[Paper Review] Amplitudes for space-like separations and causality

Sebastian P. Horvath, D. Schritt+1 more|arXiv (Cornell University)|Oct 6, 2011

Neutrino Physics Research43 references18 citations

TL;DR

This paper develops a relativistic quantum field theory formalism to describe space-like propagation of massive fermions, showing that for ultra-light particles like neutrinos, non-zero detection amplitudes can occur at space-like separations without violating causality. Applying the formalism to OPERA and ICARUS experiments, it finds that while OPERA's 60 ns early-arrival peak is not reproduced (as predictions peak on the light-cone), ICARUS data for μeV-scale neutrino masses are consistent with the model, suggesting high-precision neutrino speed experiments could measure absolute neutrino masses.

ABSTRACT

This paper explores whether quantum field theory allows the events of emission and absorption of a single particle to be separated by a space-like interval without violating Lorentz symmetries and causality. Although the answer is indeed affirmative, traditionally such effects have been considered negligible. We show that for sufficiently light mass eigenstates such processes can become significant over macroscopic length scales. A critical review of the historical literature reveals various shortcomings of the standard methods; specifically, one finds that they are restricted to states for which the expectation value of momentum vanishes. Furthermore, the results obtained here correct Feynman's analysis of this subject. A formalism is thus developed that allows the description of states with non-zero momentum, which is then applied to the OPERA and ICARUS neutrino-speed experiments. For OPERA we choose a mass in the nano electron-volt range and find that although our formalism predicts a non-zero detection probability for an early arrival time of 60 ns, the predicted event distribution is maximal on the light-cone. Consequently, our prediction does not reproduce the peak at 60 ns reported by the OPERA collaboration. Turning to the ICARUS experiment, we note that while the collaboration reported an average time of flight that is consistent with the speed of light, the event data with its associated uncertainties nevertheless indicates that some of the detection events are separated from their corresponding emission events by a space-like interval. For a micro electron-volt mass range, this is in agreement with the here reported formalism. We thus raise the possibility of employing high-precision neutrino-speed experiments to determine the absolute masses of neutrino mass eigenstates.

Motivation & Objective

• To investigate whether massive particles in quantum field theory can influence events at space-like separations without violating Lorentz invariance or causality.
• To address limitations in prior analyses that assumed zero-momentum states, which restricts applicability to realistic particle beams with non-zero momentum.
• To correct Feynman’s earlier analysis by developing a formalism that includes non-zero momentum states for massive fermions.
• To test the formalism against experimental data from OPERA and ICARUS neutrino-speed experiments.
• To explore the feasibility of using high-precision neutrino speed measurements to determine the absolute masses of neutrino mass eigenstates.

Proposed method

• The study uses a relativistic Dirac field formalism with explicit momentum-dependent spinor wavefunctions, incorporating Lorentz transformations to describe moving particles.
• It derives the amplitude for a detector to respond to a particle emitted from a source at space-like separation using the field operator Ψ†(x) acting on the vacuum state.
• The formalism includes both particle and antiparticle contributions via creation/annihilation operators, ensuring causality is preserved through the Feynman-Stueckelberg interpretation.
• The probability amplitude is calculated as a function of time and distance, with the result expressed in terms of Bessel functions and Hankel functions via integral representations.
• The model is applied to the OPERA experiment using a neutrino mass of ~10⁻² neV and to ICARUS with a mass of ~1 μeV to compare predicted event distributions with data.
• The analysis accounts for detector uncertainties and compares the shape of predicted detection probabilities with observed time-of-flight distributions.

Experimental results

Research questions

• RQ1 Can massive fermions in quantum field theory trigger detectors at space-like separations without violating causality or Lorentz symmetry?
• RQ2 Why do standard analyses fail to describe non-zero momentum states, and how can this limitation be overcome?
• RQ3 Does the proposed formalism reproduce the 60 ns early-arrival signal reported by the OPERA collaboration?
• RQ4 Is the ICARUS experiment's time-of-flight data consistent with space-like propagation for light neutrino mass eigenstates?
• RQ5 Can high-precision neutrino speed experiments be used to measure the absolute masses of neutrino mass eigenstates?

Key findings

• For a neutrino mass of ~10⁻² neV, the formalism predicts a non-zero detection probability for events arriving 60 ns early, but the peak of the distribution is centered on the light-cone, not at 60 ns, thus failing to reproduce the OPERA peak.
• For a neutrino mass of ~1 μeV, the predicted detection probability distribution from the formalism shows visual agreement with the ICARUS data, despite only seven data points being available.
• The model's predictions are consistent with the observed time delay of neutrinos from SN1987a, supporting the validity of space-like amplitudes in astrophysical contexts.
• The formalism corrects Feynman’s earlier analysis by including non-zero momentum states, which previous methods had excluded.
• The results suggest that high-precision neutrino speed experiments could serve as a probe for measuring the absolute masses of neutrino mass eigenstates, particularly in the μeV to neV range.
• The study demonstrates that for ultra-light particles, space-like propagation effects can become macroscopically significant, challenging the assumption that such effects are always negligible.