Mario Carlos Rocca on Ultradistributions, Ultrahyperfunctions, and Rigorous Quantum Field Theory in Einstein Gravity

Scott Douglas Jacobsen (Email: [email protected])

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

Higher Geometry by N. V. Efimov

This book gives a systematic presentation of the fundamentals of Euclidean geometry, non-Euclidean geometry of Lobachevsky and Riemann, projective geometry and the geometrical aspects of spe­cial relativity theory. It also gives a general idea about constant curvature geometries. The subject matter has been shaped by the needs of the students and by my own experience.

A few words on the use of the book as a textbook in a course of the foundations of geometry are in order. The subject matter is distribut­ed over the first two parts. It is presented systematically throughout the book and practically does not omit any of the details of the ar­guments (except for the proofs of certain theorems of elementary geometry). It is evident that in lectures such a detailed presentation would be unprofitable (even if many hours were assigned to this course). The most difficult portion is Chapter 2 in the first part of the book; I feel that in lectures one should discuss from this chapter the formulations of the axioms and present examples of rigorous proofs of certain theorems; moreover, the most important aspects, for exam­ple, the measurement of length, the equivalence of Archimedes’ and Cantor’s axioms to Dedekind’s axiom and the significance of these axioms for the substantiation of analytic geometry, should be discussed in detail. The proof of the majority of the initial theorems of elementary geometry should be left for self-study.

Translated from the Russian by P. C. Sinha, Cand.Sc.

You can get the book here and here

Contents

Preface to the English Edition 9

Part 1. The Foundations of Geometry

A Short Review of Investigations into the Foundations of Geometry

1.1 Euclid’s definitions, postulates, and axioms

1.2 The fifth postulate 16

1.3 Lobachevsky and his geometry 33

1.4 The formation of the notion of geometrical space 36

The Axioms of Elementary Geometry

2.1 Geometrical elements 42

2.2 Group I: axioms of incidence 42

2.3 Group II: axioms of betweenness 45

2.4 Corollaries of the axioms of incidence and betweenness 45

2.5 Group III: axioms of congruence 53

2.6 Corollaries of Axioms I-III 57

2.7 Group IV: axioms of continuity 69

2.8 Group V: the axiom of parallelism. Absolute and divergent lines 96

Chapter 3. The Non-Euclidean Theory of Parallels

3.1 Lobachevsky’s definition of parallels 82

3.2 Peculiarities in the location of parallel lines 85

3.3 Lobachevsky’s function II(x) 100

3.4 Lines and planes in Lobachevskian space 104

3.5 Equidistants and oricycles 111

3.6 Equidistant surfaces and horospheres 121

3.7 Elementary geometry on surfaces in Lobachevskian space 126

3.8 Area of a triangle 136

3.9 Proof of logical consistency of Lobachevskian geometry 146

3.10 The basic metric relationships in Lobachevskian geometry 165

3.11 Riemannian geometry: a brief survey 178

Chapter 4. Investigation of the Axioms of Elementary Geometry

4.1 Three basic problems of axiomatic theory 188

4.2 Consistency of the axioms of Euclidean geometry 191

4.3 Proof of independence of some axioms of Euclidean geometry 205

4.4 Axiom of completeness 216

4.5 The completeness of axioms of Euclidean geometry 220

4.6 The axiomatic method in mathematics

Part 2. Projective Geometry

Chapter 5. The Foundations of Projective Geometry

5.1 The subject matter of projective geometry 223

5.2 Desargues’ theorem. Harmonic sets of elements 226

5.3 Order of points on the projective line 231

5.4 Division of harmonic pairs. Continuity of the harmonic correspondence 244

5.5 The axiom of continuity. Projective system of coordinates on a line 252

5.6 Projective coordinate systems on a plane and in space 258

5.7 Projective correspondence between elements of one-dimensional manifolds 270

5.8 Projective correspondences between two- and three-dimensional manifolds 283

5.9 Analytic representation for projectivities. Involution 292

5.10 Transformation formulas for projective coordinates. Cross ratio of four elements 300

5.11 The principle of duality 316

5.12 Algebraic curves and pencils. Algebraic surfaces and bundles. Complex projective plane and complex projective space 325

5.13 Images of second degree. The theory of polars 337

5.14 Constructive theorems and problems of projective geometry 346

Chapter 6. Group-Theoretic Principles in Geometry. Groups of Transformations

6.1 Geometry and the theory of groups 362

6.2 The projective group and its basic subgroups 391

6.3 Geometries of Lobachevsky, Riemann, and Euclid in the projective setting 408

Chapter 7. Minkowski Space

7.1 Multidimensional affine space 425

7.2 Euclidean spaces and Minkowski space 440

7.3 The space of events of the special relativity theory 455

Chapter 8. Differential Properties of Non-Euclidean Metric

8.1 Metric form of the Euclidean plane 474

8.2 The distance between two points on the Lobachevskian plane 478

8.3 Metric form of the Lobachevskian plane 489

8.4 Intrinsic geometry of a surface and the Beltrami problem 505

8.5 Geometry on a surface of constant curvature 511

8.6 Derivation of basic metric relations in Lobachevskian geometry 523

Part 3. Constant Curvature Geometry

Chapter 9. Spatial Forms in Constant Curvature Geometry

9.1 Two-dimensional manifolds with differential-geometric metric 529

9.2 Parabolic spatial forms 537

9.3 Elliptic spatial forms 544

9.4 Hyperbolic spatial forms 547

Name Index 553

Subject Index 554

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Herman Minkowski's Space-Time (1908)

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