E&M in higher dimensions
February 11, 2012 4:17 PM Subscribe
Are Maxwell's laws dimension independent?
The beauty of using the gradient operator in physics was that it abstracted away the dimensions. But the cross product exists only in three dimensions and seven dimensions. What is the meaning of grad cross product vector(B) in four dimensions? Is the cross product expression replaced by the Gram-Schmidt orthogonality process for the curl of the magnetic field when in dimensions not 3 or 7?
The beauty of using the gradient operator in physics was that it abstracted away the dimensions. But the cross product exists only in three dimensions and seven dimensions. What is the meaning of grad cross product vector(B) in four dimensions? Is the cross product expression replaced by the Gram-Schmidt orthogonality process for the curl of the magnetic field when in dimensions not 3 or 7?
Best answer: In a naive dimensionally independent way, Maxwell's equations can be written:
dF = 0
d*F = J
where * is the hodge star operator on the n-dimensional space, d is the covariant derivative and F is a "two-form" which would include the electric and magnetic terms i.e. E and B. Now, the hodge star operator assumes a conformal class of metrics on the space, so while you can write this for arbitrary dimension, you are assuming the space has a (conformal class of a) metric i.e. a way of infinitesimally measuring distance (think Minkowski metric for n=4).
Now, this is naive. This is just a compact way of writing a bunch of partial differential equations that can be easily generalized to n-dimensions. It's not necessarily meaningful (especially for arbitrary conformal class) for n != 4. For n = 4 it defines the Maxwell's equations you know and love.
If you want, you can rewrite this using Clifford algebras or (for n=4) use quaternions. A nice resource for this stuff is the second volume of Bamberg and Sternberg's "A Course in Mathematics for Students of Physics." (available on the internet near you) The upshot is that the "cross product" is really a sort of mnemomic device. Technically it's the product operation for the lie algebra of SO(3) (orientation preserving orthogonal group of rotations of R^3) which just happens to look like R^3 but that's a mouthful right?
It's important to distinguish notation, which ultimately is just a matter of convenience, and the actual ideas, which are notation independent. Maxwell wrote his equations in coordinates, and had many of the ideas of modern field theory. This is a fun subject but there is no magic notation which makes a straight path to E&M...
posted by ennui.bz at 5:45 PM on February 11, 2012 [4 favorites]
dF = 0
d*F = J
where * is the hodge star operator on the n-dimensional space, d is the covariant derivative and F is a "two-form" which would include the electric and magnetic terms i.e. E and B. Now, the hodge star operator assumes a conformal class of metrics on the space, so while you can write this for arbitrary dimension, you are assuming the space has a (conformal class of a) metric i.e. a way of infinitesimally measuring distance (think Minkowski metric for n=4).
Now, this is naive. This is just a compact way of writing a bunch of partial differential equations that can be easily generalized to n-dimensions. It's not necessarily meaningful (especially for arbitrary conformal class) for n != 4. For n = 4 it defines the Maxwell's equations you know and love.
If you want, you can rewrite this using Clifford algebras or (for n=4) use quaternions. A nice resource for this stuff is the second volume of Bamberg and Sternberg's "A Course in Mathematics for Students of Physics." (available on the internet near you) The upshot is that the "cross product" is really a sort of mnemomic device. Technically it's the product operation for the lie algebra of SO(3) (orientation preserving orthogonal group of rotations of R^3) which just happens to look like R^3 but that's a mouthful right?
It's important to distinguish notation, which ultimately is just a matter of convenience, and the actual ideas, which are notation independent. Maxwell wrote his equations in coordinates, and had many of the ideas of modern field theory. This is a fun subject but there is no magic notation which makes a straight path to E&M...
posted by ennui.bz at 5:45 PM on February 11, 2012 [4 favorites]
(should say, for n=4 and the minkowski metric defines the Maxwell's equations you know and love... that's the foundation of special relativity inverted)
posted by ennui.bz at 5:48 PM on February 11, 2012
posted by ennui.bz at 5:48 PM on February 11, 2012
What is the meaning of grad cross product vector(B) in four dimensions?
An important thing to note is that the very existence of B as a vector field is actually dependent on you being in three spatial dimensions. The magnetic field is what's known as a pseudovector, which means that its definition actually relies on a correspondence between spatial two-forms and vector fields. (The dimensions of these spaces are only equal in three dimensions.) This mapping, between the space of the "wedge products" of two vector fields (one-form fields, really) and the vector fields themselves, is none other than the cross product.
So the use of the cross product is implicit in the definition of B (every way to define B given a current distribution involves a "right-hand rule", after all.) In n dimensions, the electric field would still be an n-dimensional vector, but the magnetic field would be properly defined as a two-form in n dimensions, with n(n - 1)/2 components.
posted by Johnny Assay at 9:06 AM on February 12, 2012
An important thing to note is that the very existence of B as a vector field is actually dependent on you being in three spatial dimensions. The magnetic field is what's known as a pseudovector, which means that its definition actually relies on a correspondence between spatial two-forms and vector fields. (The dimensions of these spaces are only equal in three dimensions.) This mapping, between the space of the "wedge products" of two vector fields (one-form fields, really) and the vector fields themselves, is none other than the cross product.
So the use of the cross product is implicit in the definition of B (every way to define B given a current distribution involves a "right-hand rule", after all.) In n dimensions, the electric field would still be an n-dimensional vector, but the magnetic field would be properly defined as a two-form in n dimensions, with n(n - 1)/2 components.
posted by Johnny Assay at 9:06 AM on February 12, 2012
From a model-building point of view, the common way to describe fields of all sorts in higher dimensions is by enforcing certain requirements such as gauge invariance and renormalizability when writing down the action. This offers a pretty simple recipe for how to write down E&M, the weak and strong interactions, etc...
For example, E&M is described by a U(1) gauge field.
posted by dsword at 10:06 PM on February 12, 2012
For example, E&M is described by a U(1) gauge field.
posted by dsword at 10:06 PM on February 12, 2012
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You can generalise Maxwell's equations to higher dimensions by using the language of differential geometry. Operators such as divergence and curl have natural extensions in this language. For example, curl is represented by the wedge product. So mathematically you can do it. But I don't know if it makes sense physically.
posted by lollusc at 5:41 PM on February 11, 2012