The Zen anti-interpretation of quantum mechanics (2021)
scottaaronson.blog
scottaaronson.blog
In this view, quantum probability (of finite states) is just about study of category of classical quantum maps described in Picturing Quantum Processes by Bob Coecke and Aleks Kissinger [1]. I highly recommend the book, which requires hardly any prerequisites other than linear algebra and mathematical maturity. Once we get into this framework, no philosophical interpretation is required. E.g. we can't reproduce a quantum state by classical data just means any morphism from a (non-trivial) purely quantum state to a purely classical state does not have a left inverse.
[0]: https://en.wikipedia.org/wiki/Markov_kernel [1]: https://www.cambridge.org/core/books/picturing-quantum-proce...
Classical theories don't have this problem.
And of course what gravity has to do with it.
The question of interpreting quantum mechanics is quite different. We do not have a popular mathematical framework that clearly explains why we perceive a classical world while being "inside" of a quantum wavefunction. There is plenty of vagueness and oversimplification in that last statement, but the difference is "we have an abstraction that explains mostly-perfectly EM waves and we do not care about whether people are comfortable with the abstraction" vs "we do not even have a mathematical abstraction explaining why we perceive a classical world inside of a quantum universe".
It’s pretty easy for me to imagine things bumping into each other because I see that happening all around me
Wait, what? What do they do then? My physical science education ended in high school.
The real answer is that constituent parts of bound states like atoms don't really exist as separate things. An atom is a particular very complicated field configuration, and if you want anything deeper than that you're going to have to learn QFT.
When I think of chemistry, it seems like even if there's some sense in which the bound atom is one big writhing mass of fields, the whole atom doesn't interract in a uniform way with the outside, right? The electrons interract with the outside a lot, the nucleus is harder to get to?
Is it wrong to say the layer of my onion exist as separate entities, when they seem to interract largely independently from the core of the onion, like they do in metals and chemistry?
(I guess the answer is still "you're going to have to learn QFT" :) But the next falsehood down seems fun to ask about too!)
Some of both. Valence electrons in a conductor are delocalized over the entire surface.
> When I think of chemistry, it seems like even if there's some sense in which the bound atom is one big writhing mass of fields, the whole atom doesn't interract in a uniform way with the outside, right? The electrons interract with the outside a lot, the nucleus is harder to get to?
Right, at least for stable atoms at low energy scales.
What I mean is that the mechanism by which sound travels is the same as the mechanism by which I can punch a punching bag and see it swing.
Even if the two materials never “come into contact” because it’s actually the result of charges repelling each other or whatever, the way it works in both cases is the same.
What I suspect you want, however, is some sort of substance or essential nature that makes the thing go. If that's the case, then the answer is that there are no such things. Not for light, not for ordinary classical mechanics, not for anything. Physical systems are what they do.
People have been doing this since pre-helenics, "what is the medium where everything operates" or along the questions of "what is the medium that contains vacuum".
It's not that anyone isn't trying to understand that, it's just that thinking about unprovable things is boring and uninteresting from science point of view. At that level we may as well accept we all live in the Rabbit's Hole with Magic.
> He realised that a changing electric field could create a changing magnetic field, which would then create another electric field and so on. The result would be a self-sustaining electromagnetic field, endlessly repeating, travelling incredibly fast.
https://cosmosmagazine.com/science/physics/what-is-light/
It’s generally studied in undergraduate E&M courses.
Now if you’re curious about how E or M fields propagate you’ll start getting into quantum again. Generally this leads up to quantum electrodynamics, where as I understand it everything is treated as fields and interactions as particles.
How those fields propagate is the same as asking how wave functions “travel”. We simply don’t have to tools to even ask the right questions, IMHO.
"All the electrons that are in your body are not fundamental. All the electrons that exist in your body are waves of the same underlying field."
The next question is, if there are no individual things, what does math count? http://www.katabane.com/mt/ontology.html
You can only measure (compare) something in relation to something else.
If everything is homogeneous there are no measures, no clocks, and no way you could count anything.
And if numbers represent the things we seem to perceive (discrete, fundamental things), though our science demonstrates that individual things are illusory.
The fun part is that the science which demonstrates the illusory nature of the things that math counts is dependent on the math.
All is clear and logical except the part where Aaronson endorses the opinion that Everett isn’t wrong, but grandiose (whatever this means). In justification, it seems he is pushed into further twisted locutions. He speaks of what “breathes fire” into branches of the wave function and says that the question of existence of other branches is “above the pay grade of physics”. Breathing fire and pay grade of physics?
How much simpler to just assert that yes evolution is unitary at all times and, yes, this implies many classical branches.
But I like the idea proposed by Swiss physicist Nicolas Gisin who reasoned that when you our universe is finite and inherently imprecise that “time really passes and new information is created.” which fits with having to update after quantum collapse because new information has been introduced.
https://www.quantamagazine.org/does-time-really-flow-new-clu...
Scott's answer to this seems like a cop-out. He's basically saying "I can't tell which things are epistemic or ontic, and this all seems very complicated, so I'm going to assume the distinction doesn't exist."
Scott Aaranson:
"OK, but are the quantum states “ontic” (really out in the world), or “epistemic” (only in our heads)?" ... "Bad dichotomy."
Daniel Harlow #16:
"silliness of the ontic/epistemic discussion"
Mateus Araújo #50:
"The crucial distinction between ontic versus epistemic entities is that the latter don’t have to respect the laws of physics."
"If quantum states are epistemic, the unphysical collapse is not a problem. If they are ontic, well, we can’t have them collapsing, can we?"
Scott #88:
"There’s the state that I ascribe to the system I’m measuring, which does collapse when I measure it (more epistemic), and then there’s the state that I’d hypothetically ascribe to the entire universe including myself, which never collapses (more ontic)"
Mateus Araújo #97: "Both of your examples are trivially ontic states, the state of the system in your branch of the wavefunction, and the state of the whole wavefunction." "... you can’t even understand what people mean when they say they’re epistemic."
Scott #108: ?
Mateus Araújo #124: ?
Scott #129: no-go theorems => epistemicists should give up
Mateus Araújo #138: Agrees with #129 but lists some epistemicists (including Bohmians).
Scott #187: Are Bohmians epistemicists?
Mateus Araújo #199: "I think calling them (Bohmians) epistemicists is fair, though perhaps an oversimplification."
The blog post’s Zen approach sounds like an agnostic stance, which probably isn’t a bad thing in that regard.
consider the question: is the distinction between theory A vs B testable?
if we can not yet verify that a distinction is untestable, it might still be testable, but we might just be missing the insight on how to construct the experimental setup to test a distinction.
if in fact a hypothesis eventually turns out to be testable, there is no guarantee that a single person or single group would have come up with such a test, perhaps devising the test requires careful and deliberate thought by a lot of people separated by location and or generations.
in that case, the prescription to "shut up and make testable hypotheses and then invalidate them experimentally" would be counter-productive advice, eliding the agressive "shut up" might do wonders...
as an aside, perhaps we fail to see the forest for the trees:
consider Youngs double slit experiment, the quantum mechanical version, perhaps with single photons or alternatively electrons, self-interfering with single particle detection on a screen.
according to the path integral formalism the particle is taking all possible paths, and the complex amplitudes are added together with their respective path-length-delays, i.e. "simple interference", except I only see course notes picturing the paths through slit A and B, and I never see the quantum mechanics books picturing the infinite number of topological loops: through slit A -> [back through slit B, through slit A again] x N -> to screen...
are many worlds interpretations truly untestable, or are we refusing to acknowledge their having been measured a long time ago?
what prevents a physicist from claiming all of the paths in the path integral formalism correspond to the system histories of the universe splitting over the (continuous or discrete) paths, and then merging by interference on the same final measurement point on the screen.
one can not explain the interference as long as one refuses to recognize the particle passed both slits.
if something so basic to the theory of quantum mechanics such as the single particle equivalent to Youngs double slit experiment could be recognized as not just proof of splitting histories, but also proof of merging histories, that might be seen as a paradigm shift.
what makes self-interference of a particle exploring all paths so much more palatable than simply claiming all these paths correspond to different histories of the photon, and those same histories merging through the laws of interference?
That would render the acceptance of interference into an acceptance of not just many worlds, but also the possibility of diverging histories interacting (at least in setups where interference takes effect).
> You shouldn’t confuse the Zen Anti-Interpretation with “Shut Up And Calculate.” The latter phrase, mistakenly attributed to Feynman but really due to David Mermin, is something one might say at the beginning of the path, when one is as a baby.
> but after years of study and effort you’ve returned to the situation of the baby, who just sees the thing for what it is.
So we need to explain that to avoid various very ad-hock notions of classical measurement device and similar routes.
There are attempts to explain the classical world using a notion of decoherence, but that have problems with little progress in the last, say, 20 years to address them.
Then there are ideas based on Everett works that perhaps it is a conscious itself that perceives the classical world and the real world is the wave function. But then such ideas do not explain the exact numerical probabilities.
The interesting resent suggestion was that the probabilities were not really physical but rather represent the lack of information in Bayesian sense.
In past that was used to construct various hidden variable theories, but the new take is that the classical notion that the state of a system can be fully described by its properties at some moment in time is wrong. One also needs to know some information about future to fully describe evolution of the system. This is very speculative, but at least it is fully compatible with relativity and explains probabilities. They simply represent lack of knowledge of boundary conditions in the future.
Once you can locate yourself in a system, the system is adequate as a physical theory of your phenomenological state.
As to “exactly when the universe will split into many worlds”, it happens zillions of times per second pretty much everywhere. It’s even not clear whether the branching is a discrete and countable occurrence, it could really be a continuous fanning out.
This is the problem with MWI gospel, people come and say that it solves all the math but it leaves gaping holes like this. And you didn't even solve the randomness, since as you say it is "statistical", although that is just a theory we currently have no formula for this.
Calling it a wave function collapse is much more honest since it accurately describes the situation, that we have no clue at what it is, why or when it happens. All we know how to do is calculate what possibly happens when we know that a wave function collapse will happen in a certain way.
Decoherence is a phenomenon that's been studied mathematically and experimentally, not simply an interpretation or hand-waving.
At best the decoherence can try to explain how appearance of a sufficiently big classical object may trigger appearance of other objects, but it does not explain how the initial object appears.
The point of the bath is it's a generic model of a large system with lots of states and interactions.
> I.e. why the system interacts with a particular bath and not a superposition of such baths?
The bath can be in a superposition state at the start, that's fine. (Indeed the whole model would be pretty useless otherwise).
In universe-as-a-wave-function there is no external bath. The system as whole is always coherent and is described by a single wave function.
Then the idea of the decoherence is that perhaps during evolution of this big wave function it is possible to find a subsystem that looks like it is approximately decoherent during some time interval. Then one claim that our visible universe is just such subsystem.
The trouble is that so far nobody came up with a model where such temporary approximately classical subsystem appears.
Perhaps we have not tried hard enough or this is a really difficult problem, but at this point I am rather skeptical that decoherence is the right approach at all.
They don't assume it's classical. They assume it's a system with lots of states and interactions, because if you want to have a model of "the external universe" then you have to have some kind of concept of what that looks like.
> In universe-as-a-wave-function there is no external bath. The system as whole is always coherent and is described by a single wave function.
Sure; the point is that if you split the universe into "these two entangled particles" and "the rest of the universe", because you want to understand how we observe that entanglement "decaying" as the particles interact with the rest of the universe, then you can model that as "these two entangled particles" interacting with "a thermal bath".
> The trouble is that so far nobody came up with a model where such temporary approximately classical subsystem appears.
This is nonsense? Wavefunctions generally have a lot of structure and decomposing them into smaller subsystems and functions happens all the time.
Such 2-part subsystem should explain then how the classically-looking part gets the value it measures and the nature of probabilities.
Despite attempts nobody was able to find such subsystem.
If you model it as e.g. a quantum system with a state space that describes what you're measuring, coupled to a thermal bath, then it works. What kind of model are you looking for? Real measuring instruments are probably too big to ever model the full quantum state space of all the particles they're made up of, for example.
> And the whole point of the decoherence exercise is to avoid Born rule as it implies a truly classical subsystem
What do you mean by "truly classical"? The Born rule implies that our experience is the same as if we found ourselves within one of an ensemble of possible classical-like states with a certain probability, but, well, experimentally that's what we do; any interpretation of quantum mechanics needs to be able to recover that reality.
(Analogy: imagine saying "Relativity doesn't explain how we live in a Newtonian world. If relativity is valid, it should be able to say at exactly what speed motion becomes nonrelativistic. Provide a formula for it")
Here is a better theory than MWI: We live in a simulation, the computer is optimized so when there are too complex interactions in an area it simplifies the state into some probable untangled version. This theory predicts that decoherence thus happens at certain computational complexity levels, that is something we could try to find and test experimentally making this theory more scientific than MWI. MWI doesn't lead anywhere, it isn't science, it is just nonsense. I'm not saying my theory here is a good one, but it is better than MWI which isn't a high bar.
Relativity is a real formula and the classical limit is a real phenomenon, but there's no formula for when a relativistic situation "becomes classical".
> you just say that decoherence seems to correlate with entropy and then gave a formula for entropy
If the observed system and the observer/external universe are fully entangled (i.e. their entropy of entanglement is maximal) then decoherence has occurred and all observations of the observed system by that observer will be indistinguishable from if the system were not in superposition. That is a mathematical fact that falls right out of the equations. So I don't know what else you're asking for.
As for parsimonious interpretations, what does superdeterminism add?
As far as I can see people's objection to this tends to be that they don't feel like they're in a superposition. To which my response is: what would you expect it to feel like? Bearing in mind that the wavefunction is describing all the states of your neurons etc..
Sure; I was being poetic. We could talk about when you sleep, or distant regions of the universe that you simply can't measure within a human lifetime.
> MWI isn't the only interpretation, and there isn't a current experiment which can determine which if any of the interpretations are correct. This is the same problem String Theory has had. You can't just base reality on the math.
When other "interpretations" require extra assumptions, you can, and should. Otherwise you can never rule out theories that add extra epicycles that have no observable consequences.
The problem is that the name "Many Worlds Interpretation" sounds grandiose. "Many worlds" is simply an imaginative way of describing the existence of different states of the wavefunction - a concept that is extremely well established and universally accepted. MWI just says that we should assume that Quantum Mechanics holds in all situations, including when we make measurements (i.e., there's nothing special about the measurement process).
There isn't yet any successful quantum theory of gravity, but it's universally believed in the physics community that there will eventually be a quantum description of gravity. Dark energy and dark matter are merely components of the universe whose precise character is not yet known, but again, everyone believes that whatever they are, they will be described quantum mechanically. The arrow of time is merely a consequence of entropy being low in the early Universe.
The reason why physicists universally believe that the Universe is quantum mechanical is that it's impossible for QM to describe only a part of physics. If one aspect of physics is quantum mechanical, it "infects" every other aspect of physics and forces it to be quantum mechanical. You can't have elections obey quantum mechanical laws, but the gravity they produce be classical. It's really all or nothing.
This is why after the discovery of Quantum Mechanics, there was a drive to formulate quantum versions of all fundamental theories.
Either everything is quantum, or nothing is. Given that we are pretty darn sure that most of physics is quantum, we're pretty sure the answer is that everything is quantum, not that nothing is quantum.
There are different ways of formulating quantum theories of gravity (in fact, you can derive the Einstein equations fairly straightforwardly by trying to formulate a quantum theory of a massless spin-2 boson). We don't know which of those theories is correct yet, but it's not as if there's no way to quantize gravity.
By that logic assuming that distant parts of the universe still exist is metaphysics (since it would take many years for any information from there to reach us), yet few people feel the need to call it such.
Under aforementioned requirement, of course. If some sort of MWI is true, there may be just no way to confirm it, and that’s it.
That simpler interpretation is also equivalent to a simpler implementation that doesn’t require apparently-arbitrary resolutions of quantum probabilities (MWI). It’s essentially a question of where you hide the “weird”.
It seems likely to me that they’re both wrong, and the in-principle-untestability of all this points to our mental machinery just failing to map intuitive concepts onto this space, but I’m totally agnostic to this. I do find MWI more conceptually appealing.
There are probability amplitudes in the universal wave function, and those are complex square roots of probabilities.
It is true that deriving the Born Rule purely from the unitary dynamics of the wave function is an unsolved problem. This is discussed a fair bit in the comments to Aaronson's article.
In any case, none of the crutches of the article seem contradictory to me.
What problems? Note that Aaronson, in the article (as well as a number of commenters in the comments) gives decoherence as precisely the answer to the issue you raise.
Another is that decoherence still does not explain the origin of probabilities. I.e. there are models that shows how a pure classical system may arise. But we still have no explanation how a classical device interacting with a quantum system gets the value it measures.
Outside the perturbative regime, there isn't a preferred basis because there aren't any classical observers.
consilient answered this one.
> decoherence still does not explain the origin of probabilities
The amplitudes corresponding to each decoherent branch are the right ones to give the Born rule probabilities. Which, given that the Born rule is empirically observed to work, is probably as good as we're going to get unless (as I doubt will ever happen) someone manages to find a way to derive the Born rule from unitary dynamics.
Do you have any references that give more details on this?
That seems like a very extravagant claim given that we can compute the same results without using the path integral formalism, and without having to know anything about future end points.
This looks something like the Transactional Interpretation of QM, which made use of advanced as well as retarded waves to explain things like EPR correlations. I think it's fair to say this is still a very open area of research and it's way too early to say how it will end up.
this is definitely not true and contradicts relativity. (Type-III Von Neumann algebras have no pure states.)
All this means is that you're not going to recover the state of the entire universe from local observables. Disappointing, but hardly surprising.
Especially interesting point I'd love to read more is that "Because QM is a (the?) generalization of probability theory to involve complex numbers, whose squared absolute values are probabilities. It includes probability as a special case."
The Zen Anti-Interpretation of Quantum Mechanics - https://news.ycombinator.com/item?id=26363004 - March 2021 (64 comments)
Like sure, you can just do the math and get results and not worry about it. But what's the motivation here?
When you get down to it, he seems to believe many-worlds is true, BUT holds the philosophical position that you shouldn't, like, care. Because there's no practical benefit to caring; because it doesn't help you with the calculations.
But, hey, I still care! What the real nature of reality is is an interesting question! It's fun & satisfying to think about, even if has no direct physical practical implications. Plus, it can certainly have psychological implications, dramatically impacting how we feel about the things that have happened to us and life in general.
Other than offering amusement by recreationally asking how many parallel universes can dance on the head of a pin, how should it affect anything?
It is no mercy to know there are kinder universes out there, for with them come those more cruel.
You are already a mote in the sea of infinity, what hope is offered by adding more dimensions to that ocean's depth?
Anyone wise enough to make it this far should know better than to presume that the best available theoretical model is actually the truth. We just use it out of pragmatic lack of a better one (which will likely someday come, but we are approaching the less fun side of the scientific knowledge sigmoid)
If you think there is, what better guide than our best theories of physics so far?
So of course what the nature of the universe is doesn't really affect anything, on a physical level, but our sense of what happens after we're gone does in fact change our behaviors. E.g. Most people contemplate with more horror the possibility of all humanity being wiped out than their own isolated death, even though from the individual perspective it is the same.
Somehow things that cannot possibly matter to us physically still matter to us intellectually.
> It is no mercy to know there are kinder universes out there, for with them come those more cruel.
Well, that's debatable. For example, if you're finding this universe to be particularly cruel, perhaps it is a solace to think it's just one of the universes you must endure, and in some other universe you're having a much easier time.
The Alter where you both exist and is better is not actually guaranteed to exist. In fact basically all them are guaranteed to essentially identical. Quantum splits at a macro level would basically only happen now that we have quantum random number generators to make decisions based on and I doubt that will add up to alters where things are much better.
There are those that think reality (including the universe) is just mathematics. Not modeled by mathematics, but is mathematics. To me, the mathematical universe hypothesis is starting to be the only theory that makes sense.
It might be a distinction without a difference, but I always understood the idea to be that objects we believe to be real are actually mathematical structures.
I see it as sharing some ideas from the we-are-living-in-a-simulation idea. If we were in a simulation, you wouldn’t say data structures and objects are real but would say the table in front of you is actually just data.
That's different from the world being mathematical or informational, which raises the question of what makes it real to us. What breathes life into the equations? Consciousness? Where does consciousness come into the picture?
It's neither fun nor satisfying (edit:for me) to think about. It's theoretically impossible to design and run an experiment which would either falsify Copenhagen and not-falsify Many-Worlds, or falsify Many-Worlds and not-falsify Copenhagen. Unless we can do that (we can't) then arguing about Copenhagen vs Many-Worlds is just navel gazing.
I'm fine with arguing about interpretations which are experimentally distinguishable from Copenhagen.
It's true that a decohering system will have exponentially more classical-ish parts over time - but the parts get smaller at the same rate, and the total size of the system remains the same.
When someone finds another 'world' where those properties are different, I'll be all eyes. Until then, I'll continue to suspect that some people take the current state of our mathematics too seriously. Re-normalizing? Really?
If you don't believe in fairy tales nor assume the existence of a magic mysterious force named "free will" *
They are just tools to facilitate thought. But they are not models that can be accepted as scientific "reality" (as physicists use the word), since they cannot be experimentally verified by design (they do not lead to different predictions from the accepted theory), they are merely plausible.
“Shut up and calculate”
The main issue I have with the prevailing perspective on QM (including this Zen one, which is more widely accepted than the article makes it sound), is just that it's rather pessimistic and slightly unscientific. When declaring that the world works on this foundation of pure mathematical randomness, you are effectively stating that's the end of the line, there's nothing below to explain the phenomena. Probability is a tool to meaningfully reason about incomplete information, but the information is not fundamentally unknowable.
You could for instance say that wether a child will be male or female is purely mathematically random. For all intents and purposes, it is random in the sense we cannot hope to predict it before conception in a single instance. It's not uniformly random, there are external factors that may affect the probability, but it would still be correct to model it as a random variable.
But we must never forget it is a model. We do actually (mostly) know why it looks random from a macro scale: (being very simplistic, I might be quite wrong here) we have many molecules with a certain temperature wizzing past each other with Brownian motion, and it is effectively impossible to predict which will react with which, leading to a certain gene to be selected. We cannot predict it, but we can explain it and prove it by experimental means. It's a chaotic system ruled by relatively simple rules, and we can do a lot of meaningful work with that.
I'm sure many professional physicists also treat QM as an abstraction and model, and there are people trying to peek under the rug. But the community has mostly failed to express this perspective when communicating with a wider audience, and possibly between themselves and their students, for the last century.
Physicists like to redefine the word "real" as to mean "the mathematical model that currently best fits observations", but that is not the common sense of the word. It is merely a pragmatic excuse to focus students on actual publishable research and prevent them from getting constantly philosophical. But it does also make it slightly heretical to keep researching how these phenomena emerge from underlying laws. Which, to an outsider, is kind of the whole point of physics.
To be fair, it is becoming exceedingly expensive to do any experimental work in this area. And as long as we don't have a good number of strong measurements that differ from the theory's predictions, we can only formulate plausible but mostly unprovable theoretical models. I understand why the community is pushing students away from such folly with a small sprinkling of dogmatism.
My difficulty with quantum mechanics is much earlier, that is, more elementary and basic than what Aaronson considers. I've watched MIT lectures, looked into books by experts, ..., and I get nothing I can believe in. Right in the first hour, I get questions with no answers. E.g.,
(1) Wave Function. Okay. I can't find a definition of a wave function. Nothing subtle but just the most elementary objection possible: What are the range and domain of the function? That is, for a function f, we usually write
f: A --> B
for sets A and B meaning that for each element x in set A there is an element y in set B so that f(x) = y. For function f, set A is the domain and set B is the range. Fine. Now for a wave function, what are the sets A and B? No joke -- I can't find a clean, clear, unambiguous, simple, beginning, first step definition.
(2) Everything. There is a common statement that the wave function has everything we can know about the particle, molecule, whatever. Okay: What about Schrödinger's Cat -- where in the wave function is the color of the cat? Simpler, for an electron, where in the wave function is the spin of the electron?
(3) Probability Density. We are supposed to take the complex valued wave function, take its absolute value, scale the function so that its integral is 1, and then, presto, bingo, since it is continuous, non-negative, and integrates to 1, it can be a probability density. In particular it can be the probability density of the current location of the particle. Okay, it CAN be.
Now, what is the evidence that it actually IS the probability density of the particle? That is, so far I can't find any argument, justification, reason to conclude that the probability density of the particle is not something else. E.g., the probability density of the particle might be Gaussian.
(4) Hilbert Space. I've read so many times, more than I can count, that the set of all wave functions form a Hilbert space. Tilt. The definition of Hilbert space I saw in math, e.g., W. Rudin, Real and Complex Analysis, is that a Hilbert space is a complete inner product space where complete means that every Cauchy convergent sequence converges. Well, the wave functions have to be differentiable and, thus, continuous, and it's easy enough to have a sequence of such functions converge to a discontinuous and, thus, not differentiable function and not a wave function. So, wave functions might be points in a Hilbert space, but they can't form a Hilbert space.
(5) Superposition. Some definitions say that superposition is just linearity. But if I have complex numbers a and b and wave functions f and g and form the linear combination
h = af + bg
that seems to be an example of superposition. Okay.
Now, is h actually also a wave function? Its absolute value might not integrate to 1. Okay, scale it so that we do get 1. Now what particle has wave function h? That is, what physics does this linear combination describe? What does h in
h = af + bg
have to do with physics? E.g., from the Pauli exclusion principle, maybe h can't correspond to anything in physics. What is going on with superposition?
(6) Can't Know. We are told that as an electron goes by, we can't know anything about it except its wave function unless we take a measurement at which time we change the wave function, what the electron might do, or some such.
Hmm. The electron has a negative charge and mass. As it goes by, it has to emit from its electric charge an electrostatic force, that is, a Coulomb field and since it has mass, a gravitational wave.
If that situation is not strong enough, then have the electron go past a strong Coulomb field so that it moves in a curve. Such a curve is acceleration so that the electron might generate a photon. We detect the photon and then "know" that the electron went by. And, even if we don't have a strong Coulomb field, our universe is awash in Coulomb fields that might deflect the electron. So, the wave function can't be "destroyed" merely by some Coulomb field. Then we just use a strong Coulomb field, detect the photon, and, thus, know a lot about the electron without changing its wave function. So, the claim that we can't know seems wrong.
(7) The Other Half. We take a beam splitter, such as in the Mach-Zehnder interferometer, and shoot an electron at it. The wave function splits into two parts, say, f and g. For part f, we have a detector. For the last electron we shot, we got a detection from part f. Now, what happens to part g?
No fair saying that part g does not exist since the Mach-Zehnder interferometer can bring such f and g back together and generate interference fringes that part f or g can't generate alone. So, part g does exist. So, what happens to it?
(8) Domain. For the wave function, a guess at the domain is the set of all ordered pairs (x, t) where x is a point in three space and t is time. Well, three space is a big place, and we usually believe that nothing can travel, propagate faster than the speed of light. So, all the wave functions so far are still traveling, still expanding. Right? What happens to them?
Just a few, simple, obvious, elementary, first day questions. I can't find any answers.
https://en.wikipedia.org/wiki/Dirac_equation
“There are four components in ψ because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.”
(1) The wave functions are vectors in a Hilbert space. Functions can form a vector space. The domain of these functions you might as well think of a bit like an index, but continuous. If I have a 2 component vector that's used to represent spin, it can be indexed like psi_0, psi_1. I have psi, you give me 0 or 1, you get a complex number corresponding to the probability amplitude. If I have an infinite-component vector I'm using to represent the position of a particle, you can index it with psi(x=something). I have a psi, you give me a something, I give you a complex number.
(2) Any time you have a question about QM, ask the CM question first. Where is the color of the bouncing ball I'm modeling incorporated? Well, it's not being modeled, so we needn't incorporate it. Ultimately if you have multiple properties you want to represent, you can take the tensor product of the hilbert spaces that model those properties. I have a hilbert space for the position space, and I have a hilbert space for the spin. You just stick them together with the tensor product.
(3) That's the theory and it works in the lab. I don't have a much better answer for you here.
(5) The point of superposition is that h indeed is a valid state of the system, to say nothing of how you'd actually construct it. I don't know how to say what you want "in general." All you've done is make _some_ new state.
(6) I can't figure out what this means -- I think it's not even wrong. You can never know the wavefunction of a system without having several copies available or having some strong constraints on it.
(7) When you detect at f you've collapsed the path of the electron to the path along which you've detected it traveled. If you know you've sent the electron and you get no click on g, then you know it's on path f.
(8) The wavefunction is basically in your head (according to my religion.) It doesn't really "propagate" so much as help you predict what will happen. If you like -- yes, causality propagates at the speed of light. Really we should say light propagates at the speed of causality, and the wavefunction behaves the same. In the end, once all participants are brought together to compare answers, both QM and relativity ensure their answers are consistent.
The range is a state vector specific to the system you're modelling. The domain is usually space if you're trying to model a system statically, or space and time if you're trying to model it dynamically. The terminology is certainly sloppy - people say "wavefunction" to mean both "the state as a function of time/space" and "one particular state value".
> Now, what is the evidence that it actually IS the probability density of the particle? That is, so far I can't find any argument, justification, reason to conclude that the probability density of the particle is not something else. E.g., the probability density of the particle might be Gaussian.
Sure. A priori there's no reason to assume that any of this corresponds to anything physically real. The point is that if you construct these particular mathematical functions, they turn out to correspond to our observations, and furthermore every system we've been able to measure (except perhaps where gravity is involved) has a description in these terms.
> Well, the wave functions have to be differentiable and, thus, continuous, and it's easy enough to have a sequence of such functions converge to a discontinuous and, thus, not differentiable function and not a wave function.
Can you actually construct this though? Not every differentiable function is a valid wavefunction.
> Now, is h actually also a wave function? Its absolute value might not integrate to 1. Okay, scale it so that we do get 1. Now what particle has wave function h? That is, what physics does this linear combination describe? What does h in
> h = af + bg
> have to do with physics? E.g., from the Pauli exclusion principle, maybe h can't correspond to anything in physics. What is going on with superposition?
No, that can't be true - any such h is necessarily a valid state of the system. The Pauli exclusion principle comes out of a case where the amplitude is 0. This is a central part of the theory - if you found a system where h was physically impossible, that would disprove quantum mechanics.
> Can't Know. We are told that as an electron goes by, we can't know anything about it except its wave function unless we take a measurement at which time we change the wave function, what the electron might do, or some such.
I think you've got something you were told backwards. We can't generally know the full wavefunction; when we want to learn about the electron we interact with it and that changes the wavefunction (that is, changes the current state vector - of course the actual function that governs our interaction with the electron remains fixed).
> The wave function splits into two parts, say, f and g. For part f, we have a detector. For the last electron we shot, we got a detection from part f. Now, what happens to part g?
> No fair saying that part g does not exist since the Mach-Zehnder interferometer can bring such f and g back together and generate interference fringes that part f or g can't generate alone. So, part g does exist. So, what happens to it?
This is the collapse/decoherence phenomenon that people argue about. I have my views, but I can't say there's a full consensus.
> Well, three space is a big place, and we usually believe that nothing can travel, propagate faster than the speed of light. So, all the wave functions so far are still traveling, still expanding. Right? What happens to them?
If you fire an electron off into the distance then yes, it keeps travelling. Just like if you fire a cannonball whose position is described by f(x, t) = [1 if x == 2t, 0 otherwise] then yeah, that cannonball keeps going forever. This is normal physics?
Sounds more like a cop-out