Interpretations of QM

[Genady]

On 1/1/2023 at 9:05 AM, joigus said:

Because a few decades is a long time for experiments not to end up catching up dramatically with our frontiers of knowledge, and because it's never been the case that a breakthrough in our basic understanding of scientific issues has left the technological landscape untouched. It's bound to happen sooner rather than later.

Seemingly unsolvable problems in basic science seem to be piling up, and something's got to give.

Aren't we in this situation for a few decades already? About 3 or more?

[joigus]

4 minutes ago, Genady said:

Aren't we in this situation for a few decades already? About 3 or more?

Yes. Part of the difficulty is that experiments are getting really costly. I also think that people who conceive them should spend more effort in trying to disprove quantum mechanics, or find where different interpretations could produce different predictions.

Also interesting are experiments that build particle-like analogues of quantum particles. More should be done in this direction, I think. So far it's been a bunch of mavericks. Look at what Yves Couder did:

This goes to prove that what I said --that particle-like self-sustaining bundles of the field carried along with the quantum state is not that outlandish an idea.

Bohm's theory could the precursor of a much deeper, much more complex underlying level, I suspect. One not involving point particles, of course, but dynamics of non-linear phenomena.

You must ponder the fact that what this man has done in a laboratory was once declared even impossible to conceive by great physicists.

We should be ready for surprises. But effort must be spent in looking for them.

[Genady]

38 minutes ago, joigus said:

disprove quantum mechanics, or find where different interpretations could produce different predictions.

A historical analogy doesn't prove anything of course but can work heuristically. By analogy, no efforts to disprove classical mechanics or find where different interpretations of it could produce different results have brough a breakthrough. The breakthrough rather came from different developments, i.e., EM, thermal radiation, spectroscopy. By this analogy, I'd expect the next breakthrough to come from a different development as well. Cosmology? Something else?

[Mordred]

3 hours ago, joigus said:

 

As to @Mordred's statement that he's happy with particles corresponding to just excitations in quantum fields, I agree that it's good enough for all practical purposes, including relativistic contexts in which the Born interpretation is not so easy to invoke unambiguously.

 

The reason I am happy with the field excitation view is that I have yet to encounter any form of particle interaction that QFT cannot adequately explain. Things such as electron spin, the distribution of spin statistics in the particle view would require superluminal angular momentum, where as in the field view with a greater effective radius this is easily accounted for. Particles popping in and out of existence is easily described through the creation/annihilation operators. Other factors tricky to describe with particle view where its easily described in the QFT view include Quantum tunneling, Bose and Fermi condensate, electron creation using photons (has been experimentally done) rather interesting in the method.

The above is just some examples however even given the above I still feel its wrong to consider fields as being fundamental regardless of its accuracy and range of predictive ability

[joigus]

34 minutes ago, Mordred said:

The reason I am happy with the field excitation view is that I have yet to encounter any form of particle interaction that QFT cannot adequately explain. Things such as electron spin, the distribution of spin statistics in the particle view would require superluminal angular momentum, where as in the field view with a greater effective radius this is easily accounted for. Particles popping in and out of existence is easily described through the creation/annihilation operators. Other factors tricky to describe with particle view where its easily described in the QFT view include Quantum tunneling, Bose and Fermi condensate, electron creation using photons (has been experimentally done) rather interesting in the method.

I'm aware of it. That's part of the reason why I said that DBB theory cannot be the whole story. Silicone droplets are indeed not particles, when you think about it. They are, in a manner of speaking, non-linear "excitations carried on top of the linear, or quasi-linear modes" of the field --or perhaps analogues of those.

In the last decades there has been extensive study of topological field theories. Topological field theories have no propagating degrees of freedom, even though quite a number of people who are allegedly "in the know" are not totally clear about this, or chose not to answer to that particular point:

https://physics.stackexchange.com/questions/550561/on-tqft-and-theories-without-propagating-degrees-of-freedom

Quote

Maybe not a very sensible question, but I would like to know, whether there exist topological field theories (TQFT) with propagating degrees of freedom, or, conversely, theories without propagating degrees of freedom, which are not topological?

https://math.stackexchange.com/questions/701100/topological-quantum-field-theories

Quote

So, there are no dynamics or propagation. [...]

Also, my own question @ physics.stackexchange which received no answer about the particular point I was asking (topological field theories do not propagate, therefore their degrees of freedom must be "quasi-rigidities")...

https://physics.stackexchange.com/questions/614516/criteria-to-define-a-classical-topological-field-lagrangian-conjecture?noredirect=1&lq=1

I know for sure that a number of people consider this particular aspect at the very least intriguing. But not many people, AFAIK, have thought at much depth, if any, about it. I find that especially revealing. Revealing of what? Well, revealing, if nothing else, of the fact that almost no one is interested, not of whether or not it can be done, or of whether or not it's a dead end, or irrelevant, etc. 

[Mordred]

3 hours ago, joigus said:

I'm aware of it. That's part of the reason why I said that DBB theory cannot be the whole story. Silicone droplets are indeed not particles, when you think about it. They are, in a manner of speaking, non-linear "excitations carried on top of the linear, or quasi-linear modes" of the field --or perhaps analogues of those.

In the last decades there has been extensive study of topological field theories. Topological field theories have no propagating degrees of freedom, even though quite a number of people who are allegedly "in the know" are not totally clear about this, or chose not to answer to that particular point:

https://physics.stackexchange.com/questions/550561/on-tqft-and-theories-without-propagating-degrees-of-freedom

https://math.stackexchange.com/questions/701100/topological-quantum-field-theories

Also, my own question @ physics.stackexchange which received no answer about the particular point I was asking (topological field theories do not propagate, therefore their degrees of freedom must be "quasi-rigidities")...

https://physics.stackexchange.com/questions/614516/criteria-to-define-a-classical-topological-field-lagrangian-conjecture?noredirect=1&lq=1

I know for sure that a number of people consider this particular aspect at the very least intriguing. But not many people, AFAIK, have thought at much depth, if any, about it. I find that especially revealing. Revealing of what? Well, revealing, if nothing else, of the fact that almost no one is interested, not of whether or not it can be done, or of whether or not it's a dead end, or irrelevant, etc. 

Each topological space is invariant and thus rigid in any of its geometric  degrees of freedom. Any variations of say length generates a different topological space. The transformations between topological space will vary between observers. When it comes to amplitudes in the vector space there is no propagation as H=0 within that space \[\Sigma^{d^{n-1}}\] however you will have non trivial tunneling amplitudes between spaces \[\Sigma_0^{d^{n-1}},,, \Sigma_1^{d^{n-1}}\] through an intervening manifold M

\[\partial M=\Sigma_0^\ast \cup \Sigma_1\]

hope that helps answer that question a simplistic descriptive is that there is no local degrees of freedom (local defined by the space propagation is on the global topography

Edited January 2, 2023 by Mordred

[Markus Hanke]

16 hours ago, Lorentz Jr said:

I was referring to fundamental scientific principles and alternatives to modern mainstream theory

You were originally referring to “19th century principles of science” - which were based on a Newtonian world view. And even back then, people were already aware of numerous problems and issues that didn’t fit that world view. 

16 hours ago, Lorentz Jr said:

This is ad hominem, Markus.

No, it is an honest appraisal by someone who reads a lot of papers on these subjects.

16 hours ago, Lorentz Jr said:

but it's published, so I'm going to give it the benefit of the doubt.

So are all the other theories, so you’ll have to give them the benefit of the doubt too.

16 hours ago, Lorentz Jr said:

There's also gauge theory gravity (with another article and a Wikipedia article), which, if I'm not mistaken, is virtually identical to GR at all experimentally observed field strengths, and has what seem to me to be the advantages of disallowing exotic sci-fi phenomena like wormholes and time loops.

Yes, GTG is a specific example of a gauge theory of gravity, as opposed to a metric theory such as GR. It can be shown that there are, in fact, infinitely many such theories, all of which describe propagating 2-polarisation states of gravitational radiation, and which resemble GR (no spin) or Einstein-Cartan gravity (spin) under the appropriate circumstances. Within the domain that we can experimentally test and observe, these models are generally distinguishable from GR only insofar as they don’t contain any equivalence principle - therefore testing the equivalence principle is a good first step in testing for gauge theories of gravity. At present, no violations of the equivalence principle have been observed, not even in the strong field domain (BH mergers etc). This doesn’t invalidate all gauge theories, but it does constrain the form they can realistically take.

There is another critical issue with this, however - because gravity now also couples directly to spin (unlike in pure GR), these theories introduce extra terms into the Dirac equation. These extra terms are too small for us to be able to experimentally detect them right now, but they would become important in the strong field regime. Again, I am not aware of any indications that such phenomena have been observed anywhere.

As for exotic phenomena - these gauge theories exclude the possibility of singularities, even in the classical domain, which is definitely good. Wormholes are not categorically excluded though as far as I know, where did you read this? Note sure about CTCs, but it’s possible that these don’t occur, since no ring singularity forms.

I personally like gauge theories of gravity, since the basic approach is very elegant, and there are no immediate conflicts with available data. I’d say that out of all the various alternatives (or rather: extensions) to GR, these are probably the most promising. I would say, though, that what we need isn’t an alternative to GR (unless some data becomes available that is in direct conflict with it), but rather a generalisation of it.

16 hours ago, Lorentz Jr said:

My suspicion of current quantum theory is based on thinking it may not be radical enough (being formulated in terms of particles, which are derived from classical intuition)

But that’s only (sort of) true for ordinary non-relativistic QM, which is just a simplified approximation. Full quantum field theory is based on fields and their interactions, not particles.

16 hours ago, Lorentz Jr said:

and my suspicion of relativity is based on thinking it may be unscientific (I'm not convinced by the "geometry" argument of time dilation in the twin paradox).

I have no idea what you mean by “unscientific” - both SR and GR are fully amenable to the scientific method, irrespective of the precise mathematical details of these models.

Edited January 3, 2023 by Markus Hanke

[Markus Hanke]

20 hours ago, Lorentz Jr said:

There's also gauge theory gravity

P.S. I’m genuinely curious - why does this particular model resonate with you more than GR does? They are both geometric models, but GTG relies on much more abstract underlying entities (gauge fields). At least the metric in GR directly connects to real-world measurements of times, distances and angles, which is a very “hands on” kind of thing…whereas gauge fields are really very mathematical ideas, and don’t correspond to anything even remotely as practical as aforementioned measurements. From your previous comments here I would have thought that you’re not in favour of overly mathematical concepts. Just wondering  

[Lorentz Jr]

1 hour ago, Markus Hanke said:

They are both geometric models

No, GTG is defined in flat Minkowski space. It models gravity in the way that Einstein first tried, as a gauge-invariant "index of refraction". Hestenes suggests that Einstein only resorted to using geometry because that's the only math that was available to him at the time.

I don't necessarily prefer GTG, I just use it as a counterexample to show that the geometric interpretation hasn't been proven to be "real". As I said before, I don't object to theories because my thinking is conventional and they seem "radical" to me. I object to theories that seem unscientific to me. Neither QM nor GR bothers me nearly as much as SR, i.e. the principle of relativity itself, and QM only bothers me because it assumes that quantization must be a static property of matter (rather than a dynamic property of interactions), and that seems to be forced on it by SR.

1 hour ago, Markus Hanke said:

From your previous comments here I would have thought that you’re not in favour of overly mathematical concepts.

I'll try not to give that impression in the future. I'm certainly not a mathematician, but the amount of math in theories has nothing to do with how I evaluate them.

I'm not in favor of pseudoscientific fantasies, I'm not in favor of "theories" that don't make predictions (or make 10⁵⁰⁰ predictions), and I'm not in favor of the current fashion (as I see it) of treating math as a substitute for physics. Many Worlds says energy eigenfunctions of the universe are "real", interpretations like quantum information and Von Neumann–Wigner seem to deny the reality of the universe itself, and the current consensus seems to be that thinking about ontology is nothing more than a navel-gazing exercise for philosophers. It's like the physics profession has been hijacked by mathematicians.

Quote

Wormholes are not categorically excluded though as far as I know, where did you read this?

Page 36.

Edited January 3, 2023 by Lorentz Jr

[Genady]

15 minutes ago, Lorentz Jr said:

I'm not in favor of pseudoscientific fantasies, I'm not in favor of "theories" that don't make predictions (or make 10⁵⁰⁰ predictions), and I'm not in favor of the current fashion (as I see it) of treating math as a substitute for physics.

I'm with you here 100%.

However,

16 minutes ago, Lorentz Jr said:

the current consensus seems to be that thinking about ontology is nothing more than a navel-gazing exercise for philosophers.

Isn't thinking about ontology another substitute for physics?

[Lorentz Jr]

9 minutes ago, Genady said:

Isn't thinking about ontology another substitute for physics?

No, I think it's part of physics. It's part of what defines physics as a science, as opposed to a branch of pure mathematics. As I was saying recently, ontology is useful in predicting what's likely to happen in situations that haven't been studied yet. During earthquakes, for instance, in the discussion about maps and territories.

Edited January 3, 2023 by Lorentz Jr

[Genady]

10 minutes ago, Lorentz Jr said:

physics as a science, as opposed to a branch of pure mathematics.

I think that the difference between physics and mathematics is that - paraphrasing Feynman - in physics we got a partner.

[Genady]

1 hour ago, Lorentz Jr said:

ontology is useful in predicting what's likely to happen in situations that haven't been studied yet. During earthquakes, for instance, in the discussion about maps and territories.

I don't understand how it can be useful in such situations. Could you give an example (rather than a metaphor)?

[Lorentz Jr]

41 minutes ago, Genady said:

I don't understand how it can be useful in such situations. Could you give an example (rather than a metaphor)?

How do you explain Brownian motion? You need to have some idea of what water is made of.

How do you predict the life cycles of stars? You need to have some idea of what's inside them.

How do you develop ways to travel faster than the speed of light? You need to have some idea of what matter and the vacuum are made of.

Ontology is necessary for fundamental research. Existing theory is fine for developing applications, but developing new fundamental theories requires some kind of intuition about what those theories will describe. How can you unify gravity and quantum theory without having some idea of what they describe? Without some intuition about the nature of the underlying reality, all you can do is flail around wildly with ten-dimensional fantasies that don't describe anything.

EDIT: Another good example is troubleshooting. If you want to fix a clock or an engine, or figure out what's making a person sick, you have to have some idea of what's inside the clock or the engine or the person. You can't use phenomenological models of how the things normally behave when their behavior is abnormal.

BTW, I don't get that Feynman reference.

Edited January 3, 2023 by Lorentz Jr

[Genady]

14 minutes ago, Lorentz Jr said:

some kind of intuition about what those theories [] describe

Now I understand what you mean by "ontology". I'm not sure it is a common understanding of this word. But with this understanding, I fully agree with you.

Re Feynman, doesn't matter, just a quote.

22 minutes ago, Lorentz Jr said:

I don't get that Feynman reference

 

1 hour ago, Genady said:

in physics we got a partner.

The partner is Nature.

[joigus]

15 hours ago, Mordred said:

Each topological space is invariant and thus rigid in any of its geometric  degrees of freedom. Any variations of say length generates a different topological space. The transformations between topological space will vary between observers. When it comes to amplitudes in the vector space there is no propagation as H=0 within that space

Σdn−1

however you will have non trivial tunneling amplitudes between spaces

Σdn−10,,,Σdn−11

through an intervening manifold M

 

 

∂M=Σ∗0∪Σ1

 

hope that helps answer that question a simplistic descriptive is that there is no local degrees of freedom (local defined by the space propagation is on the global topography

This is not exactly what I meant, although it does partially overlap with what I meant. The adjective "topological" I didn't mean as applied to the background manifold, but to certain classes of solutions of differential equations on those manifolds. Thus, you can have many field theories defined on a topological manifold. Among all these theories, only a very restricted class of theories are topological, and most other theories are not. Topological theories have a Lagrangian not involving the metric. One interesting feature of these theories is that they are useful tools to study topological invariants of the manifold, like Wilson loops, probably Betti numbers too, and the like. They are invariant under diffeomorphisms, and once you incorporate the constraints --se below-- with the method of Lagrange multipliers, the constrained Hamiltonian becomes identically zero on the constraint surface. I say all this just to guarantee that we're talking about the same thing.

But from the strictly dynamical point of view, topological fields are very constrained in the way the can evolve. In fact, they are maximally constrained. They --the fields-- have no local degrees of freedom, which means they do not propagate. The number of constraints exactly equals the number of degrees of freedom.

Let me be as clear as humanly possible. As a warm-up, taking a 1st-order in time point-particle theory as a particularly simple example, what I mean by having #(DoF) = #(constraints) is the following:

\[ q_{1}\left(t\right),\cdots,q_{n}\left(t\right) \]

So a set of initial conditions \( q_{1}\left(0\right),\cdots,q_{n}\left(0\right)=q_{10},\cdots,q_{n0} \) completely determines the trajectory in the configuration space.

Equivalently, if we're extremely lucky --the system is integrable--, we may manage to find a set of n integrals of motion:

\[ \mathcal{J}_{1}\left(q_{1}\left(t\right),\cdots q_{n}\left(t\right)\right)=0 \]

\[ \vdots \]

\[ \mathcal{J}_{n}\left(q_{1}\left(t\right),\cdots q_{n}\left(t\right)\right)=0 \]

This is (modulo condition of non-vanishing Jacobian) equivalent to fixing the previous n initial conditions. Integrals of motion for integrable systems depend on 1to1 on initial conditions.

Now, what are constraints? Constraints are both mathematically and physically very similar, but with a very important nuance. Assume n constraints:

\[ F_{1}\left(q_{1}\left(t\right),\cdots q_{n}\left(t\right)\right)=0 \]

\[ \vdots \]

\[ F_{n}\left(q_{1}\left(t\right),\cdots q_{n}\left(t\right)\right)=0 \]

But these relations being in place for every set of initial conditions. What does this imply mathematically? It implies nothing other than that the system cannot move at all. Cannot evolve in any meaningful way at all. It's frozen dynamically. How much? It's just a point in configuration space.

Now, what happens to a field theory under analogous strictures? We must now promote the \( q_{i}\left(t\right) \) to some \( \varphi_{a}\left(\boldsymbol{x},t\right) \), where \( a \) goes from 1 to n. If we impose n initial conditions, that means specifying the value of the \( \varphi \)'s,

\[ \left.\varphi_{1}\left(\boldsymbol{x},\tau\right)\right|_{\varSigma^{d-1}}=f_{1}\left(\boldsymbol{x}\right),\cdots,\left.\varphi_{n}\left(\boldsymbol{x},\tau\right)\right|_{\varSigma^{d-1}}=f_{n}\left(\boldsymbol{x}\right) \]

where \( \tau \) is some curvilinear coordinate that specifies the sub-manifold \( \varSigma^{d-1} \) the space-like foliations that define the Cauchy problem.

If, instead, we have a set of n constraints, and as any constraints worth their salt, they do not depend on initial conditions, but are the same for all posible initial conditions, we will have the system extremely limited in its evolution. But it is no longer true that the set of states compatible with this situation "shrinks" to a point in configuration space. It does freeze, but due to the presence of the space variables \( \boldsymbol{x} \) it does so --it must-- to a fixed function sitting on the topological space. Or perhaps to a finite or countable set of such "frozen" functions. It is in that sense that I was talking about a quasi-rigidity. Known examples of constraints in field theory (valid for all sets of initial conditions) are transversalities and Gauss or Lorentz gauge-fixing constraints. Photons are known to "inhabit" a space of configurations with 4 degrees of freedom when you give them mass by assumption. If you impose masslessness, gauge fixing, they become more and more restricted in their evolution. If you imposed further constraints, there would be no dynamical situation that implies propagation. They would be "frozen." But not to a point. Instead they would be frozen to a pretty restrictive class of what I've tried to refer to by this term "quasi-rigidity."

The previous discussion generalises trivially to a 2nd-order-in-time system by substituting 1,...,n with 1,...,n,...2n, and the \( \varphi \)'s to those plus their canonical momenta.

Now, I know these musings not to be totally out of whack, because I've posed the question (the one about quasi-rigidity) elsewhere appropriate and I know for a fact that knowledgeable people consider it a totally-non-silly conjecture at least. Or, if you want, a question worth answering. But what would come next --the relation to the interpretation of QM-- does have speculative elements on my part, and would probably require a thread of its own.

 

 

 

[TheVat]

2 hours ago, Lorentz Jr said:

Ontology is necessary for fundamental research. Existing theory is fine for developing applications, but developing new fundamental theories requires some kind of intuition about what those theories will describe. How can you unify gravity and quantum theory without having some idea of what they describe? Without some intuition about the nature of the underlying reality, all you can do is flail around wildly with ten-dimensional fantasies that don't describe anything.

I appreciated the nod to a realist interpretation.  And dig at string theory.

I think my earlier comment on cowering from probabilistic theories was confused by some -  @Mordred was one -    as me not seeing the uses of probability in physics.  Well of course I do.   What I should have said was I'm leery of acausal theories (aka nondeterministic), which seem to skirt thorny ontological problems and just tell you like a stern schoolmarm that it's all stochastic.  

Here's a lump of twenty trillion thorium-234 atoms.  Some of them will soon beta decay to protactinium-234.  Some of them won't.  Let's give each thorium atom in the lump an address.  And name.  At 221-B, there is Sherlock.  At 10, is Boris.  Either could, randomly, decay.  As it happened, Boris decayed first, before Sherlock.  At a macro scale, such an event seems to have a cause.  We have an ontology of macro scale Borises, and can understand why they decay so easily.  But the thorium atoms all seem identical.  All intuitions seem wrong.  Ontology can help.  Maybe.

[studiot]

2 hours ago, joigus said:

The adjective "topological" I didn't mean as applied to the background manifold, but to certain classes of solutions of differential equations on those manifolds. Thus, you can have many field theories defined on a topological manifold. Among all these theories, only a very restricted class of theories are topological, and most other theories are not. Topological theories have a Lagrangian not involving the metric. One interesting feature of these theories is that they are useful tools to study topological invariants of the manifold, like Wilson loops, probably Betti numbers too, and the like. They are invariant under diffeomorphisms, and once you incorporate the constraints --se below-- with the method of Lagrange multipliers, the constrained Hamiltonian becomes identically zero on the constraint surface. I say all this just to guarantee that we're talking about the same thing.

Interesting, Thank you.  +1

[joigus]

1 hour ago, TheVat said:

I appreciated the nod to a realist interpretation.  And dig at string theory.

I think my earlier comment on cowering from probabilistic theories was confused by some -  @Mordred was one -    as me not seeing the uses of probability in physics.  Well of course I do.   What I should have said was I'm leery of acausal theories (aka nondeterministic), which seem to skirt thorny ontological problems and just tell you like a stern schoolmarm that it's all stochastic.  

Here's a lump of twenty trillion thorium-234 atoms.  Some of them will soon beta decay to protactinium-234.  Some of them won't.  Let's give each thorium atom in the lump an address.  And name.  At 221-B, there is Sherlock.  At 10, is Boris.  Either could, randomly, decay.  As it happened, Boris decayed first, before Sherlock.  At a macro scale, such an event seems to have a cause.  We have an ontology of macro scale Borises, and can understand why they decay so easily.  But the thorium atoms all seem identical.  All intuitions seem wrong.  Ontology can help.  Maybe.

Perhaps ontology in a weaker sense inspired by the oft-newfangled concept of "emergent"?

In fact, it is incumbent upon us to explain why the world looks like entities moving around, or standing still. It does look like that. We must admit the world resembles exactly that: Numbers of entities having lifetimes, births and extinctions, checkpoints and rendezvous. 

1 hour ago, studiot said:

Interesting, Thank you.  +1

You're welcome. There is a very interesting series of lectures by Lee Smolin on Time in Quantum Gravity (Perimeter Institute) in which he gets hands dirty with topological theories as theories with #(DoF)=#(constraints) until you totally remove all local degrees of freedom. It's as if the field variables get as close as it gets to doing nothing. I'm struggling with intuitive ways to put it in words...

[Genady]

1 hour ago, joigus said:

In fact, it is incumbent upon us to explain why the world looks like entities moving around, or standing still. It does look like that. We must admit the world resembles exactly that: Numbers of entities having lifetimes, births and extinctions, checkpoints and rendezvous. 

The first question perhaps is what of this look is on the world and what on the beholder.

[Mordred]

5 hours ago, joigus said:

This is not exactly what I meant, although it does partially overlap with what I meant. The adjective "topological" I didn't mean as applied to the background manifold, but to certain classes of solutions of differential equations on those manifolds. Thus, you can have many field theories defined on a topological manifold. Among all these theories, only a very restricted class of theories are topological, and most other theories are not. Topological theories have a Lagrangian not involving the metric. One interesting feature of these theories is that they are useful tools to study topological invariants of the manifold, like Wilson loops, probably Betti numbers too, and the like. They are invariant under diffeomorphisms, and once you incorporate the constraints --se below-- with the method of Lagrange multipliers, the constrained Hamiltonian becomes identically zero on the constraint surface. I say all this just to guarantee that we're talking about the same thing.

But from the strictly dynamical point of view, topological fields are very constrained in the way the can evolve. In fact, they are maximally constrained. They --the fields-- have no local degrees of freedom, which means they do not propagate. The number of constraints exactly equals the number of degrees of freedom.

Let me be as clear as humanly possible. As a warm-up, taking a 1st-order in time point-particle theory as a particularly simple example, what I mean by having #(DoF) = #(constraints) is the following:

 

q1(t),⋯,qn(t)

 

So a set of initial conditions q1(0),⋯,qn(0)=q10,⋯,qn0 completely determines the trajectory in the configuration space.

Equivalently, if we're extremely lucky --the system is integrable--, we may manage to find a set of n integrals of motion:

 

J1(q1(t),⋯qn(t))=0

 

 

⋮

 

 

Jn(q1(t),⋯qn(t))=0

 

This is (modulo condition of non-vanishing Jacobian) equivalent to fixing the previous n initial conditions. Integrals of motion for integrable systems depend on 1to1 on initial conditions.

Now, what are constraints? Constraints are both mathematically and physically very similar, but with a very important nuance. Assume n constraints:

 

F1(q1(t),⋯qn(t))=0

 

 

⋮

 

 

Fn(q1(t),⋯qn(t))=0

 

But these relations being in place for every set of initial conditions. What does this imply mathematically? It implies nothing other than that the system cannot move at all. Cannot evolve in any meaningful way at all. It's frozen dynamically. How much? It's just a point in configuration space.

Now, what happens to a field theory under analogous strictures? We must now promote the qi(t) to some φa(x,t) , where a goes from 1 to n. If we impose n initial conditions, that means specifying the value of the φ 's,

 

φ1(x,τ)|Σd−1=f1(x),⋯,φn(x,τ)|Σd−1=fn(x)

 

where τ is some curvilinear coordinate that specifies the sub-manifold Σd−1 the space-like foliations that define the Cauchy problem.

If, instead, we have a set of n constraints, and as any constraints worth their salt, they do not depend on initial conditions, but are the same for all posible initial conditions, we will have the system extremely limited in its evolution. But it is no longer true that the set of states compatible with this situation "shrinks" to a point in configuration space. It does freeze, but due to the presence of the space variables x it does so --it must-- to a fixed function sitting on the topological space. Or perhaps to a finite or countable set of such "frozen" functions. It is in that sense that I was talking about a quasi-rigidity. Known examples of constraints in field theory (valid for all sets of initial conditions) are transversalities and Gauss or Lorentz gauge-fixing constraints. Photons are known to "inhabit" a space of configurations with 4 degrees of freedom when you give them mass by assumption. If you impose masslessness, gauge fixing, they become more and more restricted in their evolution. If you imposed further constraints, there would be no dynamical situation that implies propagation. They would be "frozen." But not to a point. Instead they would be frozen to a pretty restrictive class of what I've tried to refer to by this term "quasi-rigidity."

The previous discussion generalises trivially to a 2nd-order-in-time system by substituting 1,...,n with 1,...,n,...2n, and the φ 's to those plus their canonical momenta.

Now, I know these musings not to be totally out of whack, because I've posed the question (the one about quasi-rigidity) elsewhere appropriate and I know for a fact that knowledgeable people consider it a totally-non-silly conjecture at least. Or, if you want, a question worth answering. But what would come next --the relation to the interpretation of QM-- does have speculative elements on my part, and would probably require a thread of its own.

I will have to look closer at the affine connections however I understand where your coming from will closer as to how the operators are handled 

[MigL]

2 hours ago, Genady said:

The first question perhaps is what of this look is on the world and what on the beholder.

I suspect neither.
Rather, it is on the method used for 'beholding'.
Just as the wave or particle nature we behold in QM depends entirely on the experimental set-up, or method of observation.

Keep in mind that interpretations are like opinions, only loosely based on the mathematics we are trying to interpret and compare to 'real world' experiences.
Also, just like opinions, and other bodily parts, everyone has one.
Take them with a 'grain of salt'.

[Markus Hanke]

20 hours ago, Lorentz Jr said:

No, GTG is defined in flat Minkowski space

…which is a geometric model itself, with g=diag(-1,1,1,1).

GTG uses a pair of gauge fields (corresponding to translations and rotations) instead of the metric as its fundamental entity, and employs the formalism of geometric algebra to build the model. To me, that’s very much geometry - the clue is even in the name.

20 hours ago, Lorentz Jr said:

I don't necessarily prefer GTG, I just use it as a counterexample to show that the geometric interpretation hasn't been proven to be "real".

That’s fair enough. But it does bring us back to the previous point about what it actually is we are trying to do here - I maintain that in physics we simply make models of aspects of the world. GR is a model of gravity that happens to employ Riemann geometry as its language; but to me that does not imply that that aspect of the world “really is” geometry in an ontological sense. It implies only that the particular formalism employed by GR shares the same structure and behaviour as real-world gravity, and thus it is a useful model, akin to a map. It also does not imply that the standard formalism of GR is the only possible way to draw a map of gravity - it evidently isn’t. 

P.S. I always use the word “ontology” in the sense it is employed in philosophy. 

20 hours ago, Lorentz Jr said:

Neither QM nor GR bothers me nearly as much as SR

That’s strange, since your earlier comments implied that you had no objections to Minkowski spacetime as the basis for GTG. Besides, GR reduces to SR everywhere in small local regions, so saying that SR bothers you more than GR is…well, strange.

20 hours ago, Lorentz Jr said:

I'm not in favor of pseudoscientific fantasies, I'm not in favor of "theories" that don't make predictions (or make 10⁵⁰⁰ predictions), and I'm not in favor of the current fashion (as I see it) of treating math as a substitute for physics. Many Worlds says energy eigenfunctions of the universe are "real", interpretations like quantum information and Von Neumann–Wigner seem to deny the reality of the universe itself, and the current consensus seems to be that thinking about ontology is nothing more than a navel-gazing exercise for philosophers. It's like the physics profession has been hijacked by mathematicians.

I agree with most of this, except the comments on ontology. It is a serious and important discipline, but to me it is not what physics is primarily concerned about. Though of course, there is a certain amount of overlap.

20 hours ago, Lorentz Jr said:

Page 36.

Yes, this refers specifically to the Einstein-Rosen bridge that appears in standard GR in the maximally extended Schwarzschild spacetime - this feature does indeed not appear in GTG.

However, we need to remember that a “wormhole” is a general term for a class of topological constructs that lead to spacetime becoming multiply connected in some way; there are many different types of these, and not all of them require singularities. I do not believe that GTG actually guarantees spacetime to always be singly connected, but I’m open to correction on this one.

Edited January 4, 2023 by Markus Hanke

[Lorentz Jr]

1 hour ago, Markus Hanke said:

…which is a geometric model itself, with g=diag(-1,1,1,1).

GTG uses a pair of gauge fields (corresponding to translations and rotations) instead of the metric as its fundamental entity, and employs the formalism of geometric algebra to build the model. To me, that’s very much geometry - the clue is even in the name.

Right, but that's not what I meant by "geometry". I was referring to gravity as curvature of space, which I don't think is present in GTG.

1 hour ago, Markus Hanke said:

I maintain that in physics we simply make models of aspects of the world.

Yes, of course. But there's a difference between shallow phenomenological models and deeper, more robust models. The deeper ones always involve a model of what things are made of and what principles govern the behavior of the constituent parts, even though those principles aren't always obvious from the behavior of the systems in normal circumstances. Models of what goes on "under the hood", as we like to say here in the States.

So, again, a topological map of a geographical region isn't going to help you during an earthquake. And a doctor or a mechanic needs to have a deep model of the cells and tissues and organs in the human body or the mechanisms inside the machine, not just a schedule for when the person wakes up and goes to work every morning or the external size and shape of the machine.

"Deep models" is all I mean by "ontology". Maybe it's not the best word. Genady seems to think my usage of it is nonstandard.

1 hour ago, Markus Hanke said:

GR is a model of gravity that happens to employ Riemann geometry as its language; but to me that does not imply that that aspect of the world “really is” geometry in an ontological sense.

But it could be very useful to have a deeper model. We're not going to develop any warp drive or hyperdrive based on relativity, and there could also be uses for a deeper model of gravity. For instance, in unifying it with quantum theory.

1 hour ago, Markus Hanke said:

your earlier comments implied that you had no objections to Minkowski spacetime as the basis for GTG.

I take what I can get, Markus. If there were an equivalent theory without Minkowski space, I would be interested in that.

1 hour ago, Markus Hanke said:

Besides, GR reduces to SR everywhere in small local regions, so saying that SR bothers you more than GR is…well, strange.

Not really. What I mean is that the principle of relativity bothers me more than modeling gravity as the curvature of (any) space. Ilja Schmelzer's General Lorentz Ether Theory is an example of "GR without SR", so to speak.

 

On 1/2/2023 at 7:10 AM, Markus Hanke said:

my guess is we’ll be in for a complete overhaul of our understanding of what “space” and “time” mean on a fundamental level.

Yep. And I'm interested in that.

Edited January 4, 2023 by Lorentz Jr

[joigus]

16 hours ago, Genady said:

The first question perhaps is what of this look is on the world and what on the beholder.

Good point. Maybe observation carries with it this illusion of entities. It doesn't look like QM with its linear+unitary description alone can do that. The reason is that quantum superpositions evolve forever in every subspace allowable. In simple words: The "quantum curse" that everything that can happen, does happen, propagates to every degree of freedom available. Even to the degrees of freedom of the interacting system. It contaminates everything, there's no way around it. So, if you take QM to the ultimate consequences, not only both the dead cat and the living cat keep going, but also the observer that sees a dead cat, and the observer that sees a living cat, in a quantum superposition of observers in different states of observation.

We still need Copenhagen to make sense of the ordinary, even though it's so ugly and we don't know how to interpret it in a cosmological context. As to cosmology, where we really need to make sense of this, and need it badly, we can't afford to have 7-odd interpretations to choose from.