vectors, tensors, spinors, twistors...

[thigle]

is the sequence vectors, tensors, spinors, twistors finite? by what operation on what does one get these ?

in this sequence ... I imagine (not understand):

* vectors as events/entities with direction and value. a couple of parameters.
* tensors i imagine as 'free' vectors whose direction can change - is somehow free not fixed (but i don't know how and why). how many parameters does a tensor have - 4 ?
* spinors i imagine as pathways through quaternion unit sphere - as loops of sequences of rotations
* twistors... as vortex stuctures ???

what do physicists use these for ? are these concepts restricted to certain dimensions ?

analogical explanations prefered to digital ones. :wink:

[jinydu]

I have to admit, I've never heard of "spinors" or "twistors" before.

A vector is a quantity with a magnitude (also called length) and direction. The most common way of expressing vectors is in terms of Cartesian coordinates: where the vector is represented using an ordered n-tuple of numbers.

Vectors show up all over physics. Any point in n-dimensional space can be specified using a vector with n components. Velocity, acceleration, force, torque, gravitational field and electromagnetic field are all represented by vectors.

I could go on and on; there are entire university level courses on what you can do with vectors. I myself have taken one such course, and there will be more to come.

Tensors, unfortunately, I know much less about. I have heard that they should up in Einstein's General Theory of Relativity. I've also heard that, whereas a vector can be thought of as a one-dimensional array of numbers, a tensor can have any number of dimensions.

For (much) more information, visit mathworld.wolfram.com

[houserichichi]

In short:

Both one-forms and vectors are specific kinds of tensors.
Spinors are similar to tensors in that they change under coordinate transformations but under a different mapping. Tensors, when rotated 360 degrees return to their original form. Spinors, conversely, return to their original form after being rotated by 720 degrees. Twistors, as far as I can recall, are a complex 2n-dimensional version of n-dimensional spinors, though I could be wrong. I'd recommend taking a quick stab at

http://www.compsoc.net/~fedja/twistors/twistors.html

and see where you go from there.

[PWrong]

I just found this introduction to tensors. It makes a lot more sense than I first expected.
http://gltrs.grc.nasa.gov/reports/2002/TM-2002-211716.pdf

As far as I can tell, you can think of a tensor as just a group of vectors. They use the same components, except joined together, for instance:
aii+bij+cik
+dji+ejj+...

It looks like they'd be extremely useful for studying 4D.

[thigle]

actually, i was asking due to my interest in how to represent spin rotations by quaternions (i would like to use the math to script some animations of rotations in 3dsmax, which are gettiing gimbal-locked via the traditional euler interface). maxwell formulated his theory with quats, they are the natural algebra(or whatsthename) of euclidean 4-space. the unit sphere of quats forms the orthogonal group of rotations for 4-space. each quat as quadruple of numbers represent an axis through origin by 3 values and magnitude of rotation by 4th value.

so a tensor which would be a group of all vectors defined by SO(4), would be a tensor of all the possible 4d views? so bi-glomohedric prism, a terix in 6d, can be considered a tensor? now i am confused. :?

[wendy]

Tensors are used to refer to matrices with any number of subscripts. A vector and a matrix are both tensors.

Spinors are for recollection, a result (like a spin-vector).

I use 'trimex' to define a particular three-subscript matrix, such as used for defining vector multiplication.

Regarding quaterions, i am of the understanding that these are associative, and therefore can be represented by a trimex multiplication. Octonions are not associative, and therefore have no defining trimex.

[wendy]

so a tensor which would be a group of all vectors defined by SO(4), would be a tensor of all the possible 4d views? so bi-glomohedric prism, a terix in 6d, can be considered a tensor? now i am confused.

A tensor is a means of taking two vectors and making an output vector. It is not a set of vectors, just a vector with two or more subscripts. For example, the cross-product in three-dimensions is a tensor.

One often finds tensors where one has a multi-directional vector (ie the size depends on the direction). Magnetism makes use of tensors.

The bi-glomohedric prism is simply what the physicists would call a representation in a phase-space: each point of the prism represents a valid phase (here a great-arrow in 4d).

What you do with a phase-space is to plot out the kind of motion that might arise when something changes. For example, the complete phase space for rotations in 4d is a bi-glomohedric pyramid. The natural tendancy for rotations is to move to one of the bases of the pyramid. This would be represented by a path in phase-space.

It's best to think of matrices as arithmetic processes [ie something you do with numbers], and tensors as some kind of physical process [like a whirlywind]. One might describe a tensor by a matrix.

The bi-glomohedric prism and pyramid, are phase-spaces, which represent each possible state of great-arrows and rotations in general, as points on a surface. When something changes rotation-mode, then the point representing its motion moves on the phase-space.

Wendy