Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity
Scott Douglas Jacobsen (Email: scott.jacobsen2026@gmail.com)
Publisher, In-Sight Publishing
Fort Langley, British Columbia, Canada
Received: October 3, 2025
Accepted: January 8, 2026
Published: January 8, 2026
Abstract
This interview with Mario Carlos Rocca addresses the definition and use of tempered ultradistributions and ultradistributions of exponential type within quantum-field-theoretic constructions. Rocca describes a framework in which tempered ultradistributions are continuous linear functionals on a space of entire test functions, and he outlines an analytic representation using contour integrals outside strips containing singularities. He explains why a naïve convolution formula does not generally exist for arbitrary pairs of ultradistributions, and presents a regulated construction using a complex parameter (λ), analytic continuation, and extraction of the λ-independent term to define convolution. The interview also summarizes how products of distributions of exponential type are handled via Fourier-transform relations and notes that associativity is not guaranteed in general due to the algebraic structure described as a ring with zero divisors. Rocca provides explicit finite convolution expressions for Wheeler propagators in massless and complex-mass cases and comments on microcausality, loop finiteness claims in the ultradistribution/ultrahyperfunction setting, and extensions from Minkowski space to semi-Riemannian and globally hyperbolic spacetimes. The final portion includes Rocca’s high-level descriptions of Einstein gravity and Gupta–Feynman quantization, as well as formulas for graviton self-energy calculations presented in BTZ-background studies.
Keywords
Analytic continuation, Analytic representation, Associativity, BTZ gravity, Cauchy integral formula, Complex delta function, Complex mass, Convolution, Entire test functions, Einstein gravity, Exponential growth bounds, Fourier transform, Functional analysis, Gauge conditions, Gel’fand triplet, Ghost avoidance, Graviton self-energy, Gupta–Bleuler method, Gupta–Feynman quantization, Loop integrals, Microcausality, Minkowski space, Nuclear spaces, Operator-valued distributions, Propagators, Rigged Hilbert space, Schwartz distributions, Tempered ultradistributions, Ultradistributions of exponential type, Ultrahyperfunctions, Wheeler propagator, Zero divisors
Introduction
The interview concerns mathematical structures used in quantum field theory that generalize Schwartz distributions, focusing on tempered ultradistributions and ultradistributions of exponential type. In the interview text, these objects are presented through test-function spaces built from entire analytic functions with specified growth bounds, and through the rigged Hilbert space (Gel’fand triplet) approach in which distributions are realized as continuous linear functionals on a nuclear test-function space. The discussion emphasizes analytic representations of generalized functions by contour integrals in the complex plane, where the contour is chosen to avoid bands containing singularities. The interview also treats the technical problem of defining convolutions and products for generalized functions: Rocca explains that direct convolution expressions are not always well-defined, motivating the use of regulators, analytic continuation, and a prescription selecting a parameter-independent term to define a convolution in cases of interest.
In addition to these core constructions, the interview addresses downstream physics-facing topics: conditions under which commutators satisfy microcausality in the ultrahyperfunction setting, claims about the treatment (or absence) of singularities within that formalism, and how these methods are applied to propagators, loop integrals, and model calculations in gravitational settings. These topics are framed in the interview by reference to established distribution theory traditions associated with Schwartz and to ultradistribution/ultrahyperfunction developments associated with Sebastião e Silva and later work.
Main Text (Interview)
Title: Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity
Interviewer: Scott Douglas Jacobsen
Interviewees: Dr. Mario Carlos Rocca
Mario Carlos Rocca is an Argentine theoretical and mathematical physicist at the Universidad Nacional de La Plata and an Independent Researcher with Argentina’s CONICET. His work sits at the crossroads of functional analysis and high-energy/gravitation, especially the use of ultradistributions of Sebastiao e Silva tambien conocidas como ultrahyper functions (refinements and extensiones generalizadas of Schwartz distributions) in quantum field theory. He co-developed influential formalisms on the convolution of ultradistributions with the late C. G. Bollini and has continued that line with collaborators such as Angel Plastino. Recent papers push ultrahyperfunction-based methods into Einstein gravity, black holes, and dark-matter-adjacent questions, including co-authorships with Mir Hameeda and Behnam Pourhassan.
Scott Douglas Jacobsen: In your 1998 paper with Bollini and Escobar, how do you define the convolution of two tempered ultradistributions?
Mario Carlos Rocca: Explaining what a tempered ultradistribution or an exponential ultradistribution is, [1] is practically impossible with words alone. To solve the problem, I have added some sections taken from my papers in which I explain what tempered ultradistributions and exponential ultradistributions are.
Jacobsen: In that 1998 framework, how do you construct the product of two distributions of exponential type?
Rocca: The product of two exponential distributions is the Fourier anti-transform of the two tempered ultradistributions, that is, a product on a ring with zero divisors.
Jacobsen: Following from the previous question, what minimal conditions ensure existence and associativity?
Rocca: The existence of the product is established constructively, with the corresponding theorems (see the beginning of this note). The product is not generally associative, since it is a product in a ring with zero divisors. The product must be performed carefully, taking into account the physical conditions of the problem.
Jacobsen: Using that four-dimensional result, what is the explicit finite expression for the convolution of two Wheeler prop agators in massless and complex-mass cases?
Rocca:
Jacobsen: What necessary and sufficient analyticity and support conditions in the ultrahyperfunction setting guarantee microcausality for field commutators?
Rocca: This is guaranteed, since quantum fields are vector ultradistributions. Their product is defined from the usual ultradistri butions. A vector ultradistribution is a continuous functional defined on a space of test functions and taking values in a locally convex topological vector space. Operator valued distributions are a special case of vector ultradistributions.
Jacobsen: Which growth and analyticity conditions on propagators treated as ultrahyperfunctions ensure finiteness of loop integrals without renormalization?
Rocca: All propagators known so far are ultradistributions. If they are exponentially increasing propagators, they are exponential ultradistributions. This ensures the finiteness of the integral loops.
Jacobsen: What is Einstein Gravity?
Rocca: It is the geometry of space-time created by the presence of masses.
Jacobsen: What is Gupta–Feynman–based on the QFT of Einstein gravity?
Rocca: This is the gravity proposed by Gupta and Feynman by developing the graviton field into powers of the gravitational constant and quantizing it using the Gupta-Bleurer method. The best-known case is the linear approximation, which is the case I solved exactly with Mir Hameeda and Angelo Plastino.
Jacobsen: Following from the last two questions, which constraint and gauge conditions maintain S-matrix unitarity while avoiding Faddeev-Popov ghosts?
Rocca: In the case we are dealing with, the quantization was made unitary by adding a simple constraint. So far, I have had the experience that if the most general quantization method, the Feynman-Schwinger Variational Principle, is used, the ghosts do not appear in the theory being treated.
Jacobsen: How does the ultrahyperfunction-based quantization program relate to effective field theory?
Rocca: The theory of ultrahyperfunctions is used to quantize fundamental theories. However, using the same method, one can also quantize effective theories. For me, Einstein’s theory of gravity is a fundamental theory. To quantize this theory, a very rigorous mathematical theory must be used, like the one we use.
Jacobsen: What are infrared and massless regimes?
Rocca: The infrared regime corresponds to small momentums. The massless regime is obtained for massive particles outside the mass-shell.
Jacobsen: For these, how are soft and collinear singularities handled within the ultradistribution and ultrahyperfunction formalism?
Rocca: In the theory of ultrahyperfunctions, singularities do not exist.
Jacobsen: What is Minkowski space?
Rocca: It is simply a semi-Riemmannian manifold with a particular metric.
Jacobsen: Following from the previous questions, which parts of the ultrahyperfunction construction extend from Minkowski space to general globally hyperbolic curved spacetimes?
Rocca: The construction of ultradistributions for any semi-Riemmannian manifold is analogous to the construction in Minkowski space, only with another metric and other variables.
Jacobsen: In your BTZ-background studies, how is the graviton self-energy computed in 2+1 and 3+1dimensions?
Finally, I should clarify that ultrahyperfunctions are to distributions what complex functions are to real-world functions. That’s how important their role is in next-generation rigorous mathematics.
Jacobsen: Thank you for the opportunity and your time, Marco.
Methods
The interview was conducted via typed questions—with explicit consent—for review, and curation. This process complied with applicable data protection laws, including the California Consumer Privacy Act (CCPA), Canada’s Personal Information Protection and Electronic Documents Act (PIPEDA), and Europe’s General Data Protection Regulation (GDPR), i.e., recordings if any were stored securely, retained only as needed, and deleted upon request, as well in accordance with Federal Trade Commission (FTC) and Advertising Standards Canada guidelines.
Data Availability
No datasets were generated or analyzed during the current article. All interview content remains the intellectual property of the interviewer and interviewee.
References
[1] J. Sebastião e Silva. Math. Ann. 136, 38 (1958).
[2] M. Hasumi. Tohoku Math. J. 13, 94 (1961).
[3] I. M. Gel’fand & G. E. Shilov. Generalized Functions, Vol. 2. Academic Press (1968).
[4] I. M. Gel’fand & N. Ya. Vilenkin. Generalized Functions, Vol. 4. Academic Press (1964).
[5] C. G. Bollini, L. E. Oxman & M. C. Rocca. J. Math. Phys. 35, 4429 (1994).
[6] I. M. Gel’fand & G. E. Shilov. Generalized Functions, Vol. 1, Ch. 1, §3. Academic Press (1964).
[7] L. Schwartz. Théorie des distributions. Hermann, Paris (1966).
[8] R. F. Hoskins & J. Sousa Pinto. Distributions, Ultradistributions and other Generalised Functions. Ellis Horwood (1994).
[9] M. Hameeda, A. Plastino, B. Pourhassan & M. C. Rocca. “Quantum Field Theory of 3+1 Dimensional BTZ Gravity: Graviton Self-Energy, Axion Interactions, and Dark Matter in the Ultrahyperfunction Framework.” ResearchGate. https://www.researchgate.net/publication/395268698_Quantum_Field_Theory_of_31_Dimensional_BTZ_Gravity_Graviton_Self-Energy_Axion_Interactions_and_Dark_Matter_in_the_Ultrahyperfunction_Framework
[10] H. Farahani, M. Hameeda, A. Plastino, B. Pourhassan & M. C. Rocca. “Quantum Field Theory of 2+1 Dimensional BTZ Gravity: Graviton Self-Energy, Axion Interactions, and Dark Matter in the Ultrahyperfunction Framework.” ResearchGate. https://www.researchgate.net/publication/395268790_Quantum_Field_Theory_of_21_Dimensional_BTZ_Gravity_Graviton_Self-Energy_Axion_Interactions_and_Dark_Matter_in_the_Ultrahyperfunction_Framework
Journal & Article Details
Publisher: In-Sight Publishing
Publisher Founding: March 1, 2014
Web Domain: http://www.in-sightpublishing.com
Location: Fort Langley, Township of Langley, British Columbia, Canada
Journal: In-Sight: Interviews
Journal Founding: August 2, 2012
Frequency: Four Times Per Year
Review Status: Non-Peer-Reviewed
Access: Electronic/Digital & Open Access
Fees: None (Free)
Volume Numbering: 14
Issue Numbering: 1
Section: A
Theme Type: Discipline
Theme Premise: Quantum Cosmology
Formal Sub-Theme: None.
Individual Publication Date: January 8, 2026
Issue Publication Date: April 1, 2026
Author(s): Scott Douglas Jacobsen
Word Count: 3,278
Image Credits: Mario Carlos Rocca
ISSN (International Standard Serial Number): 2369-6885
Acknowledgements
The author acknowledges Mario Carlos Rocca for her time, expertise, and valuable contributions. Her thoughtful insights and detailed explanations have greatly enhanced the quality and depth of this work, providing a solid foundation for the discussion presented herein.
Author Contributions
S.D.J. conceived the subject matter, conducted the interview, transcribed and edited the conversation, and prepared the manuscript.
Competing Interests
The author declares no competing interests.
License & Copyright
In-Sight Publishing by Scott Douglas Jacobsen is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
© Scott Douglas Jacobsen and In-Sight Publishing 2012–Present.
Unauthorized use or duplication of material without express permission from Scott Douglas Jacobsen is strictly prohibited. Excerpts and links must use full credit to Scott Douglas Jacobsen and In-Sight Publishing with direction to the original content.
Supplementary Information
Below are various citation formats for Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity (Scott Douglas Jacobsen, January 8, 2026).
American Medical Association (AMA 11th Edition)
Jacobsen SD. Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity. In-Sight: Interviews. 2026;14(1). Published January 8, 2026. http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity
American Psychological Association (APA 7th Edition)
Jacobsen, S. D. (2026, January 8). Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity. In-Sight: Interviews, 14(1). In-Sight Publishing. http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity
Brazilian National Standards (ABNT)
JACOBSEN, Scott Douglas. Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity. In-Sight: Interviews, Fort Langley, v. 14, n. 1, 8 jan. 2026. Disponível em: http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity
Chicago/Turabian, Author-Date (17th Edition)
Jacobsen, Scott Douglas. 2026. “Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity.” In-Sight: Interviews 14 (1). http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity.
Chicago/Turabian, Notes & Bibliography (17th Edition)
Jacobsen, Scott Douglas. “Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity.” In-Sight: Interviews 14, no. 1 (January 8, 2026). http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity.
Harvard
Jacobsen, S.D. (2026) ‘Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity’, In-Sight: Interviews, 14(1), 8 January. Available at: http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity.
Harvard (Australian)
Jacobsen, SD 2026, ‘Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity’, In-Sight: Interviews, vol. 14, no. 1, 8 January, viewed 8 January 2026, http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity.
Modern Language Association (MLA, 9th Edition)
Jacobsen, Scott Douglas. “Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity.” In-Sight: Interviews, vol. 14, no. 1, 2026, http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity.
Vancouver/ICMJE
Jacobsen SD. Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity [Internet]. 2026 Jan 8;14(1). Available from: http://www.in-sightpublishing.com/mario-carlos-rocca-ultradistributions-ultrahyperfunctions-rigorous-quantum-field-theory-einstein-gravity
Note on Formatting
This document follows an adapted Nature research-article format tailored for an interview. Traditional sections such as Methods, Results, and Discussion are replaced with clearly defined parts: Abstract, Keywords, Introduction, Main Text (Interview), and a concluding Discussion, along with supplementary sections detailing Data Availability, References, and Author Contributions. This structure maintains scholarly rigor while effectively accommodating narrative content.
#AnalyticContinuation #AnalyticRepresentation #Associativity #BTZGravity #CauchyIntegralFormula #ComplexDeltaFunction #ComplexMass #Convolution #EinsteinGravity #EntireTestFunctions #ExponentialGrowthBounds #FourierTransform #FunctionalAnalysis #GaugeConditions #GelFandTriplet #GhostAvoidance #GravitonSelfEnergy #GuptaBleulerMethod #GuptaFeynmanQuantization #LoopIntegrals #Microcausality #MinkowskiSpace #NuclearSpaces #OperatorValuedDistributions #Propagators #RiggedHilbertSpace #SchwartzDistributions #TemperedUltradistributions #UltradistributionsOfExponentialType #Ultrahyperfunctions #WheelerPropagator #ZeroDivisors
