Causal order

The key lesson from general relativity is background independence. There is no classical background metric, but the metric is the gravitational field which participates in the dynamics. However, this poses problems for the Hamiltonian formulation of physics. 

In its simplest form, Hamiltonian physics deals with the phase space of positions and momenta at time \(t=0\), and the choice of a time coordinate makes use of the background metric. More generally, spacetime can be foliated into spacelike surfaces, but the notion of spacelikeness again requires a background metric. There is also a formulation that probably goes back to Lagrange which identifies phase space with the space of solutions to the equations of motions, but the background metric sneaks in even here. Such an orbit is typically specified by position and velocity at time \(t=0\). So orbits are parametrized by positions and velocities, but phase space is the space of positions and momenta. In order to identify the two we need to identify velocity and momentum, i.e. a contravariant vector with a covariant one. This again requires a background metric.

In QJT the problem goes away altogether, because there is no notion of spacelikeness. All objects in the theory are separated by a timelike distance, because everything lives on the observer's trajectory, both the Taylor coefficients \(\phi_m(t)\) and the expansion point \(q(t)\). This does of course not mean that points outside the observer's trajectory do not exist, but they are not covered by QJT because they can never be directly observed by the preferred observer. 

The same situation arises in mathematics. In order to make a Taylor expansion we must specify a field and an expansion point, i.e. the observer's position. We could pick a different expansion point, and there is a relation between the two sets of Taylor coefficients. But we must pick one, i.e. we must commit to one specific observer. All points on the observer's worldline are then causally ordered.

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