Thanks. Dont worry about the "maths speak". The rigour of your arguments reminds me to try to be more careful in how I phrase my arguments. Yes I enjoyed the theory of complex numbers when I was at University so I understand what you say.
Thanks. I am aware of this article and have read it but as you correctly say it is a bit abstract for me to understand. Perhaps I should tr to understand as you say it discusses general duality and fourier transforms. Perhaps if I understood it I may have a clue !
Yes I am familiar with the concepts of the Fourier Transform used here. Perhaps I am mistaken and there is no connection andI should treat the Real and Reciprocal spaces as seperate from the tangent and cotangent spaces.
Talking about "connections" - I understand a connection in the Framework of General Relativity where the Christoffel Symbols are the "connection" relating a vector defined at one point in a smooth manifold to its image defined at a second point in the manifold. Hope I got that right at least. So...
OK ! I understand that COMPLETELY ! Also, as you call yourself a "physics novice" I would call myself a "mathematics novice". I did a 4-year physics degree with 3 years applied mathematics and 2 years mathematics. However that was some 30 years ago. Today I dabble in physics for the fun of it...
I would say that my intention is to work with a smooth manifold on which an inner product is defined. Also a metric but I guess that to define an inner product we must have a metric defined (or am I missing it again?).
Sorry - that`s the "amateur physicist" in me trying to talk to the mathematician in you. I have a degree in Physics (1985) but dont work in an academic environment so my language could be very rusty. my language is certainly not as rigorous and precise as yours. I am actually a radiation...
Thank you Mathwonk for your explanation. I should have mentioned that an inner product was defined. I will need some time to study your response. It clarifies many things for me. I guess when you say ""maps in" , or vectors, are dual, or reciprocal, to "maps out" " you are speaking in the...
In trying to answer my own question, and bearing in mind that I am no expert,
We recognise that for each vector (in the tangent space) there is an associated 1-Form (in the cotangent space). One possible 1-Form is the Gradient function (I hope). Now Gradient means "rate of change over a unit...
Good day Mathwonk.
Thank you for your comment. You are correct that the term "Reciprocal Space " is not a common term in differential geometry. I am playing with ideas and concepts here. When you do this you may often try to match two things which seem very different. If we accept that the...
OK.
I may be answering my own question. We have vectors and we have 1-forms. They are in no way Fourier Transforms of one another. However the Fourier Transform, together with the Inverse Fourier Transform defines a relation between them...
Hi All.
I hope this question makes sense.
In the case of Fourier Transforms one has the complex exponentials exp(2..π i. ξ.x)
In 3-D, if we single out where the complex exponentials are equal to 1 (zero phase), which is when ξ.x is an integer, a given ( ξ1,ξ2,ξ3).deﬁnes a family ξ.x= integer...
Fresh_42. I think you understand this whole thing way better than I do and probably my questions are not making sense to you. The problem is to convince me of my ignorance. You have changed my perspective on this and I need to think deeply on my new perspective. Its late here so I am retiring...
OK perhaps, to explain, If instead of involving V with V*. If we had two vector spaces V and W being subspaces of R4 we might expect V to have dim 2 and perhaps W to have dim 2. Then, if I am correct, the Direct Sum V and W would give a vector space of dim 4. So if we could have defined a metric...
OK Fresh_42, you have REALLY got me thinking now about my original question which may well have been misguided by my ignorance.
You said earlier;
Yes, but then you have V⊗V∗≅L(V,V)≅M(n,R)V⊗V∗≅L(V,V)≅M(n,R) and dimension n2n2.
Yes this has dimension nxn rather than 2n. I need to rethink my...
Maybe I dont need to form any kind of Sum. Maybe I just need to recognise that I have a Tensor V(n,n) consisting of the n contravariant components together with the n covectors (or 1-Forms). The these n 1-Forms Map the n contravariant elements to the set of Real numbers ?
Thank you very much for your reply.
The second option (factoring out) seems a bit complicated to me. The first option, I think, at least as you have written it, seems to be the answer. I say this because you say the Direct Sum will produce a Space of Dim 2n from 2 spaces each of dim n. I...