joigus

LOL. Beta decay = "when the neutron gives up the ghost" I understand 'ghostly' as 'interacting very weakly.'

By "fundamental" I mean resting on minimal/general assumptions, although I know how slippery that concept can be. Yes, theory of the mind --understanding that others have minds probably like we feel we ourselves have minds, and acting accordingly-- is one of the most important adaptive pressures that shaped the evolution of the human brain. I don't have the sources at hand to assert this, by I know from the reasoned arguments of many scientists of human evolution I've sampled through the years. I am in no doubt that there must be a reason why Nature has kept the genes that give us autism, which at first glance looks like just a cognitive impairment. Look closer and more can be seen. From my experience, from what I know from you, and others like you: Genuinely caring individuals, who suffer when they see conflict, generally devoid of manipulative intentions. Never foul players, sincerely concerned about problems, both human and technical, their own, and those of others. Hard workers, obsessive in a way that can be very productive, given the proper outlook, focus, and advise about how letting go when the time comes. Sometimes we're discussing something and we're being petty and narrow-minded. And here comes Markus Hanke and shines his light. All the pettiness is dissolved. The problem is re-focused to what the problem is. I guess that's why you lot are here for. Don't look now, but you activate us in a direction that --always in my experience, mind you-- usually is a good one.

Thanks, Phi. Now, what do you wish to discuss concerning neutrinos, @Vette888?

Right, but not news.

I couldn't agree more. For every bundle of sensory input we receive, we build models in our minds, filled with switches and state variables, or whatever you like to call them. Right now, in my mind, I picture @Genady's opinions and ideas, criteria, etc as --somewhat loosely-- some kind of a list with states of opinion, philosophical tenets and so on, that are perfectly defined as a series of 0s and 1s, so to speak. And you me, I'm sure. We work under such assumptions, and assign states, probabilities, etc to all these things. There is no fundamental reason why this should be true. What's more, even if it happens to prove itself useful to a certain degree, there's no fundamental reason why this process could be continued till the last least little consequence for everything we experience.

Exactly. I agree 100% with your professor. I have no qualms about considering the refraction index as a complex number, and its real and imaginary parts as real numbers. What I try not to forget, ever, is that there is a corpus of theory undelying this idea. It could just be approximately right. A real number is a convenient tool that gives you room to accomodate infinite precision. What's not to like about this wonderful tool?

Interactions, class of observers to class of observers transformations, thermal equilibrium. All of these are notions that require some kind of idealisation or another that is not real in any strict sense. Interactions: Consider an electron. You want to study it in detail in the ultraviolet regime, you need to shoot something at it and, pretty soon, you excite the virtual degrees of freedom, and you have to worry about virtual photons and pairs that seem to come out of this --initially at least-- well-defined thing. Lorentz transformations: You consider classes of observers that are at rest wrt each other and fill up all of space and time. Thermal equilibrium: In order to rigorously define thermal equilibrium you need to go to the so-called thermodynamic limit. That is, all the extensive --additive-- variables are infinite. So what's interesting to me is that, in order to study reality --whatever that is-- you absolutely must to take some distance from it, go to a theoretical framework that is not real in any meaningful way --it's just a convenient abstraction--, and draw your conclusions from there.

Numbers (real numbers) are not atoms, they're not beads on a string, they're not pieces of code on a DNA strand, they're not characters on an alphanumeric string. They're abstractions. That's, I think, at the root of why you're finding it so hard to wrap your head around the ideas that define them. Same reason why people I will describe in anecdote a1) at the end of this post, completely missed the point too. So you're not alone. Axiom of completeness: (1) Semi-intuitive notion: Real numbers have no gaps or holes (2) Rigorous notion: Every nonempty subset \( X \) of \( \mathbb{R} \) that is bounded above has a least upper bound. That is, \( \sup X \) exists and is a real number. Consider this sequence --not made up of atoms, nor made up of beads, nor made of little LED lights; made up of numbers--, \[ X=\left\{ x\in\mathbb{Q}/x=1-\frac{1}{n},n\in\mathbb{N}\right\} \] In other words, \[ X=\left\{ 0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\cdots\right\} \] Now consider the following facts: A) Every number in \( X \) is less than 1. B) For every \( r \) such that every number in \( X \) is less than \( r \), \( r\geq1 \). That means that 1 is not only an upper bound to \( X \); it is, actually, the best upper bound we can find. That is, 1 is the least upper bound. Challenge: Find a number in \( X \) that gives you exactly 1. If you get to understand how you will fail to find that number, say \( x_{n}=1-\frac{1}{n} \), you will be a step closer to understanding this logical frustration. Let me finish with a couple of anecdotes: a1) I was once sitting at a Calculus class and the professor told us about the axiom of completenes in the form, "every monotonically increasing sequence in the real numbers possesses a least upper bound." A couple of students in front of me started giggling and went something like, "Doh! Why of course." Needless to say, they'd completely missed the point. a2) Our Electricity and Magnetism professor said. "Imagine an R, RC, or LRC circuit with an amperimeter. You measure the current and it gives 1.3 Amperes. What kind of number is that? Is it real, rational, imaginary? After a prolongued murmur somebody uttered: It's a real number. The professor said: "No, it's neither one of them. It's a measured number." Don't let the adjective 'real' deceive you. They're real alright. In some sense.

The \( f^{0} \) comes from the 0 component of 4-momentum by differentiating wrt coordinate time. It's not to do with the usual gamma factor in Lorentz transformations. It's to do with a time-dependent gamma factor. The energy of a particle is \( mc^{2}\gamma \), but this \( \gamma \) is a function of time. The power gain/loss for the particle is the time derivative of its energy, so that, \[ \frac{d}{dt}\left(mc^{2}\gamma\right)=qu^{\nu}\left.F_{\nu}\right.^{0}=q\left(c\gamma F_{00}+\gamma v_{k}F_{k0}\right)=q\left(c\gamma F_{00}+\gamma\boldsymbol{v}\cdot\boldsymbol{E}\right) \] \( F_{00} \) is zero, as F is a 2-rank antisymmetric tensor. Its non-diagonal elements are the electric and magnetic field. The zero component of the time derivative of 4-momentum is thereby the power. It's all beautifully wrapped up in space-time formalism. I've re-instated the gamma factors, but I might be missing some c factor and perhaps my initial definition of F (the electromagnetic tensor) had the wrong sign. I'm sorry that I'm missing the main point in relation to physics and reality at the moment. Please, let me come back tomorrow and try to catch up with the finer points.

The Lorentz force does have a time component, though. When you write down the complete --covariant-- form of the equation it gives, \[ f^{\mu}=qu^{\nu}\left.F_{\nu}\right.^{\mu} \] as a 4-vector equation. Where \( \left.F_{\nu}\right.^{\mu} \) produces all the components of \( \boldsymbol{E} \) and \( \boldsymbol{B} \). These equations decouple into, \[ f^{0}=q\boldsymbol{v}\cdot\boldsymbol{E} \] \[ f^{k}=q\left(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B}\right) \] and where I think I may be missing a gamma factor. The 0-th Lorentz equation gives you the power gain or loss, and the spatial equation is the conventional Lorentz equation.

Very small curvature in both cases, never mind Riemann tensor components or curvature scalar (some kind of average of all Riemann components). It can be estimated by means of Newtonian gravity.

It certainly doesn't cramp my style.

Equivalence relations are at the basis of categorical thinking, or Aristotelian categories. We're hardwired to think in terms of categorical thinking. When we can't place the category, when that category does not close in mathematical terms, it's loosely defined, we feel confused. I think it was Wittgenstein that worried a lot about that problem.

Correct. https://en.wikipedia.org/wiki/Self-energy Is it just a manner of speaking due to electrons not being isolated 'things' in any meaningful sense?

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel One has to be careful even with the natural numbers. The ocean of natural numbers admits arbitrarily many more drops! The real numbers are even more counterintuitive.

Just stretching/shrinking space without time being involved... I see this difficult to reconcile with known physics. Physics with matter --massive-- is not invariant under scale transformations. Time-dependent scale transformations would make this even worse. I don't see how you could save conservation of charge, for example, if space is actually shrinking at small scales... Exactly!

You mean shrinking space? Or also time? What about mass, electric --and other charges-- etc? Would they be shrinking in your picture? x-posted with @studiot

This is very deep. If I understood it correctly, it's like what Swansont said about phonons. There are other examples: Defects in a crystal, different 2-dimensional modes that live on the surface between insulators (topological insulators.) These things live in a context: Surface phenomena, modes in a lattice, etc. They are not 'real' in the sense that you can take, say, a phonon and separate it from its context, and study it in isolation from everything else. You can't take any of these instances, isolate them, and study them independently of the embedding context. There are contingencies --I think that's the right word-- that define their 'being there.' If you dissolve the contingency, you dissolve the 'thing.'

I would call it a lemma or proposition --small theorem-- that's easy to prove.

To me, the funny thing about it is that every so often the mathematical model does suggest to us that the previous model in terms of inalienable properties A, B, C, etc. does suggest that we'd better drop say, C, no matter how cherished a property of this 'reality' it is. In QM, this 'C' could be 'position' or 'momentum.' In SR, it could be position and/or velocity, and in GR it could be the concept itself of an inertial system, or coordinates by themselves. For lack of a better term, I would define this as a very ordered process of 'letting go' of anchors to what we think to be real. x-posted with @exchemist Agreed. But I think that's more an over-simplification that some people do when they don't understand time-tested principles like operationalism --the theory should have a counterpart in laboratory operations--, Ockam's razor --the theory should never create arbitrarily complicated constructs, and should be as logically simple as possible; it should produce falsifiable propositions --Popper--, etc. I recognise that as a risk, but I think an acquaintance with some principles of the philosophy of science generally operate as a good antidote. If one is well-versed in the history of science, I think one can minimise the risk.

This is probably the best starting point for this question --no pun intended. If x, y are in R, and x \( \neq\) y. Then, Either x>y or x<y. Assume x<y. A property of the real numbers is there always exists a z in R such that x<z<y if x and y are different. So there isn't such a thing as 'next number' in R. x-posted with @Genady

In the short term, yes. I've got friends who trusted me with their dog every now and then when they were away. I'm told I'm pretty good with pets when it's just for, say, a month, or some weeks. In the long term, no. I can't afford taking care of a pet on a daily basis. Having a pet is an enormous responsibility. It seems like it was yesterday when you mentioned it. CPT symmetry suggests they would. When you include magnetic monopoles in Maxwell's equations, you have both electric charges q, and magnetic charges g. For every particle with charge q there must be the corresponding anti-particle of charge -q. So for every particle with magnetic charge g there should be an anti-particle of magnetic charge -g, or else CPT symmetry would not hold. We think CPT symmetry holds. It would be a total re-think of quantum field theory if CPT symmetry failed. For more details, ask @Markus Hanke.

You're confusing a group with a representation of a group. Any Abelian group can be represented by addition of parameters. Example: \[ e^{x+y}=e^{x}e^{y} \] The Lorentz group is represented multiplicatively on wave functions, but additively on other objects. Again, take a look at, https://www.amazon.com/Theory-Application-Physical-Problems-Physics/dp/0486661814 and it will dawn on you. There's nothing else I can do for you here. I'm sorry my help wasn't enough.

Mr. @Abouzar Bahari, I'm reinstating all the negative points you're giving everybody, just out of spite on your part, apparently. You've already gotten your answers, non-scientifically. Thank you. That's all I wanted to read from you.

I understand you got a new cat. Is that right? I love pets, but I can't be trusted with them. Neither do I. I'm just wondering where we can get from just nothing. Maybe something's in store for us all.