Abstract:
This thesis is divided into two parts. The first part is a study of
the general problem of vacuum energy or so-called `zero point
fluctuations' of a quantum field on expanding spacetimes and the
interplay between the dilution of energy as a mode expands and the
generation of energy as new modes enter from below the ultraviolet
cutoff. The second deals with the phenomenon of superoscillations
and some of the consequences in quantum theory and cosmology.
Modern theoretical cosmology sits upon the theory of inflation
which assumes that the universe underwent a period of accelerated
expansion sometime in the past. Indeed, a whole new scientific
discipline was born known as high precision observational cosmology
when ground breaking detailed measurements were made in the late
1990's that confirmed some predictions of the theory of inflation.
However, inflation is essentially a classical phenomenon with the
inclusion of the quantum theory relegated to the provision of
initial perturbations of an otherwise homogeneous and isotropic
spacetime usually interpreted as the inexorable quantum fluctuations
of a (classical) scalar field coupled to the metric.
Quantum field theory on curved spacetime, on the other hand, has
some novel features in comparison to it's flat spacetime cousin. The
inclusion of some of these effects into a discussion of the novel
inflationary picture should provide some very interesting and
non-trivial insights to the very early universe and perhaps might
shed some light on the fundamental nature of the gravitational
interaction itself. Usually when studying quantum fields in curved
spacetime and the energetic interaction between gravity and the
quanta one works in the semi-classical picture where gravity remains
a classical field. This is not only because a fully consistent
quantum theory of gravity has not been constructed yet. Indeed,
there should exist a presumably quite extensive regime where the
picture of quantum fields propagating on a classical background
remains a valid approximation. In the first part of this thesis we
study this regime and some of the interesting physics that arises.
Eventually, however, we go one step further than a semi-classical
treatment and investigate the hypothesis that the dynamics of
cosmologically significant spacetimes is provided by the spacetime
dependence of the quantum vacuum energy of a scalar field on
that spacetime. Put more simply, we discuss the possibility that the
tendency for a spacetime to expand and accelerate it's expansion
reduces to the statement that it is vacuum-energetically favorable
to do so. The idea that the gravitational degrees of freedom are
induced in this way is an old one due to Sakharov and is one
represented in this thesis in simplified form and with specific
calculations and examples. We find that such an interpretation is at
least not excluded and, in fact, sits satisfactorily with the ideas
of inflation.
Along the way to our conclusive discussion of the `induced
cosmology' we discuss, after briefly reviewing inflation and quantum
field theory in curved spacetime, the general problem of vacuum
energy in curved spacetime and some simplified models of the quantum
mechanical ground state energy of a collection of harmonic
oscillators on expanding spaces including some discrete models. Our
philosophy throughout will be one of pragmatism; we assume a cutoff
on momenta (or length scales) at an unspecified energy scale and
assume our conclusions hold, if not all the way from the Planck
scale (which would presumably be subject to beyond the standard
model `quantum gravitational' effects), then at least in some
meso-scale between the Hubble scale and the Planck scale. It is
certainly true that a quantum scalar field really is a collection of
independent harmonic oscillators one for each different comoving
length scale (wavelength). The question that we seek to address in
this thesis is ``what of the vacuum energy of those modes associated
with neither the extreme ultraviolet (where ordinary field theory
breaks down) nor the extreme infrared (where ordinary general
relativity is assumed to break down)?". After discussing an infrared
divergence which we find to be present in a larger class of powerlaw
spacetimes than has been previously found, we also implement an
infrared cutoff on energies. An interpretation of the infrared
cutoff as the realization of a local expansion of a patch of an
otherwise flat and very large `ambient' spacetime is attempted and
the corresponding picture of an energetic initiation of inflation is
provided.
The phenomenon of superoscillations for bandlimited functions is the
observation that a function may approximate to arbitrary precision a
plane wave not contained in it's Fourier decomposition. In
particular, it is possible for a function with compact support in
the frequency domain to approximate with arbitrary precision a high
frequency waveform outside of this support on an arbitrary long
interval. This phenomenon has only recently begun to be studied in
the literature and as yet very few quantitative results have been
obtained. In the second part of this thesis we study the energetic
response of a classical and quantum harmonic oscillator driven by a
superoscillating driving force. We find that the oscillator indeed
responds to the `imposter' driving force as if it were real and,
dubbing the response `ghost resonance', we investigate some
consequences in quantum field theory and cosmology.