Maxwell's Equations in N Dimensions

[Hornbein]

I couldn't finish it, so I paid $35 for Alan Macdonald's Vector and Geometric Calculus. This uses geometric algebra, where vectors may be multiplied together to form bivectors, trivectors, and so forth. They are added together with abandon.

The electric field E is more or less 1D so it is unchanged. In N dimensions there is no such thing as a magnetic pole, so one needs to return to basic special relativity to see what will happen. Magnetism is 2D: the remainder of dimensions are irrelevant. It's plane is defined by three points: the location of the source charge, the location of the source charge after dt, and the location of the affected particle. Magnetism can be represented by a bivector B which defines the plane in which magnetism operates, the magnitude, and the sign. The magnetic force is calculated by taking the inner product of B with the velocity vector of a charged particle. The inner product of a vector with a bivector is the projection of that vector onto the plane rotated 90 degrees, which just so happens to be precisely what we want.

A scalar is grade zero, an ordinary monovector is grade 1, a bivector is grade 2, etc.
In GA gradient = div + curl. Taking the div lowers the grade while the curl increases the grade. Take the time derivative leaves the grade unchanged.

Everything is as usual with the monovector E. Field B is the curl of the charge current potential vector, so it is a bivector. The divergence of a bivector is a vector, while the curl is a trivector. There are no trivector terms on the other side of the equation, so it must be zero.

With no charges,

div E = 0
curl E = -dB/dt
div B = -dE/dt
curl B = 0

The grade of both sides of each of the equations is, in order,
0
2
1
3

That's all there is to it, for any N dimensions.

 

[robphy]

The electric field E is more or less 1D so it is unchanged.

So, in your title, I presume you mean N spatial or N space-time dimensions.
Here, when you refer to the electric field as "1D", you are referring to it being a vector (or covector)---that is, (in tensor notation) a one-index tensor.
And the magnetic field, as you go on to discuss, is a bivector (or a two-form)---that is, an [antisymmetric] two-index tensor. (Only in 3-spatial dimensions, since a bivector has 3 components, one often thinks of a magnetic field as a vector [actually a pseudovector].)

With no charges,

div E = 0
curl E = -dB/dt
div B = -dE/dt
curl B = 0

For the last two equations (Gauss for B and Ampere-Maxwell),
I think you mean
div B = 0
curl B = dE/dt

 

[Hornbein]

So, in your title, I presume you mean N spatial or N space-time dimensions.
Here, when you refer to the electric field as "1D", you are referring to it being a vector (or covector)---that is, (in tensor notation) a one-index tensor.
And the magnetic field, as you go on to discuss, is a bivector (or a two-form)---that is, an [antisymmetric] two-index tensor. (Only in 3-spatial dimensions, since a bivector has 3 components, one often thinks of a magnetic field as a vector [actually a pseudovector].)

A bivector or trivector or whatever is neither antisymmetric nor symmetric. Geometric algebra revolves around the geometric product of multivectors. The geometric product is the sum of a symmetric portion and the antisymmetric portion. Grad is defined as a certain geometric product. The symmetric portion is called div and the antisymmetric portion is curl, so grad = div + curl.

A bivector is grade 2. Div is lower in grade by one so divB is a monovector. Curl increases grade by one so curlB is a trivector. It cannot be equal to the time derivative of E. Taking the time derivative does not change the grade, so dE/dt is a monovector.

For the grades to be consistent and with F = E + B the only choice is

div E = 0
curl E = -dB/dt
div B = -dE/dt
curl B = 0

Isn't it interesting? Macdonald doesn't give the version with charges. I hesitate to add them in because I might get a sign wrong.

Geometric algebra is essentially Clifford algebra. I don't know what the difference is between the two, if any.

 

[robphy]

div E = 0
curl E = -dB/dt
div B = -dE/dt
curl B = 0

Can you interpret each physically?
For example, Which one is ampere's law?

 

[Hornbein]

Can you interpret each physically?
For example, Which one is ampere's law?

To show that it is necessary to add in the effect of charges.

div E = rho
curl E = -dB/dt
div B = -J -dE/dt
curl B = 0

(I'm guessing about the sign of J. )

The third equation here is equivalent to Ampere's law. B is a bivector instead of a monovector. I have multiplied out everything to ensure that it is the same. It is a bit verbose, but an entirely mechanical procedure.

In short, these four equations have exactly the same physical significance as Maxwell's equations. It is just a different formalism that is extendable to N dimensions.

 

[robphy]

page 20 of Macdonald's survey article
https://www.astro.umd.edu/~jph/GAandGC.pdf#page=20
lists the Maxwell equations in standard notation, which is in agreement with my earlier comment

For the last two equations (Gauss for B and Ampere-Maxwell),
I think you mean
div B = 0
curl B = dE/dt

I don't have a copy of the book you have.
Can you transcribe/copy/scan the relevant section from it?
Does it differ from what is written on page 20 in that survey article?
(Maybe there is a clash in the notation of "div" and "curl" applied to "B".)

 

[Hornbein]

page 20 of Macdonald's survey article
https://www.astro.umd.edu/~jph/GAandGC.pdf#page=20
lists the Maxwell equations in standard notation, which is in agreement with my earlier commentI don't have a copy of the book you have.
Can you transcribe/copy/scan the relevant section from it?
Does it differ from what is written on page 20 in that survey article?
(Maybe there is a clash in the notation of "div" and "curl" applied to "B".)

Hmmm, I'll try. My scanner hasn't worked for years. But I swear my transcription is accurate. I checked all the math too, and it works out. I couldn't find an online reference, but that is no surprise. If I could I wouldn't have paid the $35.

The Maxwell's equations you quoted is the traditional one restricted to monovectors and scalars. He's talking about using that as a base for Hestenes' 3+1 D spacetime algebra. I put some effort trying to convert that so it would apply in N spatial dimensions but so far have failed. I don't claim to understand what they are up to. Also, by using the trad Maxwell's they have to do a lot of fiddling around with +- signs, which didn't attract me. It was one of those "do I really want to go to the trouble of understanding this" things. So they get E+B = F = J. So what? The first thing you are going to do is split it back apart again. I want simplification, not another layer of complexity.

The example I quoted has the magnetic field as a bivector. That difference is the source of all other differences. I think it is quite neat that the div and curl operations seem to swap roles when operating on a bivector, a sort of duality. I don't yet understand what that "means."

 

[Hornbein]

(Maybe there is a clash in the notation of "div" and "curl" applied to "B".)

Let b be the trad magnetic field monovector. b is more or less the dual of B, the magnetic field bivector. With v the velocity vector of a charged particle, the magnetic force it sees is v x b = v . B, with . the inner product.

With D the derivative operator, D x b = D . B. The curl of b is everywhere the same as div of B.

3D only, of course.

 

[robphy]

After decoding a little of this formalism, I now see that
your expressions
div B = -J -dE/dt
curl B = 0
correspond to
Ampere's Law (written as a 1-form equation [your grade 1]) [although I have to still see where that minus sign for B comes in]... in other notations, curl b=j+de/dt
Gauss for B (written as a 3-form equation [your grade 3]... in other notations, div b=0, div(curl a)=0 or dda=0).
There's metric and/or constitutive-equation floating around in here somewhere.

I have requested Macdonald's text from interlibrary-loan.
Maybe that will be a good introduction to this geometric calculus formalism (due to Hestenes?).

There's been a so-called pre-metric formulation of electrodynamics (due to van Dantzig)...
I'm not sure how all of these formulations (vector, tensor, differential-form, geometric-calculus, pre-metric) are connected... yet.

Thanks.

 

[Hornbein]

After decoding a little of this formalism, I now see that
your expressions
div B = -J -dE/dt
curl B = 0
correspond to
Ampere's Law (written as a 1-form equation [your grade 1]) [although I have to still see where that minus sign for B comes in]

It's because GA doesn't have the "right hand rule" convention, which of course doesn't extend to N dimensions.

I haven't tackled rho and J yet, I just mashed them in there so the equations would appear more familiar, but I think they will appear different if expressed in GA.

... in other notations, curl b=j+de/dt
Gauss for B (written as a 3-form equation [your grade 3]... in other notations, div b=0, div(curl a)=0 or dda=0).
There's metric and/or constitutive-equation floating around in here somewhere.

I have requested Macdonald's text from interlibrary-loan.
Maybe that will be a good introduction to this geometric calculus formalism (due to Hestenes?).

Macdonald has another text that is a prerequisite for the vector calculus text. I believe that I was able to learn the elements of GA from free Internet sources so I didn't buy it. I thought (perhaps or perhaps not incorrectly) that it would be too basic, because I already know vector algebra and didn't want to buy another first-division undergrad linear algebra textbook.

Yes, that text is an introduction to geometric calculus, which is more or less an extension of vector calculus to N dimensions. It seems like a decent enough book.

Clifford invented GA and called it by that name back in 1890 or so. He was interested in N dimensional math. Mathematicians adopted it and call it Clifford algebra. Hestenes has applied basic Clifford algebra to physics since 1960 or so with some success. Physics teachers like to use it to teach angular momentum and so forth.

There's been a so-called pre-metric formulation of electrodynamics (due to van Dantzig)...
I'm not sure how all of these formulations (vector, tensor, differential-form, geometric-calculus, pre-metric) are connected... yet.

Thanks.

That makes two of us.

 

[olgerm]

By generalising Maxwell´s equations from electromagnetic tensor form I got:
$\begin{cases} & \sum_{i=1}^D(\frac{\partial E_i}{\partial x_i})=\frac{\partial^D q}{\partial x^D}\frac{1}{{\epsilon_0}} \\ & \frac{\partial E_a}{\partial t}=\sum_{i=1}^D(\frac{\partial B_{[i;a]}}{\partial x_i})-\frac{\partial^D q_a}{\partial t*\partial x^{D-1}} \\ & \frac{\partial B_{[a;b]}}{\partial t}=\frac{\partial E_b}{\partial x_a}-\frac{\partial E_a}{\partial x_b}\\ & \frac{\partial B_{[a;b]}}{\partial x_c}+\frac{\partial B_{[b;c]}}{\partial x_a}+\frac{\partial B_{[c;a]}}{\partial x_b}=0 \end{cases}$
Where:

• E_a is a-axis oriented electric field.
• B_[a;b] is electrimagnetic field crossing with a-axis and b-axis.