In his textbook Quantum Field Theory: A Modern Approach, Michio Kaku first describes quantum field theory as
Later on he says,
where is the number of degrees of freedom of the system. How compatible are these descriptions?
Consider the group-theoretic description of n-sided polygons given by dihedral groups. A regular polygon with n sides has 2n symmetries, half rotational and half reflective. As one intuitively expects the n-gon to approach a circle and thus would expect that the infinite dihedral group would thus describe the group of symmetries of a circle. But, if I understand this correctly, it doesn’t. In fact the infinite dihedral group is actually an isometry group of which is a line!
Given Kaku’s two definitions, then, I am impelled to ask,
- is QFT <em>really</em> the limit of QM for as ; and
- if it is, then what might we learn by instead considering the limit of large ?
Does anyone know if this has been done?