joigus

Exactly. It's always a why from a how, or a how from a further how. And an initial how --what to assume-- is always susceptible of a further why. That's why there's always a how in the last analysis. How about that?

Interesting...

Yes, I don't know what all of this has to do with interpretations of QM, but composition of collinear boosts is not linear in velocities, although it is in rapidities[?*] (which are essentially hyperbolic-angle arguments.) That's probably what @Markus Hanke meant. The part where I get lost, as I say, is what any of this has to do with QM and its interpretations. I do not wish to keep talking about this, as it has no bearing on the particular aspects I'm more interested in, but suffice it to say that linear or non-linear depends on the arguments on which the object is considered to be acting. Eg, and totally analogous to collinear boosts, rotations about the same axis are linear in the angle, but not in the trig functions the rotation depends on. * Is that the word?

Thank you, Markus. +1. As usual, you helped to take the discussion back on its track. And that's right. They are linear. Thank you for your attentive reading. +1 I must confess I must go back to @Genady and @Lorentz Jr's parallel discussion and understand the point. The argument has spilt over into different aspects of physics, and it's hard to keep up.

@Lorentz Jr, you'd be well advised to refrain from your usual hissy fits, stop using neg-reps as some kind of covering fire for your arguments, and stick to the topic. If I didn't understand your point, say it nicely and we can all get back to business. I can assure you I, and many others, think you make a valuable contribution to the forums. Lorentz transformations don't act on the v's, they're parametrised by them. That's whay I got confused. I'm sorry for having ruffled your feathers. Jeez! I've just reverted my red point.

Why do you act so childishly? Saying they are non-linear is not the standard way of referring to them, and certainly not the appropriate way. It's at best confusing. If you wish to say they are non-linear in velocities, you should specify that. That's not the way we refer to them, among other things because once you are in the domain of SR, 3-velocity is not a vector quantity on which to define linear / non-linear transformations. It's just a set of 3 parameters.

They are linear.

Absolutely no rush. This thing about topological Lagrangians giving rise to systems (quantum or not) entailing some kind of rigidity has kept me wondering for years. For trajectories of point particles, it's clear that you get the system frozen to a point. But for field theories it's not so clear. If you allow for non-commutativity (quantum) the question is even more involved. To me, it both is interesting and makes sense. But for some reason people who study topological field theories haven't put it on the front burner, for years and years. You only see it mentioned as some kind of quirk. Yeah, OK. I must confess I have to think about this harder, and read more stuff with as much attention as possible. Some of these conditions coming from constraints do look a lot like continuity equations. It's peculiar to me that conditions like the Lorentz gauge-fixing condition are formally identical to local conservation laws for a Noether charge, although they're just constraints. And in the case of gauge fixing, they seem quite arbitrary. In the case of these topological theories, they seem to appear more "naturally." At least the way I saw Lee Smolin handle them in those lectures. That's all I can say for now in the way of a reflection. It's just peculiar to me.

Yes, I think we are, to the extent that I can follow all the arguments. It's a recurring theme in Nature, I think. Simple principles, very complicated consequences.

You may be interested to know that Lee Smolin changed his mind about this question relatively recently. If you do a search of "time as fundamental" you're likely to find as many entries as for "time as emergent," which goes to prove that the question is far from settled. If you propose time as fundamental, your first order of business should be answering why it is that it appears to be a dimension, and has infinitely many versions corresponding to infinitely many observers. I'm not sure that Smolin and collaborators have got around to this question as yet. It would be nice to know if they have. I don't.

I do not expect it to be different.

That's what I like to dream. But one look at the standard model is enough to realise that the dream seems to vanish in one fell swoop. It's like the basic idea is extremely simple, but Nature uses it to make crossroads and turnarounds in any possible way: Arbitrary mixings, symmetry "offsets," apparently idle copies of the same thing with arbitrary displacements. It's as if a master engineer had made a thing of beauty, and a naughty kid had been playing with it. I agree, but I would like to add a note of optimism to it. Gravitational waves are something no ordinary primate would have dreamt of "seeing." We have this capacity to build an extraordinary prosthetics to extend the reach of our senses by cleverly arranging pieces of material.

Something we should never overlook as a possibility. We might have blind spots that are impossible to overcome. Why should the rules of the universe be written in a language that primates can understand?

To me it's not. I also think for @TheVat it isn't either --from a previous post. But QM goes further than probability. It speaks of amplitudes, which seem to be on a sub-level with respect to probabilities. What is this sub-level about? Why do we need a choice of global and local phase, but the gauge principle makes it completely irrelevant? Those are questions that keep me awake. I'm wary of interpretations that summon the brain into the big picture. I don't think it's a matter of brains. I also abhor of interpretations based on gravity, but I find a bit more difficult to explain why. I think it's more a question of classical mechanics appearing as a fully-fledged assymptotic approximation from within quantum mechanics, rather than by analogy or correspondence, plus a series of artificial assumptions.

This seems to suggest that the very process of observation involves some kind of "translation"... But the original "language" is not available to us. Pardon my quotation marks.

Razor-sharp.

How do you know?

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.

Translate to scientific language: Is the world a simulation? Or could it be in any sense? Could something like reincarnation be programmed in such a simulation? Then proceed to, How could we prove that is the case? What is the substrate on which such simulation is running? etc.

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. 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...

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.

Obviously, I meant consciousness. I'm not implying anything about the cat's morality.

It's a question having to do with quantum mechanics of open systems interacting dissipatively with billions of atomic degrees of freedom. The environment "learns" pretty fast whether the cat is dead or alive, because the environment makes the alternatives (corresponding to quantum numbers that gave rise to the fateful event) decohere with each other --in modern parlance. So it's nothing to do with conscience. It's about the system interacting in a complex way with its environment. Conscious beings are but a particular example of this occurrence. The cat is no exception, I'm sure, as it's also made up of billions of subatomic degrees of freedom, so the cat learns pretty fast too about his own health state just before he dies. There's a question pending about the so-called pointer states that's still not solved. But it has no consequences from the practical point of view. Different people embrace different interpretations of quantum mechanics in order to make a mental picture of what happens to the previous quantum states.

How do you reconcile this with charge conservation? The initial electron has charge -e. The e+ e- pair, on the other hand, has charge 0. --As pointed out by @swansont and @Mordred.

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.