scholarly journals A compendium of sphere path integrals

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Y.T. Albert Law
Keyword(s):  
Standard Procedure ◽  
Path Integrals ◽  
Symmetric Tensor ◽  
Arbitrary Spin ◽  
Tensor Fields ◽  
Arbitrary Integer ◽  
Integer Spin ◽  
Higher Spin ◽  
Detailed Derivation ◽  
Massless Fields

Abstract We study the manifestly covariant and local 1-loop path integrals on Sd+1 for general massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin s ≥ 0 in any dimensions d ≥ 2. After reviewing the cases of massless fields with spin s = 1, 2, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins s ≥ 1. Following the standard procedure of Wick-rotating the negative conformal modes, we find a higher spin analog of Polchinski’s phase for any integer spin s ≥ 2. The derivations for low-spin (s = 0, 1, 2) massive, shift-symmetric and partially massless fields are also carried out explicitly. Finally, we provide general prescriptions for general massive and shift-symmetric fields of arbitrary integer spins and partially massless fields of arbitrary integer spins and depths.

scholarly journals The σ− Cohomology Analysis for Symmetric Higher-Spin Fields

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1498
Author(s):  
Alexey S. Bychkov ◽  
Kirill A. Ushakov ◽  
Mikhail A. Vasiliev
Keyword(s):  
Minkowski Space ◽  
Dimensional Case ◽  
Arbitrary Integer ◽  
Complete Proof ◽  
Four Dimensions ◽  
Integer Spin ◽  
Half Integer ◽  
Higher Spin ◽  
Spin Fields ◽  
Massless Fields

In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions. This is achieved by calculation of the respective σ− cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in AdS4. In the d-dimensional case Hp(σ−) is computed for any p and interpretation of Hp(σ−) is given both for the original Fronsdal system and for the associated systems of higher form fields.

UNCONSTRAINED HIGHER SPINS IN FOUR DIMENSIONS

2011 ◽  
Vol 08 (03) ◽  
pp. 511-556 ◽  
Author(s):  
GIUSEPPE BANDELLONI
Keyword(s):  
Equations Of Motion ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Geometrical Meaning ◽  
Four Dimensions ◽  
Spin Field ◽  
Angular Momenta ◽  
Higher Spin ◽  
Spin Fields ◽  
The Right

The relativistic symmetric tensor fields are, in four dimensions, the right candidates to describe Higher Spin Fields. Their highest spin content is isolated with the aid of covariant conditions, discussed within a group theory framework, in which auxiliary fields remove the lower intrinsic angular momenta sectors. These conditions are embedded within a Lagrangian Quantum Field theory which describes an Higher Spin Field interacting with a Classical background. The model is invariant under a (B.R.S.) symmetric unconstrained tensor extension of the reparametrization symmetry, which include the Fang–Fronsdal algebra in a well defined limit. However, the symmetry setting reveals that the compensator field, which restore the Fang–Fronsdal symmetry of the free equations of motion, is in the existing in the framework and has a relevant geometrical meaning. The Ward identities coming from this symmetry are discussed. Our constraints give the result that the space of the invariant observables is restricted to the ones constructed with the Highest Spin Field content. The quantum extension of the symmetry reveals that no new anomaly is present. The role of the compensator field in this result is fundamental.

scholarly journals MASSIVE FIELDS WITH ARBITRARY INTEGER SPIN IN HOMOGENEOUS ELECTROMAGNETIC FIELD

2000 ◽  
Vol 15 (04) ◽  
pp. 535-552 ◽  
Author(s):  
S. M. KLISHEVICH
Keyword(s):  
Electromagnetic Field ◽  
Massive Particle ◽  
Tensor Fields ◽  
Arbitrary Integer ◽  
Integer Spin ◽  
Constant Electromagnetic Field ◽  
Electromagnetic Field Strength ◽  
Spin Fields ◽  
Homogeneous Electromagnetic Field ◽  
Dimensionality Of Space

We study the interaction of gauge fields of arbitrary integer spins with the constant electromagnetic field. We reduce the problem of obtaining the gauge-invariant Lagrangian of integer spin fields in the external field to purely algebraic problem of finding a set of operators with certain features using the representation of the high-spin fields in the form of vectors in a pseudo-Hilbert space. We consider such a construction up to the second order in the electromagnetic field strength and also present an explicit form of interaction Lagrangian for a massive particle of spin s in terms of symmetrical tensor fields in linear approximation. The result obtained does not depend on dimensionality of space–time.

THE MODIFIED BARGMANN-WIGNER FORMALISM FOR HIGHER SPIN FIELDS AND RELATIVISTIC QUANTUM MECHANICS

2011 ◽  
Vol 03 ◽  
pp. 121-132 ◽  
Author(s):  
VALERIY V. DVOEGLAZOV
Keyword(s):  
Relativistic Quantum ◽  
Standard Procedure ◽  
Symmetric Tensor ◽  
Riemann Tensor ◽  
Relativistic Quantum Mechanics ◽  
Dynamical Equations ◽  
Higher Spin ◽  
Spin Fields ◽  
Higher Spins ◽  
Dirac Formalism

On the basis of our recent modifications of the Dirac formalism we generalize the Bargmann-Wigner formalism for higher spins to be compatible with other formalisms for bosons and fermions. Connections with dual electrodynamics, with the Ogievetskii-Polubarinov notoph and the Weinberg 2(2 S +1) theory are found. Next, we proceed to derive the equations for the symmetric tensor of the second rank on the basis of the Bargmann-Wigner formalism in a straightforward way. The symmetric multispinor of the fourth rank is used. It is constructed out of the Dirac 4-spinors. Due to serious problems with the interpretation of results we generalize the standard procedure and we obtain the spin-2 relativistic equations, which are consistent with the Einstein-Hilbert equation. We introduce the dual analogues of the Riemann tensor and we derive corresponding dynamical equations in the Minkowski space. Connections with the Marques-Spehler chiral gravity theory are discussed. The importance of the 4-vector field (and its gauge part) is pointed out. The spin-3/2 case is briefly discussed too.

A BRST APPROACH TO MASSLESS GAUGE FIELDS OF INTEGER SPIN

1990 ◽  
Vol 05 (16) ◽  
pp. 3247-3264
Author(s):  
THOMAS DOMEIJ
Keyword(s):  
Wave Function ◽  
Symmetric Tensor ◽  
Flat Space ◽  
Gauge Fields ◽  
Arbitrary Integer ◽  
Massless Particles ◽  
Gauge Invariant ◽  
Function Representation ◽  
Integer Spin ◽  
Do So

We present a method based on BRST techniques for how to obtain the conventional equations for symmetric tensor gauge fields which describe free massless particles of arbitrary integer spin. We do so in a wave function representation in a flat space-time background of any dimension. A fairly straightforward way is also presented for how to obtain the equations from a gauge invariant Lagrangian of the form [Formula: see text].

scholarly journals Off-shell higher-spin fields in AdS4 and external currents

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
N.G. Misuna
Keyword(s):  
Spin System ◽  
Wave Equations ◽  
Current System ◽  
De Sitter ◽  
Arbitrary Integer ◽  
Shell System ◽  
Integer Spin ◽  
Higher Spin ◽  
Spin Fields ◽  
Higher Spin Fields

Abstract We construct an unfolded system for off-shell fields of arbitrary integer spin in 4d anti-de Sitter space. To this end we couple an on-shell system, encoding Fronsdal equations, to external Fronsdal currents for which we find an unfolded formulation. We present a reduction of the Fronsdal current system which brings it to the unfolded Fierz-Pauli system describing massive fields of arbitrary integer spin. Reformulating off-shell higher-spin system as the set of Schwinger–Dyson equations we compute propagators of higher-spin fields in the de Donder gauge directly from the unfolded equations. We discover operators that significantly simplify this computation, allowing a straightforward extraction of wave equations from an unfolded system.

POINT PARTICLE–SYMMETRIC TENSORS INTERACTION AND GENERALIZED GAUGE PRINCIPLE

2003 ◽  
Vol 18 (27) ◽  
pp. 5021-5038 ◽  
Author(s):  
ARKADY Y. SEGAL
Keyword(s):  
Symmetric Tensor ◽  
Point Particle ◽  
Gauge Fields ◽  
Symmetric Tensors ◽  
Tensor Fields ◽  
First Order ◽  
Higher Spin ◽  
Infinite Collection ◽  
Higher Rank ◽  
Particle Action

The model of a point particle in the background of external symmetric tensor fields is analyzed from the higher spin theory perspective. It is proposed that the gauge transformations of the infinite collection of symmetric tensor fields may be read off from the covariance properties of the point particle action w.r.t. general canonical transformations. The gauge group turns out to be a semidirect product of all phase space canonical transformations to an Abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. A general configuration of external fields includes rank-0,1,2 symmetric tensors, so the whole system may be truncated to ordinary particle in Einstein–Maxwell backgrounds by switching off the higher-rank symmetric tensors. When otherwise all the higher rank tensors are switched on, the full gauge group provides a huge gauge symmetry acting on the whole infinite collection of symmetric tensors. We analyze this gauge symmetry and show that the symmetric tensors which couple to the point particle should not be interpreted as Fronsdal gauge fields, but rather as gauge fields of some conformal higher spin theories. It is shown that the Fronsdal fields system possesses twice as many symmetric tensor fields as is contained in the general background of the point particle. Besides, the particle action in general backgrounds is shown to reproduce De Wit–Freedman point particle–symmetric tensors first order interaction suggested many years ago, and extends their result to all orders in interaction, while the generalized equivalence principle completes the first order covariance transformations found in their paper, in all orders.

scholarly journals MASSIVE FIELDS WITH ARBITRARY HALF-INTEGER SPIN IN CONSTANT ELECTROMAGNETIC FIELD

2000 ◽  
Vol 15 (05) ◽  
pp. 609-624 ◽  
Author(s):  
S. M. KLISHEVICH
Keyword(s):  
Electromagnetic Field ◽  
External Electromagnetic Field ◽  
Tensor Fields ◽  
Gauge Transformations ◽  
Integer Spin ◽  
Half Integer ◽  
Constant Electromagnetic Field ◽  
Higher Spin ◽  
Spin Fields ◽  
Homogeneous Electromagnetic Field

We study the interaction of gauge fields of arbitrary half-integer spins with the homogeneous electromagnetic field. We reduce the problem of obtaining the gauge-invariant Lagrangian and transformations of the half-integer spin fields in the external field to an algebraic problem of search for a set of operators with certain algebraic features using the representation of the higher-spin fields as vectors in a pseudo-Hilbert space. We consider such construction at linear order in the external electromagnetic field and also present an explicit form of interaction Lagrangians and gauge transformations for the massive particles of spins [Formula: see text] and [Formula: see text] in terms of symmetric spin-tensor fields. The obtained result is valid for space–time of arbitrary even dimension.

scholarly journals AdS (super)projectors in three dimensions and partial masslessness

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Daniel Hutchings ◽  
Sergei M. Kuzenko ◽  
Michael Ponds
Keyword(s):  
Isometry Group ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Irreducible Representations ◽  
Projection Operators ◽  
De Sitter ◽  
Arbitrary Integer ◽  
Higher Spin ◽  
Casimir Operators ◽  
Massless Fields

Abstract We derive the transverse projection operators for fields with arbitrary integer and half-integer spin on three-dimensional anti-de Sitter space, AdS3. The projectors are constructed in terms of the quadratic Casimir operators of the isometry group SO(2, 2) of AdS3. Their poles are demonstrated to correspond to (partially) massless fields. As an application, we make use of the projectors to recast the conformal and topologically massive higher-spin actions in AdS3 into a manifestly gauge-invariant and factorised form. We also propose operators which isolate the component of a field that is transverse and carries a definite helicity. Such fields correspond to irreducible representations of SO(2, 2). Our results are then extended to the case of $$ \mathcal{N} $$ N = 1 AdS3 supersymmetry.

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