Today I want to take on a topic many of you
requested, repeatedly. That is David Bohm’s approach to Quantum
Mechanics, also known as the Pilot Wave Interpretation, or sometimes just Bohmian Mechanics. In this video, I want to tell you what Bohmian
mechanics is, how it works, and what’s good and bad about it. Ahead, I want to tell you a little about David
Bohm himself, because I think the historical context is relevant to understand today’s
situation with Bohmian Mechanics. David Bohm was born in 1917 in Pennsylvania,
in the Eastern United States. His early work in physics was in the areas
we would now call plasma physics and nuclear physics. In 1951, he published a textbook about quantum
mechanics. In the course of writing it, he became dissatisfied
with the then prevailing standard interpretation of quantum mechanics. The standard interpretation at the time was
that pioneered by the Copenhagen group – notably Bohr, Heisenberg, and Schrödinger – and
is today usually referred to as the Copenhagen Interpretation. It works as follows. In quantum mechanics, everything is described
by a wave-function, usually denoted Psi. Psi is a function of time. One can calculate how it changes in time with
a differential equation known as the Schrödinger equation. When one makes a measurement, one calculates
probabilities for the measurement outcomes from the wave-function. The equation by help of which one calculates
these probabilities is known as Born’s Rule. I explained in an earlier video how this works. The peculiar thing about the Copenhagen Interpretation
is now that it does not tell you what happens *before you make a measurement. If you have a particle described by a wave-function
that says the particle is in two places at once, then the Copenhagen Interpretation merely
says, at the moment you measure the particle it’s *either* here *or* there, with a certain
probability that follows from the wave-function. But how the particle transitioned from being
in two places at once to suddenly being in only one place, the Copenhagen Interpretation
does not tell you. Those who advocate this interpretation would
say that’s a question you are not supposed to ask because, by definition, what happens
before the measurement is not measureable. Bohm was not the only one dismayed that the
Copenhagen people would answer a question by saying you’re not supposed to ask it. Albert Einstein didn’t like it either. If you remember, Einstein famously said “God
does not throw dice”, by which he meant he does not believe that the probabilistic
nature of quantum mechanics is fundamental. In contrast to what is often claimed, Einstein
did not think quantum mechanics was wrong. He just thought it is probabilistic the same
way classical physics is probabilistic, namely, that our inability to predict the outcome
of a measurement in quantum mechanics comes from our lack of information. Einstein thought, in a nutshell, there must
be some more information, some information that is missing in quantum mechanics, which
is why it appears random. This missing information in quantum mechanics
is usually called “hidden variables”. If you knew the hidden variables, you could
predict the outcome of a measurement. But the variables are “hidden”, so you
can only calculate the *probability of getting a particular outcome. Back to Bohm. In 1952, he published two papers in which
he laid out his idea for how to make sense of quantum mechanics. According to Bohm, the wave-function in quantum
mechanics is not what we actually observe. Instead, what we observe are particles, which
are guided by the wave-function. One can arrive at this interpretation in a
few lines of calculation. I will not go through this in detail because
it’s probably not so interesting for most of you. Let me just say you take the wave-function
apart into an absolute value and a phase, insert it into the Schrödinger equation,
and then separate the resulting equation into its real and imaginary part. That’s pretty much it. The result is that in Bohmian mechanics the
Schrödinger equation falls apart into two equations. One describes the conservation of probability
and determines what the guiding field does. The other determines the position of the particle,
and it depends on the guiding field. This second equation is usually called the
“guiding equation.” So this is how Bohmian mechanics works. You have particles, and they are guided by
a field which in return depends on the particle. To use Bohm’s theory, you then need one
further assumption, one that tells what the probability is for the particle to be at a
certain place in the guiding field. This adds another equation, usually called
the “quantum equilibrium hypothesis”. It is basically equivalent to Born’s rule
and says that the probability for finding the particle in a particular place in the
guiding field is given by the absolute square of the wave-function at that place. Taken together, these equations – the conservation
of probability, the guiding equation, and the quantum equilibrium hypothesis – give
the exact same predictions as quantum mechanics. The important difference is that in Bohmian
mechanics, the particle is really always in only one place, which is not the case in quantum
mechanics. As they say, a picture speaks a thousand words,
so let me just show you how this looks like for the double slit experiment. These thin black curves you see here are the
possible ways that the particle could go from the double slit to the screen where it is
measured by following the guiding field. Just which way the particle goes is determined
by the place it started from. The randomness in the observed outcome is
simply due to not knowing exactly where the particle came from. What is it good for? The great thing about Bohmian mechanics is
that it explains what happens in a quantum measurement. Bohmian mechanics says that the reason we
can only make probabilistic predictions in quantum mechanics is just that we did not
exactly know where the particle initially was. If we measure it, we find out where it is. Nothing mysterious about this. Bohm’s theory, therefore, says that probabilities
in quantum mechanics are of the same type as in classical mechanics. The reason we can only predict probabilities
for outcomes is because we are missing information. Bohmian mechanics is a hidden variables theory,
and the hidden variables are the positions of those particles. So, that’s the big benefit of Bohmian mechanics. I should add that while Bohm was working on
his papers, it was brought to his attention that a very similar idea had previously been
put forward in 1927 by De Broglie. This is why, in the literature, the theory
is often more accurately referred to as “De Broglie Bohm”. But de Broglie’s proposal did, at the time,
not attract much attention. So how did physicists react to Bohm’s proposal
in fifty-two. Not very kindly. Niels Bohr called it “very foolish”. Leon Rosenfeld called it “very ingenious,
but basically wrong”. Oppenheimer put it down as “juvenile deviationism”. And Einstein, too, was not convinced. He called it “a physical fairy-tale for
children” and “not very hopeful.” Why the criticism? One of the big disadvantages of Bohmian mechanics,
that Einstein in particular disliked, is that it is even more non-local than quantum mechanics
already is. That’s because the guiding field depends
on all the particles you want to measure. This means, if you have a system of entangled
particles, then the guiding equation says the velocity of one particle depends on the
velocity of the other particles, *regardless of how far away they are from each other. That’s a problem because we know that quantum
mechanics is strictly speaking only an approximation. The correct theory is really a more complicated
version of quantum mechanics, known as quantum field theory. Quantum field theory is the type of theory
that we use for the standard model of particle physics. It’s what people at CERN use to make predictions
for their experiments. And in quantum field theory, locality and
the speed of light limit, are super-important. They are built very deeply into the math. The problem is now that since Bohmian mechanics
is not local, it has turned out to be very difficult to make a quantum field theory out
of it. Some have made attempts, but currently there
is simply no Pilot Wave alternative for the Standard Model of Particle Physics. And for many physicists, me included, this
is a game stopper. It means the Bohmian approach cannot reproduce
the achievements of the Copenhagen Interpretation. Bohmian mechanics has another odd feature
that seems to have perplexed Albert Einstein and John Bell in particular. It’s that, depending on the exact initial
position of the particle, the guiding field tells the particle to go either one way or
another. But the guiding field has a lot of valleys
where particles could be going. So what happens with the empty valleys if
you make a measurement? In principle, these empty valleys continue
to exist. David Deutsch has claimed this means “pilot-wave
theories are parallel-universes theories in a state of chronic denial.” Bohm himself, interestingly enough, seems
to have changed his attitude towards his own theory. He originally thought it would in some cases
give predictions different from quantum mechanics. I only learned this recently from a Biography
of Bohm written by David Peat. Peat writes “Bohm told Einstein… his only hope was
that conventional quantum theory would not apply to very rapid processes. Experiments done in a rapid succession would,
he hoped, show divergences from the conventional theory and give clues as to what lies at a
deeper level.” However, Bohm had pretty much the whole community
against him. After a particularly hefty criticism by Heisenberg,
Bohm changed course and claimed that his theory made the same predictions as quantum mechanics. But it did not help. After this, they just complained that the
theory did not make new predictions. And in the end, they just ignored him. So is Bohmian mechanics in the end just a
way of making you feel better about the predictions of quantum mechanics? Depends on whether or not you think the “quantum
equilibrium hypothesis” is always fulfilled. If it is always fulfilled, the two theories
give the same predictions. But if the equilibrium is actually a state
the system must first settle in, as the name certainly suggests, then there might be cases
when this assumption is not fulfilled. And then, Bohmian mechanics is really a different
theory. Physicists still debate today whether such
deviations from quantum equilibrium can happen, and whether we can therefore find out that
Bohm was right. This video was sponsored by Brilliant which
is a website that offers interactive courses on a large variety of topics in science and
mathematics. I always try to show you some of the key equations,
but if you really want to understand how to use them, then Brilliant is a great starting
point. For this video, for example, I would recommend
their courses on differential equations, linear algebra, and quantum objects. To support this channel and learn more about
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David Bohm's Pilot Wave Interpretation of Quantum Mechanics
David Bohm was a victim of McCarthyism. It's also worth mentioning that Bohm was fired, his PhD research confiscated and classified before he could write a thesis, and shortly after publishing his theory he was exiled to Brazil and not able to return. It's difficult to to do science and defend your work when you are persecuted, forced to leave your country and live in exile, first in Brazil and then Israel.
I dunno about Bohmian mechanics but pilot wave theory itself is perfectly Lorentz invariant, actually consequence of Lorentz invariance. Why wake waves are forming around object in motion? Well just because speed of surface ripples remains independent of object speed, i.e. invariant. De Broglie waves are actually common denominator of relativity and quantum mechanics.
Finally making sense of the double-slit experiment (2017, Aharonov): "The nonlocal equations of motion in the Heisenberg picture thus allow us to consider a particle going through only one of the slits, but it nevertheless has nonlocal information regarding the other slit.... The Heisenberg picture, however, offers a different explanation for the loss of interference that is not in the language of collapse: if one of the slits is closed by the experimenter, a nonlocal exchange of modular momentum with the particle occurs....Alternatively, in the Heisenberg picture, the particle has both a definite location and a nonlocal modular momentum that can “sense” the presence of the other slit and therefore, create interference." as John Bell states: "Is it not clear from the diffraction and interference patterns, that the motion of the particle is directed by the wave?" See also:
Photon trajectories, anomalous velocities and weak measurements: a classical interpretation
Common Misconceptions Regarding Quantum Mechanics
What Non-locality Means In Quantum Mechanics?
The Rebel Physicist Angelo Bassi on the Hunt for a Better Interpretation of Quantum Mechanics
New theory of quantum mechanics shows matter is not in the eye of the observer