Theory of Critical Phenomena with Long-Range Temporal Interaction

We develop a systematic theory for the critical phenomena with memory in all spatial dimensions, including $d<d_c$, $d=d_c$, and $d>d_c$, the upper critical dimension. We show that the Hamiltonian plays a unique role in dynamics and the dimensional constant $\mathfrak{d}_t$ that embodies the intimate relationship between space and time is the fundamental ingredient of the theory. However, its value varies with the space dimension continuously and vanishes exactly at $d=4$, reflecting reasonably the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Such variations of the temporal dimension save all scaling laws though the fluctuation-dissipation theorem is violated. Various new universality classes emerge.