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Solutions to Maxwell's equations shows that the $\vec{E}$ and $\vec{H}$ component are of the same phase, which means they go to maximum and zero together, therefore the intensity of the electromagnetic wave should be 'lumpy'.

My first question is, if I have a fast enough light intensity detector, or say detecting low frequency like radio wave, can I see the fluctuation in the signal of intensity?

I always think it is reasonable to observe that, until I learned in quantum field theory, that telling me electromagnetic interactions obey the phase gauge transform invariance, phase is always relative, absolute phase is not observable.

Why absolute phase is not observable? Of course we can see the absolute phase in alternating current, so I'm confused, what is the quantum field theory trying to imply?


UPDATE:

For the second question, it looks like I'm mixing up the complex amplitude in optics and the wave function in quantum mechanics, complex amplitude $u=e^{i\varphi}$ is actually a representation for $\cos \varphi$, while wave function $\psi=e^{i\varphi}$ doesn't have a meaning because only the $|\psi|^2$ is observable.

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In theory, it should be (lumpy), however:

Even if you could observe the peaks, or nodes and antinodes of the EM vectors of a single photon, its energy would be absorbed by the instrument you employed to observe it. Fast enough doesn't cut it. If the instrument absorbs no energy from the photon, it will also not be detected.

Intensity for light is not just the amplitude of a single photon. It is the superposition average of the intensities (amplitudes) of lots of photons. Even a laser will have limitations depending on spatial and temporal coherence, which is to say, all of the photons emitted will also not go to zero amplitude at exactly the same location or time.

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Your first question (and in part, the second) reveals a possible philosophical problem you have: you think that Maxwell's equations (or any other mathematical expression) is The Absolute Law that the Nature follows. It does not. Instead, we invent models that can describe (to a certain accuracy) how stuff around us behaves. It's not the Nature that obeys our equations, it is our models are adequate enough to describe the Nature. Think about it.

As to the second part of your question - you can observe the phase (or at least the influence of it), and measure it using wavefront sensors such as Shack-Hartmann or Curvature sensor. You can see the phase inhomogeneity when looking on stars at night - they are twinkling because of the atmosphere turbulence.

Consider the wavefront as a complex matrix $U_{out} = A e^{- i \phi }$ with the amplitude $A = |U_{out}|$ in W/m$^2$ and the phase $\phi = tan^{-1} \frac{imag(U_{out})}{real(U_{out})}$. Using something like Shack-Hartmann wavefront sensor, you can estimate the phase, that is, the $\phi$ in $U_{out} = A e^{- i \phi }$.

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  • $\begingroup$ +1 Great answer - especially for the mention of the wavefront sensors. However, for completeness, it might be good to point out that measurements are invariant to the light field's scaling by any global pure phase $e^{i\,\alpha}$. $\endgroup$ – WetSavannaAnimal Sep 30 '15 at 12:18
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In principle, you should be able to detect the nodes and anti-nodes of the light, provided that:

  • you are in a classical limit, with enough photons detected per period to make the quantization irrelevant
  • your “light” is actually an electromagnetic wave of low frequency, low enough for your detector to see the individual bumps.

This ends up being quite close to your example of seeing the phase of the AC current. You can do the same thing with higher frequency waves as long as your electronics is fast enough.

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