On Sun, Jun 14, 2020 at 3:54 AM smitra <smi...@zonnet.nl> wrote:
> On 13-06-2020 02:57, Bruce Kellett wrote:
> >
> >> My point is that your arguments are
> >> totally flawed. You stated that a free field theory is analogous to
> >> a set of independent harmonic oscillators in real space, which is
> >> nonsense as they are coupled via the (nabla phi)^2 term, it's only in
> k-space
> >> that you have independent oscillators.
> >
> > Momentum space and position space are related by a Fourier transform.
> > Are you claiming that a Fourier transform violates the conservation
> > laws?
>
> By taking the Fourier transform to decouple the osscilators, you are no
> longer considering the local energy. Your argument was that because a
> free field theory is basically the same as a collection of
> noninteracting SHOs, the fact that energy is conserved means that there
> cannot be local fluctuations. But each such independent SHO in k-space
> isn't localized in real space. In real space the oscillators are coupled
> by the (nabla phi)^2 term and local energy momentum conservation does
> invilve a transfer of energy and momentum from one oscillator to
> another.
>
There seems to be something wrong with this. QFT is equivalent to a picture
in which there are independent SHOs at every point in space. If these are
all in their lowest energy state, then the field is in its lowest energy
state. Excitation of one or more of the oscillators corresponds to the
presence of more energy (and particles). The higher energy configuration
can be analysed in terms of the modes, which are the plane waves of
determinate wavelength. If you start with a smooth field with all the
oscillators in their ground state, this lowest energy configuration
persists until some energy is added from somewhere.
A simple analogy can be given in terms of the surface of a body of water,
such as a lake or the sea. If there is a "mirror calm", the surface is
completely smooth, and we have the state of lowest energy. This will
persist until some energy is added, say from the wind. This can cause
ripples which, once formed, will persist in the absence of a driving force
until friction dissipates the energy as heat. Adding more energy results in
bigger waves, until we can have the massive energy input that is seen in
storms.
The lesson I am trying to draw is that ripples on the surface of a lake
(waves) cannot arise spontaneously -- they require some energy input, say
from the coupling of the wind with the surface of the water. The same is
true in inflation. You start off with a uniformly smooth (classical)
inflaton field, with the same energy everywhere. Variations in the field,
or variations in local energy density, cannot arise spontaneously, they
require some energy input from outside. Since quantum fluctuations are
assumed to be internal to the field, they cannot add this required energy,
so the variations in energy density cannot arise from quantum fluctuations.
Quantum fluctuations, insofar as these exist as disconnected Feynman loops,
do not have any energy, and do not contribute to any physical effects.
In the Wikipedia article on quantum fluctuations, these fluctuations are
said to be a temporary change in the amount of energy at a point in space
as explained in the HUP. This is, of course, nonsense, since the
uncertainty principle does not suggest any such thing. The time-energy form
of the UP:
Delta-t*Delta-E >= hbar/2,
is an inequality. The normal story is that this allows a particle to
"borrow" and amount of energy Delta-E from the vacuum provided it pays this
amount back in the time allowed by the UP. Unfortunately, the inequality is
in the wrong direction for this to make sense. If we "borrow" and amount of
energy Delta-E, the the UP says that the uncertainty in time (the time
within which the energy must be paid back) is
Delta-t >= hbar/(2*Delta-E).
Which means that Delta-t is unbounded above -- it can take on any
arbitrarily large value and still be consistent with the UP. If true, this
would mean that energy conservation is impossible (since the vacuum could
randomly fluctuate into states of arbitrarily high energy that persisted
for arbitrarily long times.) This is contrary to observation, so one of the
input assumptions must be wrong. The only relevant input
assumption (apart from the UP) is that "borrowing" energy from the vacuum
makes sense. Since such an assumption leads to results in manifest
contradiction with experience, the assumption must be wrong.
Consequently, the usual mythology about quantum fluctuations is not true.
So quantum fluctuations cannot cause spatial variations in the field.
>
> False, this doesn't follow from the above. Also it the opposite has been
> demonstrated in a massive body of literature on this subject to which
> thousands of experts who are all extremely well versed in QFT have
> contributed to.
>
Can you point me to some recent key papers?
Bruce
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