# Quantum Mechanics – What Does This Notation for Spin Mean? $\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$

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In my quantum mechanics courses I have come across this notation many times:
$$\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$$
but I feel like I've never fully understood what this notation actually means. I know that it represents the fact that you can combine two spin 1/2 as either a spin 1 (triplet) or a spin 0 (singlet). This way they are eigenvectors of the total spin operator $$(\vec S_1+\vec S_2)^2.$$ I also know what the tensor product (Kronecker product) and direct sum do numerically, but what does this notation actually represent?

Does the 1/2 refer to the states? Or to the subspaces? Subspaces of what exactly (I've also heard subspaces many times but likewise do not fully understand it). Is the equal sign exact or is it up to some transformation?

And finally is there some (iterative) way to write a product of many of these spin 1/2's as a direct sum?
$$\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}\otimes\dots=\left(\mathbf{1}\oplus\mathbf 0\right)\otimes\mathbf{\frac 1 2}\dots=\dots$$

This is a really deep question and I urge you to go ahed and read about it in the literature i'll give at the end. I'll try to give a glimpse of what this actually means.

In physics we can construct our theories based solely upon symmetries of a system. When talking about angular momentum and spin in non relativistic quantum mechanics, we are talking about a specific set of symmetry, namely $$SU(2)$$ symmetry. $$SU(2)$$ is a lie group and, being a group, it's an abstract object. To make it something useful we use, what is called as a representation. There are many representation of $$SU(2)$$ and the one we're interested in is the spinorial representation.

The spinorial is the fundamental representation of $$SU(2)$$ since all representations can be constructed from tensor product of spinors. In physical terms this means that you can construct composite systems just by using spin $$1/2$$ particles. What you gave is how to construct a spin $$1$$ or spin $$0$$ from two spin $$1/2$$ systems $$\mathbf{\frac{1}{2}\otimes\frac{1}{2} = 0\oplus 1}$$

what do this numbers indicate? A number written in boldface gives the dimension (which is $$2j+1$$ where $$j$$ is the boldface number) of an irriducible representation of that group. What this implies is that you can decompose a composite system of two spin $$1/2$$ particles into two irriducible representation of a spin $$0$$ system and a spin $$1$$ system.

If all of this seems confusing, it's normal, it's a lot of stuff. I would suggest the following reading if you want to get a better undestanding

• Lie algebras in particle physics, Georgi
• Group Theory in a Nutshell for Physicists, Zee
• Group Theory, a physicist's survey, Ramond