# Physics Notes by Jakob Schwichtenberg

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quantum_field_theory:notions:spin_statistics_theorem

# The Spin-Statistics Theorem

A curious property of nature is that if we exchange two identical spin $1/2$ particles, the state vector stays the same except for a minus sign. In contrast, if we exchange two identical spin $0$ or two spin $1$ particles the state vector stays exactly the same.

The spin-statistics theorem tells us that spin $0,1/2,1$ particles necessarily have these properties.

## Pauli's Exclusion Principle

Take note that the fact that if we exchange two identical spin $1/2$ particles we get a minus sign, necessarily means that there never can be two spin $1/2$ in exactly the same state

This follows, because $$| \Psi_1 \Psi_2 \rangle = - | \Psi_2 \Psi_1 \rangle$$ is for $\Psi_1 = \Psi_2$ only possible if $| \Psi_1 \Psi_2 \rangle =0$

## Quotes

everyone knows the spin-statistics theorem, but no one understands it

Pauli and the Spin-Statistics Theorem by I. Duck and E. Sudarshan

The Spin-Statistics Theorem, which states that identical half-integral spin particles satisfy the Pauli Exclusion Principle and Fermi-Dirac statistics which permit no more than one particle per quantum state; identical integral spin particles do not, but satisfy Bose-Einstein statistics which permits any number of particles in each quantum state - stands as a fact of nature.

Pauli and the Spin-Statistics Theorem by I. Duck and E. Sudarshan

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.

## Proofs

Sudarshan's Proof:

Topological Marker Proof attempts:

Fermions aren't exactly the same when they are rotated by $360^\circ$, but get a minus sign.

Now, interchanging two objects is topologically the same as rotating either one of them by $360^\circ$. (To see this, first grasp the two ends of a belt, one end in each hand; then interchange the position of your hands. You have introduced a “twist” which is topologically equivalent to having rotated one end of the belt by $360^\circ$.) Thus, when fermions are interchanged, one must keep track of this “implied rotation” and the phase shift, sign change, and destructive interference to which it gives rise.

Answer to question #7 by Roy R. Gould

Weinberg's Proof: