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lark
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please see http://camoo.freeshell.org/dual_gauge.pdf"
Thanks
Laura
Thanks
Laura
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The following comment in a book was puzzling to me : ”Recall the Hodge dual of the Maxwell
tensor F. We could imagine a ’dual’ U(1) gauge connection that has F as its bundle curvature
rather than F.”
But [itex]F_ab[/itex] satisfies d[itex]F_ab[/itex] = 0, which means that [itex]F_ab[/itex] can be expressed as a derivative of a 1-form:
[tex]F_ab = \frac{\partial{_A}}{\partial{_b}}-\frac{\partial{A_a}}{\partial{x^b}}.[/tex]
and the vector potential [itex]A_a[/itex] gives you a gauge connection
[tex]\nabla_a= \frac{\partial}{\partial{x^a}} - ieA_a[/tex]
And so on.
So [itex]F_ab[/itex] is the
curvature of the connection,
[tex]\nabla_a \nabla{_b}-\nabla_a \nabla_b -ie =-ie \left(\frac{\partial{A_b}}{\partial{x^a}}-\frac{\partial A_b}{\partial x^b}\right)[/tex]
I know, but I use the built-in Latex for simple things, not for long complicated posts.jtbell said:Hi, did you know that we have inline LaTeX capabilities here?
But
[tex]d^*F_{ab}=-\frac{*j}{\epsilon 0}[/tex]
But how can *F be a gauge curvature when d*F[itex]\neq[/itex] 0? If it were in empty
space you could come up with a 1-form [itex]Z_a[/itex] so that *[itex]F_{ab}=\frac{\partial {Z_b}}{\partial{x^a}}-\frac {\partial{Z_a}}{\partial{x^b}}[/itex] , that would give
you an alternate bundle connection. Who knows what it would be a connection for!
Is the empty-space version of the Maxwell equations what you’d be using at the quantum
level? Maybe all the interactions are being described explicitly so you wouldn’t be using a
charge-current vector. What do you think?
lark said:I use the built-in Latex for simple things, not for long complicated posts.
Yes but the book didn't say anything about the field being source-free. Is my guess right, that in the application where the dual connection would be used, I guess on wavefunctions, that you wouldn't be including the local sources in the Maxwell equations, because you are looking at things at too small a scale to consider charge density, current etc.? He might be leaving this unsaid - but it seems like an obligatory aspect of considering the dual of the Maxwell tensor as a curvature. I spent a lot of time thinking about whether there'd be some way to get around that restriction, some other way of defining the covariant derivative that would give the dual as the curvature. But nothing really came up.lbrits said:There's some interest in this for Yang-Mills theories and in other contexts. Anyway I think the point is that it has to be source free. Or, approximately source free (say, in some non-simply connected region). See for example the Dirac monopole.
The Dual of Maxwell tensor is a mathematical concept in electromagnetism that describes the relationship between electric and magnetic fields. It is represented by a 4x4 matrix and is used to calculate the strength and direction of electromagnetic fields.
Gauge curvature refers to the curvature of the space-time manifold in a gauge field theory, such as electromagnetism. In the context of the Dual of Maxwell tensor, gauge curvature is a measure of the distortion of space caused by the presence of electric and magnetic fields.
The Dual of Maxwell tensor is derived from Maxwell's equations, which describe the behavior of electric and magnetic fields. It is used to simplify these equations and make them more elegant, allowing for easier calculations and predictions of electromagnetic phenomena.
The Dual of Maxwell tensor has many practical applications, including in the fields of electrical engineering, telecommunications, and quantum mechanics. It is used to analyze and design circuits, antennas, and other electronic devices, as well as to study the behavior of particles in quantum systems.
While the Dual of Maxwell tensor is a powerful tool in understanding electromagnetic phenomena, it is not applicable in all situations. It is based on classical physics and does not take into account quantum effects, such as the behavior of particles at the atomic level. Additionally, its use may be limited in highly complex systems with nonlinear interactions.