Understanding Skew Symmetric Matrices for Physics - A Helpful Guide

[kq6up]

I am a bit dense when it comes to linear algebra for some reason. I am reviewing math to prepare for a physics grad program, and I am using Mary Boas "Mathematical Methods in the Physical Sciences". She presents the idea of a skew symmetric matrix in the problem set rather than in the text. I searched the topic, and found this wiki article: http://en.wikipedia.org/wiki/Skew-symmetric_matrix it seems to be very thorough but very much written for the math types -- not physicist friendly. Can anyone recommend a good article or youtube video that unpacks this idea for physicists? Mary's treatment of most of the material is wonderful and devoid of most mathematician specific jargon and/or symbols. Something to this effect and which covers the topic would be great.

Thanks,
Chris Maness

 

[homeomorphic]

There are several different places where skew-symmetry comes up.

One is that skew-symmetric matrices represent infinitesimal rotations. To understand this, it helps to use your thumb, index finger, and middle finger to make an orthonormal basis for R^3. If you want a rotation, you can rotate this basis around to see its effect on your original standard basis vectors. If you want an infinitesimal rotation, start your basis in the standard position and rotate it a little bit. When we talk about an infinitesimal rotation, we mean each vector gets mapped to its initial velocity vector as it gets rotated.

So, for example, we could have a rotation counter-clockwise around the vertical axis, and look at the derivative at time 0. So, let's say e1 points in the direction of the x-axis, e2 points in the direction of the y-axis, and e3 points up. If we want to know where the infinitesimal rotation maps e1 to, just look at your fingers (i.e. what are the initial velocity vectors of your fingers?). Evidently, e1 maps to e2. Where does e2 go? It maps to -e1. What about e3? Well, e3 is stationary, so his velocity vector is 0. This determines the map. One thing to notice is that everything maps orthogonally to itself. This is equivalent to skew-symmetry, as the wikipedia articles points out. If you look at your fingers, though, you may notice that ei turns in the ej direction exactly opposite to the way ej turns in the ei direction. This is skew-symmetry.

So, that is one way to motivate the importance of skew-symmetric matrices. This way of thinking is especially helpful in differential geometry, which is where I learned it from.

In our example, it was clear that one vector (along the axis of rotation) mapped to 0, and therefore the determinant must be zero. This wouldn't be the case in R^2. This is because rotations in odd-dimensions have axes (eigenvectors of ordinary rotations with eigenvalue 1). There's probably some very cute argument you can do here, but I am too lazy or whatever to call it to mind. I suppose you could invoke the "hairy ball" theorem (which is semi-mathematician-specific, yet somewhat intuitive to the non-mathematician: you can't comb the hairy ball).

http://en.wikipedia.org/wiki/Hairy_ball_theorem

In physics, these infinitesimal rotations have something to do with angular momentum. If you have a path of symmetries, like rotations, in quantum mechanics, that would correspond to some unitary operator. When you take derivatives of those, you should get a (skew-)Hermitian (Hermitian times i => skew-Hermitian) operator, and therefore some sort of observable (I should probably do my mathematician's duty here and say self-adjoint, not Hermitian--oh well). Of course, the universe is a bit sneaky, so it uses SU(2), the double-cover of SO(3), as its symmetries, but the infinitesimal guys (the so-called Lie algebra of these Lie groups) are essentially the same. Incidentally, in three dimensions, you can rotate infinitesimally by taking the cross product with the axis.

I don't know of a good reference for this stuff. Personally, I've just picked it up in my mathematical travels, for example, as I mentioned, in differential geometry of curves and surfaces. Maybe I'll write another post about alternating forms or a comment or two on the spectral theory or whatever.

 

[kq6up]

Thanks for this reference. It is a bit over my head, but I would like to wrestle with it and figure out the details. I will come back to this problem after I get a little more familiarity with linear algebra. Is there something in here that I will need to tackle quantum mechanics?

Thanks,
Chris Maness

 

[homeomorphic]

I don't know about "need", but it could help to understand it better. Quantum mechanics is discussed at many different levels and from different viewpoints, and, as a mathematician, I don't have a good idea of what's standard (plus, my knowledge of it is fairly incomplete, in general).

Understanding this type of stuff could be useful if you wanted to learn GR, as well, considering the differential geometry involved.

Another place where skew-symmetry comes up in physics is with fermions. When two fermions trade places, the state vector for the combined system gets multiplied by -1, so that's another case of skew-symmetry.

Also, alternating forms are kind of secretly behind the scenes in Hamiltonian mechanics (sympletic structure of classical phase spaces).

Probably, you can learn it as it comes up, if you have to.

 

[homeomorphic]

By the way, if you have any interest in differential geometry of curves and surfaces, I'd recommend Elementary Differential Geometry, by O'Neill. He explains the stuff I was talking about in more detail, except in more of a differential geometry context, rather than a linear algebra one.

 

[homeomorphic]

Oh, and I implicitly assumed my rotation was at unit speed and then differentiated it, so if you do figure out the details, that might help clear things up a bit. Otherwise, my mapping is only correct up to multiplication by a constant. If you rotate 5 times as fast, your velocity vectors will be 5 times bigger, so you'd multiply your resulting skew-symmetric matrix by 5 in that case. I swept that under the rug because the directions are what really matters here.

 

[kq6up]

Yes, I am interested in learning GR at some point. I am currently trying to prepare myself for this course:

http://physics.fullerton.edu/~heidi/510.html

I am using Mary Boas' book and I hope it is enough to prepare for it.

Chris Maness

 

[homeomorphic]

Sounds like you're okay for the purposes of that course. I doubt skew-symmetric matrices would be a prominent thing there.