Field, plane wave and normal oscillator

[ilper@abv.bg]

Hi!
I have a problem with the energy of the plane wave as used in the
method of second quantization of a field (but prior to quantization).


In the method of second quantization a scalar field (or any other)
'f'
is represented as a sum (integral) over plane waves with all possible
wave vectors k (and --k). Than it is shown using Fourier
transformations that the energy of the field is equal to the energy
of
the normal oscillators.


I am wondering how could this be applied to the energy of a field
composed by just one plane wave. Acos(wt+kx).


1. I think that the energy of the plane wave is infinite.
Each plane wave can be replaced by an infinite system of oscillators
which are bound to each other. If we imagine that the wave was
created in one point in the infinite past from an oscillator which
has
delevered energy to his neighbours, it apparently has to consume
continuously energy from a source in order to set in motion the
infinite chain of oscillators.
2. Now this plane wave is represented in the quantization procedure
as
one oscillator (in k-space) with that same frequency 'w'. Than the
energy of the field (the lone plane wave in the case) should be equal
to the energy of that oscillator.
But it is finite.
I hope someone who understands deep the process of second
quantization
does know how to explain this.
Than you in advance.
Ilian