Question

The Pauli matrices satisfy the Clifford algebra: {σi,σj}=2δij

. For this one exercise I'm working on, I have to consider Pauli matrices with the index raised to make sense of the signs. Using the signature (+−−−) and gij=−δji we get: σi,σj}=σiσj+σjσi=gin(σnσj+σjσn)=−δin(σnσj+σjσn)=−(σiσj+σjσi)=−{σi,σj}.

However, since {σi,σj}=2δij, we get δij=−δij, which does not seem right. What am I doing wrong? My best guess is that the Clifford algebra does not apply to one raised and one lowered index in the Minkowski sense.

Answer 1

The inconsistencies arise from mixing covariant and contravariant indices without correctly applying the metric tensor in the Clifford algebra for Pauli matrices. Ensuring that operations comply with the metric used resolves the contradiction and maintains the anticommutation relation.

Step-by-step explanation:

The confusion in the application of Clifford algebra to Pauli matrices arises from the misuse of the metric tensor in the signature (+−−−), which causes an apparent contradiction. The key issue is that when dealing with Clifford algebras, it's crucial to maintain the consistency of indices, i.e., either both covariant (lower indices) or both contravariant (upper indices). Mixing indices with different variance, especially in a non-Euclidean metric such as Minkowski spacetime, without proper use of the metric tensor to raise or lower indices will lead to incorrect results. When the indices are properly handled using the metric tensor, the anticommutation relationship expressed in Clifford algebra remains consistent without yielding a self-contradiction like δij = −δij. Therefore, in the context of Clifford algebra and Pauli matrices, it's important to ensure the operations comply with the metric being used.