Activities

2025

1. Conference

   Causal Fermion Systems 2025: New Perspectives in Mathematics and Physics

   Catalina Curceanu, Claudio Dappiaggi, Felix Finster, Niky Kamran, Antonino Marcianò, Claudio Paganini, and Jürgen Tolksdorf

   Moderation, Universität Regensburg, Regensburg, Germany, Oct 2025

   Conference from 06.10 until 10.10. Participation and moderation of a discussion session on 07.10.

   Website
2. Talk

   Modular Hamiltonians of the massless and massive scalar field on the half line

   Christoph Minz and Erik Tonni

   Contributed talk, Institut für Theoretische Physik, Universität Leipzig, Germany, Sep 2025

   Presented on 17.09.

   Abs arXiv Slides Website

   We study the modular Hamiltonian of the free, massive scalar field in the vacuum state over subspace regions of a (1+1)-dimensional half space with a spatial boundary subject to a Robin (mixed) boundary condition, using a recently developed numerical approach for initial data on a time-0 hypersurface (here the real half line). For the subspace region, we consider a single interval at a generic position on the half line. When the interval is adjacent to the boundary with a Neumann or Dirichlet boundary condition, we can use the known result of boundary conformal field theory to test our algorithm. We then obtain new results in presence of a Robin boundary condition (once for the massless and once for the massive regime).
   In a second part, we move the interval subspace away from the boundary. Besides the local contribution, also non-local contributions to the modular Hamiltonian occur, which are mostly concentrated along a certain curve of conjugate points in the massless regime. For large distances to the boundary, the massless modular Hamiltonian (restricted to the subspace) approaches the Hislop-Longo double cone result. In the massive regime, the modular Hamiltonian approaches the results from a previous numerical study of the massive modular Hamiltonian of a double cone in Minkowski spacetime in the large distance limit.
   This talk is based on work in collaboration with Erik Tonni at SISSA, Trieste.

3. Seminar

   A numerical approach to modular Hamiltonians in quantum field theory

   Henning Bostelmann, Daniela Cadamuro, Markus Fröb, Christoph Minz, and Erik Tonni

   Seminar, King’s College London, UK, Sep 2025

   Presented on 15.09.

   Abs Slides

   Modular theory is a tool for von Neumann algebras and for the mathematical formulation and computation of entanglement quantities in quantum field theory. Finding analytic solutions for modular operators is challenging and only very few cases — mostly for conformal fields — are known. After giving a brief introduction to modular operators and some literature suggestions from a perspective motivated by physics, I will introduce a numerical approximation scheme that we have developed in the last years. With this scheme, we can approximate the solutions of modular Hamiltonians for local subalgebras in flat spacetime (with/without boundaries) and linear quantum fields of (massive) scalar bosons and fermions. We tested the numerical method against known results and then apply it to subspace settings for which no analytic solutions are known. Our numerical results for double cones showed that the modular Hamiltonian indeed depends on the mass parameter (and on the boundary parameter) and is ’non-local’ in most cases. These results motivated later analytic computations for fermions in the small mass regime using perturbative techniques.
   The results are from joined projects with Henning Bostelmann, Daniela Cadamuro, Markus Fröb, and Erik Tonni. First numerical results on bosons (arXiv:2209.04681) and the perturbation for fermions of small mass (arXiv:2312.04629) have been published already, while two more publications are in preparation.

4. Seminar

   Local symmetries in partially ordered sets and causal sets

   Christoph Minz

   Invited seminar, online, May 2025

   Presented on 01.05. More information on Hasse diagrams are on the project page below, including the online tool to create diagrams.

   Abs arXiv HTML Slides

   Starting from a graph of sequential growth, we group partially ordered sets (posets) into classes and characterise locally repeating structures (local symmetries) in posets. The aim is to find subsets of posets that can be enumerated using combinatorial arguments more easily. The study of local symmetries applied to causal sets may also help to distinguish between generic posets and those that are suited as discrete representations of spacetime manifolds, as discussed in this talk.
   Alongside, we will see some tools to type-set posets with LaTeX and compose them in a visual editor. I made these tools available via CTAN (to be included in LaTeX distributions and on Overleaf) and on https://c-minz.github.io/projects/poset_diagrams.

2024

1. Seminar

   On the mass dependence of modular operators over double cone regions in QFT

   Henning Bostelmann, Daniela Cadamuro, Markus Fröb, and Christoph Minz

   Invited seminar, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Italy, Nov 2024

   Presented on 19.11.

   Abs arXiv HTML

   Tomita-Takesaki modular theory is a tool in the mathematical formulation of quantum field theory, which is used in the computation of (relative) entropies between subsystems, for example. Finding analytic solutions for modular operators is challenging and only very few cases — mostly for massless fields — are known.
   I will give a brief overview on modular operators and mention some papers that give a good introduction to the topic from a perspective motivated by physics. Then I will show some results obtained with a numerical approximation scheme for the modular operator of local subalgebras in linear quantum fields of massive bosons and fermions. The numerical method is tested with the known results of right wedges in 2-dimensional Minkowski spacetime. Our numerical results for double cones showed that the modular operator indeed depends on the mass parameter (against previous expectations) and is ’non-local’. For fermions with small masses, we also developed a perturbative technique to get first order corrections to the massless operator.
   The results are from joined projects with Henning Bostelmann, Daniela Cadamuro, and Markus Fröb. First numerical results on bosons (arXiv:2209.04681) were published 2023; the analytic computations for fermions with small mass (arXiv:2312.04629) are accepted for publication; and another paper is in final preparation.

2. Journal club

   From classical to quantum fields on discrete spacetimes (causal sets)

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Journal club, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Italy, Oct 2024

   Presented on 30.10.

   Abs arXiv

   I will give a brief review of algebraic classical and quantum field theory on causal sets, a discrete approach to spacetimes in quantum gravity and a method of discretizing quantum fields. First, we review how to obtain causal sets from spacetime manifolds via a statistical Poisson process (sprinkling) and consider different ways to discretize the field equations (mostly following my first publication of my PhD, arXiv:2011.02965). Then, I summarize aspects of quantum field theory on causal sets including the Sorkin-Johnston state and quantization methods (deformation and geometric quantization), see arXiv:2207.05667. Our geometric construction suggests a natural generalization to less linear examples, such as interacting fields, which is subject of an ongoing project.
   The main results are based on my PhD project with Eli Hawkins and Kasia Rejzner.

3. Talk

   Do causal sets have symmetries?

   Christoph Minz

   Contributed talk, Aurum, the ’Gabriele d’Annunzio’ University and ICRANet, Pescara, Italy, Jul 2024

   Seventeenth Marcel Grossmann Meeting (07.07-12.07). Talk presented on 12.07.

   Abs arXiv HTML Slides Website

   Causal sets are locally finite, partially ordered sets (posets), which are considered as discrete models of spacetime. On the one hand, causal sets corresponding to a spacetime manifold are commonly generated with a random process called sprinkling. This process keeps only a discrete set of points of the manifold and their causal relations (loosing the spacetime symmetries in each sprinkle). On the other hand, the main conjecture of causal set theory is that given an ensemble of causal sets there is a corresponding spacetime manifold and the continuum symmetries of it are like all manifold properties ’reconstructible’ from the partial orders of all the causal sets in the ensemble. But most generic finite posets have very few layers (’instances of time’) in contrast to sprinkles with many layers in a sufficently large spacetime region.
   In a recent project, I investigated the automorphism groups of (finite) posets in order to identify and classify their symmetries systematically. The comparison of local symmetries of generic posets (including Kleitmann-Rothschild orders) with sprinkled causal sets may help us to find posets that can serve as discrete spacetime models in causal set theory.
   In the discussion of the results, we use the representation of posets as Hasse diagrams. I developed some simple tools (available for LaTeX and online) that will be explained alongside.

4. Talk

   Mass dependence of modular operators for double cones

   Henning Bostelmann, Daniela Cadamuro, Markus Fröb, and Christoph Minz

   Contributed talk, Strasbourg, France, Jul 2024

   XXI International Congress on Mathematical Physics (01.07-06.07). Talk presented on 05.07.

   Abs arXiv Website

   Tomita-Takesaki modular theory is a central tool in the mathematical formulation of quantum field theory and its application for the computation of (relative) entropies. However, finding analytic solutions for modular operators is challenging and only very few cases — mostly for massless fields — are known.
   In this talk, I present our numerical approximation scheme for the modular operator of local subalgebras in linear quantum fields of massive fermions. The numerical method is tested with the known results of a right wedge in 2-dimensional Minkowski spacetime. We then apply our scheme to one and two double cones on 2-dimensional Minkowski and cylinder spacetimes, where only massless solutions have been found analytically. The mathematical description that underlies the numerical implementation is analogous to our previous work on bosonic fields [arXiv:2209.04681], but now the operators split into blocks acting on the spinor components. We find that the modular operator depends on the mass and that an increase in mass leads to stronger localisation.
   This is a joined project with Henning Bostelmann and Daniela Cadamuro.

5. Journal club

   Mass dependence of modular operators for double cones

   Henning Bostelmann, Daniela Cadamuro, Markus Fröb, and Christoph Minz

   Journal club, Dublin Institute for Advanced Studies, Ireland, May 2024

   Presented on 14.05.

   Abs arXiv

   Tomita-Takesaki modular theory is a tool in the mathematical formulation of quantum field theory. It is applied, for example, in the computation of (relative) entropies between subsystems. However, finding analytic solutions for modular operators is challenging and only very few cases — mostly for massless fields — are known.
   I will give a brief overview on modular operators and mention some papers that give a good introduction to the topic from a perspectives motivated by physics. Then I will show some results from my current numerical project. In this project, we developed a numerical approximation scheme for the modular operator of local subalgebras in linear quantum fields of massive bosons and fermions. The numerical method is tested with the known results of right wedges in 2-dimensional Minkowski spacetime. New numerical results for a double cones showed that the modular operator indeed depends on the mass parameter (against previous expectations). For small masses, we also used a perturbative technique to get first order corrections to the massless operator.
   Further numeric results include fermions over two double cone regions, where cross-correlation effects occur that significantly change with the mass parameter.
   The results are from a joined project with Henning Bostelmann and Daniela Cadamuro.

6. Seminar

   From classical to quantum fields on causal sets

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Invited seminar, Dublin Institute for Advanced Studies, Ireland, May 2024

   Presented on 09.05.

   Abs arXiv HTML Slides

   In this talk, I will give a brief review of algebraic classical and quantum field theory on causal sets. There are different approaches to quantise the classical fields, of which we consider deformation quantisation and geometric quantisation over the symplectic space of the classical field data that is equipped with an inner product. For a free scalar field on a causal set, I show how to define a state associated to the corresponding quantisation map that turns out to be the Sorkin-Johnston state. The geometric construction suggests a natural generalization to less linear examples, such as interacting fields, which is subject to an ongoing project.
   The main results are based on my PhD project with Eli Hawkins and Kasia Rejzner (arXiv:2207.05667).

2023

1. Conference

   Quantum Gravity @ Raman Research Institute

   Sumati Surya and Madhavan Varadarajan

   Invited participation, Raman Research Institute, Bengaluru, India, Sep 2023

   Conference from 04.09 until 08.09. Invited participation and research visit (until 17.09).

   Website
2. Talk

   Algebras and states for quantum fields on causal sets

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Contributed talk, Radboud University Nijmegen, Netherlands, Jul 2023

   Quantum Gravity (10.07-14.07). Talk presented on 11.07.

   Abs arXiv HTML Website

   Geometric quantization is a natural way to construct quantum models starting from classical data given as a symplectic manifold and an inner product. When the quantization map (from classical to quantum observables) admits an adjoint operation (from quantum to classical observables), we may use this operation to equip the quantum algebra with a distinguished state. As an explicit example, I show the construction in the case of field theory on a causal set (locally finite, partially ordered set) where the classical data is a symplectic vector space and the constructed state coincides with the Sorkin-Johnston state. The construction suggests a natural generalization to less linear examples, such as an interacting field.
   This talk is based on joint work with Eli Hawkins and Kasia Rejzner (arXiv:2207.05667).

3. Poster

   Geometric construction of distinguished states

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Contributed poster, Instituto de Astrofísica de Andalucía, Spain, May 2023

   Quantum Field Theory in Curved Spacetimes, Workshop II (24.05-26.05). Poster presented on 24.05.

   Abs arXiv Video Poster Website

   Geometric quantization is a natural way to construct quantum models starting from classical data given as a symplectic manifold and an inner product. When the quantization map (from classical to quantum observables) admits an adjoint operation (from quantum to classical observables), we may use this operation to equip the quantum algebra with a distinguished state. As an explicit example, I show the construction in the case of field theory on a causal set (locally finite, partially ordered set) where the classical data is a symplectic vector space and the constructed state coincides with the Sorkin-Johnston state. The construction suggests a natural generalization to less linear examples, such as an interacting field.
   This talk is based on joint work with Eli Hawkins and Kasia Rejzner (arXiv:2207.05667).

4. Talk

   On the mass dependence of the modular operator for a double cone

   Henning Bostelmann, Daniela Cadamuro, and Christoph Minz

   Contributed talk, Dresden University of Technology, Germany, Mar 2023

   DPG Frühjahrestagung SMuK (20.03-24.03). Contribution MP 2.2 (Fachverband Theoretische und Mathematische Grundlagen der Physik, Quantum Field Theory I, 20.03)

   Abs arXiv HTML Website

   We present a numerical approximation scheme for the Tomita-Takesaki modular operator of local subalgebras in linear quantum fields, where the modular data are determined at one-particle level using time-0 formulation in position space. The technique is tested against the known results for the local subspace of a right wedge in 2-dimensional Minkowski spacetime, where one component of the modular operator is known to be a mass-independent multiplication operator. Applying the same technique to the unknown case of a double cone in 2 (and 4) dimensions, we find that the same component of the modular operator depends on the mass (and angular momentum).

2022

1. Seminar

   Quantization and Distinguished States from Dequantization

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Seminar, Institut für Theoretische Physik, Leipzig University, Germany, May 2022

   Presented on 17.05.

   Abs arXiv HTML

   Geometric quantization is a natural way to construct the quantum algebra starting from classical data given as a symplectic manifold. When the quantization admits an adjoint operation, referred to as dequantization, we may use this operation to equip the quantum algebra with a distinguished state determined by dequantization. As an example, I show the construction for the free scalar field over a (finite) causal set (locally finite, partially ordered set), where the starting point is a symplectic vector space with an inner product. I compare the result with the construction due to Sorkin, which starts from the same input data, and I show that the distinguished state coincides with the Sorkin-Johnson state, provided the inner product is chosen appropriately.
   This talk is based on joint work with Eli Hawkins and Kasia Rejzner.

2021

1. Seminar

   Local Structure for the Discretization of the Scalar Field Equations on Sprinkled Causal Sets

   Christopher J. Fewster, Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Invited webinar, online (hosted at the Raman Research Institute, India), May 2021

   Presented on 17.05.

   Abs arXiv

   One framework for quantum gravity is causal set theory, where the classical spacetime continuum is replaced by locally finite, partially ordered sets of events (causal sets). For the description of classical (and quantum) scalar fields on causal sets, it is necessary to find a discretization of the Klein-Gordon equation.
   In this talk, I will review a newly proposed method for discretization that does not require the dimension of the spacetime continuum as an input, but rather relies on an additional local structure derived from the rank 2 past of causal set events, which is the set of events that are two links to the past. I will show results from our most recent publication, where we investigated this local structure — called preferred past — using numerical techniques, generating ensembles of causal sets from Minkowski spacetimes by the Poisson process of sprinkling. Our results give a method that performs particularly well at uniquely selecting — with very high probability — a preferred past satisfying desirable properties.
   This talk is based on a publication in joint work with C. J. Fewster, E. Hawkins, and K. Rejzner.

2020

1. Talk

   The Sprinkling Process, Diamonds and Local Structures in Causal Set Theory

   Christopher J. Fewster, Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Invited talk, online, Aug 2020

   The Virtual Causet Meeting (26.08-28.08). Talk presented on 27.08.

   Abs arXiv Slides Website

   We summarize the construction of the probability measure for the Poisson process called sprinkling to generate causal sets from a given spacetime manifold.
   We also investigate the local structure of diamonds in sprinkled causal sets, where a ’diamond’ is the Alexandrov interval spanned by two points that are separated by two links. These diamonds play an essential role in a recently proposed discretization method for the Klein-Gordon equation on causal sets. For this discretization, we need to choose a ’preferred past’ from the two link past for each causet event. We test six criteria to reduce this choice and show numerical results for the diamonds that are spanned between a causet event and its two link past as selected by the criteria.
   This talk is based on joint work with C. J. Fewster, E. Hawkins, and K. Rejzner.

2. Talk

   Diamonds and Local Structure in Sprinkled Causal Sets

   Christopher J. Fewster, Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Contributed poster and talk, online (hosted by the Perimeter Institute for Theoretical Physics, Canada), Jul 2020

   Quantum Gravity (13.07-17.07). Poster presented on 13.07, poster price winner and giving the talk ’Sprinklings in Causal Set Theory and Local Structures to Discretize Field Propagators’ on 17.07.

   DOI arXiv HTML Video Poster Website
3. Seminar

   Sprinklings in Causal Set Theory and Local Structures to Discretize Field Propagators

   Christopher J. Fewster, Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Contributed talk, online, Jun 2020

   59th North British Mathematical Physics Seminar (22.06)

   Abs arXiv HTML

   Causal set theory is a framework for quantum gravity where the spacetime continuum is replaced by a locally finite partial ordered set of spacetime events. Given a continuum spacetime manifold, I summarize the construction of the probability measure for the Poisson process called sprinkling that generates causal sets. I present results for the discretization of quantum fields and their equations of motion on a causal set. The outcomes of this project are used to formulate quantum fields on causal sets in the rigorous framework of algebraic quantum field theory.
   This talk is based on joint work with C. J. Fewster, E. Hawkins, and K. Rejzner.

2019

1. Talk

   On the Probability Measure for Sprinkling in Causal Set Theory

   Eli Hawkins, Christoph Minz, and Kasia Rejzner

   Talk, Department of Mathematics, University of York, UK, May 2019

   Graduate Research Symposium (07.05). Award winner: KM Stott Prize

   Slides

2017

1. Seminar

   Ashtekar Variables

   Christoph Minz

   Research seminar, Technische Universiät Berlin, Germany, Feb 2017

   Seminar series: Geometrical Methods in Mathematical Physics

2016

1. Seminar

   Gravitational Waves on Exact Solutions in General Relativity

   Christoph Minz

   Research seminar, Technische Universiät Berlin, Germany, Jul 2016

   Seminar series: Geometrical Methods in Mathematical Physics

2011

1. Poster

   Statistical properties of photon modes in random arrays of ZnO nano-needles

   David Leipold, Christoph Minz, and Erich Runge

   Contributed poster, Dresden University of Technology, Germany, Mar 2011

   DPG Frühjahrestagung (13.03-18.03). Contriution HL 85.13 (Fachverband Halbleiterphysik, Poster Session II, 17.03)

   Abs HTML Poster Website

   Localization of electromagnetic waves in random media received renewed interest in the last years. Recent ultrafast optical experiments [1] indicate the existence of highly localized photon modes in a system of homogeneous, randomly distributed, vertically aligned ZnO nano-needles. In particular, hot spots in the spatial distribution of the second harmonic generation (SHG) were found.
   In this work, we discuss the optical near field, which we obtain from full 3D solutions of Maxwell’s equations of a model system in time domain. The spatial distribution of the electric near-field and the squared electric near-field intensity are investigated with statistical methods. The results are compared to the experimental findings.
   [1] We thank Manfred Maschek, Slawa Schmidt, Martin Silies and Christoph Lienau from the Carl von Ossietzky Universität Oldenburg as well as Takashi Yatsui, Kokoro Kitamura and Motoichi Ohtsu from the University of Tokyo for sharing their experimental data with us prior to publication.