  We show that the heat transport between two bodies, mediated by
electromagnetic fluctuations, can be controlled with an intermediate quantum
circuit - leading to the device concept Mesoscopic Photon Heat Transistor
(MPHT). Our theoretical analysis is based on a novel Meir-Wingreen-Landauer
type of conductance formula, which gives the photonic heat current through an
arbitrary circuit element coupled to two dissipative reservoirs at finite
temperatures. As an illustration we present an exact solution for the case when
the intermediate circuit can be described as an electromagnetic resonator. We
discuss in detail how the MPHT can be implemented experimentally in terms of a
flux-controlled SQUID circuit.
