John Baez: Fourier transform land ... it's much less crowded there

the four functions:

exp(i(-x -t))        exp(i(x - t))
exp(i(-x +t))        exp(i(x + t))

[...] span a 4-complex-dimensional subspace of the space of *all* complex-valued functions on R.

Here I'm using the *wrong* complex structure - the one where to compute i times f(x), you just multiply using ordinary complex arithmetic. With that definition, I'm sure our four functions are linearly independent.

With John's *clever* complex structure, I get a little confused. Do we really have four independent vectors, or am I seeing double?

exp(-i(wt - kx))

where w is the energy and k is the momentum, as usual. Let me Fourier transform and think of them as delta functions in (w,k) space, living at the following four points:

(w,k) = (1,1)
(w,k) = (1,-1)
(w,k) = (-1,1)
(w,k) = (-1,-1)

Now, there are two interesting complex structures on this vector space.

The *stupid* complex structure is just the obvious one: multiplication by i.

The *clever* complex structure is multiplication by i in the cases where (w,k) lives in the forwards lightcone, and multiplication by -i on the backwards lightcone:

 \    /
  \  /  <------ forwards lightcone (w >= 0)
   \/
   /\
  /  \  <------ backwards lightcone (w <= 0)
 /    \

Now, however, let us turn to the physics. We are studying the wave equation, but we have to decide what we are studying: *real* solutions of the wave equation, or *complex* solutions. The advantage of talking about *real* solutions is that it's it's obvious that we have to do something clever to get the right complex structure: ordinary multiplication by i isn't even available. So let's see what the above has to do with *real* solutions of the wave equation.

First of all, not every function in our little four-dimensional complex vector space corresponds to a real solution of the wave equation! There's a subspace of functions that do, which has *real* dimension four. It has a basis given by these functions:

leftward             rightward
---------            --------
sin(-t - x)           sin(-t + x)
cos(-t - x)           cos(-t + x)

Why?

Well, being quantum field theorists, we go to Fourier transform land to do our serious thinking... it's much less crowded there. If you take the Fourier transform of a real function f(t,x), its value at (w,k) is the complex conjugate of its value at (-w,-k). If we take a function on the lightcone with this property and hit it with the stupid complex structure, it loses this property! But if we hit it with the clever complex structure, it still has this property. QED.

That was supposed to make it all obvious, but if it didn't, let's do an example. Take one of our real solutions of the wave equation, say

f = sin(-t + x)

and write it out using our basis of complex exponentials:

sin(-t + x) = exp(i(-t + x))/2i - exp(-i(-t + x))/2i

Now let's hit it with the clever complex structure, which we call J. The first term is the Fourier transform of a guy living on the forwards lightcone, while the second is the Fourier transform of a guy living on the backwards lightcone, so we multiply the first by i and the second by -i, and we get:

Jf = exp(i(-t + x))/2 + exp(-i(-t + x))/2
   = cos(-t + x) 

That's how it always goes, even in higher dimensions. When we we take a real solution of the wave equation and hit it with the clever complex structure, it stays real. If we start with a sine or cosine wave moving in a given direction, hitting it with the clever complex structure gives a similar wave moving in the same direction, with the same energy and momentum.

We already knew that, actually: on eigenstates of the Hamiltonian, the complex structure is just evolution backwards in time a quarter period. For example, if we evolve

sin(-t + x)

backwards in time by a quarter period, we get

cos(-t + x)

[1] If you complain that a delta function is not a function, let me remind you that this is physics. Either learn to live with inaccuracy, or take "function" to mean "distribution".

  
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