Sum rules for forward virtual compton scattering.

Unitary representations of super-gauge symmetries

In this paper we give a more compact representation of the intelligent spin states defined by Aragone, Guerri, Salamo, and Tani. Using this new representation, we discuss the differences between minimum uncertainty states, coherent Bloch spin states and intelligent states. The evolution of these states under a particular time dependent Hamiltonian is studied, showing the relevance of the noncompact subgroup K of the Lorentz group. Finally we analyze the radiative properties connected with the intelligent states for a pointlike medium. The main results are: (I) they have a nonvanishing dipole moment (as the Bloch states) and (II) the proper intelligent states give a spontaneous emission intensity which is different from the one provided by the Bloch states.

On intelligent spin states

In order to construct the quantum field theory in a curved space with no “old” infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO0(n, 1) to positive mass representations of both the Euclidean and Poincare groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincare groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103858248

Contractions of representations of de Sitter groups

The scattering amplitude at vanishing four-momentum transfer is expanded by means of the matrix elements of the irreducible representations of the Lorentz group including space reflection and thePCT transformation. These representations are labelled by a complex parameter λ and other «quantum numbers». The high-energy asymptotic behavior of the amplitude is assumed to be determined by the contributions of some poles in the complex λ-plane (Lorentz poles). The contribution of a Lorentz pole is decomposed into a family of Reggepole contributions with given parities and signatures. The connection with other formalisms which give rise to families of Regge trajectories is briefly discussed. Particles with arbitrary spin and parity are considered. The selection rules for the quantum numbers of the Lorentz poles are treated in detail. The role of the internal symmetries and of the symmetry with respect to the perumation of identical particles is discussed. The general formalism is applied to the meson-nucleon and nucleon-nucleon scattering.

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An expansion of the scattering amplitude at vanishing four-momentum transfer using the representations of the Lorentz group

The unitary irreducible representations of the group SL(2C) belonging to the principal series restricted to the subgroup SU(1, 1) are decomposed into a direct integral of unitary irreducible representations of SU(1, 1) The matrix elements of the unitary operator which performs the decomposition are given explicitly and used to obtain a relation between the matrix elements of the unitary irreducible representations of the groups SL(2C) and SU(1, 1) Similar identities between the matrix elements of nonunitary representations of these groups are obtained by means of analytic continuation The relevance of these results to the theory of complex angular momentum and of high energy nearly forwardscattering is pointed out

Decomposition of the Unitary Irreducible Representations of the Group SL(2C) Restricted to the Subgroup SU(1, 1)

The relativistic two-body scattering amplitude is analysed in terms of irreducible representations of the Poincare group, which describe «objects» exchanged in the scattering process. These representations are obtained by the decomposition of the direct product of the representations, which describe an incoming and an outgoing particle, respectively. This procedure, which uses only the invariance and unitarity properties of the amplitude, is analogous to the usual partial-wave expansion, and we call it «crossed partial-wave analysis». Complex angular momentum appears as a quantum number describing the «exchanged objects». If the TCP invariance is taken into account, also the signature can be introduced in a natural way. The difficulties due to the fact that the representations considered are infinite-dimensional are discussed in detail. It is proved that the crossed partial-wave analysis can be performed for every invariant amplitude which satisfies the unitarity condition. However, the «crossed partial-wave amplitude» has to be considered as a bounded linear functional in a normed space of functions which depend on the quantum numbers which describe the «exchanged objects». Particles with arbitrary spins and parities are considered. The symmetry with respect to time and space inversions, and with respect to the exchange of identical particles, is taken into account. The role of the internal symmetry group is briefly discussed.

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On the group-theoretical approach to complex angular momentum and signature

A scattering process involving an arbitrary number of particles with arbitrary (not all vanishing) masses and arbitrary spins and parities is described in terms of amplitudes which are a generalization of the amplitudes introduced by Bali, Chew and Pignotti in their treatment of the multi-Regge model. The covariance properties of these functions are derived. It is proved that, if the boosts used in their definition are properly chosen in the complex Lorentz group, these amplitudes are free of kinematic singularities and constraints. The relevance of these results for the treatment of the kinematical details of the multi-Regge model is pointed out.

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On some properties of the multi-particle amplitude expressed as a function of group-theoretical variables.

Using the techniques developed in the first part of this work, we build a large class of families of conspiring Regge trajectories, which satisfy all the constraints of analyticity and factorization in a region of the momentum-transfer complex plane including the origin and the pseudothresholds. Equal- and unequal-mass reactions are treated in a unified way. Particles with arbitrary spins and parities are considered. The formalism automatically suggests a kind ofLS coupling scheme and gives rise to the right pseudothreshold factors. The pseudothreshold and the small-momentum-transfer behaviours of the Regge residues are given for all the daughter trajectories. The spin dependence of the high-energy forward-scattering amplitude is studied in detail.

/pdf/on-the-group-theoretical-approach-to-the-conspiracy-problem-41g7ac1u07.pdf

M. Toller

Papers

Decomposition of the Unitary Irreducible Representations of the Group SL(2C) Restricted to the Subgroup SU(1, 1)

An expansion of the scattering amplitude at vanishing four-momentum transfer using the representations of the Lorentz group

On the group-theoretical approach to complex angular momentum and signature

On some properties of the multi-particle amplitude expressed as a function of group-theoretical variables.

On the group-theoretical approach to the conspiracy problem for arbitrary masses. - II