- #36
hartmutneff
- 5
- 1
I always thought, it would be a good idea to explain tensors in terms of bra and ket vectors. So I have tried to write it up. I wonder if this is helpful in explaining the above questions.
(it is work in progress, by the way)
My opinion is, that every measurement introduces a dual structure, since you need a quantity to tell you how much of something you got, and you need a link to reality, a unit. If you change your units, you will have to change the numbers in an inverse way. Say, you measure something to be 10m long. If you replace m with (2m), then you have 5 of (2m).
Now, for vectors you will have the same structure, if an object that you want to describe with vectors is invariant, and therefore describes an object in nature.
I think, for vector spaces one has 2 such structures. A vector itself can describe a distance in nature and hence an invariant object. That is why base vectors and the corresponding coefficients transform in an inverse way. To reflect that for the notation used, base vectors get lower indices and coefficients upper indices. So, if you combine this two objects, one with lower and one with upper indices, you get something invariant.
The length of a vector is invariant too, if it describes a length in nature. So the computation of a length has to produce a dual structure as well, or in other words, one structure with upper and one with lower indices, that will be combined. This leads to the introduction of dual vectors, that combine with vectors to create a scalar. This dual vectors are different from vectors, as much as coefficients are different from base vectors.
Going back to the first example, 10 * (m) = 5 * (2m). We could give the units a lower index and the coefficients an upper index as well.
Therefore, upper and lower indices are just a reminder of what things you can combine to create something invariant and therefore independent of an observer. To make a mathematical structure invariant, you will always have a combination of 2 structures that transform in an inverse way to each other.
(it is work in progress, by the way)
My opinion is, that every measurement introduces a dual structure, since you need a quantity to tell you how much of something you got, and you need a link to reality, a unit. If you change your units, you will have to change the numbers in an inverse way. Say, you measure something to be 10m long. If you replace m with (2m), then you have 5 of (2m).
Now, for vectors you will have the same structure, if an object that you want to describe with vectors is invariant, and therefore describes an object in nature.
I think, for vector spaces one has 2 such structures. A vector itself can describe a distance in nature and hence an invariant object. That is why base vectors and the corresponding coefficients transform in an inverse way. To reflect that for the notation used, base vectors get lower indices and coefficients upper indices. So, if you combine this two objects, one with lower and one with upper indices, you get something invariant.
The length of a vector is invariant too, if it describes a length in nature. So the computation of a length has to produce a dual structure as well, or in other words, one structure with upper and one with lower indices, that will be combined. This leads to the introduction of dual vectors, that combine with vectors to create a scalar. This dual vectors are different from vectors, as much as coefficients are different from base vectors.
Going back to the first example, 10 * (m) = 5 * (2m). We could give the units a lower index and the coefficients an upper index as well.
Therefore, upper and lower indices are just a reminder of what things you can combine to create something invariant and therefore independent of an observer. To make a mathematical structure invariant, you will always have a combination of 2 structures that transform in an inverse way to each other.
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