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## Main Question or Discussion Point

One question that's been on my mind for a while is what the difference between a bra vector and a covector is. Are they the same thing? Why the difference of notation?

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One question that's been on my mind for a while is what the difference between a bra vector and a covector is. Are they the same thing? Why the difference of notation?

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Where did you encounter co-vectors in quantum theory? And what did they look like?

Cheers,

Jazz

Cheers,

Jazz

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I didn't. I'm a math guy. :tongue:Where did you encounter co-vectors in quantum theory? And what did they look like?

Cheers,

Jazz

I read a thing on how the Riemann Hypothesis is related to quantum physics, and the article used some bras and kets. I assumed, from context, that a bra vector was essentially the same as a covector. I wanted to know if I was right.

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WannabeNewton

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Hope this clears it up :)

Cheers,

Jazz

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A co-vector is a more general notion than of a bra-vector, because you needn't have a topology, nor a scalar product to speak about vectors and co-vectors, but you need to have them to speak of bra-s and ket-s.One question that's been on my mind for a while is what the difference between a bra vector and a covector is. Are they the same thing? Why the difference of notation?

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That's interesting you bring that up. There seem to be two different concepts or definitions of co-vectors in the literature, depending on whether you get there using the exterior algebra and differential forms or multilinear forms and a metric (with the metric as the defining bijection for the dual). For the diff-forms approach you only need the metric when you introduce the hodge dual and not the co-vectors.A co-vector is a more general notion than of a bra-vector, because you needn't have a topology, nor a scalar product to speak about vectors and co-vectors, but you need to have them to speak of bra-s and ket-s.

I think I understand this all pretty well, but I never found the different definitions very intuitive and the differences rather confusing. But then again, I'm only a theoretical physicist and not a real mathematician.

Any thoughts on that?

Cheers,

Jazz

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So what's your most general definition of a co-vector?

Cheers,

Jazz

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rubi

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In the case of manifolds, you automatically have the tangent spaces ##T_p M## at every point and since they are topological vector spaces (all finite-dimensional vector spaces have this property), you can form their dual ##T^*_p M##. If you have a metric, the tangent spaces are Hilbert spaces and you can use the Riesz isomorphism to identify tangent vectors with cotangent vectors.

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