joigus

1 hour ago, exchemist said:

But you still stigmatise them as ill. So ill but stable?

I didn't mean religious people --see my comments to Luc Turpin below.

Apparently schizotypals were discovered as a consequence of behaviour scientists wondering: How come an illness as detrimental as schizophrenia is so significantly present in the gene pool? --In the ballpark of 1%. Wouldn't there be a milder but related version of the illness that could be proven as advantageous under certain circumstances? The parallel was sickle-cell anemia, which can kill you, but a milder version of which can protect you from malaria. So they found a high correlation of peculiar characters in relatives of people suffering from schizophrenia.

I wouldn't dare to use the term "ill" for any of these people. AFAIK triggering of even serious form of schizophrenia only happens after environmental factors have made their appearance.

But I'm very far from being an expert here and I'd gladly accept corrections by anyone who knows more about this.

14 minutes ago, Luc Turpin said:

 

On 3/16/2024 at 7:00 PM, joigus said:

Religious types could, after all, be not much more than socially-accepted schizotipicals, that have somehow met the medium, and the way, to make their illness socially palatable.

But not this one.

Sorry, by "religious types" I didn't mean the followers of a religion. Rather, I meant the prophets, the visionaries, the people who hear voices, the people who see angels. You know, the founders of religions.

The following of a religion is a completely different matter. Some people join because they feel comforted, others because they want to fit in, others because they are folklore-motivated, etc. Who knows. At least, I don't.

5 hours ago, JosephStang said:

Geometry is not math. 

 

It is a branch of math.

https://www.britannica.com/science/geometry

Quote

Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics,

The branch is part of the tree, although the tree is not the branch.

23 minutes ago, Luc Turpin said:

My religious friends appear more mentally stable than most in society.

Schizotipical behaviour is not to do with mental stability. It's to do with delusional perception of experience (sensory or otherwise). Have you skimmed through the wikipedia article or references thereby? My emphasis in boldface in a sample from mentioned article:

Quote

1. Crespi B, Dinsdale N, Read S, Hurd P (2019-03-08). "Spirituality, dimensional autism, and schizotypal traits: The search for meaning". PLOS ONE. 14 (3): e0213456. Bibcode:2019PLoSO..1413456C. doi:10.1371/journal.pone.0213456. PMC 6407781. PMID 30849096.
2. ^ Carvalho LF, Sagradim DE, Pianowski G, Gonçalves AP (2020-10-19). "Relationship between religiosity domains and traits from borderline and schizotypal personality disorders in a Brazilian community sample". Trends in Psychiatry and Psychotherapy. 42 (3): 239–246. doi:10.1590/2237-6089-2019-0085. PMC 7879071. PMID 33084801. S2CID 224828232.
3. ^ Breslin MJ, Lewis CA (2015-03-04). "Schizotypy and Religiosity: The Magic of Prayer". Archive for the Psychology of Religion. 37 (1): 84–97. doi:10.1163/15736121-12341300. ISSN 0084-6724. S2CID 144734469.
4. ^ Byrom GN (2009). "Differential Relationships between Experiential and Interpretive Dimensions of Mysticism and Schizotypal Magical Ideation in a University Sample". Archive for the Psychology of Religion. 31 (2): 127–150. doi:10.1163/157361209X424420. ISSN 0084-6724. S2CID 143580864.

Etc.

2 hours ago, JosephStang said:

Why would I have to specify position variables while describing a 3D geometry?

The very moment you posit that your theory is local.

7 hours ago, JosephStang said:

It overcomes Bell’s inequalities with Bohmian style locality by mediating FTL

(emphasis mine)

A theory is or is not local depending on a postulated interaction, or else by way of an ad hoc postulate or axiom. Yours is neither. It is neither non-local, nor is it local. It's only named "local" by you.

And forgive me having overlooked this, but, what do you mean it overcomes Bell's inequalities? Quantum mechanics as is already overcomes Bell's inequalities, ie, it violates local realism. Bell's inequalities are a consequence of local realism.

So again, what do you even mean?

4 hours ago, JosephStang said:

Description: Each electron/proton is a ring.

Protons are nothing like electrons. We do know as much.

In what sense is this "holonomic"? "Holonomic" means integrable, exact, it goes back to itself after a loop. I don't see anything holonomic here.

I can't fathom what's Bohmian about it, or local/non-local, as the case may be, as no mention of how position variables function in the "theory" can be spotted.

Summarising, it very much sounds like word salad with no maths underpinning it. No calculation, no formal-mathematical justification.

21 minutes ago, JosephStang said:

Do you not understand the description?

What description?

On 11/12/2023 at 4:04 AM, Chris Sawatsky said:

A sphere exploded but did not expand in every possible direction simultaneously?

Picture an inflating balloon. Now suppress the space around and inside the balloon, as there is no such thing as "inside" or outside the balloon. There would be only whatever stuff makes up the balloon. Now make the balloon itself 3-dimensional, with time providing for the "history" aspect of it.

Spaces don't have to be embedded in higher-dimensional spaces. IOW, the only existing directions are those tangential to the balloon's rubber if you will.

How about StPD at the root of many, if not all, of these reports?

https://en.wikipedia.org/wiki/Schizotypal_personality_disorder

Quote

classification describes the disorder specifically as a personality disorder characterized by thought disorder, paranoia, a characteristic form of social anxiety, derealization, transient psychosis, and unconventional beliefs.

Religious types could, after all, be not much more than socially-accepted schizotipicals, that have somehow met the medium, and the way, to make their illness socially palatable.

4 hours ago, Markus Hanke said:

That’s because mass never appears in the GR field equations - what is generally called the source term here is the energy-momentum tensor. One must also remember what these equations actually say - they state a local equivalence between a certain combination of components of the Riemann tensor (the Einstein tensor) and the energy-momentum tensor. Nowhere does it claim a ‘causative relationship’, but instead it says that these two are the same thing (up to a constant of course); neither one comes first and ‘causes’ the other.

Indeed. I --and others, you among them-- have said it before elsewhere on the forums, actually. It's the energy-momentum that sources the gravitational field.

I also agree with the absence, of necessity, of any causal connection between the Einstein tensor and the energy-momentum tensor.

On 3/14/2024 at 8:55 AM, martillo said:

the temperature of an atom is [...]

There is no such thing.

Thermodynamics defines temperature based on thermal equilibrium. Statistical mechanics relates it to average kinetic energy per degree of freedom. For statistical mechanics to make the connection between both concepts through the partition function and the Maxwell distribution, we need approximations on really big numbers of molecules.

The group of symmetry of electromagnetism is U(1) (complex numbers of length 1), and electrical charges are at the centre of it.

From the POV of symmetries, conservation laws, and irreducible representations of groups (particle multiplets) QFT of electromagnetism and its brethren --weak interaction, strong interaction-- is more user-friendly by orders of magnitude. Things kinda "fall into boxes."

GR is not like that. Not by a long shot.

The group of symmetry of GR is basically just any differentiable transformation of the coordinates. Once there, after one picks a set of coordinates that locally make a lot of sense (they solve the equations easy, yay!), they could go terribly wrong globally, so that one must introduce singular coordinate maps to fix the blunder.

Because the symmetry group of GR is this unholy mess, group theory doesn't help much, if at all. The equations are non-linear, so: Are there any solutions that might help clarify divergences, and so on, that we might have missed entirely? Who knows.

In my opinion, the very fact that the set of coordinates that, locally, happens to be the most reasonable one could (and sometimes does) totally obscure the meaning of the coordinates far away from the local choice, and thereby their predictive power out there, makes the status of any parameters that the theory suggests (mass in particular) much less helpful than charge is in EM.

Mass to GR is nowhere near anything like charge is to Yang-Mills theory (our paradigm of an honest-to-goodness QFT field theory).

2 hours ago, CharonY said:

The neat thing is that one can often deduce what is meant by those words. 

Yes! It's like a tinkertoy assembly for logically compressed inflexions[?]. Whatever I mean by that... For some reason, phonetics, syllables and their frequencies, it seems to be very friendly to the forming of composite words. The end result doesn't sound awkward.

Is this (admittedly rough) understanding that I've acquired through the years correct?:

The currency of red-ox reactions is electrons

The currency of acid-base reactions is protons

Now, in a manner of speaking,

Both oxydisers and reductors can be understood in terms of "soaking up" and "giving off" electrons

Both bases and acids can be understood in terms of "soaking up" and "giving off" protons

That's the reason why so much of chemistry hinges around these two dual concepts

Other cations, even the smallest ones, like Li⁺, are "monsters" in comparison to H⁺. Orders of magnitude so much so. So even though the mean free path of a proton is sizeably higher than that of an electron, it's bound to be gigantic as compared to that of even such a small thing as Li⁺. That would qualitatively account for an extraordinarily high mobility of protons, thereby the reactiveness of anything that either gives them off or soaks them up. That's the key to the concept of Lewis acids. Is it not?

Then, for something to be a base, in its most general sense, it must be able to soak up protons. But for it to display this character, there must be some protons around to soak up. Wouldn't something like this be at the root of NH₃ not "behaving as a base" just by itself, or in the presence of chemicals that cannot give off protons?

Wouldn't it behave as a base in the absence of water, but in the presence of acids (neutralisation) like,

NH₃+A --->NH₄⁺+A^-

with A being any acid?

8 hours ago, sethoflagos said:

At least they sound more impressive than their literal English translations ('jitter motion' and 'braking radiation').

  German scientific terms are generally very precise. They feel no embarrassment in making long composite words tagging essential characteristics of the thing. Bremsstrahlung in Spanish is radiación de frenado, which is exactly 'braking radiation', but requires three words.

8 hours ago, sethoflagos said:

and when they arrived I learnt that the Malay word for water is 'air'.

Pronounced as in English, I assume.

9 hours ago, Markus Hanke said:

deSitter spacetime has non-zero Riemann tensor, so it’s not flat.

Yes, thank you. That's what I thought.

For a while I felt nervous about zitterbewegung and bremsstrahlung, but it grows on you.

2 hours ago, MigL said:

You're gonna have to elaborate on that one also.

Spatially flat and space-time flat are often conflated in the literature. I would have to review the Riemann coefficients with 0t pairings of indices (a space cannot warp in just one dimension). I'm not sure nor do I have the time (nor the energy) now to review these notions. Maybe someone can do it for all of us. Most likely @Markus Hanke. I'm sure DS space-time is often characterised as having constant curvature*. We're kind of mixing it all together as if the scalar curvature were "the thing" that says whether a manifold is flat or nor. It's  more involved. If just one R_ijkl is non-zero, the manifold is just not flat.

Calabi-Yau manifolds are another example which are Ricci-flat (R=0), but not flat.

2 hours ago, Genady said:

Aren't there many examples, at least in principle? In particular:

"Q: The information that gets lost when we go from the Riemann tensor to the Ricci tensor does not affect the energy-momentum tensor nor Einstein’s equations. What is the meaning of this lost information then?

A: It means that for a given source configuration, there can be many solutions to Einstein’s equations. They all have the same right-hand side, namely Tμν . But they simply have different physical properties. For example, the simplest case is to ask: what if this energy-momentum stuff is zero? If it is zero, does it mean that there is no gravitation, no interesting geometry at all? No. It allows gravitational waves."

Susskind, Cabannes. General Relativity: The Theoretical Minimum. 

Not according to this: homework and exercises - Non-zero components of the Riemann tensor for the Schwarzschild metric - Physics Stack Exchange

Yes. Thank you. Read my comments to @MigL on flat vs spatially flat, Ricci-flat, and so on. They're very much in the direction you're pointing. Right now I'm beat, but I promise to follow up on this.

41 minutes ago, swansont said:

Isn’t the sun’s (or earth’s) field approximately a solution to the Schwarzschild geometry? 

Yes, of course you're right. This theorem due to Birkhoff[?] that the external solution is unique as long as it's static and spherically symmetric. Schwarzschild's solution was just an unfortunate example. I know very little about exact solutions in GR. I just figure there must be solutions with not all curvatures zero with no clearly identifiable matter distribution giving rise to them.

* 

Quote

In mathematical physics, n-dimensional de Sitter space (often abbreviated to dS_n) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian^{[further explanation needed]} analogue of an n-sphere (with its canonical Riemannian metric).

5 minutes ago, swansont said:

What’s the curvature of the Schwarzschild solution?

Infinite at one point. Zero everywhere else. But you're right. It's not a good example.

De Sitter is more what I was thinking about.

2 hours ago, DanMP said:

Maybe, but the curvature depends on the local mass and/or energy ... You remove the mass/energy and the gravity is gone.

It's a bit more subtle than this, I think. You can have vacuum solutions with curvature. If you think about it, the Schwarzschild solution is a vacuum solution. De Sitter and anti-De Sitter are too. OTOH, the Einstein field equations are nonlinear, so I wouldn't rule out other exotic vacuum solutions with curvature.

3 hours ago, Genady said:

I did, and my answer was that L is invariant under translation.

Right.

A scalar provides a particular type of covariance. Rank zero. L'(x')=L(x)

That's what one must prove in this case.

10 minutes ago, Genady said:

Isn't it rather 6000? 😄

LOL forgot x10⁶ didn't I?

Coming from me lately, how could it be otherwise? fortunately more than 4000 and more than 6000 could also be more than 4.7x10⁹ yo.

Thanks

Had the Moon disappeared more than 4000 4.10⁹ ya it would have been much much worse. Most Earth scientists think it was essential in the appearance of life.

2 hours ago, Genady said:

Another ecological effect will be on organisms which use Moon for timing, for example, coral spawning.

Or SpaceX.  

10 hours ago, KJW said:

I'm perfectly aware of the modern trend of getting rid of the entire notion of coordinate systems. But I reject this trend, regarding it as throwing the baby out with the bathwater.

Ok, so you're old school. I respect that.

But mind you that coordinates could be misleading you in some respects while they're helping you in others. This observation should always be carried along.

13 hours ago, KJW said:

However, I will ask you this: In general, does ∂μ∂μϕ=∂μ∂μϕ ?

In flat coordinates, sure. In curvilinear coordinates, it's a bit more involved than that. That's called the Laplace-Beltrami operator and you have to write some metric tensors in between, and also some epsilons, if I remember correctly. I would need some time to remember all the machinery. If yours is a genuine question.

https://en.wikipedia.org/wiki/Laplace–Beltrami_operator

Also books like Gockeler-Schucker, etc. on differential-geometry methods for theoretical physics.

Are you just asking or trying to catch me again?

14 hours ago, Genady said:

I'm at ease with the Schwartz's notation by now. It is as simple as mentally substituting AμgμαBα every time he writes AμBμ .

Yes. That's what Feynman did all the time. For some reason he didn't like the g's.

On 3/1/2024 at 6:22 PM, KJW said:

The Kronecker delta tells you nothing. Its invariance guarantees that. You chose a flat space for your example. How about choosing the spacetime of a Schwarzschild black hole, where the central singularity is not removable.

This is getting farther and farther away form discussing anything substantial (let alone anything within the OP context), and more and more about you getting out of your way in order to shift the context so that I could be proven wrong in that context.

Anyway, Kronecker delta with a superindex and a subindex is an isotropic tensor. Kronecker delta with, eg, two covariant indices (like Tαβ=δαβ ) tells you much, much less, as it's a frame-dependent equality. δαβ is telling you that what you have here is just a rule to dot-multiply vectors. How much or how little does that "encode"? The dimension, and the fact that you're dealing with a scalar product? That's about all. I'll leave to you to decide how much that is telling you. There is a reason why the connection is given in terms of g−1∂g . Neither covariant indices nor contravariant ones give you the curvature, and it's an interplay between the two that does it.

There is just another isotropic tensor in every space (the ϵ tensor). It is kind of telling you about orthogonality. The Kronecker only looks standard (1's & 0's) with one index up and the other down. The epsilon tensor only looks standard (1's, 0's, and -1's) with all indices down or all up. Otherwise, they show you all kinds of misleading info, as I clearly showed you with my textbook-standard example.

Also, multilinear operators are not just "tensors" independently of a context, as you seem to imply. Multilinear operators are or are not tensors depending on the relevance of a certain group of transformations. There are such things as O(3) tensors (orthogonal tensors), U(n) tensors,... there are pseudo-Euclidean tensors (the only ones we were talking about to begin with), there are tensors under diffeomorphisms (the ones you, for some reason, want to shift the conversation to, although they have little to do with the initial discussion), etc. 

Let me point out more mistakes that you're making:

On 3/1/2024 at 6:22 PM, KJW said:

The differentials of the coordinates vector is naturally contravariant as a result of the chain rule. The dxi=(dr    r2dθ) vector is not a differentials of the coordinates vector as it is clearly seen to contain components of the metric tensor

Again, you are wrong. There's nothing special about differentials of the coordinates:

All such objects form a basis. They are called coordinate bases or holonomic bases --see below. But not all bases are made up of derivatives of coordinates. It's only when they are that they thus called. This is from Stewart, Advanced General Relativity, Cambridge 1991:

So no, not all bases are coordinate bases. And I wrote down a totally legit basis.

More on that:

https://www.physicsforums.com/threads/non-coordinate-basis-explained.950852/#:~:text=Some examples of non-coordinate basis vectors include polar basis,defined by traditional coordinate axes.

Under a different name:

https://en.wikipedia.org/wiki/Holonomic_basis

So one thing is a basis, and quite a very specific (and a distinct) thing is a coordinate basis. In still other words: A coordinate basis is made up of a set of exact differential forms and their duals. This is in close analogy to what happens in thermodynamics: You can use the Pfaffian forms of heat and work to define any change in the energy of a system, even though there are no "heat coordinate" or "work coordinate". But a basis they are: dU=TdS-PdV, even though TdS is not d(anything) and PdV is not d(anything).

Do you or do you not agree that the variational derivative @Genady was talking about should be written as,

\[ \partial^{\mu}\left(\sum_{n}\frac{\partial\mathcal{L}}{\partial\left(\partial^{\mu}\phi_{n}\right)}\partial_{\nu}\phi_{n}-g_{\mu\nu}\mathcal{L}\right)=0 \]

instead of,

\[ \partial_{\mu}\left(\sum_{n}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_{n}\right)}\partial_{\nu}\phi_{n}-g_{\mu\nu}\mathcal{L}\right)=0 \]

As far as I can tell, that was the question, your detour into differential geometry has been satisfactorily answered, and you seem to have nothing further to say that's remotely on-topic.

12 minutes ago, Genady said:

Unfortunately, I can't read this post:

and I don't know how his result is different from mine, but it seems that his EL equation is the same as mine,

<<<<<

which is different from

<<<<<<<

I disagree with the latter. We need to use the generalized EL equation, which I have already derived in this exercise:

and got the answer compatible with this:

(Euler–Lagrange equation - Wikipedia)

 

Yes, you're right. The unusual writing of the Lagrangian set me off. Sorry. That is indeed the way to generalise to higher-order derivatives. I've proven it many times, but now I had just a couple of minutes and I screwed up. There's just a coefficient difference. 

Later.

Ok. Yes, @RobertSmart is right. There is a little mistake in the constants. Let me display my calculation in detail, because his Latex seems to have been messed up by the compiling engine or whatever and I seem to find a small discrepancy with him.

Your Lagrangian,

\[ \mathscr{\mathcal{L}}=-\frac{1}{2}\phi\Box\phi+\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4} \]

I prefer to write with an index notation, which is more convenient for variational derivatives:

\[ \mathscr{\mathcal{L}}=-\frac{1}{2}\phi\left.\phi^{,\mu}\right._{,\mu}+\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4} \]

As we have no dependence on first order derivatives,

\[ \frac{\partial\mathscr{\mathcal{L}}}{\partial\phi_{,\mu}}=0 \]

we get as the only Euler-Lagrange equation,

\[ \frac{\partial\mathscr{\mathcal{L}}}{\partial\phi}=-\frac{1}{2}\left.\phi^{,\mu}\right._{,\mu}+m^{2}\phi-\frac{\lambda}{3!}\phi^{3}=0 \]

Or,

\[ -\frac{1}{2}\Box\phi+m^{2}\phi-\frac{\lambda}{3!}\phi^{3}=0 \]

Or a bit more streamlined,

\[ \Box\phi-2m^{2}\phi+\frac{\lambda}{3}\phi^{3}=0 \]

Sorry I didn't get around to it sooner.

Paraphrasing Sir Humphrey Appleby:

Is that finally final?

I hope so.

PS: BTW, this is a simplified symmetry-breaking Lagrangian. The real thing in the SM is a complex SU(2)-symmetric multiplet \( \left(\phi_{1},\phi_{2},\phi_{3},\phi_{4}\right) \).