Abstract:
When quantizing a classical system, one is typically concerned with representing the generalized coordinates q_i and conjugate momenta p_i as operators q̂_i$\,$ and p̂_j $\,$which act on some Hilbert space, and satisfy the canonical commutation relations, [q̂_i , p̂_j] = iδ_ij.  For a finite number of coordinates and momenta these representations are almost always unitarily equivalent. However, for theories with an infinite number of q̂_i’s and p̂_j ’s, such as those found in quantum field theory, there exists infinitely many inequivalent representations. 

At first glance the light-front formulation of quantum field theory does not appear to provide the expected inequivalent representations.  However, several interesting characteristics of field theories arise as a direct result of this inequivalence. The results of any experiment are independent of the formulation that is chosen, so theories built from a light-front formulation should be equivalent to those built from the usual formulation.  By describing the theory in terms of an operator algebra and a vacuum functional, we can see that the inequivalent representations are not eliminated, but rather moved from a property of the vacuum functional, to a property of the operator algebra.