High energy modifications of blackbody radiation and dimensional reduction

Quantization prescriptions that realize generalized uncertainty relations (GUP) are motivated by quantum gravity arguments that incorporate a fundamental length scale. We apply two such methods, polymer and deformed Heisenberg quantization, to scalar field theory in Fourier space. These alternative quantizations modify the oscillator spectrum for each mode, which in turn affects the blackbody distribution. We find that for a large class of modifications, the equation of state relating pressure $P$ and energy density $\rho$ interpolates between $P=\rho/3$ at low $T$ and $P=2\rho/3$ at high $T$, where $T$ is the temperature. Furthermore, the Stefan-Boltzman law gets modified from $\rho \propto T^{4}$ to $\rho \propto T^{5/2}$ at high temperature. This suggests an effective reduction to 2.5 spacetime dimensions at high energy.