I Mirrors, phase shifts, and position

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I'm confused about something regarding phase shifts but I'm not sure I've pinned down what.

I had always thought of a mirror as introducing a global phase shift of π in the position basis, but I only now realize that this makes no sense: of course -A ⊗ B = A ⊗ -B, so a global phase shift is not specific to any particular observable. This is essentially the only operation that works this way.

Now, suppose you send a photon through a beam splitter, and in the upper path you place a mirror. This effectively introduces a relative phase shift between the two path components. But I cannot think of a way to use a mirror to introduce a relative polarization phase shift (which is normally accomplished with a waveplate). I can put a mirror in one place but not another; I cannot put it "in one polarization but not the other."

Why should the unique device that imparts a global phase shift (i.e., that treats all observables the same) "treat position specially?" I'm not sure my question makes any sense, but hopefully someone recognizes the confusion that's lurking here.
 
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vanhees71

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It's as in classical electrodynamics (Fresnel's Laws): A reflected em. wave gets a phase shift by ##\pi## if it's entering a optically denser medium (i.e., a medium with larger index of refraction like, e.g., from air to glass). For a very good and careful treatment, see the excellent textbook on QT:

B. Schumacher, M. Westmorland, Quantum Processes, Systems, and Information, Cambridge-University Press (2010)
 
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It's as in classical electrodynamics (Fresnel's Laws): A reflected em. wave gets a phase shift by ##\pi## if it's entering a optically denser medium (i.e., a medium with larger index of refraction like, e.g., from air to glass). For a very good and careful treatment, see the excellent textbook on QT:

B. Schumacher, M. Westmorland, Quantum Processes, Systems, and Information, Cambridge-University Press (2010)
Sorry, I just got around to responding. Thanks for the reference, it's very helpful. But my confusion remains.

So the operation of a mirror is to introduce a phase shift of ##\pi## -- in other words, ##|\psi\rangle \mapsto -|\psi\rangle##. What's weird to me is that this operation is not specific to any observable, or indeed, any particle: if we write the universal wave function as ##|\psi\rangle \otimes |\phi_{world}\rangle##, the mirror turns it into ##-|\psi\rangle \otimes |\phi_{world}\rangle = |\psi\rangle \otimes -|\phi_{world}\rangle##. In other words, we can't tell if we phase-shifted the photon's position, its polarization, or indeed, the entire universe's state vector.

Is that correct, and if so, how should I interpret it?
 
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I'm aware that global phase is physically irrelevant, but something about this result still feels weird. I can't put my finger on why though.
 

vanhees71

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Me neither. It's just the solution of Maxwell's equations in linear-response approximation with the correct boundary conditions, and this translates to the quantized radiation field due to the linearity of the equations.
 

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