simple generalization of the dynamics of classical critical phenomena to the quantum case [4, 28–31] (which may be termed a Gaussian qcp since the fluctuation spectra is determined by the renormalized coefficient of the quadratic term of the variable describing the fluctuat- ing order parameter) is as follows: The spectral function of the fluctuations near a qcp may in general be be written in the scaling form: ξ$_{r}$ is the correlation length at T → 0 which diverges as and k is measured from the wave-vector of the symmetry breaking. d$_{M}$ = d , the physical dimension if the order parameter is that of a conserved quantity, otherwise it can be differ- ent. The correlation length in the time-direction or equivalently the frequency scale of the fluctuations is given by z is the dynamical critical exponent which scales the frequency of fluctuations to their characteristic spatial extent reflecting that the dispersion of the critical fluctuations depends on the nature of the broken symmetry and in general has the form ω ∝ k $^{z}$. Eq.(17) differs from that in classical dynamical critical phenomena in only two ways: the substitution of ( p − p$_{c}$ ) at T = 0 for ( T − T$_{c}$ ) and that the fluctuation frequencies ω also has a scale which is simply the temperature of measurement T , i.e. the distance along the frequency axis from the quantum-transition at ( p = p$_{c}$ ) and T = 0. These two ensure that there is a scale T$_{x}$ ( p ) which gives the crossover from quantum fluctuations at low T to classical fluctuations at high T and which marks the temperature below which fermi-liquid properties are expected. As opposed to classical critical phenomena, the critically-scaled behavior in physical properties is to be expected in the entire regime bounded by the transition temperature T$_{c}$ ( p ) and the crossover scale T$_{x}$ ( p ) and the upper cut-off temperature of the fluctuations, given by the microscopic energy scales (for example, the exchange splitting in a ferro- or antiferro- qcp). This is usually referred to as the quantum-critical regime of the phase diagram. It is useful to take ξ$_{r}$, ξ$_{t}$ as dimensionless, the former normalized to a lattice constant and the latter normalized to the upper cut-off of the fluctuations ω$_{c}$ . Eq. (17) has been written so that χ$_{0}$ is the spectral weight of the critical fluctuations near the critical point, by which 23 ξ$_{t}$ ∝ ξ z $_{r}$. (19) ξ$_{r}$ ∝ | p − p$_{c}$ | − $^{ν}$, (18) χ ”( q, ω, p, T ) = χ$_{0}$ξ − $^{d$_{M}$}$F 1 $^{(}$kξ, ωξ $^{z}$, ω/T $^{)}$, (17)