# Internal flavor symmetry of the $N$ left-handed complex Weyl spinors v.s. $N$ real Majorana spinors: ${\rm U}(N)$ vs. ${\rm O}(2N)$ or ${\rm O}(N)$

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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?

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart

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The symmetry depends on the Lagrangian.

Let $$\gamma^\mu$$ be a real representation of the Dirac matrices for 4d spacetime, and define $$\Gamma := \gamma^0\gamma^1\gamma^2\gamma^3$$. Then $$\Gamma$$ is also a real matrix, and $$\Gamma^2=-1$$.


This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user Chiral Anomaly
answered Apr 5, 2020 by (70 points)

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