Bruce's 'Baffled' SO4 Thread.

.Continuation of a discussion in the 'electron-assisted fusion thread' At the request of @Bruce-H. Alan.

.....Thanks for the answer.

I am uncomfortable with the scheme for the right-hand-rule/cross-product listed in my last post. This scheme seems to depend on an ordering of the spatial dimensions such that, for instance, (X1, X2)-> X3 rather than X4. But the labelling of such axes with indices should be is arbitrary and I should be able to swap the labels on X3 and X4 without affecting the cross product. Yet it is, indeed, affected. Swapping the X3 and X4 labels gives (X1, X2)-> X4, which is not part of the scheme I listed above.

In brief, my objection is that you can't have a cyclic scheme such as you suggest unless you have an a priori ordering of spatial directions ... and I don't see where such an ordering comes from. This is why you can't have a right-hand rule in 4-space as far as I know.

To get back to Figure 1 of your ResearchGate paper once again, what direction does the magnetic field point for the current loop in the x,y plane? Is it in the u direction? The v direction?

Quote from Wyttenbach

Just think about the induced Lorenz action. Only the charge flux orthogonal to the field vector is forced on a cricle like orbit (3D space!).

Of course there is no strict right hand rule if you can have 3 right hands...I can also start with any two dimensions with a ring current and the irony is. There in reality = proton is no ring current at all...

But in mathematics you just define an order relation and then stick to it. As we here see a ring structure it is clear what happens. The complete ring as said is shown for the 3D/4D flux.

I am unsure what you mean exactly. Let's take the current loop in the x,y plane shown in your Figure 1. What direction does the magnetic field of that loop point?

Quote from Wyttenbach

Bruce I gave you enough explantions either you know the Cyclotron equation and it's relation to the Lorenz force or not.

I think you best come back, when you understand why the concept (It's definition) of topological charge in the standard model is nonsense and why it works in my SO(4) sample. You can show us your explanation!

If you are not able to handle some basic topology and the related SM concepts it makes no sense to go on.

I do understand the Lorenz force and the Cyclotron equation.

You probably want to deal with the most important parts of your theory, but since am a beginner at it I naturally want to begin at the beginning! So I ask a very basic question about Figure 1 in your ResearchGate paper. Can't you point out which direction a labeled vector ("B") points? This may be elementary to you but it confuses me greatly because I don't see how you can have an established direction for this vector in 4D.

What is the point of displaying the diagram in your paper if its most basic features are confusing? I don't think I am alone in failing to understand it.

We should distinguish here between natural laws and initial/boundary conditions. I have asked a very basic question about how the Biot-Savart law is supposed to work in 4 dimensions and you have replied that if a particular structure is cyclic (and thus stable) it doesn't matter. But your answer is about a particular set of conditions, not about the law. The laws should be the same no matter what the initial/boundary conditions are.

You are positing that current flow results a magnetic field according to Biot-Savart. Good. In 3D I understand how this works ... given the magnitude and direction of the current past a point, and the direction of the displacement between the current and the point where you measure the field, the resultant field is perpendicular to both directions. In 3D there is only one direction perpendicular to any 2 independent directions so there is no problem identifying the direction the magnetic field is supposed to point. In 4 dimensions, however, there is an entire plane of perpendicular directions, so you can't identify a unique direction for the magnetic field. And ordering the various dimensions "1,2,3, and 4" doesn't solve the problem because that ordering is arbitrary and another person may order them differently, which would result in a completely different direction for the physical field. That can't be.

It is a big stumbling block that you can't or won't identify what direction the magnetic field points for either of the current loops in the figure you published.

Quote from RobertBryant

you need to meditate Bruce.. start with 1111..

the silverspoon is not free

I will make my question more specific. Take the left-hand current loop in the figure you posted (Figure 1 from Wyttenbach's ResearchGate paper). What direction does its magnetic field point?

Quote from Alan Smith

As far as dwellers in the 3-D world are concerned, everything in the $D world is effectively imaginary.

I'm not sure what this is meant to convey. 4D structures can be projected into 3D. They are not so much imaginary as distorted. But the distortions are known.

Quote from Alan Smith

Demanding to know which particular direction the field points in is actually not helpful.

On the contrary. As far as I can make out, at least some of Wyttenbach's schemes depend on it. I think that he still believes that the magnetic field is a vector field, i.e., has a single direction at each point in space. This is why he says that all action is 2->1, the "1" is a uniquely valued vector that corresponds to a force in some direction. For instance, because he wants to derive everything from electromagnetism, Wyttenbach needs some form of the Biot-Savart law that specifies the field due to a moving charge. But in 4 dimensions this law doesn't work because no unique vector field can be specified (because there is no right hand rule and no cross products in 4 dimensions).

Quote from Alan Smith

Existence is enough - not everything can be labelled with an ordinate.

I am saying that some of the structures Wyttenbach relies on (like a magnetic vector field resulting from a Biot-Savart type relation) don't exist in 4 dimensions.

Higher dimensions is not a license to say that anything goes.

Quote from Wyttenbach

Of course we have a cross product everywhere we like to have it! You just have to give the order for the transport!.

This is still problematic for me. When you talk about the "order for the transport" I think that you are talking about the cross product depending on an arbitrary choice of order for the 4 spatial dimensions. But this would result in different physical outcomes for different choices (not just different calculational outcomes depending on a choice of reference frames -- different physical outcomes). In general this should not be allowed in a physical theory.

I am still trying to clarify for myself what you mean in all this. So let me pose the following question ...

Suppose you have a sheet of paper sitting on the table in front of you and on it you have drawn 2 orthogonal axes that you have labelled "X1" and "X2". Call "X3" the vertical axis, the one that is orthogonal to both X1 and X2 in our usual 3-dimensional world. Finally, let's posit a 4 dimensional space with an axis, "X4", orthogonal to all of X1, X2, and X3. In this example, what is the cross product of X3 and X4?

Quote from Alan Smith

The usual notation is X and Y are the horizontal axes, and Z the vertical one. At least, they are in engineering.

Yup! That is absolutely fine too. I was aligning myself with a notation that Wyttenbach used earlier (post #717, here)

Quote from Wyttenbach

You still don't get what cyclic means 1,2,3 2,3,4 3,4,1, 4,1,2. The good question would be about the sign....

You're right about the sign. But I will leave that for later. For now I'll work on which axis the cross product of X3 and X4 points along.

I think, from your answer, that you calculate cross product of X3 and X4 to lie in the X1, not the X2 direction.

Let's use that cross product rule to figure out the magnetic field for a charge sited at some distance from the piece of paper in the X3 direction and moving in the X4 direction. The Biot-Savart law predicts that the magnetic field created by this moving charge is aligned along the X1 axis (according to the X3, X4 -> X1 cross product scheme). You, Wyttenbach, can now test this prediction operationally in the same way that you would test any field - by using a small magnet or a charged particle or what have you and observing how it is deflected by the field. Let's say that you do this and find that yes, the magnetic field due to the moving charge lies along the X1 axis.

But now here is my problem with your procedure. Before you drew your X1 and X2 axes on your piece of paper I asked Alan Smith to draw 2 axes on another piece of paper. On comparing his axes and yours I find that they lie in identical directions except that the one you labelled X1 he has labelled X2, and the one you labelled X2 he has labelled X1. Your axis labels are therefore switched. Now let's say that Alan, too, has a placed a charge at the same X3 position as you and also caused it to move in the X4 direction, just as you did. The physical set up so far is identical to yours. Alan also uses the Biot-Savart rule to predict the magnetic field from the moving charge and he too uses the same cross-product rule you have recommended, i.e., X3, X4 -> X1. He therefore predicts the field to lie along his X1 axis. But since his X1 is your X2, he doesn't actually expect the field to be in the same direction you predict. He thinks it should lie along what you call the X2 direction. He can check out his prediction operationally in the same way you did by looking at the deflection of a moving charge or a magnet. Doesn't he expect to see the test charge or magnet move in a physically different direction from the one you predict? But the experiments were identical. How can this be?