My research focuses primarily on the CAUSAL SET THEORY (CST) approach to Quantum Gravity. Closely linked to CST is an interpretation of quantum theory called the QUANTUM MEASURE approach, which allows us to speak of quantum systems without
recourse to external measurements and measuring devices, which are absent in closed systems and of
course in the early universe. You can read about this approach in one of Rafael Sorkin's earliest papers on this subject.
I also have a strong interest in aspects of LORENTZIAN GEOMETRY related to the role of
the CAUSAL STRUCTURE.
My current research includes:
-
- The Benincasa-Dowker
action for causal sets makes it possible to perform the traditional sum-over-histories
quantisation of causal sets. Using the 2d form of this action, we construct a 2
dimensional theory of causal set quantum gravity. Though the sample space of causal sets includes those that are
non-manifold-like, our MCMC simulations of the quantum dynamics show that there is a phase of the
theory in which the expectation values of observables computed correspond to those of a continuum
spacetime, namely flat spacetime. We find a clear phase separation between manifold-like and
non-manifold-like behaviour which suggests that the manifold-likeness is retained after analytic
continuation. Our results are
fully non-perturbative and seem to lie in a different "universality class" from other models of 2d
quantum gravity.
The expectation value of the Action as a function of the temperature $\beta^{-1}$.
At high temperatures one has a manifold-like phase and at low temperatures
a "crystalline
phase".
The results are for small causal sets and it is therefore an important question to ask whether
they survive in the thermodynamic limit of the theory. Investigations on this are
underway.
-
Simulations are being done on the quantum dynamics in the full space of causal
sets. There are several choices for the Markov chain moves, and ours involves finding suitable
pairs in the causal set between which either a causal link can be added or removed. Preliminary
results suggest that for the uniform measure (zero action) the ensemble is
dominated by the so-called
Kleitman-Rothschild(KR) phase, as expected from analytic results in the thermodynamic limit.
Instead, when a "Link" action is used, there phase seems to have a behaviour distinct from that
of the KR causal sets. Work is underway to examine the effect of the 4d BD action on the
ensemble.
- Applying these methods to a decoherence functional or quantum measure based approach to causal set theory.
- A formulation of the quantum measure as a Vector Measure was recently obtained using the GNS
based construction of the Hilbert space from histories. This construction was used to show
that in the quantum pre-measure for simple models of quantum dynamics including a large class of
unitary evolution on a lattice do not extend to a full studying.
measure. It is an open question whether these results are also true of the quantum measure
for particles in the
continuum ala
Geroch.
- For causal sets Sorkin has
suggested the need for a modified form of the extension
theorem. It is an open question whether there is a limited extension theorem for the quantum
vector measure in this case.
- It has been shown that the Hilbert spaces can be obtained from the space of histories (the
sample space) and the quantum measure via a GNS construction of Hilbert spaces. It is an
open and important question how observables natural to the histories framework give rise to
operators in the Hilbert space framework of quantum theory.
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We use an extension of the work of Meyer to find analytic expressions for the abundance of
$k$-element chains in a causal set that is approximated by a small causal diamond. This
gives us a new dimension estimator appropriate to curved spacetime. In addition, we find a new
expression for the scalar curvature in terms of the abundance of $k$-chains, which is
different from that obtained by Benincasa and Dowker. This is work in collaboration with
my students Mriganko Roy and Debdeep Sinha,
-
An analysis of the behaviour of k-element inclusive intervals as a function of k in flat
spacetime appears to give us a good indicator of flatness and hence manifoldlikeness for a
region of a 2d causal set. This gives a quantitative and purely order theoretic measure of a "small
neighbourhood" of an element in a manifoldlike causal set. The current goal is to extend
this analysis to all spacetime dimensions. This is work in collaboration with Lisa Glaser, a
student at the Neil's Bohr institute in Copenhagen.
My current collaborators and co-workers in causal set theory include