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The ordinary Fourier transform of a field is defined as

(5.5) 
The problem with using this definition for a classical scalar
field is that if the Fourier components are given fixed
values the resultant field values become dependent on the
overall size of the box within which the theory is defined. For
example

(5.6) 
which implies

(5.7) 
The size of the box does not affect the integral, except by
turning it into a discrete sum. So to keep (and by
extension all intensive quantities) independent of the box size we
define a modified Fourier transform

(5.8) 
where is the size of the box. This modified transform takes on
the same value regardless of the box size, while the actual
Fourier transform must be rescaled. Note that the units of
are while those of are .
The Fourier transform used by the program is neither of these,
however, but rather the discrete Fourier transform , related
to the usual, continuous, one by

(5.9) 
All physical quantities should be defined in terms of
and the Fourier transform used by the program
should be adjusted accordingly. For example, the initial vacuum
state
becomes
. See section 6.3.2 for more
details.
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