arXiv subjects

Matvei Libine

Publications and source records attributed to Matvei Libine.

At least 19 recordsLinked to original sources

arXiv2026

Reduction of Symmetry in Quaternionic Analysis and Invariant Trilinear Forms

In our previous papers we repeatedly emphasized the special role in Quaternionic Analysis of the conformal group SU(2,2) and other real forms of its complexification SL(4,C). In particular, the natural product map of the left and right regular functions into a larger representation that contains the doubly regular functions as a subquotient is an intertwining operator. In this paper we show, however, that the spaces of regular and doubly regular functions do not "interact" - there is no invariant trilinear form on the tensor product of these representations. To construct a natural invariant trilinear form, we reduce the conformal group symmetry to the symplectic subgroup Sp(4,R). This suggests a new approach to the Quaternionic Analysis in general, and we make the first steps in this paper. It turns out that the spaces of regular and doubly regular functions are still irreducible after the restriction to the symplectic subgroup and have a composed structure arising from the metaplectic representation of the double cover of Sp(4,R) - the metaplectic group. This also leads us to consider the double covers of the quaternionic spaces and non-trivial pairings between them. Our study of Quaternionic Analysis based on the symplectic symmetry group culminates in the construction of the invariant trilinear forms on the products of spaces of doubly regular functions and certain counterparts of regular and quasi regular functions. Additional motivation for constructing invariant trilinear forms comes from their application to spinor representations of certain quaternionic algebras based on the doubly regular functions. The latter can be viewed as the space of solutions of the Maxwell equation, and their spinor representations are of a great importance to quantum field theory. These spinor representations will be the subject of a forthcoming paper.

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arXiv2024

Conformal Invariance of Clifford Monogenic Functions in the Indefinite Signature Case

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions - called monogenic functions - are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Mobius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case.

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arXiv2024

Quasi Regular Functions in Quaternionic Analysis

We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic conformal group. However, unlike the regular functions, the quasi regular ones do not admit an invariant unitary structure but rather a pseudounitary equivalent. The reproducing kernels of these functions have an especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi regular functions and derive the reproducing kernel expansions. We also show that the restrictions of the irreducible representations formed from the quasi regular functions to the Poincare group have three irreducible components. Our interest in the quasi regular functions arises from an application to the study of conformal-invariant algebras of quaternionic functions. We also introduce a factorization of certain intertwining operators between tensor products of spaces of quaternionic functions. This factorization is obtained using fermionic Fock spaces constructed from the quasi regular functions.

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arXiv2020

n-Regular Functions in Quaternionic Analysis

In this paper we study left and right n-regular functions that originally were introduced in [FL4]. When n=1, these functions are the usual quaternionic left and right regular functions. We show that n-regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy-Fueter type reproducing formulas. Arguably, these Cauchy-Fueter type reproducing formulas for n-regular functions are quaternionic analogues of Cauchy's integral formula for the n-th order pole expressing the (n-1)-st derivative of a holomorphic function. We also find two expansions of the Cauchy-Fueter kernel for n-regular functions in terms of certain basis functions, we give an analogue of Laurent series expansion for n-regular functions, we construct an invariant pairing between left and right n-regular functions and we describe the irreducible representations associated to the spaces of left and right n-regular functions of the conformal group and its Lie algebra.

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arXiv2020

Clifford Analysis with Indefinite Signature

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. We define (p,q)-left- and right-monogenic functions by means of Dirac operators that factor a certain wave operator. We prove two different versions of Cauchy's integral formulas for these functions. The two formulas arise from dealing with singularities in distinct ways, and are inspired by the methods of [L, FL]. These results indicate the merit of these methods for dealing with singularities.

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arXiv2019

Magic Identities for the Conformal Four-Point Integrals; the Minkowski Metric Case

The original "magic identities" are due to J.M.Drummond, J.Henn, V.A.Smirnov and E.Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [DHSS]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true in the Minkowski metric. However, the Minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct. In this article we prove the magic identities in the Minkowski metric case and, in particular, specify the cycles of integration. Our proof of magic identities relies on previous results from [L1, L2], where we give a mathematical interpretation of the n-loop box integrals in the context of representations of a Lie group U(2,2) and quaternionic analysis. The main result of [L1, L2] is a (weaker) operator version of the "magic identities". No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

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arXiv2019

Quaternionic Analysis, Representation Theory and Physics II

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

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arXiv2016

The Two-Loop Ladder Diagram and Representations of U(2,2)

Feynman diagrams are a pictorial way of describing integrals predicting possible outcomes of interactions of subatomic particles in the context of quantum field physics. It is highly desirable to have an intrinsic mathematical interpretation of Feynman diagrams, and in this article we find the representation-theoretic meaning of a particular kind of Feynman diagrams called the two-loop ladder diagram. This is done in the context of representations of a Lie group U(2,2), its Lie algebra u(2,2) and quaternionic analysis. The results and techniques developed in this article are used in [L] to provide a mathematical interpretation of all conformal four-point integrals - including those described by the n-loop ladder diagrams - in the context of representations U(2,2) and quaternionic analysis. Moreover, this representation-quaternionic model produces a proof of "magic identities" in the Minkowski metric space. No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

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arXiv2016

The Conformal Four-Point Integrals, Magic Identities and Representations of U(2,2)

In [FL1, FL3] we found mathematical interpretations of the one-loop conformal four-point Feynman integral as well as the vacuum polarization Feynman integral in the context of representations of a Lie group U(2,2) and quaternionic analysis. Then we raised a natural question of finding mathematical interpretation of other Feynman diagrams in the same setting. In this article we describe this interpretation for all conformal four-point integrals. Using this interpretation, we give a representation-theoretic proof of an operator version of the "magic identities" for the conformal four-point integrals described by the box diagrams. The original "magic identities" are due to J.M.Drummond, J.Henn, V.A.Smirnov and E.Sokatchev, they assert that all n-loop box integrals for four scalar massless particles are equal to each other [DHSS]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true for the Minkowski metric case. However, the Minkowski case is much more subtle. In this article we prove an operator version of the magic identities in the Minkowski metric case and, in particular, specify the relative positions of cycles of integration that make these identities correct. No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

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arXiv2015

Geometric Properties of Conformal Transformations on $\mathbb{R}^{p,q}$

We show that conformal transformations on the generalized Minkowski space $\mathbb{R}^{p,q}$ map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when $p$ or $q$ is $0$, and that this action has exactly three orbits if $p, q \ne 0$. Then we extend these results to hyperboloids and affine planes of arbitrary dimension. These properties generalize the well-known properties of Möbius (or fractional linear) transformations on the complex plane $\mathbb{C}$.

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arXiv2014

The Second Order Pole over Split Quaternions

This is an addition to a series of papers [FL1, FL2, FL3, FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we develop split quaternionic analogues of certain results from [FL4]. Thus we introduce a space of functions ${\cal D}^h \oplus {\cal D}^a$ with a natural action of the Lie algebra $\mathfrak{gl}(2,\mathbb H_{\mathbb C}) \simeq \mathfrak{sl}(4,\mathbb C)$, decompose ${\cal D}^h \oplus {\cal D}^a$ into irreducible components and find the $\mathfrak{gl}(2,\mathbb H_{\mathbb C})$-equivariant projectors onto each of these irreducible components.

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arXiv2014

Anti De Sitter Deformation of Quaternionic Analysis and the Second Order Pole

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy's formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum polarization diagrams in four-dimensional quantum field theory. In order to add some flexibility, especially when dealing with Cauchy's formula for the second order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the five-dimensional Euclidean space. We show that many results of quaternionic analysis - including the Cauchy-Fueter formula - admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogues of Cauchy's formula for the second order pole.

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arXiv2012

Introduction to Representations of Real Semisimple Lie Groups

These are lecture notes for a one semester introductory course I gave at Indiana University. The goal was to make this exposition as clear and elementary as possible. A particular emphasis is given on examples involving SU(1,1). These notes are in part based on lectures given by my graduate advisor Wilfried Schmid at Harvard University and PQR2003 Euroschool in Brussels as well as other sources.

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arXiv2011

On a Quaternionic Analogue of the Cross-Ratio

In this article we study an exact analogue of the cross-ratio for the algebra of quaternions H and use it to derive several interesting properties of quaternionic fractional linear transformations. In particular, we show that there exists a fractional linear transformation T on H mapping four distinct quaternions q_1, q_2, q_3 and q_4 into q'_1, q'_2, q'_3 and q'_4 respectively if and only if the quadruples (q_1, q_2, q_3, q_4) and (q'_1, q'_2, q'_3, q'_4) have the same cross-ratio. If such a fractional linear transformation T exists it is never unique. However, we prove that a fractional linear transformation on H is uniquely determined by specifying its values at five points in general position. We also prove some properties of the cross-ratio including criteria for four quaternions to lie on a single circle (or a line) and for five quaternions to lie on a single 2-sphere (or a 2-plane). As an application of the cross-ratio, we prove that fractional linear transformations on H map spheres (or affine subspaces) of dimension 1, 2 and 3 into spheres (or affine subspaces) of the same dimension.

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arXiv2011

Quaternionic Analysis and the Schrodinger Model for the Minimal Representation of O(3,3)

In the series of papers [FL,FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(4,C) and its real forms. This approach has proven very fruitful and pushed further the parallel with complex analysis and develop a rich theory. In [FL2] we study the counterparts of Cauchy-Fueter and Poisson formulas on the spaces of split quaternions H_R and Minkowski space M and show that they solve the problem of separation of the discrete and continuous series on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R). In particular, we introduce an operator Pl_R, compute its effect on the discrete and continuous series components of the space of functions H(H_R) and obtain a surprising formula for the Plancherel measure of SL(2,R). The proof is based on a transition to the Minkowski space M and some pretty lengthy computations. In this paper we introduce an operator d/dR Pl_R on H(H_R) and show that its effect on the discrete and continuous series components can be easily computed using the Schrodinger model for the minimal representation of O(p,q) (with p=q=3) and the results of Kobayashi-Mano from [KM], particularly their computation of the integral expression for the operator F_C. This provides an independent verification of the coefficients involved in the formula for Pl_R. This paper once again demonstrates a close connection between quaternionic analysis and representation theory of various O(p,q)'s.

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arXiv2011

Quaternionic Analysis, Representation Theory and Physics

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.

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arXiv2011

Split Quaternionic Analysis and Separation of the Series for SL(2,R) and SL(2,C)/SL(2,R)

We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions H_R. The special role of the unit sphere in the classical quaternions H identified with the group SU(2) is now played by the group SL(2,R) realized by the unit quaternions in H_R. As in the previous work, we use an analogue of the Cayley transform to relate the analysis on SL(2,R) to the analysis on the imaginary Lobachevski space SL(2,C)/SL(2,R) identified with the one-sheeted hyperboloid in the Minkowski space M. We study the counterparts of Cauchy-Fueter and Poisson formulas on H_R and M and show that they solve the problem of separation of the discrete and continuous series. The continuous series component on H_R gives rise to the minimal representation of the conformal group SL(4,R), while the discrete series on M provides its K-types realized in a natural polynomial basis. We also obtain a surprising formula for the Plancherel measure on SL(2,R) in terms of the Poisson integral on the split quaternions H_R. Finally, we show that the massless singular functions of four-dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary Lobachevski space SL(2,C)/SL(2,R). Our results once again reveal the central role of the Minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four-dimensional quantum field theory.

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arXiv2010

An Invitation to Split Quaternionic Analysis

Six years after William Rowan Hamilton's discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions. (He called them ``coquaternions.'') In this paper we define regular functions on split quaternions and prove two different analogues of the Cauchy-Fueter formula for these functions. In the paper "Split quaternionic analysis and the separation of the series for SL(2,R) and SL(2,C)/SL(2,R)" joint with Igor Frenkel we naturally apply the methods and formulas of quaternionic analysis to solve the problems of harmonic analysis on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R).

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