Dear Rod,
I was trying to study your paper. I think your motivations are exactly right and the paper contains a lot of exciting ideas. I must admit that I have great difficulties understanding the model, though. In part, it may just be an issue of (bad) notations, but many things don’t even make sense to me on the formal level.
For instance, at the very beginning, you introduce x and x’ as space-time coordinates. Hence, Psi(x,x’) is a multi-time wave-function of two particles. I don’t even know then what you mean when you talk about the wave-functions “before” and “after” measurement, since all the wave-functions seem to be defined on the entire space-time (or multiple copies thereof).
In the following, I already don’t understand equation (1). The right-hand-side of the equation seems to depend on the time-coordinate of particle 2, whereas the LHS doesn’t. And even if you fixed a particle time t’, the expression would be strikingly not Lorentz invariant, since you integrate over one particular spacelike hypersurface in one particular frame.
My difficulties with the presentation continue from there. For instance, I’m not sure what <x|i> means if |i> is a two-particle wave-function.
I’m not even sure what you mean by the expression <f|i>. Is this a scalar product on 4-dimensional space-time or on a 3-dimensional hypersurface? In the first case, I’m not sure if it defines a transition amplitude. In the second case, the expression (and hence the modified 4-density) is not Lorentz-invariant, since initial and final states are prescribed on certain hypersurfaces.
Maybe it’s just me, as a mathematician, being unfamiliar with the notation or being to picky und unflexible about formalities. But if all of this makes sense, I think your presentation would benefit a lot form being more precise and explicit about these things since they matter in this context.
On a more conceptual level, I don’t understand what your equations of motions are supposed to describe. As far as I can see, your Lagrangian involves only field degrees of freedom. What you call the particle 4-velocity u is actually a velocity field. In so far as your theory is supposed to be inspired by (or similar to) Bohmian mechanics, you seem to confuse the guiding field with the actual velocity of Bohmian particles and/or the variable in the wave-function with the actual position of Bohmian particles.
I’m sorry if I criticize your paper out of mere ignorance, but so far, I wasn’t able to understand what you’re doing and I’d really like to, since if your theory achieved what you claim it does, it would be nothing short of brilliant.
Best, Dustin
Comments are closed, but trackbacks and pingbacks are open.