Quantum fields with noncommutative target spaces

Quantum field theories (QFT’s) on noncommutative spacetimes are currently under intensive study. Usually such theories have world sheet noncommutativity. In the present work, instead, we study QFT’s with commutative world sheet and noncommutative target space. Such noncommutativity can be interpreted in terms of twisted statistics and is related to earlier work of Oeckl [R. Oeckl, Commun. Math. Phys. 217, 451 (2001).], and others [A. P. Balachandran, G. Mangano, A. Pinzul, and S. Vaidya, Int. J. Mod. Phys. A 21, 3111 (2006); A. P. Balachandran, A. Pinzul, and B. A. Qureshi, Phys. Lett. B 634, 434 (2006); A. P. Balachandran, A. Pinzul, B. A. Qureshi, and S. Vaidya, arXiv:hep-th/0608138; A. P. Balachandran, T. R. Govindarajan, G. Mangano, A. Pinzul, B. A. Qureshi, and S. Vaidya, Phys. Rev. D 75, 045009 (2007); A. Pinzul, Int. J. Mod. Phys. A 20, 6268 (2005); G. Fiore and J. Wess, Phys. Rev. D 75, 105022 (2007); Y. Sasai and N. Sasakura, Prog. Theor. Phys. 118, 785 (2007)]. The twisted spectra of their free Hamiltonians has been found earlier by Carmona et al. [J. M. Carmona, J. L. Cortes, J. Gamboa, and F. Mendez, Phys. Lett. B 565, 222 (2003); J. M. Carmona, J. L. Cortes, J. Gamboa, and F. Mendez, J. High Energy Phys. 03 (2003) 058]. We review their derivation and then compute the partition function of one such typical theory. It leads to a deformed blackbody spectrum, which is analyzed in detail. The difference between the usual and the deformed blackbody spectrum appears in the region of high frequencies. Therefore we expect that the deformed blackbody radiation may potentially be used to compute a Greisen-Zatsepin-Kuzmin cutoff which will depend on the noncommutative parameter $\theta$.

Figure 1

The shady region corresponds to the blackbody radiation with temperature $T=1\times {10}^4\mathrm{K}$. The solid line delimits the region of the new radiation, which is given by the deformed theory for $\sigma =1\times {10}^{-15}\mathrm{m}$. The dashed line gives the radiation curves of the deformed theory with null $\sigma$. Observe that in this case, there is a divergence in the energy density at high frequencies.

Figure 2

(a) When $\sigma$ is non-null ($\sigma =1\times {10}^{-15}\mathrm{m}$ and $T=1\times {10}^4\mathrm{K}$ in both graphs), there exists a very small deviation with respect to the usual blackbody radiation at low frequencies. The graph shows the energy density curve for the new radiation with $\theta =1.7\times {10}^{-14}\mathrm{m}$ (solid line) and $\theta =1.9\times {10}^{-14}\mathrm{m}$ (dashed line). (b) A not-so-small deviation is found for high frequencies.

Figure 3

Another example that the frequency and the highest amplitude depend on $\sigma$ and $\theta$. The dependence of the highest amplitude on $\sigma$ is stronger than the dependence on $\theta$. For both graphs, $T=1\times {10}^4\mathrm{K}$.