Toward an Infinite-Component Field Theory with a Double Symmetry: Interaction of Fields

We complete the first stage of constructing a theory of fields not investigated before; these fields transform according to Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We consider only those theories that initially have a double symmetry: relativistic invariance and the invariance under the transformations of a secondary symmetry generated by the polar or the axial four-vector representation of the orthochronous Lorentz group. The high symmetry of the theory results in an infinite degeneracy of the particle mass spectrum with respect to spin. To eliminate this degeneracy, we postulate a spontaneous secondary-symmetry breaking and then solve the problems on the existence and the structure of nontrivial interaction Lagrangians.     

Mass spectra in the doubly symmetric theory of infinite-component fields

We consider the problem of the characteristics of mass spectra in the doubly symmetric theory of fields transforming under the proper Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We show that there exists a range of free parameters of the theory where the mass spectra of fermions are quite satisfactory from the physical standpoint and correspond to the picture expected in the parton model of hadrons.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 21–36, January, 2005.     

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.     

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.     

On the tensor product of representations of semisimple Lie groups

The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation.Translated from Matematicheskie Zametki, Vol. 16, No. 5. pp. 731–739, November, 1974.     

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ^??, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand?CRaikov theorem holds for topological subgroups of S ^??: for all such groups, continuous irreducible representations separate points in the group.     

Dirac equation and spin 1 representations,a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentsz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli—Gürsey symmetry operators (without the ⁵ operator) of the Dirac equation withm=0 form the same representation D(1/2, 0)D(0, 1/2) of the O(1, 3) algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations D(1/2, 1/2) and D(1, 0) D(0, 0) of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group O(1, 3) are investigated. Analogous O(1, 2) and P(1, 2) invariance algebras are established for the Dirac equation withm>0.Institute of Nuclear Research, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 388–406, March, 1992.     

On quasifree representations of infinite dimensional symplectic group

We consider an infinite dimensional generalization of metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained.     

Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz-Krieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.     

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.     

Representing probability measures using probabilistic processes

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words, under which each successive symbol is determined by a finite probabilistic choice, generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.     

Multiplicative form of the Lagrangian

We obtain an alternative class of Lagrangians in the so-called the multiplicative form for a system with one degree of freedom in the nonrelativistic and the relativistic cases. This new form of the Lagrangian can be regarded as a one-parameter class with the parameter λ obtained using an extension of the standard additive form of the Lagrangian because both forms yield the same equation of motion. We note that the multiplicative form of the Lagrangian can be regarded as a generating function for obtaining an infinite hierarchy of Lagrangians that yield the same equation of motion. This nontrivial set of Lagrangians confirms that the Lagrange function is in fact nonunique.     

Four drafts on the representation theory of the group of infinite matrices over a finite field

The history of these drafts is described in the preface written by the first author; the drafts were written in 1997–2000, when the authors studied asymptotic representation theory. Each draft is devoted to one of the main chapters of the representation theory of the group of infinite matrices over a finite field. The list of results includes the definition of the group GLB(q), which is the right analog of the group of matrices over a finite field, or a q-analog of the infinite symmetric group, the latter corresponding to q = 1; a formula for the principal series characters of the group GLB(q) and similar groups; the statistics of Jordan forms and a law of large numbers for the characters; a construction of simplest factor representations. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 5–36.     

Complex Conjugation of Group Representations by Inner Automorphisms

For finite-dimensional unitary irreducible group representations, theorems are established giving conditions under which the transition from a representation to its complex conjugate may be accomplished by an inner automorphism of the group. The central arguments are of a purely matrix-theoretical nature. Since the investigation naturally falls into two cases according to the Frobenius-Schur classification of irreducible representations, this classification is briefly discussed.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

On the calculation of the bounds of probability of events using infinite random sets

This paper presents an extension of the theory of finite random sets to infinite random sets, that is useful for estimating the bounds of probability of events, when there is both aleatory and epistemic uncertainty in the representation of the basic variables. In particular, the basic variables can be modelled as CDFs, probability boxes, possibility distributions or as families of intervals provided by experts. These four representations are special cases of an infinite random set. The method introduces a new geometrical representation of the space of basic variables, where many of the methods for the estimation of probabilities using Monte Carlo simulation can be employed. This method is an appropriate technique to model the bounds of the probability of failure of structural systems when there is parameter uncertainty in the representation of the basic variables. A benchmark example is used to demonstrate the advantages and differences of the proposed method compared with the finite approach.     

On the regular representation of a nonunimodular locally compact group

We study the discrete part of the regular representation of a locally compact group and also its Type I part if the group is separable. Our results extend to nonunimodular groups' known results for unimodular groups about formal degrees of square integrable representations, and the Plancherel formula. We establish orthogonality relations for matrix coefficients of square integrable representations and we show that the formal degree in general is not a positive number, but a positive self-adjoint unbounded operator, semi-invariant under the representation. Integrable representations are also studied in this context. Finally we show that when the group is nonunimodular, “Plancherel measure” is not a true measure, but a measure multiplied by a section of a certain real oriented line bundle on the dual space of the group.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

动力学群微分表示的计算:以Lorentz群 SO(3,1) 为例

该文给出了动力学群在群参数空间以及陪集空间上的右、左微分表示和伴随微分表示的符号计算方法.作为例子, 计算了Lorentz 群SO(3,1)的6 -参数和3 -参数的右、左及伴随微分表示,这些表示是旋转群SO(3)关于欧拉角和极角的微分表示的相对论性推广.特别,作者给出了伴随微分表示的两种不同的3 -参数形式,同时也得到了Wigner小群SO(2,1) 和 SO(3)$的6 -参数和3 -参数的相应表示.这些表示在相对论性量子陀螺的研究中可得到应用.     

Finite-dimensional algebraic representations of the infinite phylon group

The concept of phylon is introduced as a generalisation of derivative strings, differential strings and new tensors. The behaviour of phyla under change of coordinates is given by finite-dimensional algebraic representations of a very large group, the infinite phylon group. These representations are studied from both the general and the matrix points of view. Various examples of phyla are given, mainly from a statistical context. The basic structure of these representations is given.