Re: Reality of E&M (was: wave momentum)

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding James McLean's comments:
> Although I'm inclined to agree, let me play devil's advocate.  What your
> message mostly says is that it is possible, but extremely inconvenient,
> to avoid the field degrees of freedom.  I don't suppose that nature is
> required to be convenient.

This is true.  But a similar construction can be made for *any*
interacting system where some of the system's degrees of freedom can be
"integrated out".  When the equation of motion for a given degree of
freedom is linear then it can be formally solved to describe the time
dependence of that degree of freedom in terms of integrals over the
entire past history of the other degrees of freedom with which it
interacts.  This formal expression can be substituted into the other
equations of motion for all the other degrees of freedom and the
resulting equations of motion become complicated nonlocal integro-
differential equations involving only the remaining degrees of freedom
without any reference to the formally solved ones (except for possibly
reference to their initial conditions).  Similarly, the "solved"
expressions for the "solvable" degrees of freedom can be subsituted into
the original Lagrangian resulting in the Lagrangian taking on a nonlocal
character but depending only on the remaining degrees of freedom.  The
extremely nonlocal character of the theory (involving integro-
differential equations or coupled sets of infinite order dif-eqs.)
betrays the fact that the formally appearing degrees of freedom are not
all of the whole story.  Tha system simplifys tremendously when the
theory is written in terms of all of the legitimate dynamical degrees of
freedom.  In that case the theory becomes local and and all of the
equations of motion for all of the degrees of freedom are 2nd order (in
the Lagrangian formulation, and 1st order in the Hamiltonian
formulation).  Such is the situation for classical electrodynamics when
the EM degrees of freedom are integrated out.

To see an example of what I've been talking about, consider the very
simple system of 2 masses attached to the ends of an ideal spring and
the spring-mass system is allowed to move frictionlessly along only the
one dimension parallel to the spring (whose equilibrium length is L and
whose spring constant is k).  In this case the Lagrangian is simply:

L= (1/2)*m_1*(dx_1/dt)^2 + (1/2)*m_2*(dx_2/dt)^2 -(k/2)*(x_2 - x_1 - L)^2

The E-L equations of motion are:

m_1*(d^2x_1/dt^2) = -k*(x_1 + L - x_2) and
m_2*(d^2x_2/dt^2) = -k*(x_2 - x_1 - L)  .

Because it is a *linear* dif-eq the second of these equations can be
formally solved for x_2(t) as an integral functional of the entire past
history of x_1 as well as of the initial conditions for x_2 itself.  This
formal solution for x_2 can be substituted into the first E-L equation
which determines x_1.  It can also be substituted in for x_2 and its
derivative can be substituted in for dx_2/dt where they appear in the
Lagrangian.  The resulting Lagrangian is nonlocal and it generates a
nonlocal integro-differential equation for x_1.  But no reference is made
in the new theory to x_2 (except for maybe its initial contitions).  Thus
a system with intrinsically 2 degrees of freedom can be written as a much
more complicated system involving only 1 degree of freedom.  We recognize
the 1-degree-of-freedom-theory for what it is as really a
2-degree-of-freedom system because when the second degree of freedom is
reintroduced the system of EL-equations simplifies to the usual pair of
2nd order dif-eqs, and the Lagrangian becomes a simple local function of
its coodinates evaluated at the current local time.

Would you consider the complicated 1-degree-of-freedom system as more
real than the equivalent 2-degree theory?  I wouldn't.  I prefer to
consider *both* masses at each end of the spring as equally real.

> So are you basically saying that your notion of reality is based on
> obtaining simpler equations, at the expense of having more ethereal
> things (pun intended) be considered 'real'.

I guess so, maybe?

>                                             Or do you have a deeper
> reason to lump electric fields into the same category as, say, my desk.

Well, most of what makes your desk is mostly artifacts of electric fields
and their interactions with electrons and nuclei.  The only non-EM aspect
of you desk is contained in the details of the binding of the quarks and
nucleons in each of the atomic nuclei of your desk.

David Bowman
dbowman@georgetowncollege.edu