# internal flavor symmetry of the N left-handed complex Weyl spinors v.s. N real Majorana spinors: U(N) vs. O(2N) or O(N)

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Consider 4D spacetime, it seems that for massless particles, we can easily change

- the left-handed complex Weyl spinor basis (2 component in complex $\mathbb{C}$ for Euclidean spacetime Spin(4))

to

- the real Majorana spinor basis (4 component in real $\mathbb{R}$ for Euclidean spacetime Spin(4))

So naively, we can change N left-handed complex Weyl spinors to N real Majorana spinors

However, the internal flavor symmetry of the N left-handed complex Weyl spinors is $G_{Weyl}=$ U(N).

Puzzle 1: What are the internal flavor symmetry of N real Majorana spinors?  $G_{Majorana}=?$  Is that O(N) or O(2N)?

Puzzle 2: Why the internal flavor symmetry of the N left-handed complex Weyl spinors different from N real Majorana spinors?

Thanks for your attention !

asked Apr 5, 2020

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