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TitleVacuum energy in (2+1)-dimensional quantum field theory on curved spaces
AuthorWallis, Lucas Samuel
ContributorsWiseman, Toby; Science and Technology Facilities Council (Great Britain)

Relativistic quantum degrees of freedom in their vacuum state endow geometric backgrounds with an energy, as demonstrated by the Casimir Effect. We explore the vacuum energy (or free energy at finite temperature) of (2+1)-dimensional ultrastatic relativistic quantum field theories as a functional of their spatial geometry. These theories have physical realisations as, for example, the low-energy effective description of the electronic structure of graphene: four free massless Dirac fermions. We define a UV-finite unambiguous measure of free energy for these setups: the free energy difference. We compute it for the free scalar with curvature coupling and free Dirac fermion using heat kernel methods, deriving analytic expressions for perturbative and long-wavelength deformations of maximally-symmetric two-spaces (namely the plane and the round sphere) and, using a novel numerical approach, highly-accurate estimates in the case of large (axisymmetric) deformations to the sphere. We find that for these theories, as with holographic conformal field theories (CFTs) dual to vacuum Einstein gravity with a negative cosmological constant, it is universally negative for non-trivial deformations of maximally-symmetric two-spaces and can be made arbitrarily negative as the geometry becomes singular. In fact, we find that the differenced heat kernel has a definite sign — a much stronger result. We also observe a qualitative similarity between the (appropriately normalised) vacuum energies of a conformal scalar, massless Dirac fermion and holographic CFT on deformations of the two-sphere, and a remarkably close quantitive agreement between the latter two — very dissimilar in nature — theories. Finally, we show vacuum energy negativity for all perturbative deformations to Poincaré-invariant, power-counting-renormalisable theories on the plane. Our results indicate that relativistic quantum degrees of freedom universally disfavour smooth geometries and we note this effect has the potential to be measured experimentally.

Open Access

TypeThesis or dissertation; Doctoral; Doctor of Philosophy (PhD)
RightsCreative Commons Attribution NonCommercial Licence
PublisherPhysics, Imperial College London