A Measurement problem in the Ensemble interpretation

[stevendaryl]

Further, according to our current understanding, the "classicality" of macroscopic systems (including measurement devices, which are nothing special but just also macroscopic systems) is well compatible with quantum theory and nothing else than the "Law of Large Numbers", i.e., if you have $N$ degrees microscopic degrees of freedom figuring additively into a macroscopic variable $X$ (like, e.g., the total energy of a gas which consists of $N/3$ monatomic particles) one has $\Delta X/|X| \sim 1/\sqrt{N}$. If $N$ is large the fluctuations (both quantum an thermal) are small.

I don't think that reasoning is correct. What you're suggesting is that the law of large numbers by itself is enough to explain why there are never macroscopic superpositions? The law of large numbers is only valid if you have a large number of independent systems with the same distribution of values of an observable, then the averages over all the systems will have a smaller variance than the values on the individual systems. But that's not what's going on when we perform a measurement and get a definite value.

 

[Mentz114]

I don't think that reasoning is correct. What you're suggesting is that the law of large numbers by itself is enough to explain why there are never macroscopic superpositions? The law of large numbers is only valid if you have a large number of independent systems with the same distribution of values of an observable, then the averages over all the systems will have a smaller variance than the values on the individual systems. But that's not what's going on when we perform a measurement and get a definite value.

[my emphasis]
Do you know what is happening when a measurement is made ? Your statement makes at least one implicit assumption that can be challenged.

 

[vanhees71]

I don't think that reasoning is correct. What you're suggesting is that the law of large numbers by itself is enough to explain why there are never macroscopic superpositions? The law of large numbers is only valid if you have a large number of independent systems with the same distribution of values of an observable, then the averages over all the systems will have a smaller variance than the values on the individual systems. But that's not what's going on when we perform a measurement and get a definite value.

The averaging is over a large number (of order $10^24$) of microscopic observables making up a macroscopic one.

 

[bhobba]

None of those things--symmetry, conservation, transformations, etc.---is fundamental to the project of physics, which is about describing the world that we live in in a way that is precise enough (mathematically) to make predictions capable of falsification.

It's exactly what Feynman said - nothing more - nothing less:


However Noether provided an invaluable tool in making those 'guesses'.

What came first - the guess or its experimental proof? The logic of those guesses is sometimes so compelling you are shocked if it's wrong. Even Feynman realized it. He came up with some beautiful theory (ie guess) that experiment was against at the time. He decided to wait rather than abandon it. By his dictum he should have scrapped it - but it was just so beautiful. He was right - later experiments proved it.

It really is a strange thing. As one person expressed it, and even wrote a book with the title, there is fire in the equations. Trying to get to the bottom of it has led to some rather interesting views such as those of Penrose.

Me - I am with Gell-Mann:


Thanks
Bill

 

[stevendaryl]

It's exactly what Feynman said - nothing more - nothing less:


However Noether provided an invaluable tool in making those 'guesses'.

What came first - the guess or its experimental proof? The logic of those guesses is sometimes so compelling you are shocked if it's wrong. Even Feynman realized it. He came up with some beautiful theory (ie guess) that experiment was against at the time. He decided to wait rather than abandon it. By his dictum he should have scrapped it - but it was just so beautiful. He was right - later experiments proved it.

It really is a strange thing. As one person expressed it, and even wrote a book with the title, there is fire in the equations. Trying to get to the bottom of it has led to some rather interesting views such as those of Penrose.

Me - I am with Gell-Mann:


Thanks
Bill

Yeah, there have been lots of examples where looking for a compelling theory galloped way ahead of experiment, and the experiments confirmed the beautiful theory was right. Some examples:

• I think it's true that Maxwell introduced the "displacement current" because it fixed flaws in the mathematical appearance of his equations, rather than because there was any evidence for it.
• General relativity was really driven by Einstein's desire for a theory that elegantly incorporated gravity and Special Relativity, not because of evidence. The evidence came soon afterwards.
• Antiparticles and electron spin were predicted by Dirac's equation, which was motivated by an attempt to reconcile quantum mechanics with relativity (spin had already been discovered, but it comes naturally out of the Dirac equation).

Of course, in recent years, there have been quite a few counter-examples, where the pursuit of an intellectually-pleasing theory turned out not to have any empirical support. I'm thinking the various GUT theories, supersymmetry, string theory.

 

[Mentz114]

Yeah, there have been lots of examples where looking for a compelling theory galloped way ahead of experiment, and the experiments confirmed the beautiful theory was right. Some examples:

• I think it's true that Maxwell introduced the "displacement current" because it fixed flaws in the mathematical appearance of his equations, rather than because there was any evidence for it.
• General relativity was really driven by Einstein's desire for a theory that elegantly incorporated gravity and Special Relativity, not because of evidence. The evidence came soon afterwards.
• Antiparticles and electron spin were predicted by Dirac's equation, which was motivated by an attempt to reconcile quantum mechanics with relativity (spin had already been discovered, but it comes naturally out of the Dirac equation).

Of course, in recent years, there have been quite a few counter-examples, where the pursuit of an intellectually-pleasing theory turned out not to have any empirical support. I'm thinking the various GUT theories, supersymmetry, string theory.

None of this is relevant. You guys have galloped off on some by-way.

It is preposterous to say that symmetries and conserved charges and currents are irrelevant in Physics when it is on these principles that the SM is made.
No physical theory which does not conform will be any use.

I give up.

 

[stevendaryl]

None of this is relevant. You guys have galloped off on some by-way.

Well, your comments about symmetry were pretty far off-topic to start with, so it's a little strange for you to complain about relevance.

It is preposterous to say that symmetries and conserved charges and currents are irrelevant in Physics

I didn't say they were irrelevant. I said:

None of those things--symmetry, conservation, transformations, etc.---is fundamental to the project of physics, which is about describing the world that we live in in a way that is precise enough (mathematically) to make predictions capable of falsification.

 

[stevendaryl]

The averaging is over a large number (of order $10^24$) of microscopic observables making up a macroscopic one.

But the law of large numbers is not just about any situation involving large numbers. It's specifically (from Wikipedia):

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.

If we use a macroscopic device to measure, say, the spin of an electron, the 10^{24} is not the number of times we perform the experiment. So the law of large numbers is not obviously relevant.

 

[Mentz114]

Well, your comments about symmetry were pretty far off-topic to start with, so it's a little strange for you to complain about relevance.
:

I wanted to know why you keep asking 'why can macroscopic objects not form superpositions ?'. You say that the 'rules of quantum mecanics demand it'.

Which rule says that ? How was that rule derived ?

 

[stevendaryl]

I wanted to know why you keep asking 'why can macroscopic objects not form superpositions ?'. You say that the 'rules of quantum mecanics demand it'.

Which rule says that ? How was that rule derived ?

I think it's good form not use to quote symbols unless you're quoting. It's not fair to ask me to defend a statement that you just made up (such as "symmetries and conserved charges and currents are irrelevant in Physics").

Anyway, according to quantum mechanics, states obey the principle of superposition: If |A\rangle is a state, and |B\rangle is a state, then \alpha |A\rangle + \beta |B \rangle is a state. Are you asking where that rule is stated? It's part of the definition of a Hilbert space.

 

[Mentz114]

It's exactly what Feynman said - nothing more - nothing less:
[..]
Me - I am with Gell-Mann:


Thanks
Bill

If we're down to quoting authority - Steven Weinberg said

Our job as physicists is to see things simply, to understand a great many
complicated phenomena in a unified way, in terms of a few simple principles.

 

[Mentz114]

I think it's good form not use to quote symbols unless you're quoting. It's not fair to ask me to defend a statement that you just made up (such as "symmetries and conserved charges and currents are irrelevant in Physics").

Anyway, according to quantum mechanics, states obey the principle of superposition: If |A\rangle is a state, and |B\rangle is a state, then \alpha |A\rangle + \beta |B \rangle is a state. Are you asking where that rule is stated? It's part of the definition of a Hilbert space.

Apologies, I hope I didn't distort the meaning.

Ok, thanks. A purely mathematical statement then, with no support from any physical principle.

[aside]
To anyone who finds these things interesting I commend this learned paper. I don't claim to have read it all nor agree with everything the author asserts.

http://philsci-archive.pitt.edu/878/1/PSA2002.pdf

Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity

John Earman
Dept. of History and Philosophy of Science
University of Pittsburgh

 

[bhobba]

If we're down to quoting authority - Steven Weinberg said

But finding those - well let's just say even its greatest exponent, Einstein, hell even slightly lesser lights like Von-Neumann, guys likely even better than Weinberg (and that's saying something) is a slow slow process with a lot of false twists and turns. As Feynman said a trick, like say positivism, used early on by Einstein, works just once - after that everyone knows it and tries it. If it works, which it rarely does again, but if it does progress is made and everything falls away. If it doesn't, and this is usually the case, a different approach is required. Want a Nobel? Figure out the approach that works. Good luck.

But as to your question - the answer is known - symmetry. But to understand it you need quite a bit of study or simply accept what you have been told.

Thanks
Bill

 

[vanhees71]

But the law of large numbers is not just about any situation involving large numbers. It's specifically (from Wikipedia):
If we use a macroscopic device to measure, say, the spin of an electron, the 10^{24} is not the number of times we perform the experiment. So the law of large numbers is not obviously relevant.

Ok, then why do you think classical physics works so well for macroscopic matter? The ensembles you cite are Gibbs ensembles, but the law of large number says that we measure almost with certainty a definite value for a macroscopic variable given the very small relative standard deviation of this variable of order $\mathcal{O}1/\sqrt{N}$. So you can within this relatively negligible uncertainty predict the value for this macroscopic variable. That's how, in my understanding, the apparent deterministic nature of classical physics is principally explained within the statistical interpretation.

If we measure the spin of an electron (within a neutral atom ;-)) in an SG apparatus we measure its position with a photo plate, which is the macroscopic object. What we call "position", is a very coarse grained macroscopic region consisting of very many atoms/molecules. Within that resolution we can say the "electron hit the photoplate at a definite place". That the position is related to the spin observable is due to the entanglement of the spin component determined by the directdion of the magnetic field with position.

 

[stevendaryl]

Ok, then why do you think classical physics works so well for macroscopic matter?

I think the answer is complicated. I think it has to do with the fact that decoherence will spoil interference effects between macroscopic states with different values of macroscopic observables. Without interference effects, we are free to think of quantum probabilities as classical probabilities, reflecting uncertainty about which of several alternatives is actually the case. So that's why Copenhagen (or the minimal interpretation) seems to work so well, because it is consistent to think of macroscopic probabilities as reflecting ignorance in a way that it is not consistent to interpret microscopic amplitudes.

So the whole structure works, to the extent that we can make a clean distinction between macroscopic and microscopic variables. But it's somewhat schizophrenic, since we are applying a different interpretation to probabilities depending on how large the system is.

 

[vanhees71]

Hm, but you can use quantum statistics to calculate the properties of, say, an ideal gas to get the classical results in the limit, where you can approximate the Bose-Eisntein or Fermi-Dirac statistics with the modified classical Maxwell-Boltzmann statistics (with modified I mean the notorious factor $1/N!$ borrowed from the indistinguishability of particles from QT). Even if you keep the full quantum statistics, the macroscopic quantities which are indeed very coarse grained (total energy $U$, temperature, chemical potential(s),...) you get very small fluctuations due to the fact that $N$ is large. Of course you get quantum effects, if the gas becomes degenerate (low temperatures, high densities), among them early achievements of "old quantum mechanics" like the specific heat at low temperatures, the microscopic understanding of the 3rd Law and so on.

Of course, decoherence is also an important point. Here you can even get semiclassical behavior of microscopic objects when interacting with macroscopic objects, as was already realized in Mott's famous article on why $\alpha$ prticles in a clould chamber seem to run on straight-line trajectories.

 

[zonde]

It should be possible, if measuring devices were treated no differently than microscopic systems, to restate the theory without mentioning measurements or macroscopic systems.

Why do you think it should be possible?
Measuring devices are just bigger more complicated "particles". If you take away Born rule, QM can not say anything about microscopic particles. Why it should say anything about measuring devices?

 

[stevendaryl]

Why do you think it should be possible?
Measuring devices are just bigger more complicated "particles".

That's my point. If measuring devices are just complicated systems, and measurements are just complicated interactions, then the Born rule shouldn't treat measurements differently than any other interactions. But it does: The Born rule says "If you measure a quantity, you get an eigenvalue of the corresponding operator, with such-and-such probability."

 

[zonde]

That's my point. If measuring devices are just complicated systems, and measurements are just complicated interactions, then the Born rule shouldn't treat measurements differently than any other interactions. But it does: The Born rule says "If you measure a quantity, you get an eigenvalue of the corresponding operator, with such-and-such probability."

You apply Born rule to measurements only. You do not apply Born rule to other interactions. At least this is my understanding.

And my point is that Born rule treats microscopic reality differently too. Before you apply Born rule you don't describe particles. You describe modes not particles.

 

[AlexCaledin]

As long as ensemble interpretation refuses to talk about single measurements, it cannot say anything about the measurement problem.

- but what if the EI is applied to the whole spacetime-universe? Then, there is the Ensemble of "prepared" universes, each one having its definite observable history (Bohmian trajectory) - quite definite past and future - but "we" are uncertain as to what universe we are in - and we are in the process of getting knowledge about that - and then every measurement is simply a little increment of our knowledge.

 

[vanhees71]

If you want to derive macroscopic behavior you also use Born's rule, using an appropriate statistical operator like the usual equilibrium operators (microcanonical, canonical, grand canonical).

 

[stevendaryl]

You apply Born rule to measurements only. You do not apply Born rule to other interactions. At least this is my understanding.

And my point is that Born rule treats microscopic reality differently too. Before you apply Born rule you don't describe particles. You describe modes not particles.

Okay, I guess I agree with that.

 

[stevendaryl]

If you want to derive macroscopic behavior you also use Born's rule, using an appropriate statistical operator like the usual equilibrium operators (microcanonical, canonical, grand canonical).

But that's not the same. In the case of a macroscopic measurement, each measurement produces an eigenvalue of the operator corresponding to the observable being measured. In the case of an ensemble, you're talking about averages, not properties of individual systems.

An ensemble average is a kind of macroscopic observable. So again, it seems that QM treats macroscopic systems differently than microscopic systems.

 

[vanhees71]

The point is that the pointer states have very small (relative) standard deviations, so that you can treat them as if they were determined as in classical physics.

 

[atyy]

The point is that the pointer states have very small (relative) standard deviations, so that you can treat them as if they were determined as in classical physics.

That is not true. The standard deviation of the pointer states depends on the observable that is measured.

 

[Demystifier]

The point is that the pointer states have very small (relative) standard deviations

That is so only after the information update (not to use the dirty c-word). But update needs measurement, and measurement needs "classical" pointers states, so the whole explanation becomes circular.

 

[vanhees71]

What? If I measure something, of course the apparatus has interacted with the measured object, and you get a clear pointer reading. The pointer position is a macroscopic observable and should have a small standard deviation as any macroscopic observable. No dirty c needed, just statistics ;-).

 

[Lord Jestocost]

That is so only after the information update (not to use the dirty c-word). But update needs measurement, and measurement needs "classical" pointers states, so the whole explanation becomes circular.

As I am new in "PhysicsForum": What the heck is the "dirty c-word"?

 

[Demystifier]

As I am new in "PhysicsForum": What the heck is the "dirty c-word"?

Collapse.

 

[Demystifier]

If I measure something, of course the apparatus has interacted with the measured object, and you get a clear pointer reading.

Consider a Stern-Gerlach apparatus. It has two detectors, one in the upper position and the other in the lower position. When one particle is sent through the apparatus, only one of the detectors clicks. In this case, did the other detector also interacted with the particle (measured object)?

 

[vanhees71]

Obviously not, because then (taken it as a very good detector with close to 100% detection efficiency) you'd see 2 spots in any experiment with an intensity due to the $|\psi|^2$ distribution. The very fact that this is not the case rules out the original interpretation of the wave function as a classical field describing the (charge) density of the particles by Schrödinger.

 

[Demystifier]

Obviously not, because then (taken it as a very good detector with close to 100% detection efficiency) you'd see 2 spots in any experiment with an intensity due to the $|\psi|^2$ distribution. The very fact that this is not the case rules out the original interpretation of the wave function as a classical field describing the (charge) density of the particles by Schrödinger.

So at what point did the particle decide that it will go to one detector and not to the other? Did it happen at the moment of interaction with the detector, or did it happen before that? Does the question even make sense to you?

 

[vanhees71]

QT tells us that this is just random with probabilities given by Born's rule. It doesn't make sense to ask, how the particle "made a decision".

 

[zonde]

It doesn't make sense to ask, how the particle "made a decision".

The question was not "How?" but "When?"

 

[Demystifier]

QT tells us that this is just random with probabilities given by Born's rule. It doesn't make sense to ask, how the particle "made a decision".

As zonde observed, the question is when the decision is made, not how.

 

[vanhees71]

It's of course made when the detector "clicks" (or writes the information to any kind of storage). I don't know, why this is in any sense "problematic" or "mysterious".

 

[Demystifier]

It's of course made when the detector "clicks" (or writes the information to any kind of storage). I don't know, why this is in any sense "problematic" or "mysterious".

Here is why it is problematic. You simultaneously assume that
1) The measured system (particle) exists even before measurement.
2) The dynamics is local.
3) The random decision happens when the detector clicks (not before).
Indeed, each assumption by itself seems reasonable. But the problem is that they cannot all be simultaneously true. At least one must be wrong. You must give up at least one of them.

Let me explain why they cannot all be true. From 3) and 1) it follows that, immediately before the click, the system exists not only near one detector, but near both of them. But then, puff, at the time of click, the system suddenly ceases to exist near the detector that didn't click. How did this part of the system knew that the click happened near the other part? Since the two parts are spatially separated, there must have been some non-local (even if random) mechanism, which contradicts 2). Hence assumptions 1) and 3) contradict 2), which implies that it is not possible that all three assumptions are true.

And yet, you seem not be ready to give up any of the three assumptions. That's the problem.

Note that the argument above is even simpler than the Bell theorem, because the system studied above does not involve entanglement. The Bell theorem derives a contradiction by assuming 1), 2) and entanglement. The argument above derives a contradiction by assuming 1), 2) and 3).

 

[RockyMarciano]

Here is why it is problematic. You simultaneously assume that
1) The measured system (particle) exists even before measurement.
2) The dynamics is local.
3) The random decision happens when the detector clicks (not before).
Indeed, each assumption by itself seems reasonable. But the problem is that they cannot all be simultaneously true. At least one must be wrong. You must give up at least one of them.

Let me explain why they cannot all be true. From 3) and 1) it follows that, immediately before the click, the system exists not only near one detector, but near both of them. But then, puff, at the time of click, the system suddenly ceases to exist near the detector that didn't click. How did this part of the system knew that the click happened near the other part? Since the two parts are spatially separated, there must have been some non-local (even if random) mechanism, which contradicts 2). Hence assumptions 1) and 3) contradict 2), which implies that it is not possible that all three assumptions are true.

And yet, you seem not be ready to give up any of the three assumptions. That's the problem.

Right. It could even be necessary to give up both 1) and 3) to save 2).

 

[Demystifier]

Right. It could even be necessary to give up both 1) and 3) to save 2).

Not really. To save 2) it is sufficient to give up 3). Example is Bohmian mechanics, which, in absence of entanglement, is local.

EDIT: This is my 7777th post.

 

[RockyMarciano]

Not really. To save 2) it is sufficient to give up 3). Example is Bohmian mechanics, which, in absence of entanglement, is local.

Sure. Sufficient, but I was thinking of necessary if one uses the freedom to change axioms without changing the physics. I think I've read you arguing something like this but maybe it was in a different context.

EDIT: This is my 7777th post.

Nice number. Long ways from the biblical one but still.

 

[vanhees71]

Well, according to QT there's not more knowable about your particle than the probabilities for the outcome of measurements. The probabilities evolve from a given (prepared) initial to the state at the time of detection, and what we get (repeating the experiment with equal preparations) the distribution by measuring the observable we are interested in. According to the standard model of elementary particles the interactions are local. So your point (1-3) are all well included in the standard model: (1) is ensured by the conservation laws: If I prepare, e.g., some particle with a given charge, then at least this charge must exist all the time; (2) is implemented by construction in any local and microcausal relativistic QFT, (3) is just the only way to answer reasonable the "when" question. How else would you define the "time of detection"?

 

[Demystifier]

Well, according to QT there's not more knowable about your particle than the probabilities for the outcome of measurements. The probabilities evolve from a given (prepared) initial to the state at the time of detection, and what we get (repeating the experiment with equal preparations) the distribution by measuring the observable we are interested in. According to the standard model of elementary particles the interactions are local. So your point (1-3) are all well included in the standard model: (1) is ensured by the conservation laws: If I prepare, e.g., some particle with a given charge, then at least this charge must exist all the time; (2) is implemented by construction in any local and microcausal relativistic QFT, (3) is just the only way to answer reasonable the "when" question. How else would you define the "time of detection"?

I don't agree that standard QT ensures 1). The standard QT talks only about probabilities of measurement outcomes. In particular, the conservation of charge in standard QT says that if you measure charge at two times, then you will get the same number at both times. But it does not claim that charge will exist between the two measurements. It is an extra assumption which may or may not be true, but cannot be tested by experiment. It is a reasonable assumption indeed, but it cannot be strictly derived from principles of standard QT.

So, if you want to think in lines of standard QT, I think you should give up 1).

 

[RockyMarciano]

Symmetry principles aren't the goal. The goal is modeling the world. If the world obeys certain symmetry principles, then of course, we should find out what they are, but finding out symmetry principles isn't the goal.
That is the heart of the measurement problem. Why do macroscopic systems have well-defined macroscopic values?

Sorry to get back to this but aren't precisely the symmetry principles-conservation laws what makes macro systems have (approximately) well defined values by preserving measurements(i.e. measurability)?

 

[Demystifier]

(1) is ensured by the conservation laws: If I prepare, e.g., some particle with a given charge, then at least this charge must exist all the time

Another counterargument:
Suppose that we talk about barion charge (not electric charge) and suppose that, due to some GUT effects, there is a very small probability that the charge will not be conserved. The probability can be arbitrarily small, say $10^{-100}$, but it is not zero. For all practical purposes the charge can be considered conserved. In this case, would you say that charge must exist all the time? Or almost all the time? Or would you say that it comes to existence only when the detector clicks?

 

[vanhees71]

Well, then you could measure that baryon number is not conserved and then you could indeed only give a probability for still finding the same baryon number as you started with.

 

[Demystifier]

Well, then you could measure that baryon number is not conserved and then you could indeed only give a probability for still finding the same baryon number as you started with.

So you (seem to) introduce a step function. If probability of conservation is smaller than 1, then charge does not exist between measurements. If probability of conservation is strictly 1, then charge exists between measurements.

Don't you have a feeling that there should be a continuous transition, that the difference between 1 and 0.99999999 should not be so radical?

 

[Demystifier]

Well, then you could measure that baryon number is not conserved and then you could indeed only give a probability for still finding the same baryon number as you started with.

Another reason why your reasoning doesn't make sense (to me).

You essentially say that
1) Only conserved quantities exist between measurements.
2) Dynamics is local.
But the goal of dynamics is to describe the change, i.e. to describe the behavior of quantities which are not conserved. By 1), this means that the goal of dynamics is to describe the things which do not exist between measurements. So the fact that dynamics is local really means that dynamics of non-existing entities is local. It says nothing about locality or non-locality of existing entities. Therefore the fact that QFT has local dynamics is not an argument that existing entities obey local laws.

Indeed, the Bell theorem says essentially that if there are changing entities that exist between measurements, then their dynamics must be non-local. It is compatible with the conclusion above that local QFT dynamics is only dynamics on non-existing entities.

 

[Demystifier]

But the goal of dynamics is to describe the change, i.e. to describe the behavior of quantities which are not conserved.

Let me put this point forward.

As an extreme case, consider the Lagrangian
$$L(q,\dot{q})=0$$
This Lagrangian is invariant under any conceivable transformation, so by Noether theorem everything is conserved in this theory. In other words, in this theory, there is no dynamics at all.

This demonstrates my more general point that conservation laws show that some quantities are not dynamical. But the point of dynamics is to describe quantities which are dynamical. So conservation laws, although very useful, do not describe dynamics. Conservation laws describe non-dynamics; they tell us which of the potentially interesting quantities are not dynamical, so can be excluded from further dynamical considerations.

 

[RockyMarciano]

Let me put this point forward.

As an extreme case, consider the Lagrangian
$$L(q,\dot{q})=0$$
This Lagrangian is invariant under any conceivable transformation, so by Noether theorem everything is conserved in this theory. In other words, in this theory, there is no dynamics at all.

This demonstrates my more general point that conservation laws show that some quantities are not dynamical. But the point of dynamics is to describe quantities which are dynamical. So conservation laws, although very useful, do not describe dynamics. Conservation laws describe non-dynamics; they tell us which of the potentially interesting quantities are not dynamical, so can be excluded from further dynamical considerations.

How would you reconcile this with the fact that measurements must be possible in a dynamical world for science to make sense(for different local measurements to be coherent with one another) which seems to imply that at least there must be conservation laws for dynamical measuring tools?

 

[vanhees71]

So you (seem to) introduce a step function. If probability of conservation is smaller than 1, then charge does not exist between measurements. If probability of conservation is strictly 1, then charge exists between measurements.

Don't you have a feeling that there should be a continuous transition, that the difference between 1 and 0.99999999 should not be so radical?

I've not said what you seem to have understood. All I said was what holds for any unstable particle: You prepare it, and then with some probability it's decayed after a given time. That's all you can know in such cases. I still don't get, what should be a problem with that. To the contrary thanks to Q(F)T we have a theory to describe such decays very well.

If there's a conserved charge, at least you know that it will be there forever in the one or the other form. To be sure that a once prepared particle is always there, of course it must be stable, because if there is only the tiniest probability for its decay, then you can never be sure that it is still there after some time. That's why it's called unstable.