I am broadly interested in theoretical physics, but am especially keen to work on projects involving geometry, gravitation or high energies. Even better when all these topics come together! Below I have listed some topics that I enjoy thinking about.

#### Quantum geometrodynamics

I have worked at the interface between gravitation and high-energy physics during my doctoral research projects. These were based on the assumption that gravitation is fundamentally a quantum mechanical phenomenon, as is the case for all the other known fundamental interactions.

What is special about gravitation, however, is that the techniques used in the quantisation of the other interactions cannot be straightforwardly applied.
These techniques depend on various ways to describe the interactions.
One such description studies the evolution of a system at different between different moments in time (called the *Hamiltonian description*).
If a classical system is described by a Hamiltonian $H$, then the corresponding classical system is described by the *Schrödinger equation* of that system
$$\mathrm{i}\hslash \frac{\partial \mathrm{\Psi}}{\partial t}=H\mathrm{\Psi}.$$
In this equation, $\mathrm{\Psi}$ indicates the probability that the system is in a given state (frequently $\mathrm{\Psi}$ is called the wavefunction of the system).
The important takeaway from the Schrödinger equation is that the change of this state over time is related to the action of the quantised Hamiltonian on that state, and one can track these changes with the use of time as a parameter.

The issue for gravity is that, nowadays, descriptions of gravity are based on Einstein’s theory of general relativity, where time is part of a dynamical object known as spacetime.
That means that changes in space give rise to changes in time, and therefore it is not immediately obvious what the right parameter is to keep track of these changes.
However, what one can do is specify the spatial geometry at a certain moment in time, and see how that geometry evolves as time progresses.
This is called *geometrodynamics*.

In a quantised description, this would mean that the wave function does not depend on time, and instead of the Schrödinger equation one would have the *Wheeler-DeWitt equation*
$$H\mathrm{\Psi}=0.$$
Because of this absence of time, the Wheeler-DeWitt equation is not an evolution equation (like the Schrödinger equation) but a constraint equation, which is at the centre of all sorts of interesting and counter-intuitive questions.

#### Modified theories of gravity

An important aspect of gravity is its omnipresence; all object with an energy feel the gravitational interaction.
In general relativity all things experience gravity equally, irrespective of their location in space and time (this is known as the *equivalence principle*).

There are phenomena that cannot be explained purely by resorting to general relativity, such as the current accelerated expansion of the universe.
It is therefore interesting to see what happens if general relativity is slightly changed, and what that would imply for the motion of gravitating objects.
It turns out that changes to general relativity are quite restricted (a consequence of *Lovelock's theorem*), but one thing that one can do is change gravity's dependence on the equivalence principle.
One thing that one can do in particular is envision a theory where the strength of the gravitational interaction changes with time and space, so that gravitating objects experience gravity differently in different regions of space or at different moments in time.
The strength of gravity then becomes a dynamical object in its own right, and the resulting theory is called a *scalar-tensor theory*.

One interesting application of scalar-tensor theories can be found in primordial cosmology.
One of the most successful explanations of the early universe is *cosmic inflation*, where the universe underwent a phase of accelerated expansion.
This expansion is driven by a scalar field, called the *inflaton field*.
Inflation does not explain the nature of this field, and it would seems strange for the inflaton field to appear exclusively in the early universe.
If, on the other hand, gravity is truly described by a scalar-tensor theory, then it is possible for the inflaton field to fit naturally in the currently known collection of fields.
In that case it could be identified with the Higgs field, known from the standard model of particle physics.

#### Quantum cosmology

The dynamics of the universe are very well described by classical theories of general relativity.
In fact, quantum gravitational effects (if they exist) are not expected to become relevant for energies lower than the *Planck energy*
$${E}_{\mathrm{P}}=\sqrt{\frac{\hslash {c}^{5}}{G}}\approx {10}^{19}\mathrm{GeV}.$$
This energy corresponds to a mass of around 22 micrograms.
That is not much from our point of view, but for quantum mechanical systems it is enormous.
As a result, quantum gravity is not expected to be observable in most situations.

However, the extreme conditions of the very early universe may constitute one of the few environments in which effects from quantum gravity may be detectable at all. It is therefore interesting to study the Wheeler-DeWitt equation in various cosmological scenarios, and to see how it impacts the things that are measured by observations.