Numeric approach to modular operators of free, massive bosons and fermions.

My main research at the University of Leipzig is on modular operators at the one-particle level for local subalgebras of linear quantum fields.

Free, massive bosons on a double cone

In a first part of the project, we developed a new numerical scheme to approximate the one-particle modular operator. We implemented an algorithm for the free field of a massive boson (in the vacuum state) on a double cone region of (1+1) and (1+3)-dimensional Minkowski spacetime. The algorithm discretizes time-0 data of fields and field momenta in position space for a double cone centred at the coordinate origin. It is known that one component of the massless modular generator is a multiplication operator, but our results strongly suggest that this expression does no longer hold in the massive case for which it depends on the mass and is not a multiplication operator in general (Bostelmann et al., 2023).

Small mass corrections to the modular operator of free, massless fermions on a double cone

Taking a very similar approach now for free (Majorana) fermions in two dimensions, the modular Hamiltonian is again well-known as long as the fermions are massless. Using a perturbative approach, we computed first-order corrections for free fermions with a small mass. We found that the mass corrections to the modular Hamiltonian also include non-local terms, some of which were previously unknown (Cadamuro et al., 2023).

Free, massive fermions on a double cone in two dimensions

The extension of the numerical approach to the vacuum sector of free, massive, (Majorana) fermions in Minkowski spacetime requires some modifications to the previous methods. Although the modular generator can be brought into a block form like in the bosonic case, but here the blocks act on the components of time-0 spinor data. We are currently finishing a paper draft on the new results.

We present a numerical approximation scheme for the Tomita-Takesaki modular operator of local subalgebras in linear quantum fields, working at one-particle level. This is applied to the local subspaces for double cones in the vacuum sector of a massive scalar free field in (1 + 1)- and (3 + 1)-dimensional Minkowski spacetime, using a discretization of time-0 data in position space. In the case of a wedge region, one component of the modular generator is well-known to be a mass-independent multiplication operator; our results strongly suggest that for the double cone, the corresponding component is still at least close to a multiplication operator, but that it is dependent on mass and angular momentum.

Modular Hamiltonian for fermions of small mass

Daniela Cadamuro, Markus B. Fröb, and Christoph Minz

We consider the algebra of massive fermions restricted to a diamond in two-dimensional Minkowski spacetime, and in the Minkowski vacuum state. While the massless modular Hamiltonian is known for this setting, the derivation of the massive one is an open problem. We compute the small-mass corrections to the modular Hamiltonian in a perturbative approach, finding some terms which were previously overlooked. Our approach can in principle be extended to all orders in the mass, even though it becomes computationally challenging.