Lattices and Lorentz invariance

Voidset: I can't see how tetration spaces make sense. Not physical sense, anyway. Maybe you should try something simpler, such as a 3D lattice of placeholders, wherein each placeholder is small set of 6 placeholders arranged in 3 orthogonal pairs. Send a wave rippling through them, and read up on Maxwell.

 

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A lattice (I presume you really mean a 4d lattice to include time) breaks Lorentz invariance so it's not really that much good to use it to think about electrodynamics.

 

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I didn't mean a 4D lattice including time, I meant a 3D lattice with \(A^{\alpha}\) waves running through it. Call them something else if you prefer, and yes, there is more to it, but the "wave nature of matter" means you and your clocks are composed of these waves, and as a result you always measure wave speed to be the same. That's essentially why Lorentz symmetry applies.

 

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Firstly it makes no sense whatsoever to talk about the Lorentz group in 3 dimensions not including time, because the Lorentz group is exactly the isometry group of Minkowski space SO(3,1). The Lorentz group contains rotations and boosts, which you can consider as rotations that mix space and time. If we look at your "reduced Lorentz group" in 3 spatial dimensions we're simply talking about the group of rotations in 3 dimensions SO(3). A lattice of points is quite obviously not invariant under rotations so Lorentz symmetry most certainly does not apply.

 

I didn't, you're misreading what I said. The lattice represents 3D space. It is dynamical, with waves moving through it. You can derive Minkowski spacetime from this, but remember that there are are no waves running through Minkowski spacetime because it's a static all-times representation. I mentioned Lorentz symmetry, which is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". The important point to appreciate about that is that when you and your clocks and everything else is made of waves, you always measure wave speed to be the same. Think it through.

 

So lets be clear - you're talking about a three dimensional array of points (the lattice sites) that sit at the corners of a cubic honeycomb. These points have some sort of dynamical properties that allows waves to propagate through the structure. Is that what you're saying?

I would be very interested to see this derivation.

Minkowski space is static in the same way that 3 dimensional Euclidean space is static. You can put a field into either of these spaces and solve the equations of motion and get waves.

That's exactly the point - your 3d lattice is not independent of orientation because an arbitrary rotation will change the angle of the line joining lattice sites (and since we aren't talking about Minkowski space you can't do a boost at all).

I don't know about your "clocks and everything else," but most of my stuff is made of particles.

 

Yes. But it's the lattice that has the dynamical properties rather than the intersection points, and it's just a visualization tool anyway. Space isn't literally made out of a lattice.

Read Minkowski's Space and Time. A point on the lattice is a space-point. You add a time point to give yourself four coordinates x y z t and then work out spacetime intervals. Standard stuff.

Minkowski space is static, but real space isn't. Waves move through it. There are no waves moving through Minkowski spacetime. Like I said, it's an all-time view. You plot the motion of a wave moving through space as a worldline in Minkowski spacetime.

And a rotation will alter the apparent direction of wave propagation or polarization. That's what happens in real space.

You just move in the x direction and adopt a new coodinate system, that's all. You seem to be thinking of space as Minkowski spacetime here.

And those particles are made of waves. We can perform Electron diffraction, see wiki. Read the first sentence: Electron diffraction refers to the wave nature of electrons.

 

This is perhaps a minor point but surely if the links are what you are calling dynamical, then the sites will also be in motion, and in fact the motion of the sites will be equivalent to the motion of the links?

Quite, but I don't have a copy of that book so if you could provide an example of the calculation I'd be grateful.

Ok, I grant you that "real" space is not flat but it is locally flat, meaning that I can define an observer that experiences no gravitational forces. All this is one reason why physics in flat space works so well.

I must say I'm rather confused by your remarks about waves moving though space. I am free to put any field I like in Minkowski space and it will have whatever solutions it has. If you are talking about waves of space then you are talking about gravitational waves, although I rather doubt you are. Remember this forum is for mainstream physics and maths, not for your personal ideas about physics as they are not accepted or tested in experiment.

Yes, I am aware that a vector is not invariant under rotations (it is of course covariant, meaning rotations preserve the magnitude of a vector but not direction).

We aren't talking about vectors or the direction of some wave, we are talking about Lorentz invariance. The point is that using a lattice to talk about the Lorentz invariance of Maxwell's equations is not a smart thing to do, because while spacetime is Lorentz invariant, your lattice is not Lorentz invariant.

I get the feeling this is going to go badly, but here goes:

Lorentz invariance means that a particular system is invariant under the action of the Lorentz group. The Lorentz group is SO(3,1) and consists of rotations in the three spatial directions, and boosts which you can think of as a rotation between a spatial direction (it could be an arbitrary direction in x,y and z coordinates) and the time direction.

That is the Lorentz group. Now, it was shown by Coleman and Mandula that the largest symmetry group a physical system can have is the Poincare group times an internal gauge group (Let's ignore the gauge group as it isn't relevant for this discussion). The Poincare group contains the Lorentz group and translations in space and time. What you are describing is not a boost, it is a translation.

In summary, if I don't have a time direction (in other words I am considering Euclidean space) then I cannot perform a boost, as that is a rotation in one space and one time direction. I can however perform a spatial translation which is not in the Lorentz group, although it is a symmetry of Euclidean and Minkowski space.

Let me repeat: this forum is for mainstream physics and maths, not for your personal ideas about physics as they are not accepted or tested in experiment. An electron is not "made of waves," it is a particle that is an excitation of a quantum field.

 

Mod note: I have moved these posts from the thread do tetration spaces make sense? as that thread has now been moved to the cesspool.

 

Yes, no problem.

There's no calculation per se, just Minkowski developing his expressions. See this url and go to page 70.

We describe curved spacetime as being locally flat in an infinitesimal region when talking about gravity. But as previously, spacetime isn't the same thing as space. When a wave travels through space and follows a curved path, we talk about curved spacetime.

I'm not talking about anything speculative or personal here. Electromagnetic waves move through space, gravitational waves move through space, and so on. Have a look at this for mention of a lattice.

You're labouring on this and making things difficult for yourself. I mentioned Lorentz symmetry and why we see it, that's all.

You move, and you keep moving, so now you measure time and space different to me.

The time "direction" isn't the same thing as space directions. You have freedom of motion in space but not time.

Again, I'm not promoting any personal ideas. See electron diffraction:

"Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon is commonly known as the wave-particle duality, which states that the behavior of a particle of matter (in this case the incident electron) can be described by a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction".

Electron diffraction is accepted and it is tested by experiment. We fire electrons at a sample and observe the resultant interference pattern. Please do not think that everything that is unfamiliar to you is some kind of personal notion of mine. I don't make these things up.

 

So when you said "you can derive Minkowski spacetime from this," it wasn't strictly true then?

I don't particularly want to get into the whole "spacetime vs space" thing with you again because I'm pretty sure I elucidated the difference last time. It's off topic anyway.

For starters, another forum is hardly a compelling reference. Secondly QCD is studied on the lattice because it's the only way to compute quantities at strong coupling. There is a lot of machinery in lattice QCD involved with restoring Lorentz invariance - Taking the continuum limit is something that must be done to extract physical results too. You suggested that using a lattice was a good way to understand Lorentz invariance, which it is not.

On the contrary, I'm finding this thread to be one of the easier ones to understand today.

I knew this was a bad idea. Are you telling me that after all the years of proclaiming yourself to be an expert in relativity you don't know the difference between a translation and a boost?!

Here is the difference (the long version): Suppose I have an observer O with coordinates \((t,x,y,z)\). Now suppose I have a second observer O' with coordinates \((t',x',y',z')\) and \(t' = t + \Delta t,\, x' = x + \Delta x, \, y'= y + \Delta y,\, z' = z + \Delta z\) where all the \(\Delta \) indicate a shift by a constant. The coordinate systems of O and O' are said to be related by a translation.

Now suppose I have the same observer O but a third observer O'' who is moving in the x direction with some velocity v. Their coordinate systems are related by a boost, which is \(t'' = \gamma \left(t - \frac{vx}{c^2} \right), \, x'' = \gamma \left(x - vt \right), \, y'' = y, z'' = z\) (we can of course have a boost in an arbitrary direction, but I'm not that patient with tex.).

The difference is that the two observers related by a translation are the same "distance" apart, while those related by a boost are moving at a speed v with respect to each other. Please please tell me now that you understand the difference.

On a slightly tangential note, and I believe I have mentioned this before, you seem to behave like a chatbot in that you latch one to a particular word or phrase and just come out with your standard response regardless of what I actually said. The response to what you wrote is "I am well aware of the fact that the time direction is different to space directions, and specifically it is the fact that in the metric tensor it is of the opposite sign to the space directions."

Which is completely irrelevant to my point which was, if I consider Euclidean space rather than Minkowski space boosts are no longer a symmetry and the Lorentz group is replaced with the rotation group.

I'm not questioning electron diffraction or the wave nature of electrons. What I am questioning (actually, what I'm pointing out is wrong) is your statement that electrons are made of waves, which they are not. The standard quantum field theoretic description of electrons and other particles does a perfectly good job of describing their particle and wavelike behaviour.

 

That was certainly helpful to me! I have forgotten so much that I can't say how long I will retain it, but it was a good example and explanation from where I sit.

 

I know! I'm not going to deny having to look things up for this thread.

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Summary
Farsight: A three-dimensional, regular, repeating pattern of nodes in space is a good mental model to use when thinking about Minkowski space-time and Lorentz invariance.

prometheus: How can that be when Lorentz invariance is a statement that there are no preferred directions and no state of inertial motion is singled out as a preferred standard state of rest, while a lattice always has preferred directions and a cubic lattice necessarily singles out a preferred state of rest which is the only state where the angles are right angles and all the cell lengths are the same?

Farsight: Please allow me to change the subject.

prometheus: Certainly you are free to move on to other subjects, after you admit that not only is your claim non-obvious and not in the scientific literature but that no-one should expect Farsight to support his own claims because he apparently feels free to make them wily-nily without understanding the subject matter. In the alternative case that this is an uncharitable and unjustified reading of Farsight's understanding, thinking and practice, I shall provide Farsight every opportunity to retract utterly such a senseless claim so long as he does it explicitly and an opportunity to attempt to demonstrate it should he just require more time to think about the claim and why it makes no sense.

 

When you read an undergraduate anything.

No, you don't 'add a time point', you add an additional dimension to your manifold and thus obtain an additional coordinate. You also have to stipulate the new metric components in order to obtain a space-time interval. A space-time interval is not defined purely by the underlying space, it also requires the metric. \((\mathbb{R}^{4},\delta_{ab})\) is Newtonian space-time, \((\mathbb{R}^{4},\eta_{ab})\) is Lorentzian/Minkowski space-time. Standard stuff.

Are you referring to gravitational waves or things like electromagnetic waves? The former fine, but the latter applies to both 'real space' and Minkowski space-time. Both are arenas within which events and objects, including waves, can exist.

So is a general space-time describing the real space-time we find ourselves in. The 4 dimensional 'space' covers everywhere ever.

Farsight, when you have actually done some special relativity perhaps you'll be in a position to say such things in a capacity more than just "This is what some book says so I'll repeat it". Maybe one day you will read a book someone suggests and actually gain a working understanding of this stuff.

You seem to be not thinking properly about what a lattice in space (or space-time) would mean for Lorentz transforms.

The effect of introducing a lattice into your considerations is really quite interesting. In fact requiring such a periodicity, up to moduli deformations, is precisely how you construct orbifolds in the compact dimensions of string theory. Funny how that area of mainstream work you think is so useless keeps coming up with relevant and interesting things. Which is more than can be said for you.

For someone who claims to have in sight you're quite narrow in your vision. Why should the subatomic be expressible in terms of things we experience in every day life? Why must the subatomic objects be either a particle or a wave? What's wrong with a third construct, which has a mixture of properties (and some new ones)? Our language developed long before we examined quantum systems, it's not surprising we don't have a name for such things and hence the whole misguided "Well, is it a wave or a particle?!" argument.

Well done on stating one of the defining properties of a manifold. Got any other shiners you want to spit out?

Depends on what you mean by 'curved' and 'space'. As has been previously illustrated to you, by Prom specifically, how you divide up space and time in space-time is an arbitrary choice and a curved space-time can be split into a flat space and a curved time or vice versa. Plus there's multiple meanings of 'curved'. Black hole space-times have non-zero Riemann curvature but have zero Ricci curvature. FRW metrics in inflation have zero Ricci curvature too, despite involving extreme space-time warping. Calabi-Yau manifolds have zero Ricci tensor curvature too.

It's important to be precise with your definitions and until you learn the basic mathematical formalism of relativity you're always going to be stuck waving your arms and failing to be precise.

One person's motion through time is another person's motion through time and space. They aren't exactly the same but there's no clear cut division between 'the time direction' and the rest.

As he has for years.

 

A rubbish summary. I didn't say that at all. Read the thread. What I actually said was I mentioned Lorentz symmetry, which is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". The important point to appreciate about that is that when you and your clocks and everything else is made of waves, you always measure wave speed to be the same.

Nope. It was prometheus who changed the subject from waves in space and confused space with spacetime.

You do come out with some abusive garbage, rpenner. Join the discussion and have the guts and honesty to make a sincere contribution instead of sniping from the sides.

 

Take that up with Minkowski. Let me quote him from Space and Time: I will call a space-point plus a time-point i.e, a system of values x,y,z,t, as a world point.

Yep, standard stuff. You might try actually reading Space and Time before commenting next time.

Both. The important point is that waves move through real space, but they do not move through Minkowski spacetime. There's no motion going on in it.

And every when. Let me give you an analogy: real space and time is where we can watch a red ball moving from left to right. Minkowski spacetime is like filming it happening, then cutting up the frames and stacking them up to make a block, with a static red column angling up through it.

Try to refrain from abuse, Alphanumeric.

It's a lattice in space, not spacetime. If it was a lattice in spacetime, a wave moving through space would be represented by a static columnar bulge angling up through it.

I do think the failure of string theory is rather sad. It's as if a whole generation of theoretical physicists missed the turnoff and wasted decades going round in circles in the desert before they wised up. But let's talk about that on another thread.

Aren't there any moderators on this forum to prevent this sort of abuse? Let's see now, who's a moderator here?

I'd say because the public demand it, and the public pay for it. I think there's a growing mood that theoretical physicists has not delivered in recent decades, and that funding pressures will grow. To try and head this off, IMHO physics has to give them something they understand at least. If it doesn't deliver benefit or understanding, I fear the public will write off physics as erudite elitist mysticism. And that ain't going to happen. Not on my watch.

There's no real issue with particles, just point particles. You can say the electron is a particle and a wave, like the sister-daughter exchange in Chinatown. And you can talk about spherical waves and spinors. No problem. The problem comes when you take a surpasseth all human understanding line. See above.

I said We describe curved spacetime as being locally flat in an infinitesimal region when talking about gravity in response to prometheus saying Ok, I grant you that "real" space is not flat but it is locally flat, meaning that I can define an observer that experiences no gravitational forces. He demonstrated his confusion between space and spacetime again. Back me up when I correct him.

Prom doesn't even know the difference between space and spacetime, so spare me the partisan lecture. Have the courage to spit it out and tell prometheus that actually Farsight is right about this or that. If you don't when he does find out that I'm right, he'll start asking awkward questions. BY the way, 'curved time' is an abstraction. Start a thread an I'll explain what clocks actually measure.

Do they? Start a thread and let's talk about it.

Yes, it's important to be precise with your definition. And it's especially important to know the difference between space and spacetime.

That's wrong I'm afraid. There is no actual motion through time. It's merely a figure of speech, an abstraction. Things move through space, we can see it happening, and we have freedom of motion through space. But we cannot say the same for time. If I show you a motionless body in space, would you really insist that it is indeed moving? When I challenge you and say but it's motionless would you really retort Ah, but it's moving through time. I think not. As you said, it's important to be precise with your definitions.

The time direction is only an abstract direction I'm afraid. Here, I've just moved one metre in the x direction in space, then back again. Now, you try doing the same in the t direction.

Edit: sadly, I have to go. I'll catch you later.

 

What, exactly is wrong with this summary and paraphrase?
3D lattice = a three-dimensional, regular, repeating pattern of nodes in space
a visualization tool = mental model
Lorentz symmetry = Lorentz invariance

 

It's actually the second page 70 (pages 70-80) if you go by the slider at the bottom of the window. It's an English translation of the famous 1908 Minkowski lecture where he states it's stupid to think of space and time as unrelated when geometry can be used to unify them.
Slightly better URL:
http://www.archive.org/stream/theprincipleofre00einsuoft#page/70/mode/2up

This is indeed "standard stuff" but notably makes no mention of any lattice. Minkowski's "world-points" (p 71) are described in the context of analytic geometry, not a lattice.
"we shall suppose that at every place and time, something perceptible exists." (p 71) is not talking about a lattice but about the conceptual difficulty some people have giving a combination of space and time where nothing exists a name.

 

My last proper reply was all the way back in post 11, and since then Farsight has been online and posted a number of times, but has not replied to me. I wonder what the reason for this is...

Since it's been relatively quiet today for me I took the opportunity to glace back through the archives here to find the interactions I've had with farsight in the past, and there are a few things that are relevant to what has happened on this thread.

Inflation and curvature
Where I originally likened farsight to a bot
Is it true? Is the universe flat? - the thread that precedes "Inflation and curvature."
Relativity+ - Farsight's effort at a theory of everything.

I think it's pretty clear the level that farsight is at from this thread and the links given.