Let us have two electrons, 1 and 2. We imagine that they are marked, so that we can distinguish them.
Let us prepare the two electrons in some way for our experiment, at positions which initially have spacelike separations. We calculate their wave function with a path integral. Let y be the coordinate of the electron 1 and z the coordinate of the electron 2. The coordinates may include also other parameters besides the R⁴ coordinates of the Minkowski space.
Let us prepare the electron 1 with a spinor s at an initial location -x.
Let us prepare 2 at the location x. The initial wave function of 2 is obtained by rotating the wave function of 1 through 180 degrees in a plane which includes time. The spinor s is transformed in the natural, smooth way to s'.
The wave function of the two particle system is initially
R (φ) (x) (z) * φ(-x) (y),
where R (φ) performs the rotation to the spinor and the wave function, and where * denotes some kind of a "product" of the two wave functions.
Let us rotate the coordinates smoothly through 180 more degrees with R. The spinor s' is transformed in the smooth way to s''. The rotation also flips -x and x.
The wave function in the rotated frame is
R (R (φ)) (-x) (z) * R (φ) (x) (y).
For a spin 1/2 particle, the spinor rotation through 360 degrees from s to s'' flips the sign:
R (R (φ)) = - φ
We thus have
- φ(-x) (z) * R (φ) (x) (y) =
R (φ) (x) (z) * φ(-x) (y).
The roles of y and z are switched on the two lines. That flips the sign. In this special case, the wave function Φ of the two particle system has to be antisymmetric - otherwise we would break the spinor algebra.
Analysis of the proof
The idea in the proof seems to be that we must find a configuration of two electrons where the states of the electrons seem to switch through a formal coordinate transformation. We do not need to touch the electrons physically at all. It is just a coordinate transformation to another frame. We rotate our coordinate system through 180 degrees, and if we transform the wave function in a natural, smooth way, then the sign of the wave function flips.
In quantum mechanics, we are allowed to multiply the wave function by any complex number α, where |α| = 1, before starting the experiment. We are not allowed to multiply it in the middle of the experiment.
Are we allowed to flip the sign in a coordinate transformation? Then we could declare that the sign of the wave function did not flip in the proof above, after all.
Let us just ban extra sign flips as ugly.
Could there be configurations where we can switch the electrons, like above, but the wave function sign does not change?
Rotations through 180 degrees seem to flip the sign because the two spinors are rotated a total of 360 degrees.
Does the proof show that in the general case, the wave function of two electrons is antisymmetric?
First we have to decide what "switching the electrons" exactly means.
If we have two electrons, let us move to their center of mass frame.
The spinors point at arbitrary directions. Generally, we cannot switch the electrons through a 180 degree rotation.
Let us look into literature. Is there a general proof?
The Pauli exclusion principle
Our goal is to prove the Pauli exclusion principle: two electrons cannot end up in the same quantum state.
Let us assume that we prepared the two electrons like in the first section, and their wave function starts as antisymmetric.
Does the wave function stay antisymmetric? We do not know, but let us forget that problem for a while.
t
^
|
|
|
------●-------------●-------> x
e- e-
The electrons start their life in the diagram above. In our coordinates they travel forward in time, but in the rotated coordinates, backward in time.
Let us assume that the electrons end up in some stationary states in our coordinates:
|b > |c > (our coordinates).
In our coordinates, that configuration has a wave function value A_bc.
In the rotated coordinates, the state is
|c > |b > (rotated),
and the wave function value -A_bc.
What about the state
|c > |c > (our coordinates) ?
If that state has the wave function value C in our coordinates, and -C in the rotated coordinates, is C necessarily zero?
We do not see a reason why it should be.
If we could show that in our coordinates, the wave function values for
|b > |c > (our coordinates)
and
|c > |b > (our coordinates)
have the sign flipped, then |c > |c > would have a zero wave function value in our coordinates.
The proof of the Pauli exclusion principle in Wikipedia seems to assume that the particles are switched without any coordinate transformation.
Generally, a two electron wave function
Φ(y, z)
is not antisymmetric in our coordinates. The electron at y may permanently reside in a location where z never comes. Then it is not a symmetric or antisymmetric function.
Let us check the literature. Has anyone come up with a solution which proves the Pauli exclusion principle from antisymmetry?
Wikipedia says about the spin-statistics theorem:
"An elementary explanation for the spin-statistics theorem cannot be given despite the fact that the theorem is so simple to state."
That sounds ominous.
We think that the theorem is not "simple to state", because it is unclear what switching the electrons really means.
Ilya G. Kaplan (2021) writes that correctness of various proofs of the spin-statistics theorem has been under debate for the past 80 years. Wolfgang Pauli held the opinion that the Pauli exclusion principle is an empirical result and no genuine proof for it exists.
In 2018, we analyzed the correctness of various proofs of the spin-statistics theorem.
We also analyzed the conceptual difficulty in what an exclusion principle could mean. It is not clear if an interacting system has separate "states" for individual particles.
We suggested that something like the Pauli exclusion principle may follow from the extremely strong repulsion between the magnetic fields of two electrons when the spin direction is the same.
If magnetic repulsion is the reason for the empirical Pauli exclusion principle, then it has nothing to do with switching the particles.
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