Abstract
In spacetimes of any dimensionality, the massless particle states that can be created and destroyed by a field in a given representation of the Lorentz group are severely constrained by the condition that the invariant Abelian subgroup of the little group must leave these states invariant. A number of examples are given of the massless one-particle states that can be described by various tensor and spinor-tensor fields, and a speculation is offered for the general case.
Highlights
For a century physicists have speculated that our familiar four-dimensional spacetime may really be embedded in a higher dimensional continuum [1]
It is simplest and most usual to suppose that this continuum is a d-dimensional spacetime in which in locally inertial frames the laws of nature are invariant under the Lorentz group SOðd − 1; 1Þ
Since the generators −i1⁄2Γi; Γj of the SOðd − 2Þ subgroup of the little group commute with Γμkμ, and there are no other conditions on u, the massless particle states described by this field transform according to the fundamental spinor representation of SOðd − 2Þ
Summary
For a century physicists have speculated that our familiar four-dimensional spacetime may really be embedded in a higher dimensional continuum [1]. The matrix element (5) (and the free field) is necessarily the antisymmetrized second derivative of a symmetric second rank potential This is the generalization of the result for d 1⁄4 4 that the Weyl field has Lorentz transformation of type ð2; 0Þ ⊕ ð0; 2Þ and can only describe massless particles with helicity Æ2. The tensor field Cμνρσ is the second spacetime derivative of a symmetric potential hμν, but since this potential is constructed only from the creation and annihilation operators of gravitons of helicity Æ2, it cannot transform as a true tensor under Lorentz transformations, but rather as a tensor only up to a gauge transformation, such as the metric perturbation in transverse-traceless gauge From this point of view, gauge invariance is not a fundamental assumption, as it is in Labastida’s work, but is rather a consequence of the Lorentz transformation properties of fields and massless particle states
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