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There are some posts about the reps of SO, but I'm confused about some physical understanding of this.

We define types of fields depending on how they transform under a Lorentz transformation, i.e. which representation of SO(3,1) they carry.

The scalar carries the trivial rep, and lives in a 1-dim vector space.

The vector carries a rep generated by 4x4 matrices, and lives in a 4-dim vector space, i.e. it is a 4-column.

Weyl spinors and Dirac spinors carry a irreducible 2-dim rep and a reducible 4-dim rep respectively.

I don't quite understand how to think of tensors. In particular, two things:

1. What objects carry the adjoint representation? This is generated by 6x6 matrices, so should act on 6-columns?

2. I read that a rank-2 tensor can be thought of something that transforms under the tensor product of two 4x4 matrices, i.e. a 16x16 matrix. Hence it should be a 16-column? How can I reconcile with writing a tensor as usual as a 4x4 matrix?

Thanks a lot!

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# Adjoint representation of Lorentz group

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