Abstract
Massive and massless potentials play an essential role in the perturbative formulation of particle interactions. Many difficulties arise due to the indefinite metric in gauge theoretic approaches, or the increase with the spin of the UV dimension of massive potentials. All these problems can be evaded in one stroke: modify the potentials by suitable terms that leave unchanged the field strengths, but are not polynomial in the momenta. This feature implies a weaker localization property: the potentials are “string-localized”. In this setting, several old issues can be solved directly in the physical Hilbert space of the respective particles: We can control the separation of helicities in the massless limit of higher spin fields and conversely we recover massive potentials with 2s+1 degrees of freedom by a smooth deformation of the massless potentials (“fattening”). We construct stress–energy tensors for massless fields of any helicity (thus evading the Weinberg–Witten theorem). We arrive at a simple understanding of the van Dam–Veltman–Zakharov discontinuity concerning, e.g., the distinction between a massless or a very light graviton. Finally, the use of string-localized fields opens new perspectives for interacting quantum field theories with, e.g., vector bosons or gravitons.
Highlights
The purpose of this contribution is to formulate and investigate a unified setting for potentials describing both massless and massive vector and tensor bosons, that live in Hilbert space
While Fronsdal proves the positivity of the 2-point function contracted with constrained sources, most of the more recent work concentrates on Lagrangians and field equations without even addressing the crucial issues of quantization: positivity and causal localization
The massless symmetric tensor potentials A(r)(x, e) are traceless and conserved. They satisfy in addition the axial gauge condition eμA(μrμ) 2...μr (x, e) = 0. They are string-localized potentials given by the same formula Eq (3.1) for the massless field strengths associated with the Wigner representations of helicity h = ±r [42]
Summary
The purpose of this contribution is to formulate and investigate a unified setting for potentials describing both massless and massive vector and tensor bosons, that live in Hilbert space (i.e., without negative-norm states even at intermediate steps). The e-independence of the causal S matrix requires at the quantum level that the time-ordering can be defined in such a way that total derivatives are preserved These conditions impose already in lowest orders certain constraints on the possible interactions, that are all realized in the Standard Model: the Lie algebra structure of cubic couplings of several species of vector bosons [38]; the presence of a Higgs field when there are non-Abelian massive vector bosons [38,26]; and the chirality of their coupling to fermions [17]. In scalar massive QED, the cubic part of the string-local minimal coupling induces the quartic part [38], so that the full fibre-bundle-like structure of the quantum theory turns out to be a consequence of imposing e-independence of the unitary S matrix, of positivity and causality, rather than a classical local gauge symmetry We wonder whether this remarkable feature extends to higher spin and gravitational couplings, possibly demanding additional couplings to lower spin fields. It becomes clear that the role of the Higgs boson is not the generation of the mass, but the preservation of the renormalizability and locality under the constraints imposed by positivity [38,26]
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