Seminar of Theoretical Physics - Adam Hložný (29.5.2018)

Tuesday 29.5.2018 at 14:00, Lecture room F2/125

---

Adam Hložný:
Noncommutative Dirac operator

Abstract: 
We present approach of constructing non-commutative Dirac operator of spin $\frac{1}{2}$ particle, which is very important object for both non-commutative geometry and relativistic quantum field theory. We build on the fact that \emph{Poincaré algebra is sub-algebra of $su\left(2,2\right)$}. We can thus use oscillator representation of $su\left(2,2\right)$ to build representation of Poincaré algebra. During the process, we find that just fundamental representation does not suffice, since it is impossible to define massive theory. This is due to the fact that \textit{Casimir element} $\hat{m}^2 = \hat{p}_{\mu} \hat{p}^{\mu}$ of representation of Poincaré algebra is equal to zero. The same situation arises for anti-fundamental representation: $\tilde{m}^2 = \tilde{p}_{\mu} \tilde{p}^{\mu}=0$. This phenomenon is not at all uncommon, one encounters it when dealing with (spinor) representations of Lorentz algebra - it is not possible to have massive theory of just left-handed or right-handed fermions. Solution to this is analogous to the Lorentz algebra case - we take direct sum of fundamental and anti-fundamental representation. This is fundamental representation of $spin\left(4,2\right)$ algebra (or conformal algebra $so\left(4,2\right)$, since these two are isomorphic and $Spin\left(4,2\right)$ is double cover of $SO\left(4,2\right)$ at the group level). It is then natural that final momentum operator $P_{\mu}$ is given as $\hat{p}_{\mu}+\tilde{p}_{\mu}$. Dirac operator is then constructed in $p$-representation and eigenvalue problem is examined.

---