## Monday, May 10, 2021

### Quantum wavefunction is not everything

Reader Ajit argues that I am not following textbook quantum mechanics properly. He has posted Postulates of quantum mechanics, as stated by various authors.

Checking other versions of the postultes, I find:

The state of a system is completely described by a wavefunction
I wonder why this would be stated as a postulate. It is not used by the theory anywhere, and it is not true.

Sometimes it is stated for a single particle, but it cannot be true if the particle is entangled with another. Sometimes it is stated for scalar wave functions, but that cannot be true if the particle has spin.

You can correct those problems by introducing spinor-valued wave functions of several variables, but then you are still ignoring quantum fields and all sorts of other complexities.

Now you might say: Okay, but if use the whole Standard Model, or some bigger unified field that takes into account all possible interactions, and then we construct a wave function of the universe, then that would completely describe the state of the universe.

That would not be quantum mechanics. That would be some theorist's fantasy that has never been carried out.

Quantum mechanics is a theory that takes in some available info, and makes some predictions, but never achieves a complete description of the system. Nobody has any idea how any such complete description would ever be accomplished.

Take a simple example, the Schroedinger Cat. The wavefunction is a superposition of dead and alive states. Is it a complete description of the state of the system? No, of course not. The cat is either dead or alive. You can get a more complete description by opening the door and looking to see if the cat is dead. The wavefunction is most emphatically not giving a complete description.

I don't know why anyone would say that the wavefunction is a complete description of the system. Other physics theories do not start off with a postulate declaring some sort of god-like omniscient. It doesn't make sense to even say something like that.

And yet this postulate is prominent on various lists of postulates for quantum mechanics. I will have to do some further research to find out who is responsible for this silly idea.

This week's Dr. Bee video is on Einstein's spooky action at a distance. She says that the spookiness is the measurement update (ie, collapse of the wavefunction), not entanglement.

Believing that the wave function is a complete description necessarily causes these spooky concerns. Any observation affect distant parts of the wavefunction. If the wavefunction is a complete physical thing, then it is spooky.

1. FWIW, I agree with your saying that "some bigger unified field that takes into account all possible interaction" has never been carried out and that "Nobody has any idea how any such complete description would ever be accomplished." Even with dark matter, dark energy, and gravity included, how could we know for sure whether there is or isn't some other field?

For the cat, you could also ask, X:="Is the cat in an eigenstate of can-the-cat-be-resuscitated?". If D:="is the cat dead?" is a diagonal matrix applied to a 2-dimensional Hilbert space vector, then X is not diagonal and [D,X]≠0. Obviously this is somewhat whimsical, but noncommutativity is a significant aspect of the relationship between classical and quantum ways of modeling systems. The concept of quantum state tomography depends on noncommutativity. Even classically, X makes sense for a cat.

2. Dear Roger,

0. Thanks!

1. The Wiki on "Branches of physics" shows a diagram which is helpful a bit. It contrasts QM from CM by "size", which is not a fundamentally most accurate way of looking at their difference. A more accurate description is given by the "Bronstein cube", but I won't refer to it here. (I got to know of it just now!)

2. Introductory courses and text-books on QM always take "QM" to mean: the non-relativistic QM (NR-QM for short). The Postulates document I compiled is at this level.

3. Postulating the system wavefunction is necessary because QM defines *all* dynamical variables in reference to it.

4. The simplest *theoretical* system that can still show some "quantum-ness" is the NR-QM of a single, spinless, particle.

5. Next in generality might be the NR-QM of a single particle having *spin*.

See the paper by Nottale and Célérier (i.e. the last section from my Postulates document). Their Postulate 1 says: "wavefunction $\psi(r,s,t)$ is an equivalence class of complex functions" of "... any additional degrees of freedom such as spin $s$".

So, broadly speaking, the term "wavefunction" can also *include* the spin.

6. For generalization from point 4., some people ignore spin but instead go to a theoretical system of 2 or more *spinless* particles.

The system wavefunction for an $N$-particle system is defined over a $3N$-dim. configuration space, not the $3$-dim. physical space. This opens one more issue for interpretation!

You can then incorporate the spin into the description.

7. Now, increasing $N$, *and* after performing what seems to me is a distinct step of abstraction, one can say that the state of the *universe* is completely described by a single wavefunction.

8. For Foundations of QM, this level of generality is more than adequate; you don't need relativity for it.

Indeed, issues like the wave-particle duality, randomness (Born's postulate), the Heisenberg uncertainty, and tunneling, can all be discussed with a system of just *one* *spinless* particle (i.e. point no. 4)!

[Contd in the next comment...]

3. [...Contd from the previous comment]

9. Optionally, many people follow a different path. They *start* by abstractly considering only the spin aspect of a few-particle system. Thus, they leave the spinless (aka Schrodinger) wavefunction $\psi(x_1, x_2, \dots,t)$ out of an explicit consideration.

This idea makes the Hilbert space finite-dimensional---which is categorically easier to handle. This way, they say, they can better handle issues like entanglement.

However, as Dan Schroeder has pointed out, "Entanglement isn't just for spin" ( https://arxiv.org/abs/1703.10620 ).

Personally, I like to keep the wavefunction in the picture at all times---whether spin is considered or not!

10. I have no knowledge of the relativity theory (not even the SR). So I won't comment on the related issues. But people agree that: (i) GR + QM is still under research (there is no fully validated theory for it), but (ii) "quantum-ness" will be an integral part of it---itself the subject of Foundations of *QM*.

11. If my new approach is correct, then the measurement problem of QM can be satisfactorily solved using just the NR-QM of a many-particle system. IMO, Foundations-wise, the measurement problem is the only "real" problem (of *NR-QM*) that is still standing.

12. The jump from an $N$ particle system to the universe is not special to QM.

In the 19th c., the "classical" thermodynamics took a similar jump of abstraction when it extended the energy conservation principle to hold valid not only for *finite*-sized isolated systems, but also to the whole universe, via the device of imagining the universe to be an isolated system.

Their rationale was (putting it in my words): Qua a concept that subsumes all physical objects, there is nothing else left for the universe to interact with. With "absence of external interactions" as the defining feature, the universe can be regarded as an isolated system. Here, the word "isolated" means: *conceptually* isolated (in thought), and not *physically* isolated (from other physically existing objects, as by imagining perfectly rigid boundaries made of perfectly heat-insulating material).

Best,
--Ajit

4. Gee... if only we could know all the initial conditions of a system, we could predict blah blah blah...

Gee... if only a single hydrogen particle could actually be fully described in a Hilbert space we could predict blah blah blah...

Gee...if only we could fully describe three objects interacting gravitationally we could actually predict blah blah blah...

Gee...if only it was possible in GR to have more than one mass piled up in a given spacetime without pretending that Superposition does not apply since the entire solution is highly non-linear we could predict blah blah blah...

Gee...if only the standard model actually even contained gravity at all we could predict blah blah blah...

Yeah, if my aunt had balls she'd be my uncle.

It's just a cruel cruel universe ruled by the god of GIGO.

"On two occasions I have been asked, "Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?" ... I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question."
— Charles Babbage

1. Dear CFT,

>> "if only we could know all the initial conditions of a system, we could predict blah blah blah..."

1. This bit sounds as if it were coming from a faithful student of Heisenberg's.

Of course, Heisenberg's own expression was a bit different; what he said was to the effect: "if only we could *precisely* *specify* all the initial conditions...".

The "knowing"- and "observer"-related nonsense was introduced by others, first in Europe. The crystallization (i.e. formalization) in physics of this nonsense is due to the American mathematician-physicist John von Neumann. (The same guy who is wrongly credited for having invented the stored program architecture of computers.) Hippies and Indian "guru"s came later, much later.

2. Proper initial conditions (ICs for short) have to be specified in adequate detail. That's a part of problem statement. If you leave the statement of a problem incomplete, then you can't hope to solve / answer it.

3. A bit on nonlinearity.

2.1. Recall: Videos in which people play with a cat, by shining the light of a laser-pointer around it.

Take a bunch of shiny spheres, place them on the floor of a room. Holding a laser-pointer steady in a specific position and angle, shine its light at the spheres. After multiple reflections, a sharp point of light will appear somewhere on the walls of the room. Mark the spot where the bright dot appeared on the wall.

The position of the dot is repeatable, in reality. It also is predictable, in theory. The ray-tracing software uses the same idea in general. (They trace the ray in the reverse, but that's not germane here.)

Now, change the angle a slight bit, say by 5 degrees. The spot doesn't change its position proportionately. The *change* in the position of the spot is wild. Even with a 5 degree change, the spot can appear on the *opposite* wall too!

That's nonlinearity at work for you.

2.2 Nonlinearity (or chaos) doesn't mean: unreal. It also doesn't mean: acausal, random, or unpredictable.

2.3 In the above example, nonlinearity doesn't refer to just one observation. Rather, it refers to a certain *change* observed in two observations. It refers to the fact that a *small* change in the *input* conditions causally produces a *wild* change in the *output*. Happens in reality. Causally. Is predictable.

2.4. You can replace the laser-ray by the trajectory of a tiny (discrete) particle.

2.5. Nonlinearity doesn't always refer to *two* experiments (first throw light/particle from a certain angle, then throw it the second time at a slightly different angle). The two differing ICs can occur *simultaneously*, i.e., in a single experiment. E.g.: fluid flow.

The streamlines (or pathlines, or streaklines) specified at two nearby points on the upstream side of a flow often are found to be diverged on the downstream side. That's due to nonlinearity. The "*change* in the specified condition", in such a case, refers to the fact that two velocities were specified at two nearby but different points at the same time.

The remarkable difference in the trajectories is what happens *in reality*. (Even ancient artists noticed it.) The Navier-Stokes equations also reproduce the behaviour---in theory.

2. [...Contd from previous reply to CFT]

[BTW, there is a typo in the immediately above reply to CFT. The sub-point #s 2.1 through 2.5 should read 3.1 through 3.5.]

4. When Heisenberg empahsized the role of the ICs, he meant something different from their role in the pre-quantum physics.

Heisenberg was emphasizing the idea that, in QM, we should only talk of the laser-pointer and the dot on the wall, because these are the only quantities that can be measured in experiments.

(So, following Heisenberg, spheres would play the role of "hidden" variables.)

5. You can argue that the objection of the Copenhagen interpretation to the possibility of hidden variables is uncalled for. People do argue that way. (I don't, but people do.)

But you *cannot* argue against the very necessity of having to specify, adequately well, the required ICs.

To illustrate this point in the simplest manner, consider this question:

"Who was the president of the USA?"

Unless you specify a time slice (or by taking a limit, an instant) which also refers to a valid range in time, you can't answer that question.

6. So, to rebel against the necessity of having to specify the right kind of ICs (to the required level of detail) is to rebel against the very idea of physics, of science, of history, ...; why, actually against cognition itself; even against reality!

Best,
--Ajit

5. Ajit,
I was simply making very light of the conceit and hubris of those who think they can even possibly know all know ALL the initial conditions which were present billions of years ago which are required to make their precious published speculations work.

As far as science goes, I firmly believe uncertainty and inexactitude are not something you can handwave away like an unpleasant odor from a flatulent dog whenever it gets in the way of publication, personal politics, funding, ego, or petty Nobel prize grubbing.

Reality is the Devil in the details keeping score.

1. CFT,

1. I don't want to quibble, though I still don't think that your first line even remotely suggests what you say was in your mind. ... But it's OK. If you want to crack jokes on people like those, I am perfectly fine with that.

2. Yes, data come with statistical fluctuations (as also under the influence of unknown/unaccounted for factors) and hence, data must be reported with error bars---always. ... But no, experimental uncertainties don't prevent us from reaching certainty in knowledge.

3. I don't know why physicists (and their employers (like universities), and even their national agencies/administrations and compatriots) go so crazy about the Nobel prizes.

It has been a long time that I have been thinking differently about the whole thing.

I only ask myself if some prominent work is of enough quality so as to fall somewhere within the band of the Nobel prize or not. That's the question I ask, not whether this person "is Einstein" or not. And I ask the first question even while *studying* physics, whether the work in question actually got the Nobel or not.

For determining the answer, I find that the question becomes easier if it is rephrased thus: "Can this work rightly be *nominated* for a Nobel? Would *I* do that? why (or why not)?".

Then, I like to stop right there.

If a work is worth a nomination, then it falls in that band, and that's all that matters to me.

The rest is the limitations of the rules and the mechanics it requires, other factors like how many deserving people are still alive and waiting in the queue, human factors like the workings of committees, and of course, the indirect "persuasion" tactics and "optics" (in which Americans / Western countries are so damn good!).

Nothing of this last list matters to me. Not even one bit.

In short: "Worth nominating? yes? no?" and "why / why not?". That's where the matter stops. Ideally, IMO, that's where it should. But then, you know, Americans and Westerners / people from rich countries *are* different... (Also some people from India like IITians and all...)

Anyway, bye for now...

Best,
--Ajit