When you calculate the path and speed of a photon through a block of glass, you must do a calculation at the boundaries of the glass block, because the conditions change (ie glass has a higher refractive index than air), and this affects the speed of the photon. It does not matter whether you are using classical optics or

infinite-dimensional Hilbert spaces to do the calculation.

You are mixing classical and quantum concepts here. If you want to talk of Maxwell's equations, then forget the word "photon".

The fact that QED is able to unify the behaviour of photons & electrons makes it better than classical physics; maybe one day we will have a theory that includes gravity too, and that will be even better than QED.

But the existence of a "better" theory does not necessarily render the old theory invalid. The new theory is only considered valid because in the "classical limit", it predicts the same results as the experiments which had already been successfully explained by the classical theory.

So in selecting a method to solve a particular problem, it is important to know how accurate an answer do you need?

All design calculations are approximations; will the approximations of a given method yield an answer which is "good enough" for the problem at hand? This means you must understand the domain of applicability of the different theories:

- Where they will give
**essentially identical **answers

- Where they will give answers that are "
**close enough**", and you can build a prototype and tweak it to correct the discrepancies that occur because real objects are never *exactly* the same as even the *best* theoretical models.

- Where they give
**divergent **answers, and you must use the more complex theory (but you still need to build the prototype!).

In cases 1 & 2, the selection is essentially made on pragmatic grounds like simplicity and accuracy (the proverbial "back of the envelope calculation" may be all you can afford in the middle of a discussion).

For myself, my maths classes did not go beyond eigenvectors on finite dimensional vector spaces, so I don't have all the mathematical tools needed for determining the eigenvalues of infinite-dimensional Hilbert spaces.

However, I would argue that you do not need these advanced mathematical tools to calculate the path and speed of a photon through a glass block, because this is in the domain of classical optics. The answers are "close enough", provided the size of the block of glass is much

*larger* than the wavelength of light, and the size of the atoms in the glass is far

*smaller *than the wavelength of the light. You just need to recognize that the classical term "a ray of light" is an approximation to "a stream of photons" in quantum theory.

Undoubtedly, a grand unified theory which includes gravity (or even a small extension of current theories which is able to explain why calculations on Feynman diagrams work

*so well*) will require additional, even more complex mathematical tools in your toolkit.

You don't always need a sledgehammer to crack a nut...