Why don't more physicists subscribe to pilot wave theory?
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"It's been demonstrated experimentally that you can't have a local, deterministic, real, definite theory (i.e. that would lead to demonstrably false conclusions). People intuitively expect all of those things to be true, but we can prove that it isn't the case. The various interpretations all try to give up one of "local", real", "definite", or "deterministic" in order to preserve the others, thus preserving at least a modicum of their intuition.
The pilot wave idea gives up on locality"
And I will add:
Multiple worlds gives up on definiteness.
"Zero worlds" (my personal favorite) gives up on realism.
And, of course Copenhagen gives up on determinism.
Take your pick. Or go with decoherence, which kind of lets you turn a knob to give up a little bit of all four and dial in whatever setting you like.
See https://www.youtube.com/watch?v=dEaecUuEqfc or http://www.flownet.com/ron/QM.pdf for a detailed explanation.
Many worlds basically says "the wave function is real". You just take what the math says at face value, and the math says there's a blob of amplitude where the cat is alive, and another blob of amplitude where the cat is dead, and those blobs do not interact.
Copenhagen adds something on top of the math: a kind of "collapse" where the blob you did not observe gets mysteriously zeroed out. There's only one universe, which looks simpler, but the theory itself is more complex, because you just added that collapse.
Pilot Wave (of which I know nothing) seems to add a similar complexity. There's no collapse, but there's this additional non-local "wave" that's laid out on top of everything, and determines which of the blobs is real (the dead cat blob or the live cat blob).
Think Chess vs Go. Chess has rather complex rules, with an initial position, moves for 6 different pieces, and a couple special cases. Go's rules on the other hand can fit on one page. So, Go is simpler. However, in terms of possibilities, the universe generated by the Go rules is orders of magnitudes bigger than Chess'. Simpler rules can lead to more diverse possibilities. Quantum physics interpretations are similar: Many Worlds have the simplest rules, but it also describes the biggest universe.
Imagine 2 pearls on a thread. You have 2 ways to represent them: the obvious one is 2 points on a line. A less obvious (but just as valid) is a single point on a plane: the X axis would represent the first pearl, and the Y axis would represent the second pearl. Similarly, 2 billiard balls on a billiard can be represented by a single point in a 4 dimension configuration space. And the entire universe require many many more dimensions than that.
There are many more subtleties. I suggest you read the Quantum Physics Sequence for a comprehensible explanation of all this mess. http://lesswrong.com/lw/r5/the_quantum_physics_sequence/
Furthermore, I don't have a problem with accepting non-locality from a theoretical perspective either, now that I know about the Holographic Principle. If the universe is a hologram, then the apparatus that is projecting the hologram can be connected in ways that aren't obvious in the hologram.
The computer program runs inside the computer. The computer is outside the program. But nevertheless the computer's hardware is causally connected to the program. Further still, the programmer is outside the program, but he very much can influence it. Indeed, he brought it into being.
A computer program is not a universe independent of the programmer. They both exist in the same universe because they are causally connected. If they were in different universes, they must be causally disconnected. You're whole argument rests on the assumption that the program and the programmer reside in different universes, which is clearly not the case.
True that.
Who cares? Experiments are a tie: those different interpretations all make the same observable claims.
From there, which is best depends on what you care about most: is the first interpretation we came up with best, because "science"? Is the simplest interpretation best, because "Bayes"? Do stuff like locality, determinism etc. matter?
Me I side with Bayes, and that means Many Worlds/decoherence, which is simplest. Others would make another choice.
You're probably thinking of "instrumentalism" aka "shut up and calculate". Copenhagen is not instrumentalism.
Copenhagen can't actually explain where the quantum speedup actually comes from.
Other interpretations try to explain it, but are similarly problematic. For instance, in many worlds, Everett tried to argue that pieces of the computation are shuffled off to other worlds and then brought back, but this is problematic because information isn't supposed to be shared with other worlds, and since other worlds also shuffle off an equal amount of computation to those worlds, how is overall speedup achieved exactly?
In pilot wave theories, quantum computing has a simple story: because the world is deterministic, the entire history of time leading up to your computation was predetermined and part of the computation. "Programming" a quantum computer simply creates the conditions where you can read out the answer.
The apparent quantum speedup is actually part of an illusion that we and our equipment are separate from the rest of the universe, but quantum computation is really a classical computation that had lots of time and plenty of resources (all particles in the universe) to run.
Note: I understand that a wave function of just 3 photons or photons that travel in straight lines could not cause an interference pattern.
The idea was originally known as the Theory of the Universal Wave Function, which makes a lot more sense as long as you're unafraid of mathematics.
There is no "splitting off" of universes; I agree with you that that wouldn't make much sense. Instead, the observer simply becomes entangled with the observed physical system during the measurement process. This entanglement is a gradual process, though it certainly happens very quickly.
This basically pushes all the weirdness out of quantum mechanics and into the fact that we just don't understand consciousness very well. Why don't we perceive the full linear combination of quantum states? And can we somehow map our subjective experience of probability and statistics to what happens in the Universal Wave Function? It's much easier to ignore that weirdness, since we're used to ignoring it in our daily lives anyway. It's also a more parsimonious interpretation than the others, because it doesn't postulate anything special about us (the "observers").
(This does not mean that consciousness is a quantum or even meta-physical phenomenon. Quite the opposite, actually: I find that the mystery lies more in why consciousness is unable to perceive quantum states despite existing in a universe that has quantum physics.)
My personal belief is that it is because consciousness is a classical information-processing phenomenon. In other words, we can only directly perceive things that can be described as real numbers rather than complex numbers because we are Turing machines, and Turing machines are classical.
This fits with a Hofstadter-like perspective that consciousness is about "strange loops", where we somehow repeatedly evaluate simplified models of the world including yourself. Doing such repeated evaluations requires (at least partial) "cloning" of the state of the world outside for the purpose of evaluation, and cloning quantum states is impossible, hence consciousness must be a classical phenomenon.
I like that line of argument.
Note: I wouldn't state it as being unable to perceive things that can be described as complex numbers, but rather complex linear combinations.
But there is a version of it which does make sense: https://arxiv.org/abs/0903.2211
The basic question for all theories is, "what is fundamentally real?" and the wave function of quantum mechanics just doesn't work in that role. This is why Bohmian mechanics adds particles; that is what is real in that theory.
But to make a MW theory, one can integrate out the wave function to create a mass density function on 3-space. In any given instance, this is a mess and useless. But if you watch it evolve, then you could see multiple different stories evolving. And those are the different "worlds".
There is no splitting, just regimes that are no longer relevant.
There's little to understand: at its core, Many Worlds is real close to "shut up and calculate". The only significant assumption is that the wave function as described by the math is real. And when the math says there are 2 non-interacting blobs, well, this translates to 2 universes.
That's about it.
As for why people seems to reject it instinctively, it's probably because our subjective experience is linear —or mono-threaded. However, linear histories aren't contradictory with trees. Imagine a Git History that never merges: each leaf is the product of a linear set of modifications, a perfectly clean history. From that final commit's perspective, other branches might as well not exist. You'd need merges to break that illusion, and our universe doesn't have macro-merges —we only observe them at a very small scale, for instance with photon interference.
But what does that mean? What is the mapping to my experience? Is it that if the wave function is non-zero at a configuration point, then the configuration is realized as an actual universe? One big problem with this is that wave functions can be non-zero at all configuration points and the dispersive nature of the Laplacian tends to make that happen.
So then it becomes one of size of |psi|^2? Is there a cutoff for considering that to be real? Having a probability of something is needed to deal with this, but it is not clear to me what the probability would correspond to here.
In BM, the probability is that of finding particles in some region. In collapse theories, it is the probability of collapsing to a specific state.
As for the Git History, my above concern could be phrased as a history in which all possible streams of texts are possible though maybe some have a large font-size. This is library of babel kind of stuff. How is the evolution of the text that I consider myself to be a part of handled? How are nearby configurations connected by the evolution of a wave function?
Also, macro-mergers are not theoretically ruled out; it is practically ruled out.
Let's say that I convinced you that 2 non-interacting blobs did not objectively evolve to occur, but rather a smearing over all. Would this break your interpretation?
If I understand correctly, what you speak of should actually be observable —you could make falsifiable predictions about it. That would break more than my interpretation, I think.
That is to say, where is the magnitude of the wave function being used in this interpretation?
That is a really excellent analogy. It is also important to understand that there is a reason we don't have "macro merges", and that is because some irreversible process is required to create classical information, which is what we (the entities having this conversation) are actually made of. See:
http://blog.rongarret.info/2014/10/parallel-universes-and-ar...
Newton's fluxions and Leibniz's differentials provided great guidance, and in a lot of cases they even worked and gave correct results, but they were not really rigorously defined, and later mathematicians started running into serious problems with that approach. Those were resolved with the approach that we all now known and love (loath?): limits and the epsilon/delta approach to them.
NSA extends the real field to contain a new kind of number that is positive but is less than 1/N for any integer N, giving a new field called the hyperreals. You can then do things like define the derivative of f at x to be (f(x+h)-f(x))/h where h is an infinitesimal, work out that using ordinary algebraic manipulations, and then just drop any terms that are infinitesimal.
NSA goes far beyond just calculus. You can use these ideas all through analysis.
Seems pretty cool, and a lot clearer than a limits approach. So why has it not taken over?
I believe that a big part of it is that it has been shown that NSA and standard analysis are equivalent. Anything that you can prove in one can be proven in the other. Perhaps more important, one doesn't seem any better than the other at leading to deep insights. If you can't solve a problem in standard analysis, you probably aren't going to figure out the solution if you use NSA, and vice versa.
So while it might be overall nicer if we were starting from scratch to use NSA instead of standard analysis everywhere, so much is already done in standard analysis that switching now would be too painful.
However Bohmian Mechanics is a different interpretation, not a different formalism. You can accept the interpretation, but you still end up solving the Schrödinger equation in the same way, using the same mathematical methods.
The analogy works better for matrix mechanics as compared to Schrödinger's wave mechanics. These are mathematically quite different even though they give the same results.
Pilot Waves, on the other hand, are pretty much always superfluous. Please link me, if you have seen a situation where using Pilot Waves simplifies a calculation.
I would think that the 'default' interpretation would be excluding them, since (clearly) they're not necessary (evidence being: interpretations without them exist). That would be the definition of superfluous, right? Unless they give different experimental results somewhere, which I'm pretty sure in't the case.
The wave equation is always there, or you wouldn't get quantum behaviour. There exist no-go theorems demonstrating that the wave function can't just be a reflection of our ignorance, it must be "ontic", ie. exist, in some real way. What's disputed is whether the particles need to be given separate existence from the waves that are necessarily there.
> I would think that the 'default' interpretation would be excluding them, since (clearly) they're not necessary (evidence being: interpretations without them exist). That would be the definition of superfluous, right?
No, because the interpretations you speak of can't actually explain the measurement problem without positing additional axioms which then make them less plausible. It's well known that pilot wave theories are more axiomatically parsimonious, meaning that they require fewer assumptions overall to explain all of our observations.
For example, the Born rule must be assumed by most interpretations of QM, with little rhyme or reason other than we know it's empirically valid. But because pilot wave theories posit real physical entities with well understood properties, we can actually derive the Born rule. This is just one example that demonstrates how pilot wave theories positing additional real entities can make for an overall simpler set of assumptions.
Also: aren't we to the point where measurement makes perfect sense (besides the values of the probabilities, as given by the Born rule), via entanglement between experiment+lab frames, and decoherence of unrelated Degrees of Freedom? Cause I thought we were. (see, for example, http://www.preposterousuniverse.com/blog/2014/06/30/why-the-...)
Also, pilot wave fails badly in relativistic extensions, for the obvious reason that a universe-wide pilot wave function is hard to make covariant. There are attempts at fixing this but last I heard none of them are doing a good job. So that's another strike against it, in my book.
Except it's not, because you also have to posit the measurement postulates in orthodox QM. This so-called "extra crap" reproduces the measurement postulates and the Born rule, thus replacing a large set of assumptions with a much smaller set.
Many-Worlds is indeed much simpler than orthodox QM, but it's still not simpler than pilot waves. They are roughly comparable, with many-worlds still having unresolved conceptual difficulties surrounding probabilities, among other issues [1]. Which is more parsimonious between many-worlds and pilot waves is hotly debated among philosophers of science.
> Also: aren't we to the point where measurement makes perfect sense (besides the values of the probabilities, as given by the Born rule), via entanglement between experiment+lab frames, and decoherence of unrelated Degrees of Freedom?
"Measurement now makes perfect sense" is an interpretation-specific claim. Measurement still doesn't make sense in Copenhagen, measurement mostly makes sense in Many-Worlds, modulo some of the difficulties I mentioned earlier [1].
> Also, pilot wave fails badly in relativistic extensions, for the obvious reason that a universe-wide pilot wave function is hard to make covariant.
It's actually pretty trivial if you're willing accept a preferred foliation of space-time, as long as the preferred frame is unobservable. This seems aesthetically unappealing, hence why people perpetuate this myth of "difficulty", but it's not a priori wrong.
Fortunately, a preferred foliation can actually be derived from the wave function itself, which means this foliation exists in every interpretation of QM [2].
This is the kind of surprising result that probably no one would have even bothered looking for, and I think it proves John Bell's position that non-locality is the unresolved problem of QM [3]. Other interpretations just let you paper over it, to our detriment IMO.
[1] https://plato.stanford.edu/entries/qm-manyworlds/#6
[2] https://arxiv.org/abs/1307.1714
[3] John Bell, who was a big fan of pilot waves by the way.
A more appropriate analogy to the SA v NSA would be GRW formulation vs Bohmian or even a well-defined many worlds (that does exist, but it does not involve splitting of worlds which is as problematic as collapse). These are different theories and they can lead to different generalizations.
In particular, GRW seems more amenable to being relativistic without adding a foliation or some other structure.
Bohmiam mechanics, on the other hand, does requires using a foliation though there are possibilities to use within the existing structures: https://arxiv.org/abs/1307.1714
I also think NSA has a naming problem. By its very name, how can it ever be standard?
I think the issue is also that the limit fits in with numerical analysis in that it talks about errors. NSA sounds like it skips the error analysis and is only concerned with stuff at the limit which is certainly useful in many instances, but not always.
"Nonstandard" isn't referring to how people view the field, but to additional elements you add to the model. Just to throw out a weird example, here's Edward Nelson's text on the subject:
https://web.math.princeton.edu/~nelson/books/1.pdf
"Theorem 4. There is a finite set that contains every standard object."
That is, in Nelson's system, the set of standard natural numbers is finite. That finite number is a nonstandard natural number, larger than any standard natural number.
Mixing "nonstandard"(in the sense of an undefined predicate added to the base logic) with "nonstandard"(in the sense of some professors thinking it's weird) is a type error.
Someone correct me if I'm wrong here, but logically, standard theory and pilot theory can't BOTH be right, can they?
Here's their recent episode on pilot wave theory:
Could the stuff be dark matter or dark energy?
That's pilot wave theory, where this term is sometimes called the "quantum potential" and is governed by Schroedinger's equation. As you can see, this extra term goes to zero when quantum effects become negligible and we recover classical mechanics straight away. The difficulty is the obvious nonlocality and what that means for special relativity.
That's all very accurate and extremely successful so anything that tries to explain why these models are the way they are is going to be pretty much the same as any other explanation if you tilt your head to the side and squint because they all have to produce the same very well specified set of mathematical models.
(IANAP)
"It's wrong"
A student held up his hand and asked what Bohmian mechanics was. Before the lecturer could answer
"Don't worry, it's wrong"
It was difficult to contain myself.
It can be hard to switch between the two (as a chemist we are taught Schroeder and I have a hard time wrapping my head around Heisenberg)
I think you mean that the function is a point in the vector space. More precisely, it is a point in a particular separable Hilbert space, i.e., a complete vector space equipped with an inner product that has countable, dense basis.
In addition, the ideas of Bohmian mechanics has led to a new direction in dealing with the divergences in QFT, that is, in finding a version of it that is mathematically well-defined: https://arxiv.org/abs/1506.00497
But the pilot wave is just the wave function of quantum mechanics, albeit on a universal scale. It is in all the theories. And just because one has a "collapse", which is always approximate anyway (no delta functions for a position measurement, for example), one still has a wave function as the only object in standard QM and that is an expanding object defined on 3*number of particles in universe. But we do not experience that. We experience 3 dimensional space with rather point-like objects moving about.
Pilot wave theory starts with "We have particles moving about. How do they move?" That's the question it starts with and it answers it in the simplest way possible given the wave function: the wave function tells the particles how to move using Bohm's equation which is a very easy thing to derive. Indeed, you can derive it even more quickly than Schrodinger's equation from the same basic facts of Einstein's light quanta hypothesis and de Broglie's hypothesis.
It was a choice to inject mysticism into quantum mechanics. If you stick with trying to describe the evolution of particles, Bohm is natural. If you want to describe something else whose evolving configuration would give us our experience, then that is fine, but you have to say what that is. The standard interpretation does not as it is solely concerned with experiments and measurements with no definition of them in a fundamental way. It is more like they were trying to do regression fitting on experimental data without any understanding of what the underlying stuff was nor any interest in such a question.
The other difficulty is spin. Spin is trivial if you put the spin degrees of freedom in the value space of the wave function. But many think of spin as real as position. It is not. It matters what the experiment is; the spin value for a particle is not defined independent of the experiment.
Bohmian mechanics gives a theory while standard quantum mechanics gives a computational formalism. Depending on your goals, the latter can be sufficient. The former tells us why stuff happens and also allows us to derive that computational formalism.
This channel and Sixty Symbols are awesome for insight into physics/quantum mechanics without the math.
A particle's 'Zitterbewegung' motion generates wake fields in the vacuum energy, causing the particle to generate its own pilot wave as energy fluctuations in the vacuum.
https://en.wikipedia.org/wiki/Zitterbewegung
https://en.wikipedia.org/wiki/Vacuum_energy
http://phys.org/news/2013-10-strange-behavior-pilot-wave-dyn...
This makes a lot of sense if you think of it that way: Pilot wave theory gives exactly the same results as any other decent interpretation of quantum mechanics. So the work necessary to simulate it will also be exactly the same. Simulating quantum mechanics is not hard because we don't understand it. It is hard because it is an inherently hard problem (otherwise, how could quantum computers be faster than classical computers?).
So replace "it is an inherently hard problem" by "it is most likely an inherently hard problem".
However, there are techniques that allow one to compute the wave function from the Bohmian trajectories. It is just a mathematical trick that certainly does not make BM philosophically more relevant, but it rather neat. This is explored in quantum chemistry contexts. Spin is an issue that stops it from being generally useful, but depending on the context, it is possible.
https://www.crcpress.com/Quantum-Trajectories/Chattaraj/p/bo...
What's interesting is that this actually explains the source of quantum speed up, something no other interpretation can satisfactorily do at this time.
why?
And no, you can't say that about anything. In particular, you can't say that about non-deterministic interpretations of QM (most of them) because causal chains can only be traced back to the most recent non-deterministic event. In fact, pilot wave theories are among the only deterministic interpretations of QM.
That said, the phase space of a double pendulum is bounded.
Suppose you know the exact initial state (dx/dt, x) of the pendulum.
What bounds on K and T are required such that the sequence of numbers is asymptotically indistinguishable from a purely random sequence?
So, because of a silly thought experiment, one might think there is very simple order to the apparent randomness. But, as Pauli would say, "it's not even wrong"...