I don't think we can jump to that yet, as there are still some math issues to discuss.
Demystifier said:
Do you agree that this extra information (not present in nonrelativistic BM) is encoded in the initial conditions?
And if you do, do you agree that this means that equations of motion are relativistic covariant?
Unfortunately I cannot answer these with yes/no, as I feel we don't even agree on what those questions mean currently.
First, this extra information was present in the "nonrelativistic BM" as well, in the form of a preferred inertial frame. Also, these questions are starting to get away from the math and get into terminology issues: you use terms differently from the way mainstream physics does. That being said, here's a first attempt at answering your questions fully.
your Q1] If we take the dBB state to be the wavefunction and particle positions, then no. If we take the dBB state to be the wavefunction and particles positions AND the simultaneity pairing structure, then in this bizarre sense yes. I have strong reservations on this though, as it is strikingly similar to choosing a preferred frame and then adding some scalar fields with values set to the coordinate labels, and then claiming this is not a choice of frame but initial conditions for a scalar field. These 'set once' features are best described as parameters or external structure required by the theory, and not an initial conditions. Like a constant lorentz violating vector in some Lorentz violating theories in literature. These are coupling parameters or an external structure, not initial conditions.
your Q2] The easiest way to answer this is just no.
I'm not sure what to consider "s" really as a geometric entity. I once referred to it as a scalar field, but this isn't really true since the "simultaneity pairing" structure cannot always be reduced to such (for example if particle paths cross). However, it doesn't seem unreasonable to claim this s-structure can be interpreted as a geometric object of some kind (however, as noted above, even coordinate systems can be interpreted as such if taken as values on spacetime). So by introducing that s-structure, and if you constrain the metric, it seems okay to consider your tensor notation like equations as coordinate system independent. If that is all you meant (some kind of 'coordinate system independent' way of writing the equations), then yes.
However, you said "relativistically covariant", which I think you were trying to ask: does this have Lorentz invariance? There is absolutely no debate here, the answer is no.
Question for you:
Do you agree that writing an evolution equation in tensor notation can make it coordinate system independent but does not automatically imbue it with Lorentz symmetry?
For consideration, note that even Newtonian gravity can be written in tensor notation (Newton-Cartan). And if we start allowing promoting of coordinate systems to scalar fields, we can bastardize the notation to make anything written in one coordinate system to be in tensor notation.
Even if I hold my nose and say the s-structure is part of the state and therefore specified in the initial conditions, the answer is still no. Because the very existence of the s-structure break Lorentz invariance. It's not like the theory has Lorentz invariance, and then choosing a particular initial condition breaks it ... the very existence of the s-structure in the theory breaks the Lorentz invariance.
Similarly, in an aether theory, one could claim the choice of rest frame of the aether is an initial condition. The use of the term "Lorentz symmetry" by the mainstream does NOT consider such theories as having Lorentz symmetry.
For an example of mainstream use of the phrase, I searched for vector field lorentz violations and picked one with lots of citations:
Spacetime-varying couplings and Lorentz violation
http://arxiv.org/abs/astro-ph/0212003
cite: 97 times
If the lorentz violating terms are non-zero the theory is called Lorentz violating. If the theory is only an effective theory, it is possible that these violating terms arise from a more fundamental theory which has a lorentz invariant action, in which case the Lorentz violating low-energy effective theory is said to have dynamically broken lorentz symmetry. In your case we are not generating a lower energy effective theory from your theory ... so all we need to worry about is simple Lorentz symmetry yes or no. The existence of the lorentz violating s-structure makes this a simple no. It is not an issue of initial conditions.
Demystifier said:
Good.
So I assume you understand now why your other claims about removing s-dependence using similar reformulations are wrong as well, such as:
Demystifier said:
Actually yes, because, as long as the only goal is to calculate the trajectories in spacetime, the parameter s can be eliminated from the equations. In this sense, s is only an auxiliary parameter. If you don't see how s can be eliminated, see
http://xxx.lanl.gov/abs/quant-ph/0512065
Eq. (30).
That formulation still requires a simultaneity pairing structure.
I'm also worried about other issues. For example it is well known that the Klein-Gordon equation has serious issues interpreting the wavefunction as a probability distribution, but you went right ahead and did so anyway.