Bohmian mechanics assumes that particle trajectories are continuous. Also, it claims the random outcome of certain experiments (like the double-slit experiment) to be due to the random initial particle positions at the beginning of the universe.

However, there are some conjectures that space and time are quantized at the plank scale. Look at this link for example: https://www.scientificamerican.com/article/is-time-quantized-in-othe/

If we assume that such conjectures are true and space and time are quantized, doesn't this rule out Bohmian mechanics? Isn't this in conflict with the continuity of trajectories? More importantly, if the initial particle positions are quantized with a minimum distance between possible positions, I think it would be very unlikely that their random distribution would be the same as the outcome of experiments. My reason is that the evolution of trajectories is chaotic and if we cannot make the distances between possible initial positions arbitrarily small, the future would be so different for different initial distributions such that the experiment will not occur at all for all of the initial distributions except the one which has been realized (maybe even the humans would exist only for this initial distribution).

Am I true? My field is not physics and I might have made a mistake in my argument, I'm just curious.

  • $\begingroup$ The connection you draw here has nothing to do with Bohmian mechanics actually. Every theory of quantum mechanics, all usual QFTs in the Standard model, use a continuous space-time. So they all would be "wrong" if it turned out that space and time are quantized. But "wrong" is not really a good word here, it's the same sense in which Newtonian mechanics is wrong: There are better theories out there that explain more phenomena. $\endgroup$
    – Luke
    Oct 5 '18 at 8:40
  • $\begingroup$ As you know, in Bohmian mechanics they say that the uncertainty seen in the outcome of experiments is due to uncertainty in the initial particle positions at the beginning of the universe. I have no idea how much the uncertainty in initial positions is, but I think it should be larger that the quanta of length (plank length, 10^-33 m) in order for the uncertainty to be meaningful, because distances less than this are not possible. This was mainly what I wanted to know. Is it equal to the uncertainty we see in the outcome of the experiment? $\endgroup$ Oct 5 '18 at 20:32
  • $\begingroup$ "Bohmian mechanics they say that the uncertainty seen in the outcome of experiments is due to uncertainty in the initial particle positions at the beginning of the universe" Sorry, you got it wrong. The uncertainty is a statistical one, each particle still has a definite position at any time, nothing `unsharp'. We can't know it perfectly well (statistics!) but it's still sharp. $\endgroup$
    – Luke
    Oct 8 '18 at 11:27
  • $\begingroup$ @Luke You told we cannot know the initial position exactly although it is sharp. I have no problem with this, my problem is that the quantized space doesn't let the statistical distribution of the position of the particle to be smooth. The quantized space would only allow certain positions, so why can't we be sure that the position is one of the allowed ones within the distribution? This way, our uncertainty will decrease or even vanish (because we know that the position can only be one of the allowed positions), while it shouldn't. What's the problem here? $\endgroup$ Oct 16 '18 at 2:03
  • $\begingroup$ Nothing will happen that should not. Of course, some details of uncertainty considerations will change in a quantized space. Physics will be different in some details. Sure. But nothing that sounds fatal. $\endgroup$
    – Luke
    Oct 17 '18 at 12:00

The equations of Bohmian mechanics consist of the Schrodinger equation for the overall wavefunction, and a classical trajectory which follows the quantum probability current of that wavefunction. Normally these are differential equations for which space and time are continuous.

If space and time are instead discrete, there can still be a wavefunction, but its time evolution would consist of a series of discrete unitary transformations, rather than a smooth unitary evolution; and the classical trajectory would have to consist of a series of stochastic (probabilistic, nondeterministic) jumps between classical configurations, rather than a smooth deterministic evolution.

But there is no barrier here to reproducing the basic features of Bohmian mechanics, and in many cases it would even be possible to embed this discretized system in a continuous one. The real problems for Bohmian mechanics lie elsewhere, such as the assumption of a universal time coordinate, which conflicts with the philosophy (though not necessarily the calculations) of special relativity.

  • $\begingroup$ Thanks for your response. Could you please clarify why it is necessary for Bohmian mechanics to be indeterministic in the case that the spacetime is quantized? That is, why do the jumps between classical configurations "have to" be stochastic and not deterministic? $\endgroup$ Oct 2 '18 at 1:20
  • $\begingroup$ The idea is that the discretized version will in effect be clustering together continuum configurations whose trajectories diverge. If we divide space into cells, and consider the usual Bohmian theory of a single particle, particles that start at different locations within the same cell will in general exit the cell at different times, will end up in different cells in the long terms, etc, and so a physics which only talked about cell-to-cell transitions would not be deterministic. $\endgroup$ Oct 6 '18 at 0:48
  • $\begingroup$ That said, you could have a discrete Bohmian mechanics that is still deterministic, the question is whether it is able to resemble reality. Gerard 't Hooft's work on cellular automata shows a quantum evolution which has a deterministic discrete approximation. On the other hand, John Bell's "Beables for quantum field theory" shows an attempt to make a Bohmian theory for fermions, and he felt compelled to make it nondeterministic. $\endgroup$ Oct 6 '18 at 0:49
  • $\begingroup$ The quantization of which I was talking was the quantization of the space at the plank scale (10^-35 m). It means that we cannot divide the space into portions smaller that this. Can particles be in "different locations" in such cells? What do people talk about when they say that the spacetime is quantized at the plank scale? Does this mean that particles can only take certain quantized positions which differ by the plank length? Do particles "fill" these plank-scale cells, or as you told they take a certain continuous position "inside the cells" ? $\endgroup$ Oct 8 '18 at 7:27
  • $\begingroup$ I was kind of assuming that the discretized theory still mimics continuum behavior. You can hide the discreteness, and get the appearance of smooth evolution, with probabilistic leakage between cells of space-time. But if space and time are overtly discretized, you can't have smooth rotations or relativistic "boosts", and you get aliasing-type effects that do not show up experimentally. $\endgroup$ Oct 8 '18 at 7:30

does quantized spacetime rule out Bohmian mechanics?

The answer is no. From the loop quantum gravity, the minimum quantum of length is planck length(pl). To visualise, if 1pl is about a size of tree then the minimum size of a elementary particle is of the order of size of the whole solar system. In no way the position of a particle is discreate, it is always continuos with respect to pl. So the trajectory of a travelling particle is always continuous. And Bohmian mechanics is not ruled out.

In Bohmian mechanis, the trajectory of a particle is govern by the "Path guiding equation" which tells us that the exact path of a particle is well defined by the hidden variables(non-local) of the Bohm's wave function. And there is no conflict with quantized spacetime.

  • $\begingroup$ As I understand you, the uncertainty in the initial positions of particles (which leads to uncertainty and randomness in experimental results) is much larger that the plank length, and hence even if the spacetime is quantized, slight differences in the initial positions will lead to the same future universe but with a slight difference in the result of some measurement in it, am I right? $\endgroup$ Sep 28 '18 at 1:18
  • $\begingroup$ Would you clarify? You seem to talk about uncertainty of position & momentum which is inherent in QM but in Bohmian mechanics(BM), there is no inherent uncertainties to a particle. The position and momentum is well defined like in classical mechanics and the randomness is because we are unable to produce an ensemble of a system which is having exactly same state in each copy of system due to quantum potential. Perhaps in BM, it's possible for highly advanced civilization or god to produce such ensemble and to end up without any randomness. $\endgroup$
    – Aman pawar
    Sep 28 '18 at 5:44

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