I feel like I’m trying to watch — and fully understand — an episode of Heroes after missing the first three years of the show.

Dr. Susskind’s intent with this class is to impart to ordinary mortals a deeper understanding of elementary particles. While his goal has been to explain it clearly to all of us, he recognizes the necessity of some math — and not just high school math — that will inevitably leave some people behind.

I am struggling not to be one of those left behind.

I pursued engineering in college (before dropping out), took several courses in calculus, physics, chemistry and so forth — so some of this higher math is not new to me. But that was such a long time ago, and those logarithms, sines, cosines, matrix algebra, integration and derivatives and so forth are now vague. I’ve been skimming various textbooks, trying to refresh my memory, but it’s not happening fast enough.

We’re now trying to understand the quantum field. Essentially, how do we represent mathematically what is going on at the particle level? How do we formulate equations that will allow us to make accurate predictions? Imagine a billiard table cluttered with balls, and you want to know what the result will be if one ball collides with another. Assuming there are numerous ricochets, that can get complicated very fast. And that’s just classical mechanics.

Now suppose we add “quantum” characteristics to the problem: the balls are all in motion, and they’re behaving like waves as well as balls, and the table itself is fraught with waves, and balls can be created and/or annihilated as a result of collisions, and some balls attract each other and some repel, and oh-by-the-way all the balls are nebulous, so you can’t really tell for sure where they’re at and how fast they’re going.

The mathematics quickly takes on a statistical nature and becomes very very complex.

The question for me becomes (using my earlier analogy), if I’m to understand this episode of Heroes, how many previous episodes must I watch — and which ones? Well, right now we’re studying creation and annihilation operators (formulated as “bra” and “ket” pairs) modeled on a simple harmonic oscillator. Why a simple harmonic oscillator? Well, it simplifies the math (avoiding infinities) and makes for a good model of all the vibrations occurring among the particles. We’re also getting into exponentials, with a growing dependency on imaginary numbers, where “i” is the square root of -1. So I’ve been trying to refresh my memory on imaginary numbers (Penrose gives a good introduction to them in The Road to Reality), on harmonic oscillators, matrix algebra, angular velocity, acceleration and momentum, vectors — and it’s just taking too long.

I’ve tried reading wikipedia articles on various topics, thinking this might be a faster road to grokdom than the textbook route. But there seems to be no shortcuts to rewiring your brain to comprehend all this math. You simply need to put in the time, changing your gray matter bit by bit.

It reminds me of the time I read Kafka’s Metamorphosis in German. I was constantly stopping to look words up, and by the time I’d finished a sentence, I’d forgotten some of the words and would need to look them up again.

At the start of the elementary particles class, Professor Susskind posed the question, Why is this subject so hard to teach?  His conclusion was that there is just so very much to learn.  I would add to that the difficulty of the concepts, and the radical paradigm shift required to understand words we thought were familiar but are suddenly foreign

Energy.  Waves.  Work.  Units.  Dimensions.  We encounter these words in everyday life, and their meaning is easy enough to grasp.  But when these same words are used in the quantum world, they turn squirrelly.  A discussion of the Planck Constant will help show what I mean.

Constants are usually simplicity itself.  The speed of light, for instance, is just a speed.  It’s fast, true, but easy to understand.  Even Avogadro’s number can be easily understood as the number of gas molecules in a given volume at a fixed pressure and temperature.  But then you encounter the Planck Constant.  What the heck is it?  Sure, you can say things like, It’s the granularity of energy in the universe.  Or it’s the proportionality constant between the energy of a photon and the frequency of its electromagnetic wave.  I visualize an egg timer that you’re winding up.  Each little ratchet sound is the smallest unit of time you can set.  The Planck Constant is a ratchet for energy, defining the smallest unit of energy you can add or subtract from a photon.

Sort of.

Because the Planck Constant doesn’t have units of energy.  Rather, it’s joule seconds.  The value is 6.626×10^?34 joule seconds (J·s).  (One joule roughly equals the energy required to lift a small apple one meter upwards.)  Or you could use electron volt seconds (eV-s), or erg seconds (erg-s).  The important point is that it’s the product of energy and time (energy * time), which is much different than energy/time, which would be a rate.  In fact, work = energy * time, so the Planck Constant has the dimensions of work.

Or the dimensions of angular momentum — which is momentum multiplied by distance. Yes, in particle physics you’re on a chameleonic adventure as units and dimensions get converted all over the place in the pursuit of answers. All of this gets very confusing, making it hard to achieve or maintain a clear understanding of the essence of the Planck Constant.

Even if we grasp that the Planck Constant has the dimensions of energy * time, what exactly is the “energy” we’re talking about?  In the classical world, when we think of energy, we see a glowing light bulb, or a power plant, or a windmill — something where work is being done to create energy.  But energy as it applies to the photon is a bit different.  For one, there’s no work being done to maintain the energy.  A photon’s energy is somehow locked into its propogation through space and persists undiluted as long as it doesn’t interact with matter.  A photon could travel for 12 billion light years, traversing the universe, and still have the energy it started out with.  Of course, not all photons have the same energy — and it has nothing to do with their speed. After all, they all travel at the same light speed.  Rather, it has to do with their wavelength.  The shorter the wavelength, the greater their energy.  This isn’t too hard to envision — until you try to visualize the actual “waves.”  The waves are electromagnetic in nature — an electrical component meanders in one plane while a magnetic component meanders in another plane at right angles to the electrical.  Even this statement oversimplifies the reality.  It’s just too complicated to easily grasp.

Which brings us to an old joke.  Two guys in a car drive across railroad tracks.  One of them says, “A train just went by.”  The other asks, “How do you know that?” The first says, “I saw the tracks.”  Well, that brings up an interesting question about the photon.  Does it leave a track?  When we talk about a photon’s electromagnetic wave, does it extend along the photon’s path through space, persisting behind it like a boat’s wake?  Or does it exist only in the present moment, and the only reason we call it a wave is because a graph of its electromagnetic state over time resembles a wave?  To imagine that the photon leaves a wake behind it, or that it somehow resembles a plucked guitar string that reverberates all along its path, is I think erroneous.  The photon and its associated electromagnetic “wave” exist only in the now, leaving no trace in the ether behind.

Then again, I could be wrong.

Energy.  Waves.  Work.  Units.  Dimensions.  Each of these words is easy to grasp in the everyday world, but in the quantum world they turn squirrelly as nuts.