Deriving Exclusion Principle: Symmetric & Antisymmetric Wave Functions

[exponent137]

Exclusion principle of fermions is a consequence that wave functions are only anti-symmetric. Wave functions of bosons are only symmetric. Those two possibilities are consequence that bosons (and fermions) are not distinct.

But how to derive, that particles with integer spins have symmetric wavefunctions and that particles with integer and half spins have antisymmetric wave functions. How to derive that particles of the same type are not distinctable? Symmetric and anti-symmetric wave-functions are the only options for not distinctable functions, this is understandable.

I read, for instance Feynmans "QED: The strange theory of light and matter", but, I think, he assumes the above facts, not derive them?

 

[alxm]

http://en.wikipedia.org/wiki/Spin-statistics_theorem"

 

[exponent137]

In your link is also a link to Baez's explanation. He begins with topological explanation. But I am not sure what precisely he thinks. Maybe it is somewhere drawn with his strip explanation?
So, this is not moebious strip, but double twisted moebious strip?
Where to find more visualized his explanation?

 

[maverick_starstrider]

In your link is also a link to Baez's explanation. He begins with topological explanation. But I am not sure what precisely he thinks. Maybe it is somewhere drawn with his strip explanation?
So, this is not moebious strip, but double twisted moebious strip?
Where to find more visualized his explanation?

FYI the spin-statistics theorem comes from quantum field theory, there's no way to "understand" it using only quantum mechanics (I don't know if you know QFT or not). In other words it's a relativistic quantum effect.

 

[exponent137]

FYI the spin-statistics theorem comes from quantum field theory, there's no way to "understand" it using only quantum mechanics (I don't know if you know QFT or not). In other words it's a relativistic quantum effect.

Yes, it is relativistic effect. I found some articles. But, they have large derivations, where I have not find essence of this, I think, elementary and I believe simple effect. Maybe, Duck's and Sudarshan's article is more pedagogical?

But, Baez's explanation seems cool to me, as beginning for further articles. But visualisation of his twisted strip fails.

I read QFT somewhere. I try to understand and visualize it. Maybe also understanding of spin-statistics will help.

 

[alxm]

But, Baez's explanation seems cool to me, as beginning for further articles. But visualisation of his twisted strip fails.

http://www.youtube.com/watch?v=Nwe04f8on-s")

 

[exponent137]

alxm very thanks for these links.

Now I try to better understand rotatation of electron for 360°. I read
http://en.wikipedia.org/wiki/Spinor
So precisely, this means rotation of spinor.
What is then sense of rotation of spinor for any angle, let us say for 41,4°? Maybe for intererence of two electrons?
Is then the matrix \gamma only half of a rotational matrix? Or, does \gamma^2 on a common vector means rotation?
Let us say that I will understand factor -1 at rotation of electron: what I need to understand that swaping of two electron means factor -1?One joke question: what means "skcn".

 

[exponent137]

Derivation of exclusion principle - more concrete questions

THE "DOUBLE SLIT" QUESTION:
Let us execute a double slit experiment with electrons, or with photons. Is here any phase difference between interference lines in both experiments? Let us assume, that momentum of electron and photons is the same, so the distance between interference lines is equal inside(!) of both experiments. If this is true this can be a now visualization of difference between bosons and fermions.

At the above experiment one electron or photon flies at determined time. So, let us make a different experiment. Namely, let us assume that two electrons fly through two slits every time. (They have the same spin and the same momentum.) At the center position, where distance for both electrons is the same, here is an interference minimum? If the same experiment is done with two photons, here is interference maximum. (Polarisation and energy of both photons are equal.)

At common double slit experiment, one electron flies through the both slits the same time. Does exculsion principle is valid for those two "halves" of electron? So I suppose that is similarly as at two electrons? I am not sure.

THE "SPACE-LIKE" QUESTION:
For derivation of Pauli exclusion principle we need space-like distance between two electrons. But this distance exists only in relativistic mechanics, so this derivation needs relativistic physics. Is here any other reason for introduction of relativistic physics. Space-like distance is important at swaping of two electrons and at comutation rule.

 

[exponent137]

I think that a basic question is, why a spinor rotation for 360° gives factor -1.

I found http://en.wikipedia.org/wiki/Spinor

where it is also written:
"An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:

\gamma(u) = \gamma u \gamma^* = \gamma^2 u\,.

On the other hand, comparing with the action on spinors γ(φ) = γφ, γ on ordinary vectors acts as the square of its action on spinors."

But:
1. Vector is compared with spinor. Is spinor not a vector?
2. Rotation of vector is described as rotation of matrix??
3. Here is not a direct formula which can describe rotation for, let us say, 37°?
4. What here means 2 dimensions, 3 dimensions, I am interested in rotation in common 3D space?