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A systematic method for the computation of finite temperature ($T$) crossoverfunctions near quantum critical points close to, or above, their upper-criticaldimension is devised. We describe the physics of the various regions in the $T$and critical tuning parameter ($t$) plane. The quantum critical point is at$T=0$, $t=0$, and in many cases there is a line of finite temperaturetransitions at $T = T_c (t)$, $t < 0$ with $T_c (0) = 0$. For the relativistic,$n$-component $\phi^4$ continuum quantum field theory (which describes latticequantum rotor ($n \geq 2$) and transverse field Ising ($n=1$) models) the uppercritical dimension is $d=3$, and for $d<3$, $\epsilon=3-d$ is the controlparameter over the entire phase diagram. In the region $|T - T_c (t)| \ll T_c(t)$, we obtain an $\epsilon$ expansion for coupling constants which then areinput as arguments of known {\em classical, tricritical,} crossover functions.In the high $T$ region of the continuum theory, an expansion in integer powersof $\sqrt{\epsilon}$, modulo powers of $\ln \epsilon$, holds for allthermodynamic observables, static correlators, and dynamic properties at allMatsubara frequencies; for the imaginary part of correlators at realfrequencies ($\omega$), the perturbative $\sqrt{\epsilon}$ expansion describesquantum relaxation at $\hbar \omega \sim k_B T$ or larger, but fails for $\hbar\omega \sim \sqrt{\epsilon} k_B T$ or smaller. An important principle,underlying the whole calculation, is the analyticity of all observables asfunctions of $t$ at $t=0$, for $T>0$; indeed, analytic continuation in $t$ isused to obtain results in a portion of the phase diagram. Our method alsoapplies to a large class of other quantum critical points and their associatedcontinuum quantum field theories.Comment: 36 pages, 4 eps figure

Theory of finite-temperature crossovers near quantum critical points close to,or above, their upper-critical dimension