Planck's constant has units of action (joule-seconds) instead of units of energy (joules), the quantum kernel is | k | 2 + m 2 {\displaystyle {\sqrt {|k|^{2}+m^{2}\,}}\,} instead of 1 2 ( | k | 2 + m 2 ) {\displaystyle {\scriptstyle {\frac {1}{2}}}(|k|^{2}+m^{2})} (the quantum kernel is nonlocal from a classical heat kernel viewpoint, but it is local in the sense that it does not allow signals to be transmitted),[citation needed] the quantum vacuum state is Lorentz invariant (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz invariant, but the Gibbs probability density is not a Lorentz invariant initial condition).