Why is Schrodinger's Equation complex?

*Dickfore*

Quote by Phrak

The number of Maxwell's equations can be reduced to zero where all that is left is the assignment of variables to measurable things, which is already implied in the original equations anyway.

Please justify this statement.

Quote by Phrak

However, once you introduce complex fields, how do you expect to make the magnetic monopoles go away to correspond with know physical reality, except by demanding the phase in the complex plane is fixed?

The phase of what?

*Phrak*

Quote by Dickfore

Please justify this statement.

I think I did that in post #32. If there is anything there that needs clarification, I might supplement. But I think what Delta² brought up in post #26 is more interesting and to the topic at hand.

The phase of what?

The phase of the complex 4-potential. Going to a complex 4-vector potential we can gauge fix the phase of the vector to be real valued over all spacetime. This is a global gauge fixing. Then the complex equations reduce to Maxwell's equations and magnetic charge is fixed to zero, everywhere (and everywhen).

I believe a global phase factor is an unmeasurable quantity. A = A' <-- A e^phi

arkajad

Quote by Phrak

Again, with the wild cards. Why don't you tell why you think the Lorentz gauge is relevant. Recall that the expressions I gave above are differential not integral.

Lorentz gauge is not necessarily relevant in general. But here we are discussing Maxwell equations in the context of wave equations. Wave equations, similar to Schrodinger's equation, usually describe quantum mechanics of elementary particles. Photon supposedly is a spin 1 particle. Without Lorentz gauge we also have longitudinal spin zero states.

*Dickfore*

Quote by Phrak

I think I did that in post #32. If there is anything there that needs clarification, I might supplement. But I think what Delta² brought up in post #26 is more interesting and to the topic at hand.

Quote by Phrak

For all this to hang together a subtle ontological distinction is made. It is tacitly assumed that no equivalence relationship is implied between distinct physical elements such as J==d*dA, but that current and charge density are simply aspects of the vector potential.

I'm afraid you had mistaken the cause and effect. J IS the source for d*dA, not the other way around. All currents are produced by moving charged particles in a unique manner. Consequently, external fields act on these particles with a Lorentz force. Your model fails when you try to write an equation of motion for a charged particle in an external field.

Quote by Phrak

The phase of the complex 4-potential. Going to a complex 4-vector potential we can gauge fix the phase of the vector to be real valued over all spacetime. This is a global gauge fixing. Then the complex equations reduce to Maxwell's equations and magnetic charge is fixed to zero, everywhere (and everywhen).

I believe a global phase factor is an unmeasurable quantity. A = A' <-- A e^phi

If you had looked at the equations more carefully, you would have noticed that the sourceless equations (Gauss' Law and Faraday's Law) now have a non-zero right hand side (due to monopoles). This is why the concept of electrodynamic potentials in the usual sense of the word does not have a straightforward generalization.

*Phrak*

Quote by Dickfore

I'm afraid you had mistaken the cause and effect. J IS the source for d*dA, not the other way around. All currents are produced by moving charged particles in a unique manner. Consequently, external fields act on these particles with a Lorentz force. Your model fails when you try to write an equation of motion for a charged particle in an external field.

It is convenient to begin with a distribution of charge and work out the fields. This convenience does not imply a physically measurable cause and effect such as "charge causes electromagnetic fields," or that "the electromagnetic fields cause the vector potential." I'm open to proof or motivation to this conjecture.

If you had looked at the equations more carefully, you would have noticed that the sourceless equations (Gauss' Law and Faraday's Law) now have a non-zero right hand side (due to monopoles). This is why the concept of electrodynamic potentials in the usual sense of the word does not have a straightforward generalization.

No, a complex vector potential admits magnetic monopoles.

*Dickfore*

Quote by Phrak

It is convenient to begin with a distribution of charge and work out the fields. This convenience does not imply a physically measurable cause and effect such as "charge causes electromagnetic fields," or that "the electromagnetic fields cause the vector potential." I'm open to proof or motivation to this conjecture.

But, it's not the question whether electromagnetic fields cause a vector potential, but whether electromagnetic fields cause charges and currents. If you don't think this stance is absurd, I don't have what to say anymore.

Quote by Phrak

No, a complex vector potential admits magnetic monopoles.

Please show your work.

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