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Re: Imaginary reality



I wrote:
>
> An imaginary piece to the index of refraction is an _imperfect_ shorthand
> for representing absorption. It is easy to see how it produces an
> imaginary part to the wavenumber (k), which leads to a wavefunction that
> decays in magnitude as it goes along.
>
> It is also easy to see that this representation cannot possibly be correct
> in detail. It suggests that the equation of motion is non-unitary. It
> doesn't conserve phase space. It violates the fluctuation-dissipation
> theorem. It violates the 2nd law of thermodynamics. It violates the
> uncertainty principle. But if you don't look too closely, you might not
> notice the violations.
>

Then at 08:10 PM 4/22/00 -0500, Jack Uretsky wrote:
>
Huh? The index is complex if the wave is traveling in an absorbing
medium. What's to conserve?

Surely you're not suggesting that absorbing media should be exempt from the
2nd law of thermodynamics. Things that don't conserve phase space
more-or-less automatically violate the 2nd law.

I don't know what John meant by his caveat "in detail".

In this case "in detail" means including thermal fluctuations and quantum
fluctuations.

1) Given a real (i.e. noncomplex) index of refraction, you can write a
self-consistent sensible description of propagation through the
nonabsorbing medium.

2) If you want to describe an absorbing medium it does _not_ suffice to add
an imaginary part to the index. You must also add some sort of source term
to represent the injection of thermal and/or quantum noise from the heat bath.

========

The same caveat applies to ordinary elementary electric circuits, although
if you think in terms of reactance the convention is reversed: the
energy-conserving elements like inductors and capacitors have been assigned
to the pure-imaginary part of the reactance, while the dissipative elements
have been assigned to the real part of the reactance.

To make an explicit analogy between index of refraction and RLC circuits:

1) Given real (i.e. noncomplex) inductances and capacitances, you can write
a self-consistent sensible description of an undamped LC filter.

2) If you want to described a damped filter, it does _not_ suffice to add
an imaginary part to the capacitance or inductance (i.e. a real part to the
reactance). You must also add some sort of source term to represent the
injection of thermal and/or quantum noise from the heat bath.

==============

Bottom line: In general, if you have a real-number formula that describes
a piece of good physics, you cannot assume that analytically continuing the
formula to complex numbers also represents good physics.