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We study the quantum Ising model in D dimensions with the equation of motiontechnique, in combination with the Majorana representation for spins. Thedecoupling scheme used for the Green's functions is based on the hierarchy ofcorrelations in position space. To lowest order this method reproduces the wellknown mean field phase diagram and critical exponents. When correlationsbetween spins are included, we show how the appearance of thermal fluctuationsand magnons strongly affect the physical properties. We demonstrate that in 1Dand for B=0, thermal fluctuations completely destroy the ordered phase, andthat near the quantum critical point, the quantum model displays differentbehavior than its classical counterpart. We discuss the connection with theDyson's equation formalism and the explicit form of the self-energies.

Hierarchy of correlations for the Ising model in the Majorana representation