Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no “hostile confusion” or “flaw,” there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren’t just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can’t give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all “pre-given.” Makes zero sense. You have not pointed out a “flaw,” you just ramble and declare victory, throwing personal attacks calling me “confused” like this is some sort of competition or something when you have not even made a single coherent point. Attacking me and downvoting all my posts isn’t going to somehow going to prove that you cannot decompose any complex-valued operations into real numbers, nor is it going to prove that complex numbers somehow don’t have to have their properties and operations on them postulated just like real numbers.

And you can also just write it out using real numbers if you wish, it’s just more mathematically concise to use complex numbers. It’s a purely subjective, personal choice to choose to use complex-valued notation. You are trying to argue that making a personal, subjective, arbitrary choice somehow imposes something upon physical reality. It doesn’t. There isn’t anything wrong with the standard formulation, but it is a choice of convention, and conventions aren’t physical. If I describe my losses in a positive number, and then later change convention and describe my winnings with a negative number, the underlying physical reality has not changed, it’s not going to suddenly transmute into something else because of a change in convention in how I describe it.

The complex numbers in quantum theory are not magic. They are also popular in classical mechanics as well, and are just quite common in wave mechanics in general (classical or quantum). In classical wave mechanics, in classical computer science, we use the Fourier transform a lot which is typically expressed as a complex number. It’s because waves have two degrees of freedom, and so you could describe them using a vector of two real numbers, or you could describe them using complex numbers. People like the complex-valued notation because it’s more concise to write down and express formulas in, but at the end of the day it’s just a convention, a notation created by human beings which many other mathematically equivalent notations can describe the same exact thing.

I am having genuine difficulty imagining in your head how you think you made a point here. It seems you’re claiming that given if two vectors have the same symbols between them, they should have identical output, such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di), or complex numbers are not reducible to real numbers.

You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be (ac-bd,ad+bc) and now it is mathematically well-defined and gives identical results.

Negative numbers are just real numbers with a symbol attached. Yes, that’s literally true. In computer code we only ever deal with 0s and 1s. We come up with a convention to represent negative numbers, they are still ultimately zeros and ones but we just say “zeros and ones in this form represent a negative number,” usually just by having the most significant bit 1. They are no physical negative numbers floating out there in the world like in a Platonic sense. What we call “negative” is contextual. It depends upon how we frame a problem and how we interpret a situation. You can lose money at a casino and say your earnings are now negative, or you can say your losses are now positive. Zeus isn’t going to strike you down for saying one over the other. There is nothing physically dictating what convention you use. You just use which convention you find most intuitive and mathematically convenient given the problem you’re trying to describe.

Yes, when we are talking about how computers work, we are talking about how numbers actually manifest in objective, physical reality. They are not some magical substance floating out there in the Platonic realm. Whenever we actually go to implement complex numbers or even negative in the real world, whenever we try to construct a physical system that replicates their behavior and can perform calculations on a physical level, we always just use unsigned real numbers (or natural numbers), and then later establish signage and complexity as conventions combined with a set of operations on how they should behave.

I’m not sure your point about fractional numbers. If you mean literally a/b, yes, there is software that treats a/b as just two natural numbers stitched together, but it’s actually a bit mathematically complicated to always keep things in fractional form, so that’s incredibly rare and you’d only see it in very specialized math software. Usually it’s represented with a floating point number. In a digital computer that number is an approximation as it’s ultimately digital, but I wouldn’t say that means only digital numbers are physical, because we can also construct analogue computers that can do useful computations and are not digital. Unless we discover that space is quantized and thus they were digital all along, then I do think it is meaningful to treat real numbers as, well, physically real, because we can physically implement them.

uh… broski… you do realize a vector of two real numbers can be rotated… right? Please give me a single example for a supposed impossible operation to do on a vector of two real numbers that you can do on complex numbers. I can just define v² where v is a vector (a,b) as (a,b)²=(a²-b²,2ab). Okay, now I’ve succeeded in reproducing your supposedly mathematically impossible operation. Give me another one.

A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.

In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.

A lot of people go into physics because they want to learn how the world works, but then are told that is not only not the topic of discussion but it is actively discouraged from asking that question. I think, on a pure pragmatic standpoint, there is no problem with this. As long as the math works it works. As long as the stuff you build with it functions, then you’ve done a good job. But I think there are some people who get disappointed in that. But I guess that’s a personal taste. If you are a pure utilitarian, I guess I cannot construct any argument that would change your mind on such a topic.

I’m not sure I understand your last question. Of course your opinion on physical reality doesn’t make any different to reality. The point is that these are different claims and thus cannot all be correct. Either pilot wave people are factually correct that there are pilot waves or they are wrong. Either many worlds people are factually correct that there is a multiverse or they are wrong. Either objective collapse people are factually correct that there is an objective collapse or they are wrong (also objective collapse theories make different predictions, so they are not the same empirically).

If we are not going to be a complete postmodernist, then we would have to admit that only one description of physical reality is actually correct, or, at the very least, if they are all incorrect, some are closer to reality than others. You are basically doing the same thing religious people do when they say there should be no problem believing a God exists as long as they don’t use that belief to contradict any of the known scientific laws. While I see where they are coming from, and maybe this is just due to personal taste, at the end of the day, I personally do care whether or not my beliefs are actually correct.

There is also a benefit of having an agreement on how to understand a theory, which is it then becomes more intuitive. You’re not just told to “shut up and calculate” whenever someone asks a question. If you take a class in general relativity, you will be given a very intuitive mental picture of what’s going on, but if you take a class in quantum mechanics, you will not only not be given one, but be discouraged from even asking the question of what is going on. You just have to work with the maths in a very abstract and utilitarian sense.

No, it’s the lack of agreement that is the problem. Interpreting classical mechanics is philosophical as well, but there is generally agreement on how to think about it. You rarely see deep philosophical debates around Newtonian mechanics on how to “properly” interpret it. Even when we get into Einsteinian mechanics, there are some disagreements on how to interpret it but nothing too significant. The thing is that something like Newtonian mechanics is largely inline with our basic intuitions, so it is rather easy to get people on board with it, but QM requires you to give up a basic intuition, and which one you choose to give up on gives you an entirely different picture of what’s physically going on.

Philosophy has never been empirical, of course any philosophical interpretation of the meaning of the mathematics gives you the same empirical results. The empirical results only change if you change the mathematics. The difficulty is precisely that it is more difficult to get everyone on the same page on QM. There are technically, again, some disagreements in classical mechanics, like whether or not the curvature of spacetime really constitutes a substance that is warping or if it is just a convenient way to describe the dispositions of how systems move. Einstein for example criticized the notion of reifying the equations too much. You also cannot distinguish which interpretation is correct here as it’s, again, philosophical.

If we just all decided to agree on a particular way to interpret QM then there wouldn’t be an issue. The problem is that, while you can mostly get everyone on board with classical theories, with QM, you can interpret it in a time-symmetric way, a relational way, a way with a multiverse, etc, and they all give you drastically different pictures of physical reality. If we did just all pick one and agreed to it, then QM would be in the same boat as classical mechanics: some minor disagreements here and there but most people generally agree with the overall picture.

it isn’t scientifically accurate…

There are plenty of simple ways to understand QM on a more ontological level than just the maths. The literature is filled to the brim with them these days. The problem is not so much that it’s difficult, but that there is no agreement. So discussions regarding it just lead to arguments that can’t be settled, and so professors get tired of it and tell people to just shut up and calculate.

Many-worlds is nonsensical mumbo jumbo. It doesn’t even make sense without adding an additional unprovable postulate called the universal wave function. Every paper just has to assume it without deriving it from anywhere. If you take MWI and subtract away this arbitrary postulate then you get RQM. MWI - big psi = RQM. So RQM is inherently simpler.

Although the simplest explanation isn’t even RQM, but to drop the postulate that the world is time-asymmetric. If A causes B and B causes C, one of the assumptions of Bell’s theorem is that it would be invalid to say C causes B which then causes A, even though we can compute the time-reverse in quantum mechanics and there is nothing in the theory that tells us the time-reverse is not equally valid.

Indeed, that’s what unitary evolution means. Unitarity just means time-reversibility. You test if an operator is unitary by multiplying it by its own time-reverse, and if it gives you the identity matrix, meaning it completely cancels itself out, then it’s unitary.

If you just accept time-symmetry then it is just as valid to say A causes B as it is to say C causes B, as B is connected to both through a local causal chain of events. You can then imagine that if you compute A’s impact on B it has ambiguities, and if you compute C’s impact on B it also has ambiguities, but if you combine both together the ambiguities disappear and you get an absolutely deterministic value for B.

Indeed, it turns out quantum mechanics works precisely like this. If you compute the unitary evolution of a system from a known initial condition to an intermediate point, and the time-reverse of a known final condition to that intermediate point, you can then compute the values of all the observables at that intermediate point. If you repeat this process for all observables in the experiment, you will find that they evolve entirely locally and continuously. Entangled particles form their correlations when they locally interact, not when you later measure them.

But for some reason people would rather believe in an infinite multiverse than just accept that quantum mechanics is not a time-asymmetric theory.

cuz I would immediately die?

Speaking of predicting outcomes implies a forwards arrow of time. As far as we know, the arrow of time is a macroscopic feature of the universe and just doesn’t exist at a fundamental level. You cannot explain it with entropy without appealing to the past hypothesis, which then requires appealing to the Big Bang, which is in and of itself an appeal to general relativity, something which is not part of quantum mechanics.

Let’s say we happen to live in a universe where causality is genuinely indifferent to the arrow of time. This doesn’t mean such a universe would have retrocausality, because retrocausality is just causality with an arrow facing backwards. If its causal structure was genuinely independent of the arrow of time, then its causal structure would follow what the physicist Emily Adlam refers to as global determinism and an "all-at-once* structure of causality.

Such a causal model would require the universe’s future and past to follow certain global consistency rules, but each taken separately would not allow you to derive the outcomes of systems deterministically. You would only ever be able to describe the deterministic evolution of a system retrospecitvely, when you know its initial and final state, and then subject it to those consistency rules. Given science is usually driven by predictive theories, it would thus be useless in terms of making predictions, as in practice we’re usually only interested in making future predictions and not giving retrospective explanations.

If the initial conditions aren’t sufficient to predict the future, then any future prediction based on an initial state, not being sufficient to constrain the future state to a specific value, would lead to ambiguities, causing us to have to predict it probabilistically. And since physicists are very practically-minded, everyone would focus on the probabilistic forwards-evolution in time, and very few people would be that interested in reconstructing the state of the system retrospectively as it would have no practical predictive benefit.

I bring this all up because, as the physicists Ken Wharton, Roderick Sutherland, Titus Amza, Raylor Liu, and James Saslow have pointed out, you can quite easily reconstruct values for all the observables in the evolution of system retrospectively by analyzing its weak values, and those values appear to evolve entirely locally, deterministically, and continuously, but doing so requires conditioning on both the initial and final state of the system simultaneously and evolving both ends towards that intermediate point to arrive at the value of the observable at that intermediate point in time. You can therefore only do this retrospectively.

This is already built into the mathematics. You don’t have to add any additional assumptions. It is basically already a feature of quantum mechanics that if you evolve a known eigenstate at t=-1 and a known eigenstate at t=1 and evolve them towards each other simultaneously until they intersect at t=0, at the interaction you can seemingly compute the values of the observables at t=0. Even though the laws of quantum mechanics do not apply sufficient constraints to recover the observables when evolving them in a single direction in time, either forwards or backwards, if you do both simultaneously it gives you those sufficient constraints to determine a concrete value.

Of course, there is no practical utility to this, but we should not necessarily confuse practicality with reality. Yes, being able to retrospectively reconstruct the system’s local and deterministic evolution is not practically useful as science is more about future prediction, but we shouldn’t declare from this practical choice that therefore the system has no deterministic dynamics, that it has no intermediate values and when it’s in a superposition of states it has no physical state at all or is literally equivalent to its probability distribution (a spread out wave in phase space). You are right that reconstructing the history of the system doesn’t help us predict outcomes better, but I don’t agree it doesn’t help us understand reality better.

Take all the “paradoxes” for example, like the Einstein-Podolsky-Rosen paradox or, my favorite, the Frauchiger–Renner paradox. These are more conceptual problems dealing with an understanding of reality and ultimately your answer to them doesn’t change what predictions you make with quantum mechanics in any way. Yet, I still think there is some benefit, maybe on a more philosophical level, of giving an answer to those paradoxes. If you reconstruct the history of the systems with weak values for example, then out falls very simple solutions to these conceptual problems because you can actually just look directly at how the observables change throughout the system as it evolves.

Not taking retrospection seriously as a tool of analysis leads to people believing in all sort of bizarre things like multiverses or physically collapsing wave functions, that all disappear if you just allow for retrospection to be a legitimate tool of analysis. It might not be as important as understanding the probabilistic structure of the theory that is needed for predictions, but it can still resolve confusions around the theory and what it implies about physical reality.

According to our current model, we would probably observe un-collapsed quantum field waves, which is a concept inaccessible from within the universe, and could very well just be an artifact of the model instead of ground truth.

It so strange to me that this is the popular way people think about quantum mechanics. Without reformulating quantum mechanics in any way or changing any of its postulates, the theory already allows you to recover the intermediate values of all the observables in any system through retrospection, and it evolves locally and deterministically.

The “spreading out as a wave” isn’t a physical thing, but an epistemic one. The uncertainty principle makes it such that you can’t accurately predict the outcome of certain interactions, and the probability distribution depends upon the phase, which is the relative orientation between your measurement basis and the property you’re trying to measure. The wave-like statistical behavior arises from the phase, and the wave function is just a statistical tool to keep track of the phase.

The “collapse” is not a physical process but a measurement update. Measurements aren’t fundamental to quantum mechanics. It is just that when you interact with something, you couple it to the environment, and this coupling leads to the effects of the phase spreading out to many particles in the environment. The spreading out of the influence of the phase dilutes its effects and renders it negligible to the statistics, and so the particle then briefly behaves more classically. That is why measurement causes the interference pattern to disappear in the double-slit experiment, not because of some physical “collapsing waves.”

People just ignore the fact that you can use weak values to reconstruct the values of the observables through any quantum experiment retrospectively, which is already a feature baked into the theory and not something you need to add, and then instead choose to believe that things are somehow spreading out as waves when you’re not looking at them, which leads to a whole host of paradoxes: the Einstein-Podolsky-Rosen paradox, the Wigner’s friend paradox, the Frauchiger-Renner paradox, etc.

Literally every paradox disappears if we stop pretending that systems are literally waves and that the wave-like behavior is just the result of the relationship between the phase and the statistical distribution of the system, and that the waves are ultimately a weakly emergent phenomena. We only see particle waves made up of particles. No one has ever seen a wave made up of nothing. Waves of light are made up of photons of light, and the wave-like behavior of the light is a weakly emergent property of the wave-like statistical distributions you get due to the relationship between the statistical uncertainty and the phase. It in no way implies everything is literally made up waves that are themselves made of nothing.

The decision that your brain’s decisions are due to chemical reactions, which itself would be due to chemicals reactions, is self-referential but not circular reasoning.

That’s a classical ambiguity, not a quantum ambiguity. It would be like if I placed a camera that recorded when cars arrived but I only gave you information on when it detected a car and at what time and no other information, not even providing you with the footage, and asked you to derive which car came first. You can’t because that’s not enough information.

The issue here isn’t a quantum mechanical one but due to the resolution of your detector. In principle if it was precise enough, because the radiation emanates from different points, you could figure out which one is first because there would be non-overlapping differences. This is just a practical issue due to the low resolution of the measuring device, and not a quantum mechanical ambiguity that couldn’t be resolved with a more precise measuring apparatus.

A more quantum mechanical example is something like if you apply the H operator twice in a row and then measure it, and then ask the value of the qubit after the first application. It would be in a superposition of states which describes both possibilities symmetrically so the wavefunction you derive from its forwards-in-time evolution is not enough to tell you anything about its observables at all, and if you try to measure it at the midpoint then you also alter the outcome at the final point, no matter how precise the measuring device is.

Let’s say the initial state is at time t=x, the final state is at time t=z, and the state we’re interested in is at time t=y where x < y < z.

In classical mechanics you condition on the initial known state at t=x and evolve it up to the state you’re interested in at t=y. This works because the initial state is a sufficient constraint in order to guarantee only one possible outcome in classical mechanics, and so you don’t need to know the final state ahead of time at t=z.

This does not work in quantum mechanics because evolving time in a single direction gives you ambiguities due to the uncertainty principle. In quantum mechanics you have to condition on the known initial state at t=x and the known final state at t=z, and then evolve the initial state forwards in time from t=x to t=y and the final state backwards in time from t=z to t=y where they meet.

Both directions together provide sufficient constraints to give you a value for the observable.

I can’t explain it in more detail than that without giving you the mathematics. What you are asking is ultimately a mathematical question and so it demands a mathematical answer.

I am not that good with abstract language. It helps to put it into more logical terms.

It sounds like what you are saying is that you begin with something a superposition of states like (1/√2)(|0⟩ + |1⟩) which we could achieve with the H operator applied to |0⟩ and then you make that be the cause of something else which we would achieve with the CX operator and would give us (1/√2)(|00⟩ + |11⟩) and then measure it. We can call these t=0 starting in the |00⟩ state, then t=1 we apply H operator to the least significant, and then t=2 is the CX operator with the control on the least significant.

I can’t answer it for the two cats literally because they are made up it a gorillion particles and computing it for all of them would be computationally impossible. But in this simple case you would just compute the weak values which requires you to also condition on the final state which in this case the final states could be |00⟩ or |11⟩. For each observable, let’s say we’re interested in the one at t=x, you construct your final state vector by starting on this final state, specifically its Hermitian transpose, and multiplying it by the reversed unitary evolution from t=2 to t=x and multiply that by the observable then multiply that by the forwards-in-time evolution from t=0 to t=x multiplied by the initial state, and then normalize the whole thing by dividing it by the Hermitian transpose of the final state times the whole reverse time evolution from t=2 to t=0 and then by the final state.

In the case where the measured state at t=3 is |00⟩ we get for the observables (most significant followed by least significant)…

• t=0: (0,0,+1);(+1,+i,+1)
• t=1: (0,0,+1);(+1,-i,+1)
• t=2: (0,0,+1);(0,0,+1)

In the case where the measured state at t=3 is |11⟩ we get for the observables…

• t=0: (0,0,+1);(-1,-i,+1)
• t=1: (0,0,+1);(+1,+i,-1)
• t=2: (0,0,-1);(0,0,-1)

The values |0⟩ and |1⟩ just mean that the Z observable has a value of +1 or -1, so if we just look at the values of the Z observables we can rewrite this in something a bit more readable.

• |00⟩ → |00⟩ → |00⟩
• |00⟩ → |01⟩ → |11⟩

Even though the initial conditions both began at |00⟩ they have different values on their other observables which then plays a role in subsequent interactions. The least significant qubit in the case where the final state is |00⟩ begins with a different signage on its Y observable than in the case when the outcome is |11⟩. That causes the H opreator to have a different impact, in one case it flips the least significant qubit and in another case it does not. If it gets flipped then, since it is the control for the CX operator, it will flip the most significant qubit as well, but if it’s not then it won’t flip it.

Notice how there is also no t=3, because t=3 is when we measure, and the algorithm guarantees that the values are always in the state you will measure before you measure them. So your measurement does reveal what is really there.

If we say |0⟩ = no sleepy gas is released and the cat is awake, and |1⟩ = sleepy gas is released and the cat go sleepy time, then in the case where both cats are observed to be awake when you opened the box, at t=1: |00⟩ meaning the first one’s sleepy gas didn’t get released, and so at t=2: |00⟩ it doesn’t cause the other one’s to get released. In the case where both cats are observed to be asleep when you open the box, then t=1: |01⟩ meaning the first one’s did get released, and at t=2: |11⟩ that causes the second’s to be released.

When you compute this algorithm you find that the values of the observables are always set locally. Whenever two particles interact such that they become entangled, then they will form correlations for their observables in that moment and not later when you measure them, and you can even figure out what those values specifically are.

To borrow an analogy I heard from the physicist Emily Adlam, causality in quantum mechanics is akin to filling out a Sudoku puzzle. The global rules and some “known” values constrains the puzzle so that you are only capable of filling in very specific values, and so the “known” values plus the rules determine the rest of the values. If you are given the initial and final conditions as your “known” values plus the laws of quantum mechanics as the global rules constraining the system, then there is only one way you can fill in these numbers.

“Free will” usually refers to the belief that your decisions cannot be reduced to the laws of physics (e.g. people who say “do you really think your thoughts are just a bunch of chemical reactions in the brain???”), either because they can’t be reduced at all or that they operate according to their own independent logic. I see no reason to believe that and no evidence for it.

Some people try to bring up randomness but even if the universe is random that doesn’t get you to free will. Imagine if the state forced you to accept a job for life they choose when you turn 18, and they pick it with a random number generator. Is that free will? Of course not. Randomness is not relevant to free will. I think the confusion comes from the fact that we have two parallel debates of “free will vs determinism” and “randomness vs determinism” and people think they’re related, but in reality the term “determinism” means something different in both contexts.

In the “free will vs determinism” debate we are talking about nomological determinism, which is the idea that reality is reducible to the laws of physics and nothing more. Even if those laws may be random, it would still be incompatible with the philosophical notion of “free will” because it would still be ultimately the probabilistic mathematical laws that govern the chemical reactions in your brain that cause you to make decisions.

In the “randomness vs determinism” debate we are instead talking about absolute determinism, sometimes also called Laplacian determinism, which is the idea that if you fully know the initial state of the universe you could predict the future with absolute certainty.

These are two separate discussions and shouldn’t be confused with one another.

In a sense it is deterministic. It’s just when most people think of determinism, they think of conditioning on the initial state, and that this provides sufficient constraints to predict all future states. In quantum mechanics, conditioning on the initial state does not provide sufficient constraints to predict all future states and leads to ambiguities. However, if you condition on both the initial state and the final state, you appear to get determinstic values for all of the observables. It seems to be deterministic, just not forwards-in-time deterministic, but “all-at-once” deterministic. Laplace’s demon would just need to know the very initial conditions of the universe and the very final conditions.