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 Wondering if there is any connection between the concepts of vectors and 1forms 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 3D, 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 1form 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 1Form 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 xvectors in 3space (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 1Form from the book "Gravitation" ?
I hope this question makes sense.
In the case of Fourier Transforms one has the complex exponentials exp(2..π i. ξ.x)
In 3D, 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 1form 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 1Form 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 xvectors in 3space (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 1Form from the book "Gravitation" ?
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