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I have read about $bc$ ant $\beta\gamma$ system. These are the fermionic and bosonic ghost system respectively. My question is

  1. Why these systems are called ghost system?

  2. What is the spin for these two systems?

  3. I mean is there spin integer or half-integer?

Qmechanic
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sahi
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  • I have read in Polchinski's vol 1 second chapter, section 2.5 . The action for the $bc$ system is given in equation (2.5.4) – sahi Apr 21 '17 at 17:48
  • Related (in the sense that the answers there contain your answer): https://physics.stackexchange.com/q/197573/50583 – ACuriousMind Apr 21 '17 at 18:08

1 Answers1

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In $d=2$ Conformal Field Theory a holomorphic field $\mathcal{O}(z)$, i.e. $\partial_{\bar{z}}\mathcal{O}(z)=0$, with conformal weight $h$ has also spin $h$. Just apply the rotation $z\rightarrow e^{i\theta}z$ to convince yourself. The $bc$ ghost system has conformal weights $h_b=2$ and $h_c=-1$, while the $\beta\gamma$ ghost system has conformal weight $h_\beta=3/2$ and $h_\gamma= -1/2$. Note that those fermions has integer spin while the bosons has semi-integer spins, the opposite of the spin-statistic, characterizing them as ghosts (if you assume that your theory obey spin-statistic for the physical states).

Nogueira
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