# Zee's use of Kronecker Product in “QFT in a Nutshell” to represent Dirac matrices

In his book Quantum Theory in a Nutshell (2nd edition, p. 94), Zee describes the Dirac gamma matrices and lists a representation using Pauli matrices and the identity matrix. For example he writes

$$\gamma^0 = \begin{pmatrix}I&0\\0&-I\end{pmatrix}= I\otimes \tau_3,$$

where (I assume that) $\tau_3 = \sigma_3 = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$ is the third Pauli matrix. Yet, when I look up the definition of the Kronecker Product, $\otimes$, or ask wolframalpha for the product this should rather be

$$\gamma^0 = \begin{pmatrix}I&0\\0&-I\end{pmatrix}= \tau_3\otimes I .$$

Since Zee is consistent in reversing the order of the operands on the next page too, I assume that it is not just a typo.

Can someone explain why (it looks like) Zee reverses the order of the operands? Is this a different convention for the definition of $\otimes$ or has it to do with him using $\tau_i$ instead of $\sigma_i$ for the Pauli Matrices?

The ordering of tensor product factors in matrix representations is ultimately a convention: there is no single canonical way to order the tensor product basis (does $|{\uparrow\downarrow} \rangle$ come before or after $|{\downarrow\uparrow} \rangle$?) and the different orderings will produce different matrix representations.

You've found the two possibilities, and they are both equally reasonable (though in fields more directly connected with quantum information, the computational basis does tend to have a canonical ordering by mapping it into binary numbers, i.e. $|01 \rangle$ before $|10\rangle$). When in doubt, refer to the basis ordering in use.

In Zee's case, the basis ordering gets uniquely established as $\{|{\uparrow\uparrow}\rangle, |{\downarrow\uparrow}\rangle, |{\uparrow\downarrow}\rangle, |{\downarrow\downarrow}\rangle\}$ (i.e. opposite to the computational basis, which flips the right-hand side first) at the beginning of the Clifford Algebra section (eq. (3) of chapter II in the 2003 edition), via the equation you quote, and it then continues with that consistent notation throughout that section. Moreover, the text also explicitly confirms that $\sigma$ and $\tau$ both denote the standard Pauli matrices; this seems to be a notational device used for clarity (which is nonstandard as far as I'm aware), as the text restricts $\sigma$ to the left side of the tensor product and $\tau$ to the right-hand factor.

• I may be misunderstanding your answer, but you address the fact that the $\gamma$s can be defined this way or another, right?. Yet my question is rather about how the $\otimes$ is actually defined and as far as I understand it, $I\otimes \tau\neq (\begin{smallmatrix}I& 0 \\ 0&- I\end{smallmatrix})$ contrary to what he writes. – Harald Jul 1 '17 at 20:17
• @Harald It sort of depends on whether you already have an existing four-dimensional vector space that you want to factor out as a tensor product, or whether you're just laying out the structure of the product of known factors, but ultimately it's all the same: it's not so much in defining the tensor product in abstract space, but in mapping it to a specific matrix structure. The abstract tensor product is unique, but the matrix representation isn't (as noted above), so if you want to define the tensor product of matrices, there are two equivalent ways. – Emilio Pisanty Jul 1 '17 at 20:29

You are correct that the Kronecker product is non-commutative (though the two are permutation equivalent), and commonly defined such that $\tau_3 \otimes I$ is the appropriate ordering.

Having checked the errata for Quantum Field Theory in a Nutshell, it has either not been included as it has not been pointed out, or an alternate convention has been adopted implicitly.

Other sources agree with the ordering in Mathematica as well as other sites, such as Schafer's An Introduction to Nonassociative Algebras.