:) 4387:) 4352:) 4330:) 4310:) 4280:) 4255:) 4225:→ 4196:→ 4193:σ 4174:) 4148:) 4118:→ 4115:σ 4103:→ 4100:σ 4094:⋅ 4088:→ 4065:) 4031:→ 4013:→ 4010:σ 3998:→ 3995:σ 3989:⋅ 3983:→ 3924:) 3909:) 3890:) 3871:) 3847:) 3793:M∧ 3772:) 3761:do 3727:M∧ 3716:) 3696:) 3647:M∧ 3608:δ 3604:− 3590:δ 3571:δ 3554:δ 3547:− 3535:δ 3520:δ 3499:σ 3451:M∧ 3428:) 3413:No 3402:) 3359:M∧ 3356:. 3343:) 3298:M∧ 3284:) 3242:M∧ 3226:) 3187:ħi 3169:) 3151:) 3135:, 3127:, 3123:, 3119:, 3115:, 3111:, 3107:, 3084:) 3067:) 3039:= 3005:) 2963:M∧ 2947:"? 2939:) 2923:= 2893:= 2847:, 2834:su 2820:) 2812:. 2760:− 2719:− 2701:− 2658:− 2631:− 2606:→ 2603:σ 2597:⋅ 2591:→ 2570:) 2540:) 2509:) 2464:) 2460:• 2434:) 2417:) 2413:• 2381:) 2361:) 2342:) 2311:σ 2291:∑ 2287:≡ 2250:) 2229:) 2204:) 2183:) 2164:) 2123:σ 2101:σ 2088:σ 2061:γ 2041:σ 2017:σ 1984:σ 1977:− 1955:− 1945:− 1903:− 1887:× 1873:− 1836:σ 1832:× 1823:σ 1793:σ 1786:− 1764:− 1702:× 1688:− 1651:σ 1647:× 1638:σ 1608:σ 1548:− 1520:× 1494:− 1469:σ 1465:× 1456:σ 1426:σ 1419:− 1385:− 1335:× 1309:− 1284:σ 1280:× 1271:σ 1241:σ 1208:− 1178:− 1150:× 1102:σ 1098:× 1089:σ 1054:− 1014:σ 981:− 965:× 917:σ 913:× 904:σ 845:σ 829:σ 813:σ 797:σ 790:↔ 716:σ 703:σ 651:− 626:σ 576:σ 563:σ 542:0} 500:σ 452:,σ 386:) 4484:( 4469:( 4449:( 4442:. 4424:( 4402:( 4383:( 4348:( 4326:( 4306:( 4276:( 4251:( 4222:x 4170:( 4144:( 4109:) 4085:a 4079:( 4061:( 4028:a 4022:= 4019:] 4004:) 3980:a 3974:( 3971:[ 3967:r 3964:t 3958:2 3955:1 3920:( 3905:( 3886:( 3867:( 3843:( 3810:И 3806:ε 3804:ħ 3801:c 3795:Ŝ 3768:( 3744:И 3740:ε 3738:ħ 3735:c 3729:Ŝ 3712:( 3692:( 3664:И 3660:ε 3658:ħ 3655:c 3649:Ŝ 3628:. 3623:) 3615:3 3612:j 3597:2 3594:j 3586:i 3583:+ 3578:1 3575:j 3561:2 3558:j 3550:i 3542:1 3539:j 3527:3 3524:j 3513:( 3508:= 3503:j 3468:И 3464:ε 3462:ħ 3459:c 3453:Ŝ 3424:( 3398:( 3376:И 3372:ε 3370:ħ 3367:c 3361:Ŝ 3339:( 3315:И 3311:ε 3309:ħ 3306:c 3300:Ŝ 3280:( 3274:j 3259:И 3255:ε 3253:ħ 3250:c 3244:Ŝ 3222:( 3216:ħ 3212:i 3205:i 3200:z 3198:L 3193:z 3191:J 3165:( 3147:( 3080:( 3069:√ 3062:δ 3060:+ 3055:δ 3050:x 3048:J 3036:z 3034:J 3030:a 3026:j 3018:ħ 3001:( 2980:И 2976:ε 2974:ħ 2971:c 2965:Ŝ 2935:( 2929:3 2927:σ 2925:i 2921:3 2918:u 2914:2 2912:σ 2910:i 2906:2 2903:u 2901:, 2899:1 2897:σ 2895:i 2891:1 2888:u 2883:i 2879:σ 2874:i 2870:u 2866:3 2863:u 2859:3 2856:u 2852:2 2849:u 2845:1 2842:u 2816:( 2794:) 2787:z 2784:+ 2781:1 2772:y 2769:i 2766:+ 2763:x 2753:( 2729:) 2722:z 2716:1 2707:y 2704:i 2698:x 2691:( 2667:) 2661:z 2653:y 2650:i 2647:+ 2644:x 2637:y 2634:i 2628:x 2623:z 2617:( 2612:= 2588:n 2566:( 2536:( 2505:( 2456:( 2430:( 2409:( 2377:( 2357:( 2338:( 2315:i 2305:i 2301:a 2295:i 2284:a 2246:( 2225:( 2200:( 2179:( 2173:i 2160:( 2127:z 2119:i 2116:2 2113:= 2110:] 2105:y 2097:, 2092:x 2084:[ 1988:x 1980:i 1974:= 1969:) 1963:0 1958:i 1948:i 1940:0 1934:( 1929:= 1924:) 1918:0 1913:i 1906:i 1898:0 1892:( 1882:) 1876:1 1868:0 1861:0 1856:1 1850:( 1845:= 1840:y 1827:z 1797:y 1789:i 1783:= 1778:) 1772:0 1767:1 1757:1 1752:0 1746:( 1741:= 1736:) 1730:0 1725:1 1718:1 1713:0 1707:( 1697:) 1691:1 1683:0 1676:0 1671:1 1665:( 1660:= 1655:x 1642:z 1612:x 1604:i 1601:= 1596:) 1590:0 1585:i 1578:i 1573:0 1567:( 1562:= 1557:) 1551:1 1543:0 1536:0 1531:1 1525:( 1515:) 1509:0 1504:i 1497:i 1489:0 1483:( 1478:= 1473:z 1460:y 1430:z 1422:i 1416:= 1411:) 1405:i 1400:0 1393:0 1388:i 1379:( 1374:= 1369:) 1363:0 1358:1 1351:1 1346:0 1340:( 1330:) 1324:0 1319:i 1312:i 1304:0 1298:( 1293:= 1288:x 1275:y 1245:y 1237:i 1234:= 1229:) 1223:0 1218:1 1211:1 1203:0 1197:( 1192:= 1187:) 1181:1 1173:0 1166:0 1161:1 1155:( 1145:) 1139:0 1134:1 1127:1 1122:0 1116:( 1111:= 1106:z 1093:x 1063:) 1057:1 1049:0 1042:0 1037:1 1031:( 1026:i 1023:= 1018:z 1010:i 1007:= 1002:) 996:0 991:i 984:i 976:0 970:( 960:) 954:0 949:1 942:1 937:0 931:( 926:= 921:y 908:x 878:2 869:2 849:3 841:z 838:+ 833:2 825:y 822:+ 817:1 809:x 806:+ 801:0 793:t 787:} 784:z 781:, 778:y 775:, 772:x 769:, 766:t 763:{ 751:C 720:3 712:, 707:1 672:] 666:0 661:i 654:i 646:0 640:[ 635:= 630:2 580:3 572:, 567:1 536:0 530:t 504:2 462:R 458:3 454:2 450:1 382:( 351:. 344:] 301:. 196:. 95:: 52:.

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