Abstract
The problem of accounting for the quantum degrees of freedom in passing from massive higher-spin potentials to massless ones, and the inverse problem of “fattening” massless tensor potentials of helicity ±h to their massive s=|h| counterparts, are solved – in a perfectly ghost-free approach – using “string-localized fields”.This approach allows to overcome the Weinberg–Witten impediment against the existence of massless |h|≥2 energy–momentum tensors, and to qualitatively and quantitatively resolve the van Dam–Veltman–Zakharov discontinuity concerning, e.g., very light gravitons, in the limit m→0.
Highlights
In relativistic quantum field theory, the quantization of interacting massless or massive classical potentials of higher spin (s ≥ 1) either violates Hilbert space positivity which is an indispensable attribute of the probability interpretation of quantum theory, or leads to a violation of the power counting bound of renormalizability whose maintenance requires again a violation of positivity
In order to save positivity for those quantum fields which correspond to classically gauge invariant observables, one usually formally extends the theory by adding degrees of freedom in the form of negative metric Stückelberg fields and “ghosts” without a counterpart in classical gauge theories. The justification for this quantum gauge setting is that one can extract from the indefinite metric Krein space a Hilbert space that the gauge invariant operators generate from the vacuum
We have identified string-localized potentials for massive particles of integer spin s on the Hilbert space of their field strengths, that admit a smooth massless limit to decoupled potentials with helicities h = ±r, r ≤ s
Summary
In relativistic quantum field theory, the quantization of interacting massless or massive classical potentials of higher spin (s ≥ 1) either violates Hilbert space positivity which is an indispensable attribute of the probability interpretation of quantum theory, or leads to a violation of the power counting bound of renormalizability whose maintenance requires again a violation of positivity. The second is the DVZ observation due to van Dam and Veltman [25] and to Zakharov [28], that in interacting models with s ≥ 2, scattering amplitudes are discontinuous in the mass at m = 0, i.e., the scattering of matter through exchange of massless gravitons (say) is significantly different from the scattering via gravitons of a very small mass Both problems can be addressed, without being plagued by the positivity troubles of gauge theories, with the help of “stringlocalized quantum fields” defined in the physical Hilbert space. A point-localized massive spin s potential can be split up into a string-localized potential that has a massless limit, and derivatives of one or more so-called “escort fields” The role of the latter is to separate off derivative terms from the interaction Lagrangean or from conserved currents, that do not contribute to the S-matrix or to charges and Poincaré generators, respectively. The presentation of this solution is the principal aim of this letter, including the opposite direction, sometimes (in connection with the Higgs mechanism) referred to as “fattening”
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