* what is phase space? It is a curious eight-dimensional world that merges our familiar four dimensions of space and time and a four-dimensional world called momentum space.
* When you look at the world around you you don’t ever observe space or time – instead you see energy and momentum. When you look at your watch, for example, photons bounce off a surface and land on your retina. By detecting the energy and momentum of the photons, your brain reconstructs events in space and time.
* For observers in a curved momentum space, however, even space-time is relative
Relative Locality – if you are 10 billion light years from a supernova and the energy of its light is about 10 gigaelectronvolts, then your measurement of its location in space-time would differ from a local observer’s by a light second. That may not sound like much, but it amounts to 300,000 kilometres
We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature
The crucial idea underlying and unifying all these developments turns out to be one that is a direct consequence of the curvature of momentum space: the relativity of locality. Even apart from fundamental physics, there are situations in condensed matter physics, where it is convenient to understand excitations as living in a curved momentum space. The considerations of this paper may be relevant for those cases. Or, to put it the other way, just as some condensed matter or fluid systems provide analogues for relativity and gravity, it may be that condensed matter systems with curved momentum spaces may give us analogues to the physics of relative locality.
So look around. You see colors and angles, i.e. you are seeing into phase space. The idea that underlying it is an energy independent, invariant spacetime geometry could be an approximation, reliable only to the extent that we measure the geometry with quanta small compared to the Planck energy and we neglect phenomena of order of [formula]. Whether this is correct or not is for experimental physics to decide. If it turns out to be correct, then a new arena opens up for experimental physics and astronomy, which is the measurement of the geometry of momentum space
NASA’s Fermi gamma-ray space telescope has confirmed some aspects of relative locatility but more observations need to confirm that the difference in locality gets bigger from objects farther away and scales with distance.
If the delay is a property of the explosion, its length will not depend on how far away the burst is from our telescope; if it is a sign of relative locality, it will. Amelino-Camelia and the rest of Smolin’s team are now anxiously awaiting more data from Fermi.
The questions don’t end there, however. Even if Fermi’s observations confirm that momentum space is curved, they still won’t tell us what is doing the curving. In general relativity, it is momentum and energy in the form of mass that warp space-time. In a world in which momentum space is fundamental, could space and time somehow be responsible for curving momentum space?