Re: [Phys-l] The "why" questions

[John Denker <jsd@av8n.com>]

On 11/30/2010 10:07 AM, Stefan Jeglinski wrote:

> 1. Hand in hand, discussions of KK always are connected to causality. 

Yes.  More generally, anything involving dissipation will 
sooner or later lead to questions about causality.  Ohm's 
law is another example in the same category.

> I haven't in a while tried going thru the detailed explanations of 
> this and haven't here either, so I'm a bit at a disadvantage, but the 
> discussions seem restricted to E&M (or at least most appropriate 
> there). I don't know if that is an artificial restriction or not. Why 
> is there not some KK-like relationship in Lagrangian/Hamiltonian 
> mechanics, for example?

Thanks for the question.  It's an interesting question.
And as a matter of style, it is much nicer for everybody
when there are questions that can be answered, as opposed
to rash assertions that need to be contradicted.

It turns out that the K-K relation is not entirely correct,
not even for E&M, and that's why nobody is in a big hurry
to extend it to other spheres of activity.  You can tell
it's not correct, because 
 -- It violates the first law of thermodynamics, i.e. the
  law of conservation of energy.  The wave just "dissipates"
  and there is no way of knowing where the energy went.
 -- It violates the second law of thermodynamics.  At the
  same time (and for mostly the same reasons) it violates 
  the fluctuation/dissipation theorem, violates Liouville's 
  theorem, violates the unitarity of the equations of 
  motion, violates Heisenberg's uncertainty principle,
  violates the optical theorem....... You get the idea.

  The same can be said of Ohm's law.

You can repair Ohm's law by adding a fluctuation term 
to the equation.  The classic (and classical) analysis 
[reference: Nyquist and Johnson] can be extended to the
Lagrangian/Hamiltonian/quantum regime [reference Yurke
and Denker] whereupon we learn that quantum fluctuations
and thermal fluctuations are just two sides of the same
elephant.

Having done all that, one has an equation that is still
symmetric with respect to time-reversal.  You can then break 
the symmetry -- by choice -- by making some assumptions 
about the initial conditions, especially about the initial 
conditions inside the heat bath.

So this winds up being quite satisfactory:  Given a heat 
bath that can dissipate energy /and entropy/ we can define 
an arrow of time, pointing in the direction of increase of 
entropy.

Since the K-K analysis does not include a heat bath, you 
pretty much know from the get-go that its description of 
dissipation cannot withstand serious scrutiny.  Sometimes
you can get away with ignoring the heat bath and ignoring 
the fluctuations ... but sometimes you can't.  When trying
to answer foundational questions about causation and/or 
the arrow of time, you can't.  Over the years a steady
stream of witch doctors have tried it.  Many have thought
they succeeded, but in fact none of them did.

The same goes for Ohm's law.

Similar remarks apply to the convolution kernel that describes 
an electrical RC filter.  The shape of the kernel is usually
/chosen/ to make the response causal ... but there are other
choices that satisfy the equation equally well, so you cannot 
use the equation to establish an arrow of time.  Meanwhile, 
the whole approach ignores the heat bath and ignores the 
fluctuations, so you know from the get-go that it cannot 
withstand serious scrutiny.