Ingemar Bengtsson

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.

The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension d remains largely open. In a previous publication we gave a conjectural algorithm for how to construct a SIC if d = n2 + 3 = p, a prime number. Perhaps the most surprising number-theoretical aspect of that algorithm is the appearance of Stark units in a key role: a single Stark unit from a ray class field extension of a real quadratic field serves as a seed from which the SIC is constructed. The algorithm can be modified to apply to all dimensions d = n2 + 3. Here we focus on the case when d = n2 + 3 = 4p, p prime, for two reasons. First, special measures have to be taken on the Hilbert space side of the problem when the dimension is even. Second, the degrees of the relevant ray class fields are “smooth” in a sense that facilitates exact calculations. As a result the algorithm becomes easier to explain. We give solutions for 17 different dimensions of this form, reaching d = 39 604. Several improvements relative to our previous publication are reported, but we cannot offer a proof that the algorithm works for any dimensions where it has not been tested.

We show that ultra-massive spacetimes exist also in 2 + 1 dimensions with a positive cosmological constant Λ > 0 . They can be created through the collapse of a spherical null dust shell. The exterior of the shell is then a Mess spacetime, that is to say a locally de Sitter spacetime that cannot be obtained as a quotient of de Sitter space.

It is possible to solve the Einstein constraint equations as an evolutionary rather than an elliptic system. Here we consider the Gauss constraint in electrodynamics as a toy model for this. We use a combination of the evolutionary method with the gluing construction to produce initial data for an electromagnetic pulse surrounded by vacuum. It turns out that solving the evolutionary form of the constraint is straightforward, and explicitly yields the desired type of initial data. In contrast, proving the existence of a solution to the same problem within the elliptic setting requires sophisticated arguments based on functional analysis

The notion of SIC-POVMs comes from quantum theory, and they were not on the horizon when I was Lars Brink’s student in the early 1980s. In the summer of 2022, I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension [ ], and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars’ student, and of what his reaction to our construction was.

In Newtonian gravity it is a moot question whether energy should be localized in the field or inside matter. An argument from relativity suggests a compromise in which the contribution from the field in vacuum is positive definite. We show that the same compromise is implied by Noether’s theorem applied to a variational principle for perfect fluids, if we assume Dirichlet boundary conditions on the potential. We then analyse a thought experiment due to Bondi and McCrea that gives a clean example of inductive energy transfer by gravity. Some history of the problem is included.

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension d, there is good evidence that there always exists an aligned SIC in dimension d(d − 2), having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension d(d − 2) that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the d dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.