|Jun18-08, 09:17 AM||#1|
Maxwell equations in higher dimensions
My main question is if the Maxwell equations have been generalized
to include extra dimensions in an generally accepted form,
or is it still under investigation?
I've already read
but I didn't quite like the add-hoc assertion
We assert that in all the traditional (3 + 1) D curl operations and cross products in
physics [...] one of the vectors involved is actually an antisymmetric second rank tensor.
Moreover, even we accept the above assertion, the resulting equations loose the aesthetic symmetry of classical Maxwell equations, a fact that in physics is always a warning message of something wrong.
Furthermore the following article
makes things even more confusing as it states that Feynman derived the first set of Maxwell (relativistic) equations starting from the non relativistic Newton's second law!
Since I am a Physician only by hobby (my real occupation is Electrical Engineer) I can not totally follow the advanced mathematical/physics concepts in the relative articles I am asking your help for clarifying these things.
|Jun18-08, 03:44 PM||#2|
There are some useful generalizations of cross products using clifford algebras (those can be applied in 5D conformal modelling of 3D space in computer graphics), but I can't imagine what these maxwell equation generalizations would be used for.
If such generalizations were to have an accepted form, I would have to think there would need to be a physical application. Are there applications for such generalizations in physics?
|Jun19-08, 02:58 AM||#3|
Moreover, If the string theories and the M-theory are correct then the Maxwell equations ought to have a multi-dimensional form.
|Jun19-08, 10:56 AM||#4|
Maxwell equations in higher dimensions
The extension used in the first paper is standard, while the second paper you quote is doing something speculative by assuming that relativity might break down at high energy.
As long as you want relativity to hold and extend naturally to extra dimensions, you need to use the formalism discussed in the first paper, where the Maxwell equations are expressed in a relativistically covariant form.
While it's true that the relativistic form of the Maxwell equations seems to break a symmetry between E and B, it's not terribly hard to show that this analogy is pretty strained to begin with. All we need to do to see this is to consider parity. If we invert parity (that is, if we reverse the direction of our coordinate axes), the physical direction of any vector should be unchanged. But, since our axes are now reversed, the vector, by remaining unchanged, is now pointing in the opposed direction with respect to each axis. (If it starts out pointing in the positive x-direction, reversing the axis means that it's now in the negative x-direction, and so on.) This means that under a parity reversal, any vector should pick up a minus sign.
Now, consider any relation involving a cross-product between two vectors. If, under the parity switch, each vector picks up a minus sign, these cancel, and the resulting "vector" is unchanged under parity. Physically, this means that the resulting vector will flip along with the parity. This, then, can't be a true vector. Instead, it must be a "axial vector" or "pseudovector." In fact, any cross-product relation must involve an odd number of pseudovectors in order to be consistent under parity.
Now, consider the Maxwell equations. It is not too hard to show that the gradient operator is a vector under parity. Then, by considering the total parity transformation of each of the four equations, we can see that the electric field must act as a vector, while the magnetic field acts as a pseudovector. In three dimensions, a pseudovector is equivalent to an antisymmetric second rank tensor. So, this is the form that must be generalized to higher dimensions, leading to the standard relativistic form.
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