## On tensor product generators of the gamma matrices.

Posted by peeterjoot on June 20, 2011

# Motivation.

In [1] he writes

The Pauli matrices I had seen, but not the matrices, nor the notation. Strangerep in physicsforums points out that the is a Kronecker matrix product, a special kind of tensor product [2]. Let’s do the exersize of reverse engineering the matrices as suggested.

# Guts

Let’s start with . We want

By inspection we must have

Thus . How about ? For that matrix we have

Again by inspection we must have

so

This one is also just the Pauli matrix. For the last we have

Our last tau matrix is thus

Curious that there are two notations used in the same page for exactly the same thing? It appears that I wasn’t the only person confused about this.

# The bivector expansion

Zee writes his wedge products with the commutator, adding a complex factor

Let’s try the direct product notation to expand and . That first is

which is what was expected. The second bivector, for is zero, and for is

# References

[1] A. Zee. *Quantum field theory in a nutshell*. Universities Press, 2005.

[2] Wikipedia. Tensor product — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 21-June-2011]. http://en.wikipedia.org/w/index.php?title=Tensor_product&oldid=418002023.

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