Abstract
Crossing symmetry asserts that particles are indistinguishable from antiparticles traveling back in time. In quantum field theory, this statement translates to the long-standing conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this work we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta one can smoothly interpolate between pairs of crossing channels without encountering singularities or violating mass-shell conditions and momentum conservation. The analytic continuation can be realized using two types of moves. The first one makes use of an $iϵ$ prescription for avoiding singularities near the physical kinematics and allows us to adjust the momenta of the external particles relative to one another within their light cones. The second, more violent, step involves a rotation of subsets of particle momenta via their complexified light cones from the future to the past and vice versa. We show that any singularity along such a deformation would have to correspond to two beams of particles scattering off each other. For planar Feynman diagrams, these kinds of singularities are absent because of the particular flow of energies through their propagators. We prescribe a five-step sequence of such moves that combined together proves crossing symmetry for planar scattering amplitudes in perturbation theory, paving a way towards settling this question for more general scattering processes in quantum field theories.
Highlights
Particles are indistinguishable from antiparticles with the opposite energy and momentum [1]
In order to meaningfully formulate the question of crossing symmetry, we must assume that the amplitude exists in the first place, i.e., it can be defined in some open set of the physical region of interest
The only secondtype singularity for this diagram corresponds to collinear kinematics, yielding stu 1⁄4 0, which demarcates boundaries of the physical regions
Summary
Particles are indistinguishable from antiparticles with the opposite energy and momentum [1]. In order to convert this statement into an observable in quantum field theory, we can phrase it as measuring the two scenarios in a scattering experiment At this level, crossing symmetry states that on-shell scattering amplitudes for processes involving the particle and the antiparticle are boundary values of one and the same function, regardless of the number and type of the remaining particles it interacts with. It is a fundamentally Lorentzian notion thought to be a reflection of the compatibility of quantum theory with physical principles such as causality, locality, or unitarity. While there is nothing wrong with making simplifying assumptions, in view of the author, axioms without a clear physical meaning should not form a basis for a physical theory
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