Let me pose a puzzle. Suppose you are in a completely dark room. You know that at some point in the future, someone will turn on a light in that room for a few minutes, and then turn it off later. Being a mathematically sophisticated person, it occurs to you that you could think about the time dependence of the electric field in the room. It's zero for a while, oscillating (b/c that's what happens when there is light there) for a few minutes, and then zero again. Being clever, you think about Fourier transforming that time dependence, and thinking about all the frequencies in there - the fact that the room right now is dark is actually because of the amazing cancellation of a whole bunch of frequency components! Therefore, you should be able to put on glasses that are frequency-filtering, block out some of those components, and suddenly be able to see in a dark room! Except that totally doesn't work, even in a completely classical world without photons. Why not?
Think about a material placed in a time-varying (say, harmonically varying, because that's what physicists like) electric field. The material responds in some way - electrons rearrange themselves within the material in response to that electric field; if the field is slow enough, atoms or groups of atoms can even shift their positions. The result is a polarization density (electric dipole moment per unit volume) \(\mathbf{P} \equiv \chi_{e}\mathbf{E}\). Here \( \chi_{e}\) is the electric susceptibility (generally a tensor, meaning that \(\mathbf{P}\) and \(\mathbf{E}\) don't have to point in the same direction). The dielectric function of a material is defined \(\epsilon \equiv \epsilon_{0}(1 + \chi_{e})\). In general, the response of the material depends on the frequency \(\omega\) of the electric field, and it can be out of phase with the external electric field. This is described in mathematical shorthand by considering \(\epsilon(\omega)\) to be complex, having real and imaginary components.
The Kramers-Kronig relations are fairly intimidating looking integral expressions that describe relationships that have to be obeyed between the real and imaginary components of \(\epsilon(\omega)\). These relationships come from the fact that \(\mathbf{P}\) now can only depend on \(\mathbf{E}\) in the past, up until now. This restriction of causality, plus the properties of Fourier transforms, are what leads to the K-K integrals. The wikipedia page about this actually has a very nice description here. So, while the math is not something that most people would think of as obvious, the basic idea (electromagnetic fields influence materials in a causal way, and that places constraints on how materials can respond as a function of frequency) is not too surprising.
8 comments:
In the decades since I first learned about the K-K relations (which always seemed disturbingly abstract) I have read many times that the (non-statistical) laws of physics don't know about the arrow of time. Seems like I am missing something.
That helps, thanks! Would you also be able to do the Hilbert Transform at some point?
Don, as far as I can tell, invoking causality here is akin to choosing to keep the retarded solutions to Maxwell's equations while tossing out the advanced solutions. We make the decision to do this because empirically it works, while the equations themselves are time-reversal invariant (leaving aside magnetic materials). It does bother me. Perhaps some more mathematically sophisticated reader can comment on this (and the role of advanced and retarded greens functions in QFT). For what it's worth, there is a nice essay about these issues that doesn't really clear it up for me here.
Anzel, unfortunately I have no particular way of thinking about the Hilbert transform - wikipedia knows considerably more than me about it.
Invoking Kramers-Kronig to solve the Carvallo paradox sounds like a neat idea, but I see a couple of problems with this approach:
- it doesn't cover the period after the pulse has passed.
- the K-K relations are purely temporal: one can simply refute your argument by saying that, after the filter, both E and P have existed forever. No causality problem here!
I suspect the sleight of hand of the paradox occurs at the filter, but I don't know how to expose it.
"Therefore, you should be able to put on glasses that are frequency-filtering, block out some of those components, and suddenly be able to see in a dark room!"
This statement sounds problematic. If we are talking about a Fourier component of a time dependent electric field, we should integrate the signal at a given frequency over all of time. In this case, there will indeed be a non-zero value at all frequencies (for a pulse of field). As soon as we are talking about the time dependence of a given frequency component, we are talking about some sort of short-time Fourier transform, and then the rules of the game are different.
" (non-statistical) laws of physics don't know about the arrow of time"
I feel that it is easier to think about the KK relation simply as the math of the response function - for an impulse that begins at $t=t_0$, the response function is defined to be zero for $R(tt_0)=0$, the analyticity would be in the lower half plane and the KK relation would have an extra minus sign.
The equivalent in classical mechanics would be a force that turns on at a certain time. In this case, the response would happen only after this time. If you time-reverse everything, it would work out just fine, with a different KK relation.
sorry, bottom paragraph should read:
" (non-statistical) laws of physics don't know about the arrow of time"
I feel that it is easier to think about the KK relation simply as the math of the response function - for an impulse that begins at t=t0, the response function R is defined to be zero for times less than t0. This leads to analyticity in the upper half plane and hence the KK relation. If instead we talk about a "finish function F" where an impulse ends at t0 and F is zero for times greater than t0, the analyticity would be in the lower half plane and the KK relation would have an extra minus sign.
The equivalent in classical mechanics would be a force that turns on at a certain time. In this case, the response would happen only after this time. If you time-reverse everything, it would work out just fine, with a different KK relation.
Causality means that there can't be anything before switch on time, which can be achieved by saying that half of the light is an even function and the other half is an odd function, so that both cancel before time zero, and add up afterwards.
If you Fourier transform both components, you see that this is achieved because for every single frequency a certain relation holds between imaginary and real part.
Consequently, the required cancelling for the dark room being dark occurs frequency-wise and filtering out frequencies can not make you see in a dark room.
''This restriction of causality, plus the properties of Fourier transforms, are what leads to the K-K integrals'' What is this restrictions???
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