The many-worlds interpretation would be better. If my "subject", that is, I as an observer, is located in a certain "branch" of possible world histories, then no collapse is required. It just happens that I as a subject am in a certain branch. Then all the other laws of physics can be time-symmetric. If we further assume that the path of the subject through the tree is determined by a time-symmetric method, then we have a model which is completely time-symmetric, and does not destroy information. This would be desirable.
A freely readable .pdf file:
The de Broglie-Bohm model might provide us with a deterministic path for the observing subject, if we can extend it to cover the creation of new particles. The classic Bohm model is limited in that that it only covers a fixed number of particles, and uses the non-relativistic Schrödinger equation.
What constitutes a branch in the many-worlds interpretation?
The definition of a branch is a question which comes up regularly on the Physics Stack Exchange.
Suppose that a single photon hits a charge-coupled device (CCD), and starts a cascade which registers the location which the photon hit. A quantum mechanical object photon "decoheres" into a macroscopic object or into a macroscopic process.
Many people seem to believe that this explains what a branch means in many-worlds. It is a local process of decoherence. However, this definition of a branch is not exact, nor mathematical. If the photon happens to hit the CCD at a slightly (1 micrometer) different location, is that another branch? Do branches actually form a continuum?
The de Broglie-Bohm model might give us a mathematical definition of a branch: it simply is the time development for fixed initial hidden variable values.
We might define a branch with certain initial hidden variable values the "true state of the material world". Albert Einstein wrote that "God does not play dice". The branch would be deterministic, and Einstein would be happy. Does this approach lead to any contradictions?
de Broglie-Bohm as a flow of probability?
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--->
incoming double diffracted wave
wave slit and interference
Let us think about the double-slit experiment. The incoming wave can be understood as a probability distribution, where the probability density is the squarw of the wave function value.
We may imagine a flow of probability, a flux, which takes a "probability parcel" forward from the double slit. That would give us a deterministic "path" for an incoming particle? As is well known, a de Broglie-Bohm particle will move along a strange, curvy path.
In a Feynman path integral, also fully unrealistic paths of a particle are calculated and added. If a de Broglie-Bohm path can be designated as the "true world", what about an unrealistic Feynman path?
Werner Heisenberg writes in his 1958 book Physics and philosophy that the de Broglie-Bohm model has problems handling stationary states. The flow of probability is zero in a typical stationary state, say, for a particle in a box. That suggests that a particle would remain in a fixed place.
A particle in a box, intuitively, has two flows of probability, to opposite directions. The sum of these flows is zero.
wave packet
<-- ~~~~~~~~~~
~~~~~~~~~~ -->
wave packet
Let us have a very long wave packet which describes a single particle. We use a beam splitter to divide the packet into two wave packets and let the packets collide so that they travel to opposite directions.
The flow of probability in the overlapping part of the two packets is zero. But the natural many-worlds interpretation for the process is that a single particle is moving either to the left or to the right in the diagram?
No. In the many-worlds interpretation, "branching" means a process where the waves for various alternatives will not interfere much. The branching has not happened yet when the wave packets overlap.
Do other branches "exist"?
If we designate one de Broglie-Bohm branch as the "true world", the other branches still exist, in the sense that they interfere with the true world, and affect the probabilities how the true world develops in time. This is quite different from, say, newtonian mechanics.
Certain people have criticized the fact that all branches exist, also those that are not the "true world". But we cannot help it: if we prune some branches, we lose information, and the model no longer corresponds to quantum mechanics.
Generalizing to relativistic wave functions and creation and destruction of particles: quantum field theory
The de Broglie-Bohm model is based on the Schrödinger equation, which is nonrelativistic. A generalization to relativistic particle physics requires that we develop a de Broglie-Bohm model for quantum field theory.
Quantum field theory, in its present form, contains dubious methods like regularization and renormalization. It is unlikely that we can develop a satisfactory deterministic model for quantum field theory unless we can remove regularization and renormalization. In fact, a de Broglie-Bohm model might help us to solve the problems.
Hrvoje Nikolić (2009) describes a model where particles can be created and destroyed.
Nikolić assumes that a particle can have a finite lifetime. In the diagram, x⁰ is the time coordinate. "Virtual particles" pop up and disappear in a very short time. They are denoted with dots.
For the model to be deterministic, the position of those dots, (x⁰, x¹) should depend on the initial conditions (data) at some earlier time. Does it?
We can, of course, formally define that the locations of the dots are decided in the initial data, but that looks ugly.
Flow of probability when new particles are created
~~~~~~~~~~~ real photon
/
e- ----------------------------------
|
| virtual photon
e- ----------------------------------
Let us consider a collision of two electrons. Photons and pairs can be created. We can draw Feynman diagrams and calculate the probability of each outcome. Above is a diagram for bremsstrahlung.
In the de Broglie-Bohm model of the double-slit experiment, there is a beautiful way to define the "path" of the particle, along the flow of probability in the Schrödinger equation. The path is meandering, but it is defined in a beautiful way.
But for a Feynman diagram, there is no such beautiful way. The real photon which appears above "breaks" the system into more degrees of freedom. If we have a probability flow in 6 spatial dimensions, there is no obvious way to make it to branch into a flow in 9 dimensions. What would decide the path for an arbitrary initial configuration of two electrons? How does the path suddenly branch to more dimensions?
In our blog we have several times written about the problem of "breaking" into more degrees of freedom. This seems to be associated with regularization and renormalization problems for Feynman diagrams.
Let us assign an arbitrary (irrational) real number r for the initial two electrons. We could design an ad hoc way to reproduce the correct Feynman probability distribution from the infinite number of decimals in r, and obtain a deterministic model. But that would be a very ugly solution.
Schrödinger's cat
The de Broglie-Bohm model for the Schrödinger equation solves a paradox associated with Schrödinger's cat, for the treatment with the Schrödinger equation.
The cat always is either dead or alive, in the "true world" determined by the hidden variables. Opening the box does not change anything. The cat does not suddenly "collapse" into either state.
Let us assume that we find the cat alive.
What remains of the paradox is the "empty" branch in which the cat is dead. The branch does exist in the de Broglie-Bohm model, though that branch is not "occupied" by the values of the hidden variables.
Schrödinger's cat involves a radioactive decay process which decides if the poison is released inside the box. Such a decay must be treated in quantum field theory, since new particles are created. And the de Broglie-Bohm model has no beautiful extension there. Thus, the paradox remains.
Conclusions
The de Broglie-Bohm model is a beautiful way to introduce determinism into the Schrödinger equation. We can define in a beautiful way what is a branch in the many-worlds interpretation. We avoid the "measurement problem" and many paradoxes in quantum mechanics.
But extending the de Broglie-Bohm model into quantum field theory depends on if we can find a beautiful model quantum field theory. Presumably, divergences, regularization, and renormalization should be removed. The problem of breaking into more degrees freedom may be a mathematical problem. Can we solve it?
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