What Is the Physics Intuition Behind a Complex Vector in a Linear Space?

[Shing]

Would anyone be kind enough shed some light on the physics Intuition of a vector in a linear space over a complex field for me?

Furthermore, what does z^z mean?

 

[mathwonk]

a complex structure on a real vector space is just a linear operator J such that J^2 = -Id. It is usually thought iof as a 90degree rotation counterclockwise. but you have tod ecide what plane to rotate in.

so whereas a real vector is thought of as an arrow, i.e. having both length and direction, a complex vector also has an associated perpendicular direction.

If you think of an arrow as a real arrow that you shoot, notice it has feathers and if it has a traditional native american arrowhead, it also has a sharp flat blade. That blade determines a 2 dimensional plane containing the arrow. If that blade is also painted half red and half blue, then you can think of the blue side as determining the counterclockwise direction in that plane.

so that allows you to rotate the arrow 90 degrees in the plane of the blade, and towards the blue half of the arrowhead.I don't know where in physics this concept comes up naturally, (maybe particles with "spin"?) but that is the data

that it determines.

 

[Shing]

Thank you so much for your reply,
I am sorry that I can't understand it completely (since English is my 2nd language)
so my understanding is:
A vector with complex structure can be thought of a arrow(vector itself) with feathers (the complex number stays with the vector). While arrow pointing position, the feather itself no meaning of position but will affect the direction of the arrow.
(and that is indeed a brilliant idea!)

I would like to discuss a further question:
since both 2-D vector and a complex number are kinds of ordered pair, (but with different operations), if given another set of operation, I can create another "whatever" numbers too?
so the spirit of algebra is really about operation?

Thanks for your reading :)

 

[homeomorphic]

My physics intuition for a finite-dimensional complex vector space is that it is like the space of wavefunctions on a finite set. To each point in the set, you associate an angle and a magnitude. So, for example, you might think of each element of the set as a little spring. The magnitude is the amplitude of the oscillation and the angle just tells you what phase of the oscillation you are in (like sin (ωt + θ)). Or it could be the wave function of an electron.

I don't know where in physics this concept comes up naturally,

Wave functions are complex-valued functions on R^3. They form an infinite-dimensional complex vector space. If you want a 2-d complex vector space, you can consider a quantum mechanical system with two states, like an electron, which can be spin up or spin down. The spin up state and spin down state form a basis, and linear combinations are quantum superpositions of the spin up and spin down state (they can also be thought if as spin in some other direction via what turns out of be the Hopf map from the S^3 sitting in that complex vector space, to S^2, the set of directions in R^3). Disjoint union of physical systems corresponds to tensor product (or maybe symmetric or exterior product for bosons and fermions, respectively), so you can cook up some more physically interesting complex vector spaces that way.