Can a causal filter without phase shifts exist?

When I was studying dispersion of refraction index in semiconductors and dielectrics, my professor tried to explain that if a filter (like a dielectric absorbing some light frequencies, or an electric RC-filter) removes some frequencies, then the remaining ones must be phase shifted to compensate for those frequencies (which are infinitely spread in time as usual monochromatic signals) being subtracted from the whole signal, to preserve causality.

I intuitively understand what he was talking about, but what I'm not sure of is whether his argument is really justified - i.e. whether there can exist a non-trivial filter, which absorbs some frequencies and leaves remaining ones not shifted, but still preserving causality. I can't seem to construct one, but can't prove it doesn't exist as well.

So the question is: how can it be (dis)proved that a causal filter must shift phases of frequencies relative to each other?

Answer

Suppose that a linear filter has impulse response $h(t)$ and frequency response/transfer function $H(f) = \mathcal F [h(t)]$, where $H(f)$ has the property that $H(-f) = H^*(f)$ (conjugacy constraint).

Now, the response of this filter to complex exponential input $x(t) = e^{j2\pi f t}$ is $$y(t) = H(f)e^{j2\pi f t} = |H(f)|e^{j(2\pi f t + \angle H(f))}$$ and if we want this filter to cause no phase shift, it must be that $\angle H(f) = 0$ for all $f$.

How about if, instead of no phase shift, we are willing to allow a fixed constant phase shift for all frequencies? That is, $\angle H(f) = \theta$ for all $f$ is acceptable to us where $\theta$ need not be $0$? The extra latitude does not help very much, because $\angle H(-f) = -\angle H(f)$, and so $\angle H(f)$ cannot have fixed constant value for all $f$ unless that value is $0$.

We conclude that if a filter does not change the phase at all, then $H(f)$ is a real-valued function, and because of the conjugacy constraint, it is also an even function of $f$. But then its Fourier transform $h(t)$ is a an even function of time, and thus the filter cannot be causal (except in trivial cases): if its impulse response is nonzero for any particular $t > 0$, then it is also nonzero for $-t$ (where $-t < 0$).

Note that the filter need not be doing any frequency suppression, that is, we did not need the assumption that some frequencies are "removed" by the filter (as the OP's professor's filter does) to prove the claim that zero phase shift is not possible with a causal filter, frequency suppressor or not.