Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+\lambda W)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on ${\mathbb R}^n$, $\lambda\in\mathbb C$ is a coupling constant, and $W$ is a regular integral operator with compactly supported kernel. In particular, $W$ can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small $|\lambda|$, the hyperbolic character of $D$ is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in $\lambda$, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of $W$ on the space of solutions of $D$. 
It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.

Linear hyperbolic PDEs with non-commutative time

Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+\lambda W)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on ${\mathbb R}^n$, $\lambda\in\mathbb C$ is a coupling constant, and $W$ is a regular integral operator with compactly supported kernel. In particular, $W$ can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small $|\lambda|$, the hyperbolic character of $D$ is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in $\lambda$, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of $W$ on the space of solutions of $D$. It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.

Non-local in time perturbations of linear hyperbolic PDEs

Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form (D+λW)f=0 are studied, where D is a normal or prenormal hyperbolic differential operator on ℝn, λ∈ℂ is a coupling constant, and W is a regular integral operator with compactly supported kernel. In particular, W can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small |λ|, the hyperbolic character of D is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in λ, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of W on the space of solutions of D. 

It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.

pdf/linear-hyperbolic-pdes-with-non-commutative-time-d71ifk0f8x.pdf

The time coordinate is a common obstacle in the theory of noncommutative (nc.) spacetimes. Despite that, this work shows how the interplay between quantum fields and an underlying nc. spacetime can still be analyzed, even for the case of nc. time. This is done for the example of a general Moyal-type external potential scattering of the Dirac field in Moyal-Minkowski spacetime. The spacetime is a rare example of a Lorentzian non-compact nc. geometry. Elements of the associated spectral function algebra are shown to be operationally involved at the level of quantum field operators by Bogoliubov’s formula. Furthermore, a similar task is attacked in the case of locally nc. spacetimes. An explicit star-product is constructed by a method of Kontsevich. It implements a decay of non-commutativity with increasing distance. This behavior should benefit the technical side – diverse interesting formal attempts are discussed. It is striven for unification of several toy models of nc. spacetimes and a general strategy to define quantum field operators. Within the latter one has to implement the usual quantum behavior as well as a new kind of spacetime behavior. It is shown how this two-fold character causes key difficulties in understanding. Studies and scientific history of the author: 10/2001-09/2003 Study of physics at the Technical University of Chemnitz with subsidiary subject Information Technology 09/2003 Intermediate diploma 10/2003-09/2005 Main studies at the University of Leipzig with focuses • general relativity • quantum field theory • mathematics (differential geometry and functional analysis) 10/2005-10/2006 Diploma thesis with topic “Dirac-Feld und nichtkommutative Raumzeit”, finished as diploma physicist (grade: 1.1) Since 01/2007 PhD student in the field of “Quantum Field Theory on Non-commutative Spacetimes” 2006-2010 Participation in the 18th till 25th workshop “Foundations and Constructive Aspects of QFT” 11/2006 Seminar talk “Fermi-Fockraum und freies Dirac Feld” 01/2007-12/2008 Financial support within the scope of the DFG project “Nichtkommutative Geometrie und Elementarteilchen” 01/2007 Talk “Non-commutative potential scattering for Dirac fields on Moyal spacetime” at the abovementioned 19th workshop in Goettingen 02/2008 Participation in Spring School “Quantum Structures” in Leipzig 03/2008 Talk “Dirac Field on Moyal-Minkowski spacetime” at the DPG Spring Conference in Freiburg 07/2008 Participation in “CRCG Workshop on Quantum Fields and Non-commutative Geometry” in Goettingen 12/2008 Paper “Dirac field on Moyal-Minkowski spacetime and non-comm. potential scattering” in collaboration with Prof. R. Verch appears as preprint arXiv:0812.0786 [math-ph] 12/2008 Scientific guest stay to Jr. Prof. D. Bahns in Goettingen 01/2009-06/2010 Financial support by the IMPRS 06/2009 Seminar talk “Dirac field on Moyal-Minkowski spacetime” 01/2010 Abovementioned paper appears in Commun. Math. Phys. 293 (2010) 399, cf. [6] 02/2010 Talk “Dirac Field and Non-commutative Spacetimes” at IMPRS-Workshop in Leipzig

Quantum Field Theory on Non-commutative Spacetimes

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Markus Borris

Author Tools

Chat about Author

Papers

Quantum Field Theory on Non-commutative Spacetimes