"Why is it that particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign, whereas particles with integral spin are Bose particles whose amplitudes add with the positive sign? We apologize for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments on an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved. For the moment, you will just have to take it as one of the rules of the world."
- Richard P. Feynman, in the Feynman lectures on physics.
http://www.feynmanlectures.caltech.edu/III_04.html
We could, and maybe should, adopt as an axiom the Fermi-Dirac statistics of a spin 1/2 particle. There are various proofs of the statistics on the Internet. We try to analyze in the following what the proofs actually show.
The Pauli exclusion principle is the very concrete consequence of the statistics. Though, we need to analyze in what sense Fermi-Dirac statistics really bans two fermions in the identical quantum state.
In our blog post Nov 17, 2018 we developed a model for the electron spin.
The energy 511 keV of the electron travels at the speed of light in a circle of radius 2 * 10^-13 m. The electron only does 1/2 of the Compton wavelength in a rotation. There would be a destructive interference which would prevent such state in a single loop, but there apparently is some hidden degree of freedom which makes the rotation actually to happen in a "coil" with two loops, or, equivalently, along the edge of a Möbius strip.
We can make a coil with two loops by taking a loop, like "O", bending it to an "8", and then bending the two loops on top of each other, resulting in a double loop "o".
If the spin is rotation of the electric field, what is the hidden degree of freedom? A possibility is that the electron jumps back in time before completing a full 360 degree rotation.
Suppose that besides the electron, some unknown particle is rotating in the same loop at 1/2 the speed of light. Then that unknown particle would be the extra degree of freedom which would cause the the wave function return to the original value when the electron rotates 720 degrees.
https://en.wikipedia.org/wiki/Spin–statistics_theorem
In the spin-statistics theorem, in the Julian Schwinger proof, it is unclear what exactly the 360 degree rotation in it means. It cannot be a spatial coordinate rotation at a fixed time t_0, because then the wave function necessarily returns to the original value after a 360 degree rotation.
If the time coordinate is the same, then the state has to be the same when the spatial coordinate axes point to the original direction - unless it is a multiverse where a 360 degree rotation takes us to another universe. That is actually an interesting hypothesis. Suppose that we live in a double universe where in most cases the two sides are identical. An exception would be spin 1/2 particles, whose wave function can be defined antisymmetric relative to an arbitrary "center line" of the double universe.
The rotation topology of the double universe is like the double coil, or the edge of a Möbius strip.
For bosons, the two copies of the Möbius universe are identical, but for fermions, the wave function on the other side has the sign flipped.
Maybe it is too complicated to introduce such a double universe. Can we accomplish the same with the double particle model?
A triple particle model might work better: in the same circle with the main electron, we have an extra electron going at 1/2 of the speed of light to the same direction, and an extra positron going in the opposite direction at 1/2 the speed of light. That would explain why the magnetic moment is double of what we would expect classically.
The triple model explains the angular momentum and the magnetic moment, but how do we explain that the whole triple system has just the energy of a single electron? Can we assume that the extra electron-positron pair is tunneling with zero energy?
The electron in this model is really two electrons plus a positron. The extra particles are observable only in the spin and the magnetic moment of the electron.
Something called the Compton "scattering radius" of the electron is roughly double the radius of the circle in our model. Is that a coincidence?
What do the 180 and 360 degree rotations mean for the spin?
https://en.m.wikipedia.org/wiki/Spin_(physics)
In Wikipedia, the authors consider the probability amplitudes in a given state, for spin_z being up or down. If we do an infinitesimal rotation, we can calculate the amplitudes for spin_z up or down in the new coordinates.
If we let the amplitudes change only infinitesimally, that is, the mapping to new amplitudes does not contain big jumps, then apparently, a 360 degree rotation does flip the sign of the probability amplitudes.
How to interpret that? The physical system itself cannot change when we just turn our coordinate system.
The Pauli exclusion principle is a very real law for a real physical system. Why would it follow from something we prove for a coordinate system?
Maybe we should assume that we measure the spin of the electron repeatedly with a physical device, turning the device slightly after each measurement. We calculate the path integral for various outcomes after our device has made a 360 degree rotation. The measurements do a real change to the physical system. Let us try to repeat Schwinger's proof after this clarification.
We calculate the vacuum expectation value
< 0 | R ϕ(x) ϕ(-x) | 0 >.
…
What does it mean for two particles to be "in the same quantum state"?
The Pauli exclusion principle states that two half-integer spin particles cannot exist in the "same quantum state". If we have an atom with several electrons, what does it mean that electrons A and B are in the "same quantum state"? The wave equation has to be solved for all particles simultaneously. The wave function exists in a space with lots of spatial dimensions and maybe also many time dimensions. Since electrons are indistinguishable, it is not even clear what we mean by "an electron" or "two electrons" being in a certain quantum state.
Consider the classical particle in a box. The ground state is well-defined and clear. If we put two particles in the box, and the particles have repulsion, then most probably they cannot occupy the wave function of the ground state of a single particle. What do we mean by the statement that "the two electrons now have different quantum states"? We probably cannot measure one electron without disturbing the other.
We should describe a realistic physical measurement where we measure both electrons, and what we claim about the measurement outcome. For example, if we measure the spin of one of the electrons of the helium atom and then remove it by some method, what does a spin measurement give for the remaining electron?
Is there a classical analogue for the Pauli exclusion principle? A setup where the wave gets disturbed if its amplitude exceeds some limit? Then the wave energy has to be divided among various low-amplitude waves.
John Baez's ribbon proof
The ribbon proof contains a concrete statement: if we have an electron, and it is subsequently annihilated by a positron in a pair, then the pair electron is like the original electron rotated by 360 degrees. We assume that there is no extra disturbance in the annihilation or pair creation, which could turn the spins.
511 keV
photons
\ / ^
\/ /
/\ /
/ \ /
/ \ / e-
/ e+ \/
/ /\
/ /\
t ^ / \
^ e- 511 keV photons
|
----------> x
|
----------> x
The Feynman way of interpreting the above path is that there is just one electron which scatters first back in time, and then forward in time. It is scattered from the 511 keV photons in the diagram.
Is the above path equivalent to a 360 degree rotation in some sense? If time and space are fully symmetric at very short distances, then scattering back in time might be equivalent to bouncing back from a spatial wall. What happens to the electron spin when it is reflected from a potential? Probably not much because the electron in the hydrogen atom gets reflected all the time and keeps its spin.
In pair production, the electron and the positron often have opposite spins. Positronium annihilates much faster if the spins are opposite.
In the above diagram, we may assume that in the laboratory frame, the handedness of the electron's linear movement versus its rotation (chirality) stays the same.
In pair production, the electron and the positron often have opposite spins. Positronium annihilates much faster if the spins are opposite.
In the above diagram, we may assume that in the laboratory frame, the handedness of the electron's linear movement versus its rotation (chirality) stays the same.
What does the CPT symmetry mean in this setup? The charge changes the sign at the sharp points and time changes the direction. The parity transformation means flipping the sign of one spatial coordinate. If we change the sign of both t and x, it means a 180 degree rotation of the t, x -plane.
If t and x are completely symmetric at short distances, we may as well assume that x is the time axis and t is the space axis. If we use t as the parameter for the rotation of the electron, the the positron line segment has spin_z flipped.
But if we use x as the (time) parameter of the rotation, then spin_z stays the same sign throughout the journey. If x is the time axis, then the path describes two electron bounces from two photons. If an electron scatters from a photon, both have to keep their spin_z, since for the electron its absolute value is 1/2 and for the photon 1.
The journey seems uneventful for spin_z. Is there a reason why the probability amplitude of the electron wave function would flip sign? Does it flip the sign when an electron scatters from a photon when t is the time coordinate?
If the electron scatters from a low-energy photon, then it would be strange if it would change the phase of the electron wave function significantly. What about a high energy photon in the 511 keV range?
Generally, in wave equations, reflection from an optically dense material causes a 180 degree phase shift. Reflection from thinner material does not cause a phase shift. What kind of a reflection is it when an electron bounces from an energetic photon? If we interpret that the electron is reflected from a steep electric potential barrier, then there is a 180 degree phase shift in the electron wave function. But we can decide if the phase shift is +180 or -180 degrees. This does not prove anything.
Scattering does change the momentum and the wavelength of the scattered particle. This is the reason why measuring the particle path spoils the interference pattern in the double slit experiment.
If t and x are completely symmetric at short distances, we may as well assume that x is the time axis and t is the space axis. If we use t as the parameter for the rotation of the electron, the the positron line segment has spin_z flipped.
But if we use x as the (time) parameter of the rotation, then spin_z stays the same sign throughout the journey. If x is the time axis, then the path describes two electron bounces from two photons. If an electron scatters from a photon, both have to keep their spin_z, since for the electron its absolute value is 1/2 and for the photon 1.
The journey seems uneventful for spin_z. Is there a reason why the probability amplitude of the electron wave function would flip sign? Does it flip the sign when an electron scatters from a photon when t is the time coordinate?
If the electron scatters from a low-energy photon, then it would be strange if it would change the phase of the electron wave function significantly. What about a high energy photon in the 511 keV range?
Generally, in wave equations, reflection from an optically dense material causes a 180 degree phase shift. Reflection from thinner material does not cause a phase shift. What kind of a reflection is it when an electron bounces from an energetic photon? If we interpret that the electron is reflected from a steep electric potential barrier, then there is a 180 degree phase shift in the electron wave function. But we can decide if the phase shift is +180 or -180 degrees. This does not prove anything.
Scattering does change the momentum and the wavelength of the scattered particle. This is the reason why measuring the particle path spoils the interference pattern in the double slit experiment.
How do we prove the Pauli exclusion principle from the "interchange" property of the wave function?
In quantum field theory, we decide in which order we apply creation operators to the vacuum. We can decide to create the "first" a particle at a spacetime point x and "after" that at y. If x and y are spacelike separated, then there is no temporal order for x and y. The order of creation is just a property of the formal model which we are using, it is not a physical thing.
There is no isolated creation operator in nature. An electron-positron pair is always created together.
How do we prove the Pauli exclusion principle in our formal model? We should show that the two electrons in the helium atom ground state cannot both have the spin up.
Suppose that we have two electrons x and y far away from an alpha particle. How do we prove that the electrons cannot end up in the same quantum state around the alpha particle?
The usual reasoning seems to go like this:
1. We can describe the initial setup either as the x electron being created the "first" and the y electron created the "second". But we can as well describe the setup with y created the first.
2. Since the wave functions in these two descriptions have the sign flipped, then the sum of the two wave functions in the hypothetical final state around the alpha particle is zero.
3. Alternatively, we can "interchange" the electrons in the final state. Because the sign of the wave function is flipped, but the physical setup stays exactly the same, we conclude that the wave function has to stay the same. The wave function has to be zero.
The reasoning above is flawed. If we have just one electron, we can describe it with a wave function Ψ or -Ψ, or any
exp(i φ) Ψ,
where φ is a real number. If the electron ends in a quantum state, the sum of the two alternative wave functions is zero. Did we show that the electron cannot end up in any quantum state? No. The reasoning is wrong. We cannot use two alternative descriptions for the same physical setup and later sum those two descriptions to prove something.
The whole concept of the wave function flipping sign in an "interchange" is nonsensical. Quantum mechanics is agnostic about changing the phase of the wave function. We are always allowed to flip the sign of any wave function by multiplying it by -1, and we get an equivalent wave function. There is no need to flip the sign when we perform an "interchange". The interchange itself is not a physical operation but just a change of the description of the system. If we change the description in one way, what prevents us from changing the description again by multiplying by -1? Then there is no flip of the sign.
What about interchanging electrons x and y physically? Suppose that we have the electrons in a closed box and we describe the initial state with a wave function. After some time has passed, we have no way of knowing which electron is which, because they are indistinguishable particles.
Then we let these two electrons approach the alpha particle and form a helium atom. Can we prove that the electrons cannot have the same quantum state in the end? When the electrons approach the alpha particle, they will emit photons. We will have a system of at least 4 particles, probably many more.
Is there a destructive interference which prevents the two electrons ending up in the same state?
Suppose that we have two electrons x and y far away from an alpha particle. How do we prove that the electrons cannot end up in the same quantum state around the alpha particle?
The usual reasoning seems to go like this:
1. We can describe the initial setup either as the x electron being created the "first" and the y electron created the "second". But we can as well describe the setup with y created the first.
2. Since the wave functions in these two descriptions have the sign flipped, then the sum of the two wave functions in the hypothetical final state around the alpha particle is zero.
3. Alternatively, we can "interchange" the electrons in the final state. Because the sign of the wave function is flipped, but the physical setup stays exactly the same, we conclude that the wave function has to stay the same. The wave function has to be zero.
The reasoning above is flawed. If we have just one electron, we can describe it with a wave function Ψ or -Ψ, or any
exp(i φ) Ψ,
where φ is a real number. If the electron ends in a quantum state, the sum of the two alternative wave functions is zero. Did we show that the electron cannot end up in any quantum state? No. The reasoning is wrong. We cannot use two alternative descriptions for the same physical setup and later sum those two descriptions to prove something.
The whole concept of the wave function flipping sign in an "interchange" is nonsensical. Quantum mechanics is agnostic about changing the phase of the wave function. We are always allowed to flip the sign of any wave function by multiplying it by -1, and we get an equivalent wave function. There is no need to flip the sign when we perform an "interchange". The interchange itself is not a physical operation but just a change of the description of the system. If we change the description in one way, what prevents us from changing the description again by multiplying by -1? Then there is no flip of the sign.
Is there a flip of the sign of the wave function when electrons are interchanged physically?
What about interchanging electrons x and y physically? Suppose that we have the electrons in a closed box and we describe the initial state with a wave function. After some time has passed, we have no way of knowing which electron is which, because they are indistinguishable particles.
Then we let these two electrons approach the alpha particle and form a helium atom. Can we prove that the electrons cannot have the same quantum state in the end? When the electrons approach the alpha particle, they will emit photons. We will have a system of at least 4 particles, probably many more.
Is there a destructive interference which prevents the two electrons ending up in the same state?
The exchange interaction
The Wikipedia article above explains the Pauli exclusion principle in a simple setup through the energies of the states. Spins pointing in the same direction make the energy of the state higher. The magnetic repulsion of the two tiny magnetic needles (electrons) is negligible, but there is another effect.
Looking at energies of the states is a concrete way of deriving the Pauli principle and should not suffer from the conceptual error which we pointed out in the previous section. Does the argument in Wikipedia derive the result without making the conceptual error in the ordinary proofs of the Pauli principle?
Other proofs
Streater and Wightman have an axiomatic proof in their 1964 book. We will look at it.
Wolfgang Pauli has a heuristic proof in a 1940 paper in Phys. Rev.
Pauli shows that for fields of an integer spin, it is impossible to define a particle density in a satisfactory way, and for fields of a half-integer spin, it is not possible to define an energy density which is positive definite.
Pauli concludes that field theory cannot be interpreted as a one-body problem. Pauli says that relativistic electromagnetic theories have charged particles which are created and absorbed in pairs, and uncharged particles which are emitted and absorbed singly.
Pauli argues that half-integer spin fields change sign if we "interchange" the particles. In that, he seems to follow the typical path in the proof.
http://www.feynmanlectures.caltech.edu/III_04.html
Richard P. Feynman has presented an argument based on scattering.
Pauli shows that for fields of an integer spin, it is impossible to define a particle density in a satisfactory way, and for fields of a half-integer spin, it is not possible to define an energy density which is positive definite.
Pauli concludes that field theory cannot be interpreted as a one-body problem. Pauli says that relativistic electromagnetic theories have charged particles which are created and absorbed in pairs, and uncharged particles which are emitted and absorbed singly.
Pauli argues that half-integer spin fields change sign if we "interchange" the particles. In that, he seems to follow the typical path in the proof.
http://www.feynmanlectures.caltech.edu/III_04.html
Richard P. Feynman has presented an argument based on scattering.
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