Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
Feynman Slash Notation
Klein-Gordon Field
Dirac Field
Grassman Variable
Conformal Field Theory
Countdown to the Standard Model
From a framework to a model Yang-Mills Theory
Quantum Electrodynamics
Quantum Chromodynamics
Electroweak Theory
Higgs Mechanism
Standard Model
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Problems with the Standard Model
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
Related De Donder-Weyl Theory

Spin is a Quantum "analog" of Angular Momentum. It can be inserted by hand into Quantum Mechanics through Pauli Theory, or it can be seen as a direct result of the Dirac Equation. The total angular momentum (which is conserved in interactios, etc., etc.) is then the sum of the orbital angular momentum and the intrinsic spin angular momentum.

In Quantum MechanicsEdit this section

In Quantum Mechanics, the Spin Quantum Number is given by the number (integer or half-integer) such that the Spin Angular Momentum is given buy:

$ S=\hbar\sqrt{s\left(s+1\right)} $

The Spin OperatorEdit this section

Spin obeys Commutation Relations analogous to those of the Angular Momentum Operator.

$ \left[S_i, S_j \right] = i \hbar \epsilon_{ijk} S_k $

where $ \epsilon_{ijk} $ is the Levi-Civita Symbol. This implies that:

$ S^2 | s,m\rangle = \hbar^2 s\left(s + 1\right) | s,m\rangle $ $ S_z | s,m\rangle = \hbar m | s,m\rangle. $

The spin raising and lowering operators operate on these eigen vectors as follows:

$ S_\pm | s,m\rangle = \hbar\sqrt{s(s+1)-m(m\pm 1)} | s,m\pm 1 \rangle $


$ S_\pm = S_x \pm i S_y $

Unlike in the case of Angular Momentum, the eigenvectors are not Spherical Harmonics, so they are not just properties of the angles.

We will also include half-integer values of $ s $ and $ m $, which as we will see, will lead to Fermions.

The Spin Quantum Numbers are then quantised as follows:

$ \sigma \in \{-s\hbar , -(s-1)\hbar , \cdots ,+(s-1)\hbar ,+s\hbar\}.\,\! $

Bosons have integer Spins, whereas Fermions have half-integer Spins.

Spin RepresentationsEdit this section

The Spins can also be interpreted as representations of $ SO(N+1) $ and $ SO(N+1/2) $ for a boson and a fermion respectively, where $ N $ is the Spin Quantum Number.