QFT thought experiment in 1+1 dimensions

[Berislav]

First off let me start by stating that I am aware that you must get a great amount of "snake-oil theories" on this board and that I have no interests in "peddling". I am merely a high-school student, very interested in physics, who was lucky enough to stumble across these forums. I had an idea which is probably based on fallacious postulates, is flawed mathematically, or has something else that makes it invalid; but I would still like for it to be examined by a physicist so that they could inform me where I made (a) mistake/s, which in turn would cause me to learn something new.

Thought experiment:

Take an 1+1 dimensional Minkowski spacetime. In this spacetime we have one particle. For the sake of simplicity, assume that we can define the length of this particle to be [itex ]a[/itex] without taking quantum effects into consideration. The particle is in a vacuum, and it divides the spacetime into two vacuum fields, which are described by a typical Klein-Gordon equations each. The fields are, of course, linear. Assuming that quantum tunneling through the particle is impossible the only way that the two fields can interact is via affecting the particle itself. The solution to both fields is obtained by imposing the following Dirichlet boundary conditions:
$$\phi_\textit{l}(t,\frac{a}{2})=0$$
and
$$\phi_\textit{r}(t,\frac{-a}{2})=0$$
where the indices l and r denote the "left" and "right" field, respectively.

We can then derive the stress-energy tensor for each field. Using bra-ket and an improper integral,

$p_\textit{l}=\int_{a/2}^{\infty}$ <0|$T_\textit{x0}$|0>$dx$

$p_\textit{r}=\int_{-\infty}^{-a/2}$ <0|$T_\textit{x0}$|0>$dx$

we get momentums of the vacuum states, which are two sums, which are infinite but opposite in sign. Now as it is typically done we should cancel out the divergences. Instead let's observe that the momentum acting on the particle is not infinite, it is undefined (since the total momentum is the sum of both momentums).

If it were possible to formulate a quantum field theory by some other than the standard approach; i.e, as it is done in Robert Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, using the fact that a linear quantum field is practically the same as a collection of harmonic oscillators (with infinitely many degrees of freedom). I have not had a chance to read the book, but it seems to me that if this approach yields no divergences, then in this thought experiment it is possible that the particle shows chaotic motion when both fields influence it (since this is a very complicated system). We could first use Dirac's delta function to find the momentum and energy of the fields at the points at which they "touch" the particle. Then we could introduce the concept of iterations, as it is done in chaos theory, and find the Lyapunov exponent to see whether the particle behaves chaotically. If it does in fact behave in such a manner then it should be possible to define the attractor as a probability measure, thus deriving the behavior of the particle through probability. If this probabilistic behaviour is the same as that which is defined by the square of the wave function in quantum mechanics then we would have a little model which could possible be expanded into a theory, demonstrating why quantum mechanics is probabilistical.


I thank You for taking the time to read my post.


-Berislav

P.S.
Sorry for the state the equations are in, LaTeX is very unfamiliar to me.

 

[Berislav]

Browsing through your forums I found out about quantum entanglement. I read a short article about it on Wikipedia (I didn't find anything about it on physicsworld.wolfram.com). Applying the phenomenom to my "thought experiment" would change the outcome drastically (even if we only measure the momentum of the test particle).
Does quantum entanglement naturally generalize to quantum fields?
If so then picking hermitian operators corresponding to momenta of the fields and then combining the systems through a tensor product of two Hilbert spaces (identifing the basis of the spaces with eigenvectors) would detriment what the model tries to accomplish (i.e, derive quantum phenomena without using the notion of them in the first). Even if I used the harmonic oscillator interpretation, (which as I remind the reader that I know nothing about except what I already stated) it seems to me that I wouldn't be able to bypass this problem.

This also got me thinking that since we only can measure observables, the only logically consistent way to introduce iterations would be by making measurements at regular time intervals. This would make the probability of a certain measurement a function of the previous one, which isn't consistent with what we know from quantum mechanics.

Are there any other errors I overlooked in my proposed model?