In 2d spacetime you have log-Sobolev inequalities that control the strength of the quantum field's potential energy in terms of its kinetic energy. Most of the success in 2d constructive quantum field theory is based on these; try this book for details:
• John Baez, Irving Segal and Zhengfang Zhou, Introduction to Algebraic and Construtive Quantum Field Theory.
In higher-dimensional spacetimes these inequalities don't apply, so we need more sophisticated methods.
Essentially, as we go to higher and higher dimensions it's possible for a field to undergo larger and larger fluctuations without much cost in kinetic energy (or, alternatively, action). Understanding Sobolev inequalities and how they work in different dimensions is a good way to start getting a feeling for this. The increased difficulty in higher dimensions due to this effect shows up in all work on analysis, not just quantum field theory. For example, the quantum mechanics of atoms and molecules (Schrödinger's equation with Coulomb interaction) would be badly behaved if there were an extra dimension of space.
This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user John Baez