[Paper Review] Massless Particles in QFT from Algebras without Involution

G.E. Johnson|arXiv (Cornell University)|May 19, 2012

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper extends a revised formulation of the Wightman axioms in quantum field theory (QFT) to include massless particles by modifying the algebra of test function sequences to exclude negative energies and ensure convergence of divergences. The key result is that massless particles are consistently incorporated in four or more spacetime dimensions, with scattering amplitudes computable despite LSZ asymptotic states not being tempered functions, provided the spacetime dimension satisfies $ d \geq 4 $.

ABSTRACT

The explicit realizations of quantum field theory (QFT) admitted by a revision to the Wightman axioms for the vacuum expectation values (VEV) of fields includes massless particles when there are four or more spacetime dimensions.

Motivation & Objective

• Address the challenge of including massless particles in a revised Wightman framework for quantum field theory.
• Overcome divergences in vacuum expectation values (VEVs) that arise when massless particles are included, particularly at $ E=0 $ and $ p^2=0 $.
• Ensure Poincaré covariance, microcausality, and spectral support are preserved in the presence of massless particles.
• Establish conditions under which continuous linear functionals and scattering amplitudes remain well-defined for massless fields.
• Identify the minimal spacetime dimension $ d \geq 4 $ required for the mathematical consistency of massless particle realizations.

Proposed method

• The paper revises the Borchers-Uhlmann algebra $ \mathcal{A} $ by introducing a subalgebra $ \mathcal{B} $ of test functions that vanish for all negative energies, using a smooth cutoff function $ h(E) $ that vanishes and all its derivatives at $ E=0 $.
• Generalized functions are constructed via integration against $ \delta(E \pm \omega) $, with $ T(\mathbf{p}) \in S'({\mathbf{R}}^{d-1}) $, ensuring support is excluded near the origin.
• Massless particles are accommodated by requiring $ \tilde{g}(p) $ and all its derivatives to vanish at $ E=0 $, removing singularities in $ \omega_j $ at $ \mathbf{p}_j^2 = 0 $.
• Scattering amplitudes are evaluated using LSZ states defined via $ \tilde{\ell}(p_k) = (\omega_k + E_k)e^{i\omega_k t}\tilde{f}(\mathbf{p}_k) $, even though these are not in $ \mathcal{B} $.
• The continuity of the Wightman functional is ensured by requiring $ d \geq 4 $, which makes $ R^{-2} $ locally summable in polar coordinates for the singular neighborhood of vanishing gradients.
• Energy-momentum conservation and gradient singularities are analyzed in the neighborhood $ V $ of points where $ \nabla \Phi = 0 $, with constraints $ \mathbf{e}_j^2 = -2s_j \mathbf{u}_1 \cdot \mathbf{e}_j $ preserving unit vector norms.

Experimental results

Research questions

• RQ1 Can massless particles be consistently incorporated into a revised Wightman framework for QFT without violating Poincaré covariance or microcausality?
• RQ2 How do divergences at $ E=0 $ and $ p^2=0 $ affect the continuity of vacuum expectation values when massless particles are included?
• RQ3 What is the minimal spacetime dimension $ d $ that allows for the existence of continuous linear functionals in the presence of massless particles?
• RQ4 Are scattering amplitudes still computable for massless particles when LSZ asymptotic states are not tempered test functions?
• RQ5 Under what conditions on the algebra of test functions does the generalized function $ \delta(P_{k(0)}) $ remain well-defined in the singular neighborhood of vanishing gradients?

Key findings

• Massless particles are consistently incorporated into the revised Wightman framework by modifying the algebra $ \mathcal{A} $ and its subalgebra $ \mathcal{B} $ to exclude negative energies and ensure smoothness at $ E=0 $.
• The inclusion of massless particles requires $ d \geq 4 $ for the generalized function $ \delta(P_{k(0)}) $ to be well-defined, as $ R^{-2} $ is locally summable only in $ d \geq 4 $.
• Scattering amplitudes for massless particles are computable using LSZ states even though these states are not in $ \mathcal{B} $, and the results match those in the finite mass case.
• The gradient singularity in the phase function $ \Phi $ vanishes only when all masses are zero or all are non-zero; for all-zero masses, the singularity is scale-invariant and parametrized by $ \mathbf{e}_j $ with $ \|\mathbf{e}_j\| < \epsilon $.
• Energy-momentum conservation and the vanishing gradient condition are decoupled in the massless case, with solutions parameterized by $ \mathbf{u}_j = s_j \mathbf{u}_1 + \mathbf{e}_j $, where $ \|\mathbf{e}_j\| $ is small.
• Continuous linear functionals exist for massless particles when $ \mathcal{B} $ is defined via $ \tilde{\varphi}[f_n] = \prod_{k=1}^n h(E_k/\beta_k) \tilde{f}_n((p)_n) $, with $ h(E) = e^{-1/E} $ for $ E>0 $, $ h(E)=0 $ otherwise.