Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier and Lapla
ce transforms. The new methods also yield new perturbative expansions when the integrals cannot be solved analytically. Here, we add new results, for example, on expressing the Laplace transform and its inverse in terms of derivatives. The new methods can be used to express path integrals in terms of functional differentiation, and they also suggest new perturbative expansions in quantum field theory.
We derive new all-purpose methods that involve the Dirac Delta distribution. Some of the new methods use derivatives in the argument of the Dirac Delta. We highlight potential avenues for applications to quantum field theory and we also exhibit a con
nection to the problem of blurring/deblurring in signal processing. We find that blurring, which can be thought of as a result of multi-path evolution, is, in Euclidean quantum field theory without spontaneous symmetry breaking, the strong coupling dual of the usual small coupling expansion in terms of the sum over Feynman graphs.
EPR-type measurements on spatially separated entangled spin qubits allow one, in principle, to detect curvature. Also the entanglement of the vacuum state is affected by curvature. Here, we ask if the curvature of spacetime can be expressed entirely
in terms of the spatial entanglement structure of the vacuum. This would open up the prospect that quantum gravity could be simulated on a quantum computer and that quantum information techniques could be fully employed in the study of quantum gravity.
To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at least in th
e case of the euclidean signature, is the discipline of spectral geometry. It aims at describing curved manifolds in terms of the spectra of their canonical differential operators. As an immediate benefit, this would offer a clean gauge-independent identification of the metrics degrees of freedom in terms of invariants that should be ready to quantize. However, spectral geometry is itself hard and has been plagued by ambiguities. Here, we regularize and break up spectral geometry into small finite-dimensional and therefore manageable steps. We constructively demonstrate that this strategy works at least in two dimensions. We can now calculate the shapes of 2-dimensional objects from their vibrational spectra.
Fields in spacetime could be simultaneously discrete and continuous, in the same way that information can: it has been shown that the amplitudes, phi(x_n), that a field takes at a generic discrete set of points, x_n, can be sufficient to reconstruct
the field phi(x) for all x, namely if there exists a certain type of natural ultraviolet (UV) cutoff in nature, and if the average spacing of the sample points is at the UV cutoff scale. Here, we generalize this information-theoretic framework to spacetimes themselves. We show that samples taken at a generic discrete set of points of a Euclidean-signature spacetime can allow one to reconstruct the shape of that spacetime everywhere, down to the cutoff scale. The resulting methods could be useful in various approaches to quantum gravity.
High temperature superconductors have in common that they consist of parallel planes of copper oxide separated by layers whose composition can vary. Being ceramics, the cuprate superconductors are poor conductors above the transition temperature, T_c
. Below T_c, the parallel Cu-O planes in those materials become superconducting while the layers in between stay poor conductors. Here, we ask to what extent the change in the Casimir energy that arises when the parallel Cu-O layers become superconducting could contribute to the superconducting condensation energy. Our aim here is merely to obtain an order of magnitude estimate. To this end, the material is modelled as consisting below T_c of parallel plasma sheets separated by vacuum and as without a significant Casimir effect above T_c. Due to the close proximity of the Cu-O planes the system is in the regime where the Casimir effect becomes a van der Waals type effect, dominated by contributions from TM surface plasmons propagating along the ab planes. Within this model, the Casimir energy is found to be of the same order of magnitude as the superconducting condensation energy.
We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on
the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe.
