Dr. Christoph Stephan

wissenschaftlicher Mitarbeiter

Kontakt
Raum:
2.09.0.04
Telefon:
+49 331 977-1662


consultation hours: Thursday 10:00-11:30am

 

...

Lecture course differential geometry

Monday 08:30-10:00, Room 2.10.0.26
Dienstag 14:15-15:45, Raum 2.09.0.12

In the lecture Differential Geometry we learn the basic concepts of the geometry of curved spaces. We define the measurement of lengths and angles with the help of semi-riemannian metrics. We introduce a covariant derivative for vector fields and study locally shortest connections between two points, so-called geodesics. We then deal with various notions of curvature. This lecture is useful for students who want to understand the mathematical foundations of general relativity.

since 2022: permanent faculty member at the Institute for Mathmetics

2017 - 2022: Consultant at d-fine with focus on capital markets, IT-Projects and regulatorics of the banking sector

2018 - 2020: Master of Science in Financial Mathematics, University of Oxford

2006 - 2017: Postdoc at the Institute for Mathematics, University of Potsdam

2005 - 2006:  Postdoc ath the Centre de Physique Théorique (CNRS), Marseille

2002 - 2005: Doctorate in theoretical physics, Christian Albrechts-Universität Kiel & Université de Provence (Marseille)

1999 - 2000: Master of Philosophy in theoretical physics, University of Cambridge

1996 - 2002: Diploma in physics, Christian Albrechts-Universität Kiel

2016 | Noncommutative Geometry and the Physics of the LHC Era | Christoph StephanVerlag: Birkhäuser BaselBuchtitel: F. Finster, J. Kleiner, C. Röken, J. Tolksdorf (eds.): Quantum mathematical physicsSeiten: 471-495Link zur Publikation

Noncommutative Geometry and the Physics of the LHC Era

Autoren: Christoph Stephan (2016)

Noncommutative geometry allows to unify the basic building blocks of particle physics, Yang-Mills-Higgs theory and general relativity, into a single geometrical framework. The resulting effective theory constrains the couplings of the standard model and reduces the number of degrees of freedom. After an introduction of the basic ideas of Noncommutative geometry, I will present its predictions for the standard model and the few known models beyond the standard model based on a classification scheme for finite spectral triples. Most of these models, including the Standard Model, are now ruled out by LHC data. But interesting extensions of the standard model which agree with the presumed Higgs mass, predict new particles (Fermions, Scalars and Bosons) and await further experimental data.

Verlag:
Birkhäuser Basel
Buchtitel:
F. Finster, J. Kleiner, C. Röken, J. Tolksdorf (eds.): Quantum mathematical physics
Seiten:
471-495

2014 | A Dark Sector Extension of the Almost-Commutative Standard Model | Christoph StephanZeitschrift: Int. J. Mod. Phys. AVerlag: World ScientificSeiten: Art. ID 1450005Band: 29, no. 1Link zur Publikation , Link zum Preprint

A Dark Sector Extension of the Almost-Commutative Standard Model

Autoren: Christoph Stephan (2014)

We consider an extension of the Standard Model within the frame work of Noncommutative Geometry. The model is based on an older model [St09] which extends the Standard Model by new fermions, a new U(1)-gauge group and, crucially, a new scalar field which couples to the Higgs field. This new scalar field allows to lower the mass of the Higgs mass from ~170 GeV, as predicted by the Spectral Action for the Standard Model, to a value of 120-130 GeV. The short-coming of the previous model lay in its inability to meet all the constraints on the gauge couplings implied by the Spectral Action. These shortcomings are cured in the present model which also features a "dark sector" containing fermions and scalar particles.

Zeitschrift:
Int. J. Mod. Phys. A
Verlag:
World Scientific
Seiten:
Art. ID 1450005
Band:
29, no. 1

2013 | Chiral Asymmetry and the Spectral Action | Frank Pfäffle, Christoph StephanZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 283-310Band: 321, no. 2Link zur Publikation , Link zum Preprint

Chiral Asymmetry and the Spectral Action

Autoren: Frank Pfäffle, Christoph Stephan (2013)

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
283-310
Band:
321, no. 2

2013 | Noncommutative Geometry in the LHC-Era | Christoph StephanBuchtitel: Proceedings of the XLVIIIth Rencontres de Moriond EW sessionSeiten: 451-458Link zum Preprint

Noncommutative Geometry in the LHC-Era

Autoren: Christoph Stephan (2013)

Noncommutative geometry allows to unify the basic building blocks of particle physics, Yang-Mills-Higgs theory and General relativity, into a single geometrical framework. The resulting effective theory constrains the couplings of the particle model and reduces the number of degrees of freedom. After briefly introducing the basic ideas of noncommutative geometry, I will present its predictions for the Standard Model and the few known models beyond the Standard Model. Most of these models, including the Standard Model, are now ruled out by LHC data. But interesting extensions of the Standard Model which agree with the presumed Standard Model Scalar mass and predict new particles are still very much alive and await further experimental data.

Buchtitel:
Proceedings of the XLVIIIth Rencontres de Moriond EW session
Seiten:
451-458

2012 | On Gravity, Torsion and the Spectral Action Principle | Frank Pfäffle, Christoph StephanZeitschrift: J. Funct. Anal.Verlag: ElsevierSeiten: 1529-1565Band: 262, no. 4Link zur Publikation , Link zum Preprint

On Gravity, Torsion and the Spectral Action Principle

Autoren: Frank Pfäffle, Christoph Stephan (2012)

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.

Zeitschrift:
J. Funct. Anal.
Verlag:
Elsevier
Seiten:
1529-1565
Band:
262, no. 4

2011 | The Holst Action by the Spectral Action Principle | Frank Pfäffle, Christoph StephanZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 261-273Band: 307, no. 1Link zur Publikation , Link zum Preprint

The Holst Action by the Spectral Action Principle

Autoren: Frank Pfäffle, Christoph Stephan (2011)

We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
261-273
Band:
307, no. 1

2010 | The Spectral Action for Dirac Operators with skew-symmetric Torsion | Florian Hanisch, Frank Pfäffle, Christoph StephanZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 877-888Band: 300, no. 3Link zur Publikation , Link zum Preprint

The Spectral Action for Dirac Operators with skew-symmetric Torsion

Autoren: Florian Hanisch, Frank Pfäffle, Christoph Stephan (2010)

We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in presence of torsion from the Chamseddine-Connes Dirac operator.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
877-888
Band:
300, no. 3

2009 | Das Yang-Mills-Problem. Die mathematische Zähmung des Standardmodells | Christian Bär, Christoph StephanZeitschrift: Spektrum der WissenschaftVerlag: Spektrum der Wissenschaft VerlagSeiten: 66-73Band: 5Link zur Publikation

Das Yang-Mills-Problem. Die mathematische Zähmung des Standardmodells

Autoren: Christian Bär, Christoph Stephan (2009)

Die moderne Theorie der Elementarteilchen, die Quanten-Yang-Mills-Theorie, vermag die experimentellen Befunde mit unerhörter Genauigkeit wiederzugegen, es fehlt ihr jedoch bisher ein solides mathematisches Fundament. Über dessen Gestalt gibt es nur sehr nebelhafte Vorstellungen.

Zeitschrift:
Spektrum der Wissenschaft
Verlag:
Spektrum der Wissenschaft Verlag
Seiten:
66-73
Band:
5

2009 | On a classification of irreducible almost-commutative geometries V | Jan-Hendrik Jureit, Christoph StephanZeitschrift: J. Math. Phys.Verlag: American Institute of PhysicsSeiten: 72301Band: 50, no. 7Link zur Publikation , Link zum Preprint

On a classification of irreducible almost-commutative geometries V

Autoren: Jan-Hendrik Jureit, Christoph Stephan (2009)

We extend a classification of irreducible, almost-commutative geometries whose spectral action is dynamically non-degenerate, to internal algebras that have six simple summands. We find essentially four particle models: An extension of the standard model by a new species of fermions with vectorlike coupling to the gauge group and gauge invariant masses, two versions of the electro-strong model and a variety of the electro-strong model with Higgs mechanism.

Zeitschrift:
J. Math. Phys.
Verlag:
American Institute of Physics
Seiten:
72301
Band:
50, no. 7

2009 | New Scalar Fields in Noncommutative Geometry | Christoph StephanZeitschrift: Phys. Rev. DVerlag: American Physical SocietySeiten: 65013Band: 79, no. 6Link zur Publikation , Link zum Preprint

New Scalar Fields in Noncommutative Geometry

Autoren: Christoph Stephan (2009)

In this publication we present an extension of the Standard Model within the framework of Connes' noncommutative geometry [1]. The model presented here is based on a minimal spectral triple [7] which contains the Standard Model particles, new vectorlike fermions and a new U(1) gauge subgroup. Additionally a new complex scalar field appears that couples to the right-handed neutrino, the new fermions and the standard Higgs particle. The bosonic part of the action is given by the spectral action [1] which also determines relations among the gauge couplings, the quartic scalar couplings and the Yukawa couplings at a cut-off energy of ca. 10^(17) GeV. We investigate the renormalisation group flow of these relations. The low energy behaviour allows to constrain the Higgs mass, the mass of the new scalar and the mixing between these two scalar fields.

Zeitschrift:
Phys. Rev. D
Verlag:
American Physical Society
Seiten:
65013
Band:
79, no. 6

2009 | Krajewski diagrams and the Standard Model | Christoph StephanZeitschrift: J. Math. PhysVerlag: American Institute of PhysicsSeiten: 43515Band: 50, no. 4Link zur Publikation , Link zum Preprint

Krajewski diagrams and the Standard Model

Autoren: Christoph Stephan (2009)

This paper provides a complete list of Krajewski diagrams representing the standard model of particle physics. We will give the possible representations of the algebra and the anomaly free lifts which provide the representation of the standard model gauge group on the fermionic Hilbert space. The algebra representations following from the Krajewski diagrams are not complete in the sense that the corresponding spectral triples do not necessarily obey to the axiom of Poincare duality. This defect may be repaired by adding new particles to the model, i.e. by building models beyond the standard model. The aim of this list of finite spectral triples (up to Poincare duality) is therefore to provide a basis for model building beyond the standard model.

Zeitschrift:
J. Math. Phys
Verlag:
American Institute of Physics
Seiten:
43515
Band:
50, no. 4

2009 | Beyond the Standard Model: A Noncommutative Approach | Christoph StephanBuchtitel: Proceedings of the XLIVth Rencontres de Moriond: Electroweak Interactions and Unified TheoriesLink zum Preprint

Beyond the Standard Model: A Noncommutative Approach

Autoren: Christoph Stephan (2009)

During the last two decades Alain Connes developed Noncommutative Geometry (NCG), which allows to unify two of the basic theories of modern physics: General Relativity (GR) and the Standard Model (SM) of Particle Physics as classical field theories. In the noncommutative framework the Higgs boson, which had previously to be put in by hand, and many of the ad hoc features of the standard model appear in a natural way.
The aim of this presentation is to motivate this unification from basic physical principles and to give a flavour of its derivation. One basic prediction of the noncommutative approach to the SM is that the mass of the Higgs Boson should be of the order of 170 GeV if one assumes the Big Desert. This mass range is with reasonable probability excluded by the Tevatron and therefore it is interesting to investigate models beyond the SM that are compatible with NCG. Going beyond the SM is highly non-trivial within the NCG approach but possible extensions have been found and provide for phenomenologically interesting models. We will present in this article a short introduction into the NCG framework and describe one of these extensions of the SM. This model contains new scalar bosons (and fermions) which constitute a second Higgs-like sector mixing with theordinary Higgs sector and thus considerably modifying the mass eigenvalues.

Buchtitel:
Proceedings of the XLIVth Rencontres de Moriond: Electroweak Interactions and Unified Theories

2009 | The Inverse Seesaw Mechanism in Noncommutative Geometry | Christoph StephanZeitschrift: Phys. Rev. DVerlag: American Physical SocietySeiten: 65007Band: 80, no. 6Link zur Publikation , Link zum Preprint

The Inverse Seesaw Mechanism in Noncommutative Geometry

Autoren: Christoph Stephan (2009)

In this publication we will implement the inverse Seesaw mechanism into the noncommutative framework on the basis of the AC-extension of the Standard Model. The main difference to the classical AC model is the chiral nature of the AC fermions with respect to a U(1) extension of the Standard Model gauge group. It is this extension which allows us to couple the right-handed neutrinos via a gauge invariant mass term to left-handed A-particles. The natural scale of these gauge invariant masses is of the order of 10^17 GeV while the Dirac masses of the neutrino and the AC-particles are generated dynamically and are therefore much smaller (ca. 1 GeV to 10^6 GeV). From this configuration a working inverse Seesaw mechanism for the neutrinos is obtained.

Zeitschrift:
Phys. Rev. D
Verlag:
American Physical Society
Seiten:
65007
Band:
80, no. 6

2008 | Finding the standard model of particle physics: A combinatorial problem | Jan-Hendrik Jureit, Christoph StephanZeitschrift: Comput. Phys. Commun.Verlag: ElsevierSeiten: 230-247Band: 178, no. 3Link zur Publikation , Link zum Preprint

Finding the standard model of particle physics: A combinatorial problem

Autoren: Jan-Hendrik Jureit, Christoph Stephan (2008)

We present a combinatorial problem which consists in finding irreducible Krajewski diagrams from finite geometries. This problem boils down to placing arrows into a quadratic array with some additional constrains. The Krajewski diagrams play a central role in the description of finite noncommutative geometries. They allow to localise the standard model of particle physics within the set of all Yang-Mills-Higgs models.

Zeitschrift:
Comput. Phys. Commun.
Verlag:
Elsevier
Seiten:
230-247
Band:
178, no. 3

2008 | On a classification of irreducible almost-commutative geometries. IV | Jan-Hendrik Jureit, Christoph StephanZeitschrift: J. Math. Phys.Verlag: American Institute of PhysicsSeiten: 33502Band: 49, no. 3Link zur Publikation , Link zum Preprint

On a classification of irreducible almost-commutative geometries. IV

Autoren: Jan-Hendrik Jureit, Christoph Stephan (2008)

In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Krajewski diagrams [5]. Furthermore we will show that any real finite spectral triple in KO-dimension 6 is automatically S 0 -real. This work has been inspired by the recent paper by Alain Connes [6] and John Barrett [7].
In the classification we find that the standard model of particle physics in its minimal version fits the axioms of noncommutative geometry in the case of KO-dimension six. By minimal version it is meant that at least one neutrino has to be massless and mass-terms mixing particles and antiparticles are prohibited

Zeitschrift:
J. Math. Phys.
Verlag:
American Institute of Physics
Seiten:
33502
Band:
49, no. 3