Continuous Spin Representations Research Articles

Continuous spin representation from contraction of the conformal algebra

In this paper, we discuss the Inönü–Winger contraction of the conformal algebra. We start with the light-cone form of the Poincaré algebra and extend it to write down the conformal algebra in d dimensions. To contract the conformal algebra, we choose five dimensions for simplicity and compactify the third transverse direction to a circle of radius R following the Kaluza–Klein dimensional reduction method. We identify the inverse radius, 1/R, as the contraction parameter. After the contraction, the resulting representation is found to be the continuous spin representation in four dimensions. Even though the scaling symmetry survives the contraction, the special conformal translation vector changes and behaves like the four-momentum vector. We also discuss the generalization to d dimensions.

Superfield continuous spin equations of motion

We propose a description of manifestly supersymmetric continuous spin representations in 4D,N=1 Minkowski superspace at the level of equations of motions. The usual continuous spin wave function is promoted to a chiral or a complex linear superfield which includes both the single-valued (span integer helicities) and the double-valued (span half-integer helicities) representations thus making their connection under supersymmetry manifest. The set of proposed superspace constraints for both superfield generate the expected Wigner's conditions for both representations.

BRST approach to Lagrangian construction for bosonic continuous spin field

We formulate the conditions defining the irreducible continuous spin representation of the four-dimensional Poincaré group based on spin-tensor fields with dotted and undotted indices. Such a formulation simplifies analysis of the Bargmann–Wigner equations and reduces the number of equations from four to three. Using this formulation we develop the BRST approach and derive the covariant Lagrangian for the continuous spin fields.

Infinite (continuous) spin fields in the frame-like formalism

In this paper we elaborate on the gauge invariant frame-like Lagrangian description for the wide class of the so-called infinite (or continuous) spin representations of Poincaré group. We use our previous results on the gauge invariant formalism for the massive mixed symmetry fields corresponding to the Young tableau with two rows Y(k,l) (Y(k+1/2,l+1/2) for the fermionic case). We have shown that the corresponding infinite spin solutions can be constructed as a limit where k goes to infinity, while l remain to be fixed and label different representations. Moreover, our gauge invariant formalism provides a natural generalization to (Anti) de Sitter spaces as well. As in the completely symmetric case considered earlier by Metsaev we have found that there are no unitary solutions in de Sitter space, while there exists a rather wide spectrum of Anti de Sitter ones. In this, the question what representations of the Anti de Sitter group such solutions correspond to remains to be open.

Infinite Spin Fields in d = 3 and Beyond

In this paper, we consider the frame-like formulation for the so-called infinite (continuous) spin representations of the Poincare algebra. In the three-dimensional case, we give explicit Lagrangian formulation for bosonic and fermionic infinite spin fields (including the complete sets of the gauge-invariant objects and all the necessary extra fields). Moreover, we find the supertransformations for the supermultiplet containing one bosonic and one fermionic field, leaving the sum of their Lagrangians invariant. Properties of such fields and supermultiplets in four and higher dimensions are also briefly discussed.

A comment on continuous spin representations of the Poincaré group and perturbative string theory

AbstractWe make a simple observation that the massless continuous spin representations of the Poincaré group are not present in perturbative string theory constructions. This represents one of the very few model‐independent low‐energy consequences of these models.

A Riccati type PDE for light-front higher helicity vertices

This paper is based on a curious observation about an equation related to the tracelessness constraints of higher spin gauge fields. The equation also occurs in the theory of continuous spin representations of the Poincar\'e group. Expressed in an oscillator basis for the higher spin fields, the equation becomes a non-linear partial differential operator of the Riccati type acting on the vertex functions. The consequences of the equation for the cubic vertex is investigated in the light-front formulation of higher spin theory. The classical vertex is completely fixed but there is room for off-shell quantum corrections.

BRST theory for continuous spin

Some puzzling aspects of higher spin field theory in Minkowski space-time, such as the tracelessness constraints and the search for an underlying physical principle, are discussed. A connecting idea might be provided by the recently much researched continuous spin representations of the Poincar\'e group. The Wigner equations, treated as first class constraints, yields to a four-constraint BRST formulation. The resulting field theory, generalizing free higher spin field theory, is one among a set of higher spin theories that can be related to previous work on unconstrained formulations. In particular, it is conjectured that the unconstrained higher spin theory of Francia and Sagnotti is a limit of a continuous spin theory. Furthermore, a simple analysis of the constraint structure reveals a hint of a physical rationale behind the trace constraints.

The continuous spin limit of higher spin field equations

We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin field equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the spin to infinity with their product being fixed. The result allows to transform the Wigner equations to a gauge invariant Fronsdal-like form. We also give the generalisation of the Wigner equations to higher dimensions with fields belonging to arbitrary representations of the massless little group.

Publisher's Note: “Continuous spin representations from group contraction” [J. Math. Phys. 46, 053515 (2005)

Publisher's Note: “Continuous spin representations from group contraction” [J. Math. Phys. 46, 053515 (2005)

Continuous spin representations from group contraction

We consider how the continuous spin representation (CSR) of the Poincaré group in four dimensions can be generated by dimensional reduction. The analysis uses the front-form little group in five dimensions, which must yield the Euclidean group E(2), the little group of the CSR. We consider two cases, one is the single spin massless representation of the Poincaré group in five dimensions, the other is the infinite component Majorana equation, which describes an infinite tower of massive states in five dimensions. In the first case, the double singular limit j, R→∞, with j∕R fixed, where R is the Kaluza–Klein radius of the fifth dimension, and j is the spin of the particle in five dimensions, yields the CSR in four dimensions. It amounts to the Inönü–Wigner contraction, with the inverse Kaluza–Klein radius as contraction parameter. In the second case, the CSR appears only by taking a triple singular limit, where an internal coordinate of the Majorana theory goes to infinity, while leaving its ratio to the Kaluza–Klein radius fixed.

Continuous spin representations of the Poincaré and super-Poincaré groups

We construct Wigner’s continuous spin representations of the Poincaré algebra for massless particles in higher dimensions. The states are labeled both by the length of a spacelike translation vector and the Dynkin indices of the short little group SO(d−3), where d is the space–time dimension. Continuous spin representations are in one-to-one correspondence with representations of the short little group. We also demonstrate how combinations of the bosonic and fermionic representations form supermultiplets of the super-Poincaré algebra. If the light-cone translations are nilpotent, these representations become finite dimensional, but contain zero or negative norm states, and their supersymmetry algebra contains a central charge in four and ten dimensions.

MODULAR LOCALIZATION AND WIGNER PARTICLES

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

A classical theory of continuous spin and hidden gauge invariance

We present a classical higher derivative point particle theory whose quantization gives Wigner's continuous spin representation of the Poincaré group. Although the theory is not reparametrization-invariant in the usual sense, it does possess a hidden gauge invariance that provides a non-local representation of the reparametrization group. The Hamiltonian of the theory does not vanish and its value is the continuous spin parameter. The theory presented here represents the simples example of a wide class of higher derivative theories possessing a hidden gauge invariance.

Realizations of generators of complex angular momentum

We use the generators of complex angular momentum in complex c3 space and derive the realizations of the homogeneous Lorentz group for nonzero real mass, zero mass, and imaginary mass systems. We use the appropriate little group for different systems to calculate the modifications in the spin contribution to angular momentum and the unphysical continuous spin representations are shown to be eliminated. We diagonalize the helicity operator in c3 space and obtain the generators of complex angular-momentum operators, which are shown to lead, in a general manner, to the standard helicity representations of the Poincare group for timelike and spacelike systems.

Nuclearity of the differentiable vectors of continuous spin representations

It is shown for continuous spin, mass zero, and unitary irreducible representations of the Poincare group, the space of differentiable vectors is nuclear.

Physical Significance of the Topology of the Bondi-Metzner-Sachs Group

It has been proposed that the Bondi-Metzner-Sachs group affords a possible explanation for the discreteness of elementary-particle spins. This discreteness result depends on the topology given to the group. A careful analysis of representations in the nuclear topology indicates that discrete-spin representations (those subsumed by the original topology) describe bound gravitating systems (elementary particles) whereas the new continuous-spin representations, if physically realistic, describe scattering states of gravitating systems.

Structure of the Bondi-Metzner-Sachs Group

A simple example of an aymptotic symmetry group in two dimensions is described. The structure of the corresponding group for asymptotically flat (four-dimensional) space-times, the BMS group, is given explicitly. The recent result that all induced representations of the BMS group have discrete spins is explained in terms of the relationship between this group and the Poincaré group. In fact, it is shown that the BMS group is, in a sense, the smallest generalization of the Poincaré group which eliminates the (physically embarrassing) continuous spin representations of the Poincaré group.

On Representations of the Conformal Group Which When Restricted to Its Poincaré or Weyl Subgroups Remain Irreducible

Unitary irreducible representations of the conformal group which, when restricted to its Poincaré or Weyl (= Poincaré group extended by dilatations) subgroups, remain irreducible are found. In particular it is proved that the continuous spin representations (pμ·pμ = 0) of the Poincaré group cannot be extended to the conformal group and that, on the other hand, a known extension in the discrete spin case is unique (up to a unitary equivalence). Similar results hold for Weyl group for which, in addition, extensions also exist in the case pμ·pμ ≠ 0. Namely, each unitary irreducible representation of the Weyl group characterized by the sign of pμ·pμ (≠0) and by invariants of the corresponding little group can be extended to a one-parameter family of irreducible representations of the conformal group. Finally, it is shown that, besides the above mentioned extensions of unitary irreducible representations of the Weyl group, there are no others.

Remarks on Lightlike Continuous Spin and Spacelike Representations of the Poincaré Group

Certain classes of unitary representations of the lightlike continuous spin and the spacelike cases are constructed. The generators in these representations involve operators forming the ``intrinsic'' algebras (i.e., commuting with the orbital parts) E3 and O(3, 1) for P2 = 0 and P2 < 0, respectively. The parallel construction for P2 > 0 involving an intrinsic O4 algebra is indicated. Equivalence relations with certain other forms are given through a unitary transformation. The physical significance of the ``translation'' generators of E3 is brought out in terms of the projections of W orthogonal to P. Corresponding results for the O(3, 1) and O(4) algebras are given. For the continuous-spin case, these operators are shown to provide a basis with extremely simple transformation properties, related to a certain symmetric toplike behavior even under Lorentz transformations. With a view to future use, the matrix elements of W on the energy-rotation basis are calculated in a unified manner for all the three cases, P2⋛0. A deformation formula leading from zero-mass continuous-spin representations to spacelike ones is studied. Certain types of nonunitary representations are briefly introduced at the end.