A Connection between 1-Forms and Fourier Transform

  • Thread starter Phinrich
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Summary
Wondering if there is any connection between the concepts of vectors and 1-forms AND the Fourier Transform.
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 of parallel planes (of zero phase) in (x1,x2,x3)-space.

The normal to any of the planes is the vector ξ = ( ξ1,ξ2,ξ3).


By contrast, the Book, “Gravitation” by Misner, Thorne and Wheeler, talks about “vectors” and “1 Forms” (page 53).

It states that vectors are well known geometric objects – Agreed. They then introduce the “1 Form” as a new geometric object. It is further stated that physics associates a de Broglie wave with each particle. The 1-form is then defined as the pattern of surfaces being surfaces of equal integral phase of the de Broglie waves.

We are then told to regard the 1-Form as “a machine” into which vectors are inserted and from which numbers emerge. Then <K.V> equals the number of surfaces (of equal integral phase) pierced by the vector v.

Is there any link between the x-vectors in 3-space (x1,x2,x3), the vectors in frequency space,( ξ1,ξ2,ξ3) and the surfaces defined by ξ.x = integer (being surfaces of zero phase) and the concept of the 1-Form from the book "Gravitation" ?
 

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Vectors and 1-forms are isomorphic when there is a non-degenerate quadratic form associated with the vector space.
 
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Thank you for this. This is clear now.
 

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