On the role of dimensionality in many-body theories of magnetic long-range order

Starting from general considerations concerning phase transitions and the many-body problem, a pedagogical introduction to the Mermin-Wagner theorem is given. Particular attention is paid to how the Mermin-Wagner theorem fits in with more general methods in many-particle physics. An attempt at an assessment of its validity for approximative methods is made by contrasting the results obtained within Onsager reaction-field theory and those obtained by the well-known Tyablikov method.