I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum systems, operators can be expressed in an oscillator representation using creation and annihilation operators. However, this doesn't seem to be the case for ${\cal N}=4$ SYM theory.
Why is there no oscillator representation for operators such as the Hamiltonian or the dilatation operator in planar ${\cal N}=4$ SYM theory?
What are the requirements for an operator to have an oscillator representation?
From my understanding, having a discrete spectrum is one of the requirements for an oscillator representation. However, ${\cal N}=4$ SYM theory, with its complex gauge dynamics and supersymmetry, does not naturally lend itself to such a description.
Any insights into these points or references that discuss the criteria for oscillator representations in quantum field theories would be greatly appreciated!
