I appologize if this topic has been delt with here before,though I
cannot find a satisfactory elementary definition in the google
archives.
What does it mean to say a field has " an infinite degrees of
freedom"?
I will include a small passage from A.O. Barut's " Electrodynamics
and classical theory of fields and particles" to start.
" The infinite number of degrees of fredom of the field must be
described by continuous indices. Instead of the coordiates q_1,q_2 ...
,the dynamicalvariables of the field will be a set of functions
psi^a(X,t), a=1,.....N where (X,t) are now parameter which,together
with a,label the degrees of freedom of the system."
Now in classical mechanics I understand that the number of
variables in the Lagrangian of the system is not synonomous with the
number of degrees of freedom because one can come up with different
configuation space variables. The electromagnetic field in vacuum can
be described by either six functions(three of E and 3 of B) or 4
function A^u which are not unique. The free electromagnetic field may
be decribed by several different Lagrangians all with a finite number
of field variables e.g. E and B, or A^u and by a finite number of
field equations. So where does the "infinite number of degrees of
freedom com from? Cann someone deliniate exactly what the difference
is in the definition of "degrees of freedom" as it pertains to the
classical point particles formulation and classical field theory?

(jack) wrote in message . com...

Cann someone deliniate exactly what the difference
is in the definition of "degrees of freedom" as it pertains to the
classical point particles formulation and classical field theory?


A single particle moving in three dimensions is specified at any time
by 6 real values (position, momentum), hence 3 degrees of freedom. A
field is specified at any time by infinitely many real values (it's
values at every point in space), hence infinite degrees of freedom.
The space of all real functions constitutes a (rather nasty, I have
been told) vector space. In QM the wavefunction isn't actually an
element of that space but of a much nicer space in which you don't
need the uncountable number of reals you need for a full blown
function, but instead we divide out the subspace of all functions with
integral zero (for example the function that is zero everywhere except
at finitely many locations were it takes finite values), and we get a
Hilbert Space with countable infinite dimensionality.
When text books speak of infinitely many degrees of freedom what they
usually think of is some kind of continuum limit.
You start with a linked chain, each member of the chain with a
velocity and a position (2 degres of freedom), take the limit of
infinitely many memebers and you get a continous string (Field on the
Real line) with infinit degrees of freedom, the precise mathematical
situation is often rather subtle though.

c

frank.

jack wrote:

What does it mean to say a field has " an infinite degrees of
freedom"?

it means that the field is an arbitrary element of an infinite-dimensional
manifold, and hence needs infinitely many real numbers for its precise
description (by means of continuous operations).

Now in classical mechanics I understand that the number of
variables in the Lagrangian of the system is not synonomous with the
number of degrees of freedom because one can come up with different
configuation space variables.

Not if you don't allow constraints. If you do allow constraints,
you must subtract the number of degrees of freedoms fixed by the
constraints - then the result is description invariant.

The underlying mathematics is the implicit function theorem,
which gives nondegeneracy conditions under which a n-dimensional manifold
constrained by k conditions results in a (n-k)-dimensional manifold.
(Think of solving systems of linear equations, or of intersecting
3D surfaces. Exclude degenerate situations.)


Arnold Neumaier