From time to time I find myself looking back at my class notes from my undergraduate studies, just to brush up on a topic or two (usually when I am taking the graduate class on the subject matter). And I’ve begun to notice a trend while comparing my undergraduate and graduate notes. I’ve gotten lazier.
That is, the notation I use to describe mathematics has gotten simpler. I think the reason for this is because there has been simply more material to write down, and less time for me to do it. I’ve seen professors at least double (sometimes triple) my age move faster with a white board marker then I can move on a treadmill. I have a tough time keeping up. So to keep up with them (aside from nagging them to slow down) I’ve started adopting a shorthand notation.
But unlike the late Dr. Feynman, I’ve not come up with my own short hand notation [1]. Instead I’ve just tried to incorporate what’s known as four-vector notation.
Four Vectors
Four vector notation is the notation of choice for quantum field theory. It allows a great simplification in how much you have to write (once you know the rules).
Let’s start with a simple example. Four vector notation allows me to describe a point in space-time (with respect to some reference frame), take the point:
(ct, x, y, z)
I can write this as:
Well that’s not astonishing in the least bit, I could have just as well labeled the point P.
Let’s take a second example. I can combine a scalar and a vector together in four-vector notation. For instance, if I wanted to describe a particle’s energy and it’s momentum (again, with respect to some reference frame) I could use a four-vector:
We can even go a bit more abstract and use four-vectors as mathematical operators:
Here we have a partial derivative with respect to time and the “del” operator (sometimes referred to as a nabla).
Now suppose I wanted to multiply two four-vectors, how would I do this? The product of two arbitrary four-vectors goes like this:
Notice how A and B have either a super-script or a sub-script in the above equations. In one case we have a contra-variant four-vector (super-script); and in the other we have a co-variant four-vector (sub-script). However, their components are always labeled with super-scripts. Notice how the product of four-vectors A & B is described by a “dot-product like” operation in which their respective components are multiplied together; but the last three are assigned a minus sign.
In fact I can only ever take the product of a contra-variant with a co-variant (nothing else); but the order in which one comes first doesn’t matter, their product is left invariant. I should also point out the name of the game is “summation over repeated index.” This means if I toss a third four-vector into the mix, if it has a different index (sub- or super-script) it’s ignored:
Notice how A & B have index μ and C has index ν. The μ is the “repeated” index, and the four-vector product acts between A & B. I realize this isn’t a true summation because there is a minus sign involved, but that’s just what the process is referred as.
Maxwell’s Equations – The Lazy Way
Now let’s dive into a serious example to really show the power of four-vector notation. And let’s go outside the realm of quantum field theory, instead let’s take Maxwell’s Equations:
With these four equations-and appropriate boundary conditions-I can describe all phenomenon in classical electrodynamics (I chosen to work in Heaviside-Lorentz units as opposed to the standard SI system, this causes the pesky μ’s & ε‘s to drop out. Remember I’m lazy!!).
These are four coupled first order differential equations that relate two vector fields (electric & magnetic). But from the theory of classical electrodynamics I can write these two vector fields as originating from a scalar and a vector potential (note, I did not say potential energy, which is very different from potential):
With this I can actually express Maxwell’s four first order equations as two second order equations:
Of course this is an awful mess when you look at it. Why on earth would anyone want to do this!? There are so many more terms and derivatives all over the place.
But, in physics there is something known as the “Lorentz Condition,” sometimes also called the Lorentz Gauge [2], which says:
(When I put this into Heaviside-Lorentz units the μ & ε again drop away).
Which simplifies the above two equations rather nicely:
Now this is truly enticing, these equations are almost identical! Suppose I made a set of four-vectors:
Notice how the last two are mathematical operators, one is a co-variant and the other is a contra-variant. They are just begging to be multiplied, so let’s do just that:
This is actually a new mathematical operator known as the d’Alembertian Operator, its usually represented by a square, but I don’t know the LaTeX command to make that. =(
But, with this set of four-vectors and the two equations above I can write mankind’s sum knowledge of all electromagnetic theory in one line:
Let’s pause on this for a moment. I think this is really an astonishing miracle that physicists over the years have figured out how to write so much information about the natural world in such a small space (one line)! Some of you might remember the Standard Model Lagrangian, which is conveniently written on a coffee mug should you forget. That coffee mug contains A LOT of information, but it definitely cannot fit on one line, at least with my handwriting (maybe someone someday will come up with some ingenious notation of their own?!).
But, just like that four vector notation has allowed physicists to simplify Maxwell’s Equations (all four of them) in a single concise statement. Talk about saving space on your final exam’s equation sheet! So hopefully you’ve come to appreciate the power of four-vector notation.
Until Next Time,
-Brian
References
[1] Richard Feynman routinely used his own notation for trigonometric functions, logarithms and other common functions in mathematics, he did this because it was simpler & faster for him to write in such a fashion. For more details and other great stories, see Feynman’s own “Surely You’re Joking Mr. Feynman,” W. W. Norton & Company, Inc. 1985.
[2] See for instance D. J. Griffiths, “Introduction to Electrodynamics,” 2nd ed., Prentice-Hall, Inc., 1989.
