Let us test this BB with a simple mathematical problem: Can a circular field be equal (point for point) to a conservative field. Alternatively stated can a circular field be directly proportional to a conservative field?

Can you repost this using fewer words and making more obscure references? This question has too much verbiage and far too much clarity of intent.

Hi JB,

Its like Morgan said. We need a little clarification here.

Generally there are four equivalent ways to express the fact that a (vector) field is conservative:
1) A path integral over simple closed curve is zero;
2) Any path integrals with the same end points gives the same value;
3) The field can be expressed as the gradient of a scalar function; and
4) The curl of the field is zero, i.e., it is irrotational.

These statements are mathematically equivalent.

I can think of numerous fields that might be regarded as "circular". So the question is: what do you mean by a circular field? A particular example would be nice.


Dr. R.

Wonderful!! Your summary is spot on!! A circular field is a field for which the curl is not zero. Alternatively stated; for a circular field an integral around a closed path is not zero. So is this enough clarification?

Not unless you can integrate your clarification into your original question. I still have no idea what you are asking.

But then again I can't understand why you are asking it here rather than to your advisor or a colleague.
A desire for non-authoritative answers?

JB,

It seems that you have answered your own question, i.e., "Can a circular field be equal (point for point) to a conservative field?" If a circular field is one with a non-zero curl, then it is non-conservative. This follows from item (4) of my previous remark.

Your not really making much sense on this. Try again.

I'm off to do some flying this weekend - I'll check back on Monday.

Dr. R.

Thank you you gave me the answer I wanted us to agree on before I proceed with the rest of my argument.

Let us go to the Ginsberg-Landau approach when calculating superfluidity or superconduction in terms of their so-called "order parameter" or macro wave function. When calculating the current flow by applying the momentum operator and doing a suitable gauge transformation; they obtain that the current density is proportional to the gradient of a scalar field. This relationship also lies at the heart of the BCS theory of superconduction. So far it seems as if there is no problem; HOWEVER:

What is to me a serious problem though is that this same equation is used to model the Meissner effect and to explain flux quantization. In these cases the current field is circular. This means that superconduction is modelled by equating a circular field to the gradient of a scalar field!! The latter is clearly a conservative field. Thus nonsense is calculated to model superconduction. How do they derive the quantized amount of flux: by taking a circular integral of the conservative field and equating it to n times pi. Again nonsense because such an integral MUST ALWAYS be zero; as can be proved directly by applying straightforward mathematics. So can these theories on superconduction then be correct?

Has nobody got a comment on the analysis above?

Dr. R. wrote:
"It seems that you have answered your own question"

What additional comment are you looking for? That the wallpaper is the wrong color? That the lawn needs edging?

The additional comment I am looking for is whether scientists agree that any theory based on equating a circular field to a conservative field must be wrong. This would imply that the macroscopic mechanism that forms the basis to explain superconduction (both by Landau and Ginsberg, as well as Bardeen, Cooper, Schrieffer) must be wrong. Do you agree?

JB, the argument is similar to finding eigenfunctions in QM. In this case the field must be continuous.

If the "current field" is "circular", then there must be a singularity somewhere. The line integral is then zero if the interior region of the contour does not contan that singularity.

Thanks for responding; but I still do not understand how a current (that defines a circular field) can be proportional (point by point) to a conservative field under any circumstance. It violates the rules of simple mathematics.

After all the circular integral of the current must give you the magnetic field generated by the current. But with the current proportional to a gradient of a scalar field, then taking the circular integral on both sides gives the circular integral of the current under all circumstances (where-ever you have the sigularity) also equal to zero. Furthermore, I cannot see how a conservative field within a material which has a hole in it (singularity) will give a non-zero circular integral when you integrate along a path around the hole.