Is unitarity only true for timelike coordinates? For spacelike coordinates can't we generalise unitarity?

Question

Can we define a 2+1 submanifold/slice ("instant of space") analogous to the Cauchy surface such that we can use this as a boundary condition and find the solution for the remaining full spacetime? Assume that this 2+1 submanifold is physically permissible (we can get such a slice from a known spacetime solution and try to reconstruct the remaining full spacetime).

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More info: In QFT the following is the unitary operator for a spacetime translation by $a$ $$T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar)$$

All symmetries transformations are described by unitary operators but in this question, we only consider spacetime translation symmetry, specifically unitary operators associated with spacelike coordinates.

For the case of time, we can take Cauchy surfaces to define "an instant of time". Unitarity is the statement that once we know the physics of a single Cauchy surface we can evolve and find all the dynamics.

Can't we define generalized Cauchy surfaces as some codimension 1 Lorenztian submanifolds such that unitary translation along the remaining spatial direction gives the entire dynamical information? (Coordinate singularities along this direction can always be removed by changing the coordinates. Physical singularities cannot be present since those will be naked/spacelike singularities and require $\infty$ energy)

The only special property time has is that if we calculate coarse-grained entropy along the Cauchy surfaces it increases, so there is a preferred direction. But I don't see why unitarity needs a preferred direction since unitarity anyway gives information on both sides of a Cauchy surface.

Is there any arxiv paper where they tried this? Are timelike coordinates more special than what I am thinking?

Clarifying question: Can we get the complete dynamical information by only knowing the physics of a "2+1 dimensional submanifold"?

It is like an "instant of space". Like $z=0$ in Minkowski spacetime with coordinates $(t,x,y,z)$. Then can we use the unitary $z$ translation operator to find the complete dynamics of the full $3+1$ Minkowski spacetime? My intuition is saying that if we take the solution on $z=0$ as the boundary condition we can probably get the entire information of a part of the spacetime that is causally connected (i.e not separated by horizons).

Answer 1

In quantum field theory, there is no such thing as the "physics of a '2+1 dimensional submanifold'": The objects in our theory are states and operators on Hilbert spaces and depending on your formalization there's either a single such Hilbert space or one such Hilbert space for every time slice, or even only two Hilbert spaces - one in the infinite asymptotic past and one in the infinite asymptotic future.

In any case, you cannot in general localize the information in the states in this Hilbert space with respect to spacetime position - there are no good relativistic position operators, see e.g. this answer by Valter Moretti, so in this context it makes no sense to talk about information living on non-spacelike slices. QFT does not, in general, have wavefunctions as functions of space(time) that describe its states - states are functionals of the fields, if you want to write them as functionals of something, but not of position.

Non-quantumly, where the values of all observables on any surface are in principle at least a meaningful thing to talk about, this idea also fails - data on surfaces that are not Cauchy surfaces does not suffice to predict what happens on the entire spacetime. In your example in Minkowski space, timelike curves with constant $z$ those constant is unequal to the constant of your "instant of space" do not intersect that surface at all, and hence motion of something along these trajectories is not necessarily detectable by any data confined to that surface. Just because that trajectory is causally connected to the surface doesn't mean stuff that travels along it actually does anything that leaves a trace there. The crucial point of a Cauchy surface is not mere causal connectivity - it is that every inextendible timelike curve intersects it, nothing that exists within the spacetime has a chance to "miss" the Cauchy surface.

Altogether, there is a misunderstanding of the nature of "unitarity" here: All symmetry operators are unitary, and so a time and space translation-invariant physical theory has unitary time and space translation operators. But this doesn't mean that space translation somehow allows you to recover the entire theory from one slice - all that "unitarity" means is that the operator preserves the probabilities between two states when applied to both of them (that is how Wigner's theorem derives that symmetries have to be unitary, after all).

The theory that governs the collision of two particles is translation-invariant, that's why momentum is conserved. But that doesn't mean that we can take a snapshot (every instant into the infinite past and future) of some empty space where none of the particles ever cross through and somehow deduce the existence of colliding particles in some part of space where we didn't look by applying the translation operator - empty space translated just remains empty space.