Lattices and Lorentz invariance

Discussion in 'Physics & Math' started by Farsight, Oct 22, 2011.

  1. rpenner Fully Wired Valued Senior Member

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    That aphorism only applies to trial lawyers (barristers) who are eliciting testimony in the courtroom. Human beings and other honest peoples I have traveled with, contrastingly ask questions when they want to learn the answers.

    What is the "question" that you judge to be "valid?"
     
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  3. OnlyMe Valued Senior Member

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    Children often do! I have! And sometimes we think we know or have an idea and are seeking conformation, without risking an opinion, we are not sure of.

    And then there are those who have an answer, usually a unique and often unsupported notion, who use a question as means to involve others in a discussion where they can then, engage in a debate, about their answer rather than the question.
     
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  5. Farsight

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    You haven't exposed my mistakes, Alphanumeric. Now do try to stick to the physics so we can all enjoy a civil discussion.
     
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  7. Reiku Banned Banned

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    My answer was a reply to mirror your irrational judgement on the thread. You seem to not see that someone asks a question first to obtain an answer, not the other way around.
     
  8. AlphaNumeric Fully ionized Registered Senior Member

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    I typed that on my phone, which has the annoying habit of zooming into a single word so I can't see the text I'm typing. As such I have to wing it when typing on the tiny keyboard. On my PC I have auto-spell turned on so it highlights when I get something wrong and I correct it. Unlike a fair few, I generally make an effort to be coherent. It's also the reason I type lengthy replies, to explain myself. Those replies you always fail to respond to, you just make excuses.

    Nice to see that when you can't retort my criticism you go to spelling. Bottom of the barrel?

    Why didn't you respond to my post where I point out you're contradicting yourself in regards to thinking Euclidean space-time is dynamics but Minkowski is not. Either they both are or neither are. This is a physics point but you refuse to engage in a discussion about it with me. Yet you're willing to engage Prom in a discussion about space-time intervals because you have a few talking points.

    Can't you do anything other than be evasive?
     
  9. rpenner Fully Wired Valued Senior Member

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    Whose irrational judgment? How is that demonstrated? And where exactly is the "valid question" to which you refer? Are you writing for an audience that had more information about the nature of your thinking than is presented on this thread? Please elucidate.
     
  10. prometheus viva voce! Registered Senior Member

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    I'm not sure I agree with this. Having something that's not in motion is quite different from having the situation where time doesn't exist. With no motion you certainly would not be able to measure time, but just because you can't measure it directly doesn't mean it doesn't exist - see for example the Casimir effect.

    Again, I take issue with your treatment of 3 dimensional space + time as "real" but 4 dimensional (flat) spacetime as mathematical. Physics is concerned with constructing models of natural phenomena and in that sense both Newtonian dynamics in 3d and relativistic dynamics in 4d are models of what we observe. It turns out that 4d spacetime is a more accurate model as it better predicts what we observe at high energies while agreeing with Newton at low energy so we say relativity is "right" but it is still a model (note, not only a model).

    My mistake. I misread your post and now I see you wrote "the Lorentz factor..." Apologies.

    Sorry to labour this, but I feel it's quite an important issue: Do you or do you not think that particles are made of waves?

    I think we are going to disagree fundamentally about this. The reason I say that spin is really angular momentum is that one can show that, in quantum mechanics there is a set of spin operators \(S^2, S_x, S_y ,S_z \) and when one computes their commutation relations one finds they are exactly the same as those of angular momentum \(L^2, L_x, L_y ,L_z \). Spin is a fundamentally quantum mechanical effect, and trying to visualise it as "something going round in a circle," is a very good way to misunderstand it. The Stern-Gerlach experiment tells you that although spin is angular momentum it certainly does not behave like classical angular momentum.

    I think you've missed the point of dimensional analysis. Without the correct units for c I have no idea what it's value is.

    I have a few remarks:

    Firstly a technical point: If you set c = 1 the interval you produce is invariant. Since c = 1, why write c explicitly in the Lorentz factor below it?

    You assert that there is no flow of time, however you said previously "you can't have time without motion" (I don't necessarily agree with this but lets go with it). If I have a clock consisting of parallel mirrors and light bouncing back and forth then I do have motion, even if I'm stationary and therefore I do have time.

    I strongly disagree with your assertion that one may not measure proper time, as that is exactly what the light clock is measuring.

    You misunderstand again what the point of the twin paradox is: The twin that goes away and comes back is younger than the one that stayed behind on his return, however, they do not disagree about the time when the first twin returns - the do disagree about the time elapsed between the twin leaving and returning. Remember that time dilation applies to time intervals, not a particular point in time. You seem to have some remarks on length contraction which I suspect will contain a similar misunderstanding.

    I (and others) may have some more to add to this.
     
  11. Farsight

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    I wouldn't say time does not exist. Instead I'm on record as saying time exists like heat exists. Heat is real, it burns you, and in a similar sense time is real, it makes you old. It just doesn't exist as something you can literally move through or as something that literally flows.

    The Casimir effect is due to vacuum fluctuations, which is a kind of motion, and the plates move together. There's always motion involved somewhere.

    It's the old hold your hands up thing. Hold your hands up a foot apart. The gap between them is real, that's space. Now waggle them. The motion is real. Space and motion are real. You really can move through 3D space. But you can't move through 4D Minkowski spacetime because motion is precluded in an all-times static view. It's a "block universe", and in terms of reality, IMHO it isn't on a par with space and motion through it.

    Agreed. It's a model.

    No probs.

    Yes. We can diffract electrons. "The Dirac equation is a relativistic quantum mechanical wave equation", "Schrödinger decided to find a proper wave equation for the electron", and so on. See this bit from the wiki article on QFT:

    In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta – necessarily chunked ripples in a field, or "excitations", that 'look like' particles. Fermions, like the electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and field", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta).

    What's a ripple in a field? A wave.

    I'm not pulling your leg about this. Remember that the Einstein–de Haas effect demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics. Take a look at at an old version of the Stern-Gerlach article on wiki and see the bit that says:

    "If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 2.8 nm (the classical electron radius), its surface would have to be rotating at 2.3×10^11 m/s. The speed of rotation at the surface would be in excess of the speed of light, 2.998×10^8 m/s, and is thus impossible.[2] Instead, the spin angular momentum is a purely quantum mechanical phenomenon".

    So a real experiment tells us that spin angular momentum is like ordinary angular momentum, and some guy's billiard-ball spinning-like-a-planet inference says it isn't. I'm afraid it's the wrong inference.

    The dimensionality has been taken out because it's dealing with quasi-spherical harmonics and ratios. Just go with the usual numbers, it isn't some numerological trick:
    4π = 12.566370
    c = 299792458
    c^½ = 17314.5158177
    4π / c^1½ = 12.566370 / (299792458 * 17314.5158177)
    λ = 2.420910 x 10ˉ¹² m
    The actual λ is 2.426310 x 10ˉ¹² m, the slight difference being due to binding energy. This isn't my work, it's by a guy called Andrew Worsley.

    Mere convention. That's how you usually see the Lorentz factor written. See for example the twins section in the wiki article on proper time.

    No problem.

    That's what you call it, but look closely, there is no actual proper time flowing in this scenario, just light moving.

    I don't see any issue with that. Their clocks show different readings, the stay-at-home twin has more grey hairs, etc. The "paradox" part of the twins paradox is the symmetry wherein each twin sees the other one's clocks going slower, this symmetry being broken by the turnaround. I don't understand why you think I misunderstand this. The important issue is the invariance and why it's there. I could send rpenner and alphanumeric out and back on different journeys, travelling slower than you. When we all meet up everybody's light path is the same length.

    No problem.

    Again I can't see why you think I'm showing misunderstanding.

    OK noted.
     
  12. Farsight

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    I was going to respond to it, and then you came up with another abusive post and I thought Why bother? It'll only disrupt the thread. OK, OK I'll respond to it. But later.
     
  13. prometheus viva voce! Registered Senior Member

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    I have a big problem with this analogy. If I burn myself I can run cold water over the affected area and it reduces the heat and prevents a serious burn. Time is something altogether different from this.

    At this point I'd like to tell you what time is, but there is a problem here in that I really am not completely sure about it. One of the first thing one learns in physics is that cause must precede effect, but this assumes I have time and we move through time in a linear fashion (we cannot go backwards or "sideways.") which we call the "flow of time." Since you're assuming time exists at the first stroke it's quite awkward to define it, although at least two decent definitions exist - the first is using the second law of thermodynamics which I find a little bit contrived. The second and my favourite one is that time is a dimension where spacetime is mathematically represented as a manifold with pseudo Riemannian signature.

    Just because I define time in this way doesn't mean there is no motion, or that everything is static. Motion is when some object has non zero speed, or that \(\frac{dx}{dt} \neq 0\). I can make life as complicated for myself as I like, since any manifold that satisfies the Einstein equations \(R_{ab} + \frac{1}{2}R g_{ab} = T_{ab} \) is a legitimate spacetime, and in general the universe will be composed of "patches" where particular solutions to the Einstein field equations will be a good approximation to what is observed (for example, the Schwarzschild metric is a good approximation when we are close to the sun, but not when we are in a cloud of gas and dust.). I can put my object into any spacetime I like and it will have some non trivial motion in general

    In summary, I think you are going to have a very hard time coming up with an analogy that does justice to time.

    The point of bringing up the Casimir effect is that it is a non trivial effect that is a result of something we cannot measure directly (virtual particles in quantum field theory).

    Now you're telling me 3d space is real again when I thought we'd just agreed Euclidean and Minkowski spaces are both models that are used in an effort to understand what we observe. I will repeat: It turns out that 4d spacetime is a more accurate model as it better predicts what we observe at high energies while agreeing with Newton at low energy so we say relativity is "right."

    Good, I can see what the problem is here now. You say the Dirac equation is a "relativistic quantum mechanical wave equation" which it is, but one can show that when one treats the Dirac equation as an equation for a single electron in the same way as one would consider a low energy spinless quantum particle using the Schrodinger equation then we run into a problem: With the Schrodinger equation you can show that the energy of a state will always be positive (this comes from the fact that the Schrodinger equation is essentially a restatement of the non relativistic \(E = T + V\).). The energy of a solution to the Dirac equation is not positive definite which is a problem as energy certainly should be positive definite. The solution to this conundrum is to treat the Dirac equation as a field equation rather than a wave equation for a quantum particle, which is what your quote about QFT is saying: particles are excitations of a quantum field. This is not to say that particles are made of waves. That implies to me you are trying to say particles are classical objects which they are clearly not.

    So you're saying that particles are little balls that spin faster than the speed of light?! If you believe that relativity is correct this picture of particles with spin simply cannot be so. As I tried to say before, the Stern Gerlach experiment tells us that quantum particles with spin do not behave like classical particles with spin: With the former you see particles deflected to a finite number of points (2 points for electrons) and for classical particles you would expect to see a continuum of deflections.

    I'm still rather suspicious of this. I did a little research into Andrew Worsley. He appears to be a medical doctor. I found this thread and this essay by him. I found a couple of errors in understanding straight away and I find his use of the title Dr. rather disingenuous because he isn't clear from what he writes that he doesn't have a PhD in physics. We should probably not discuss this guy without him being around though.

    Quite. That was more of a nitpick really, but I would not object to seeing \(\gamma = \frac{1}{\sqrt{1-v^2}}\)


    Again, the light will have a velocity that is given by \(c = \frac{dx}{dt}\).

    What do you mean by their light paths? If you mean their spacetime intervals they are categorically not the same.

    Let's leave length contraction for another time. I would rather see your explanation as to why a lattice of points is a good model of Lorentz transformations.
     
  14. prometheus viva voce! Registered Senior Member

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    Another thing: Talking about a quantum mechanical wave equation is fine as long as you understand that it describes a quantum particle and not a wave. The wavefunction, which is the solution to the wave equation is essentially a measure of probability. For example, suppose I have a low energy spinless quantum particle and I stick it in a quadratic confining potential. The result will be the quantum mechanical harmonic oscillator and the solutions to the Schrodinger equation are well known: \(\psi_n \sim e^{-\frac{m \omega x}{2 \hbar}} H_n \left( \sqrt{\frac{ m \omega}{\hbar}} x\right) \quad n = 1,2,3,\ldots \)

    Plotting the lowest wavefunctions:

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    Which are obviously wavelike. In general a wavefunction will be given by a normalised linear combination of these eigenfunctions: \(\Psi = c_1 \psi_1 + c_2 \psi _2 + \ldots \). If I make a measurement of a particle in this potential I will not see a wave, I will see a particle with some particular position and momentum (within the limits of the uncertainty principle). In fact, quantum mechanics states that when I make a measurement the system goes from being in the state \(\Psi\) to being in an eigenstate \(\psi_n\) and the eigenvalues of this eigenfunction are what is observed, for example the momentum of the particle is given by the eigenvalues of the momentum operator \(\hat{p} \psi_n = i \hbar \frac{\partial}{\partial x}\psi_n\) and the energy is given by the eigenvalue of the Hamiltonian \(\hat{H} \psi_n \) which you can show in this case is \(\left(n+\frac{1}{2}\right)\hbar \omega \)
     
  15. Farsight

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    Sorry not to have been around much. We're having the house rewired and we had new windows this week. I've been having to do work on the house, all sorts of stuff.
     
  16. BWE1 Rulers are for measuring. Registered Senior Member

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    Wait, what? I think you don't mean to say that there are no continuums, commonly referred to as dimensions do you? I think you add axes or dimensions and plot your points. It's been a while so maybe I just don't understand what you are saying.
     
  17. Farsight

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    Next time check what Minkowski said before you criticise me for repeating it.

    This is abuse rather than informed discussion.

    I pay close attention.

    You've explained nothing to me. And nobody is buying string theory, so spare me the lecture.

    More abuse. No discussion of the topic as yet.

    And so it goes on.

    I know. You're brimming with featherspitting outrage and abuse, and you're a moderator. Quis custodiet ipsos custodes?

    Still no discussion of the topic.

    Ditto.

    Oh shut up. I've read the rest of your post. It's your usual ranting and raving diatribe with no discussion of the physics. Let's see what you said in your next post:


    Right, it's static. So there's no motion through it. So a gravitational wave moving through space at c is represented by a static bulge angling up through Minkowski spacetime.

    Check back through the thread. Prometheus referred to Euclidean space repeatedly on page 1 while I referred to real space. In response to him to get the point across I said consider dynamical Euclidean space with waves running through it. When we're talking gravity there's no curvature innate in this space like there is in curved spacetime, instead there's inhomogeneity that causes curved motion through it. This merely reiterates what Einstein said:

    "This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic...".

    Minkowski spacetime is definitely static, real space definitely has waves moving through it.

    Now you're confusing space and spacetime. How many times do I have to make it clear that there is no motion through spacetime, there's only motion through space.

    You're descending into outraged ranting raving abuse again.

    Ditto.

    Ditto.

    Prometheus, you're a moderator, do your job. I'm not wasting any more time on this guy.
     
  18. Farsight

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    I'm not clear on your double negative there. I was just using Minkowski's language from space and time. He talked about space points and time points.
     
  19. Me-Ki-Gal Banned Banned

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    Really ? How old is your house ?
     
  20. Farsight

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    Yes it's different. When you measure temperature in the ideal sense you're measuring an average kinetic energy, an average motion as it were. You can reduce this with the cold water. You can't similarly reduce the cumulative measure of motion displayed by a clock.

    Just look at whata clock actually measures. It doesn't matter whether its a quartz wristwatch clocking up the oscillations of a crystal, a pendulum clock, a parallel-mirror light clock, or the NIST caesium clock, there's always some kind of regular motion being clocked up and displayed as "the time".

    It might be the first thing you learn, and I have no issue with cause and effect, but this thing about moving through time is just a figure of speech. We don't really move through time and time doesn't really flow. We just move through space like everything else, and while we're doing it other, local motion through space is clocked up by clocks and displayed as the time. That's all there is to it. A World Without Time: The Forgotten Legacy of Godel and Einstein is worth reading.

    It's more concerned with entropy I suppose. See the bit on wiki that says In classical thermodynamics, the second law is a basic postulate applicable to any system involving measurable heat transfer. That heat transfer is essentially motion transfer, from something moving rapidly to something moving slower, and as a result the motion tends to even out.

    IMHO that "defines" it in terms of abstract things and doesn't help you to see the importance of motion.

    But again that motion is through space, not through spacetime. This is a crucial point.

    It's a big subject in its own right. The important thing is to focus on what clocks actually do.

    OK.

    I champion relativity, I don't challenge it. I'm just trying to get you to appreciate the distinction between real space with waves moving through it and Minkowski spacetime which is an all-times view. We draw worldlines in it to represent the motion of a particle through space over time, but the particle doesn't move through Minkowski spacetime.

    Agreed. Actually I got the impression that Dirac got a bit too much credit for the positron, but again that's one for another day.

    You're putting up too much resistance to this wave thing. There's not much difference between a wave and a field, see for example Evanescent wave. A wave such as an electromagnetic wave is typically some "travelling excitation", a field such as an electromagnetic field is a "standing excitation". But you can still diffract electrons.

    Just accept the wave nature of matter. That's all you have to do. It's nothing revolutionary.

    Hell no! No way are particles little balls. I've been saying they're waves remember. Electromagnetic waves propagate at c. But they relate to displacement current, which is a time-varying electric field.. Try to envisage what you end up with if one wave propagating at c moves through a time-varying electric field. What happens if a wave moves through itself?

    The baggage that comes with Stern-Gerlach is little balls spinning like a planet, which is where that wrong inference comes from.

    Nobody's perfect, but don't forget those expressions.

    No problem.

    You define your time using the local motion of light. And your distance. So you always measure \(c = \frac{dx}{dt}\) to be the same old value.

    I mean their light path lengths. You and I are the twins with identical parallel-mirror light clocks. It's January 1st 2011. I stay on earth whilst you take a trip out into space and back, and we meet back up on January 1st 2012. We compare notes and reason that my light has been moving like this || whilst yours has been moving like this /\/\/\/\. We get our calculators out and we work out that the total length of each of our light paths was one light year. And our spacetime intervals are the same. Whilst you were travelling, you measured space and time different to me. You were subject to time dilation and length contraction, you didn't see your light as zigzagging at the time, and by your calendar it's only December 1st 2011. But the spacetime interval between the event of our separation and the event of our meeting back up was invariant. Because our light path lengths were the same.

    I said when you're made of waves you always measure wave speed to be the same, and that this underlies Lorentz symmetry. But what else would you like me to talk about? I think it might have to be length contraction. Have you ever thought about why you see things as looking shorter when you're travelling fast towards them? You see a star looking flattened, like a Babybel cheese. Then you go even faster and it looks like a pancake. But you know that you can't really make a massive star instantly change shape just by putting the pedal to the metal. So if the star didn't change, who did? Think about it.
     
  21. Farsight

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    It was built in 1958 I think. Crittal windows, yellow and black bathroom tiles, boiler on the floor in the kitchen, that kind of thing.
     
  22. Farsight

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    That's wave function and wave equation. And you can still diffract electrons.

    That's what they say. See wiki and the bit in bold:

    A wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time. The wave function behaves qualitatively like other waves, like water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation.

    If you say so. Sorry, what's a spinless particle?

    Harmonics. Do you know much about spherical harmonics?

    You're measuring like with like. There is no point particle in there following some mystical laws of probability.

    If you say so. It's late, I'm too tired to look this up. Have a read of this old wiki HUP article, maybe it'll help:

    http://en.wikipedia.org/w/index.php?title=Uncertainty_principle&oldid=388452122

    "The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum). Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there can be no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum".
     
  23. Me-Ki-Gal Banned Banned

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