In four dimensions, left and right chiral fermion can be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\psi_L=

\begin{pmatrix}

\psi_+\\

0

\end{pmatrix},\qquad

\psi_R=

\begin{pmatrix}

0\\

\psi_-

\end{pmatrix},

[/tex]

respectively, where [itex]\psi_+[/itex] and [itex]\psi_-[/itex] are some two components spinors(Weyl spinors?). In this representation, the chirality operator [itex]\gamma_5[/itex] is written as

[tex]

\gamma_5=

\begin{pmatrix}

\mathbb{1}&0\\

0&-\mathbb{1}

\end{pmatrix}.

[/tex]

In five dimensions, the fifth [itex]\gamma[/itex]-matrix [itex]\Gamma^4[/itex] can coincide with the four dimensional chiral operator [itex]\gamma_5[/itex]. The other matrices are same as four dimensional ones:

[tex]

\Gamma^\mu=\gamma^\mu\qquad\mu=0,1,2,3\\

\Gamma^4=\gamma_5.

[/tex]

In this representation, a five dimensional fermion [itex]\Psi=\psi_L[/itex] seems to be "chiral" if I define the chiral operator as [itex]\Gamma_6=\Gamma^4=\gamma_5[/itex].

However in general, there is no notion of chirality in odd dimensions. Why the above [itex]\Psi[/itex] cannot be a chiral fermion?

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# Why chiral fermions don't exist in odd dimensions?

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