Wikipedia:Reference desk/Archives/Mathematics/2015 March 24

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 24

Pauli Matrices

Defining a vector $\sigma _{i}$ for $i=0,1,2,3$ where $\sigma _{0}$ is the $2\times 2$ identity and the other elements are the pauli matrices. I believe there is a simple form for the rank-4 tensor given by $T_{ijkl}={\frac {1}{2}}\mathrm {tr} \left[\sigma _{i}\sigma _{j}\sigma _{k}\sigma _{l}\right]$ but I can't seem to find it. If anybody is able to point me in the right direction I would be quite grateful.

Since pinged, I could point out the first boxed eqn of Pauli matrices leads directly to, I think, $T_{ijkl}=\delta _{ij}\delta _{kl}-\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}$, which appears to have the right symmetries. Cuzkatzimhut (talk) 13:38, 24 March 2015 (UTC)

I think this would be correct if the indices only cycle over $i=1,2,3$, but the inclusion of the identity as $\sigma _{0}$ makes things more complicated. — Preceding unsigned comment added by 128.40.61.82 (talk) 14:23, 24 March 2015 (UTC)

Apologies, yes, I took only indices 1,2,3. If 3 indices are 0, the expression vanishes. If two, say k,l, it is a delta of the other two. If one, l=0, it is i ε^ijk. Cuzkatzimhut (talk) 15:13, 24 March 2015 (UTC)

In a related question if there is similarly a closed form for the orthogonal matrix $O_{ij}={\frac {1}{2}}\mathrm {tr} \left[\sigma _{i}A\sigma _{j}A^{-1}\right]$ in terms of operations in the 4-space involving the vector $a_{i}={\frac {1}{2}}\mathrm {tr} \left[\sigma _{i}A\right]$ where $A$ is a general invertible complex $2\times 2$ matrix it would be very useful to know.

Thank you. — Preceding unsigned comment added by 128.40.61.82 (talk) 11:42, 24 March 2015 (UTC)

Hmm. I don't know, but I might know the right person to ping. YohanN7 (talk) 12:33, 24 March 2015 (UTC)

So, utilizing the above, you should first take out the inert determinant of A killed by the inverse A in your expression, and supplant it with −|a| the length of your 3-vector a, and the determinant of your now normalized A, which is thus now reduced to the mere Pauli vector a⋅σ. The inverse of A is just the similar Pauli vector with $a_{i}/a^{2}$ instead of $a_{i}$, and using the above expression, $O_{ij}=(-\delta _{ij}+2a_{i}a_{j}/a^{2})$ orthogonal, all right. Cuzkatzimhut (talk) 14:22, 24 March 2015 (UTC)

This I think is correct if we restrict only to the case that A is traceless.

Of course! take the trace out as an identity piece, if you like. Otherwise, you have to do the calculation with a A= bI + i a⋅σ, invert it, etc... In "customary" applications, A is not arbitrary, but a unitary, rotation matrix, whose form is then substantially constrained. Cuzkatzimhut (talk) 15:09, 24 March 2015 (UTC)

I was looking at similarity transformations on a system of spin-${\tfrac {1}{2}}$ particles, and so any spectrum preserving transformation is relevant. For what its worth, writing A= a₀ I + i a⋅σ, I obtained $O_{ij}={\frac {1}{a_{0}^{2}-a^{2}}}\left[\left(a_{0}^{2}+a^{2}\right)\delta _{ij}-2a_{i}a_{j}+2ia_{0}\epsilon _{ijk}a_{k}\right]$. Thanks anyway.

Logarithmic number

Has there been any proof that pi is not a logarithmic number?? (This means a number that can be written as b in a^b = c where a and c are natural numbers and a > 1.) It is known that all non-negative rational numbers are logarithmic, as are many irrational numbers. Georgia guy (talk) 19:37, 24 March 2015 (UTC)

Since no-one yet has answered your question, and I'm uncertain if I've understood it correctly, could you (or someone else) give an example of an irrational number b and two natural numbers a and c that satisfy the equation a^b = c? --NorwegianBlue ^talk 12:27, 25 March 2015 (UTC)

There is a definition of "logaritmic number" at mathworld.wolfram.com. I'm having trouble seeing that that definition and Georgia guy's definition are the same. --NorwegianBlue ^talk 12:38, 25 March 2015 (UTC)

They're not the same. The mathworld definition only includes rational numbers. For an irrational example of Georgia guy's notion, consider $\log _{2}3$. It's irrational, and $2^{\log _{2}3}=3$.--80.109.80.31 (talk) 12:52, 25 March 2015 (UTC)

Thanks! --NorwegianBlue ^talk 13:20, 25 March 2015 (UTC)