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?