Equivalence of light-front quantization and instant-time quantization

Abstract

Commutation or anticommutation relations quantized at equal instant time and commutation or anticommutation relations quantized at equal light-front time not only cannot be transformed into each other, they take completely different forms. While they would thus appear to describe different theories, we show that this is not in fact the case. By looking not at equal times but at unequal times, we show that unequal instant-time commutation or anticommutation relations are completely equivalent to unequal light-front time commutation or anticommutation relations. Light-front quantization and instant-time quantization are thus the same, with it being only the restriction to equal times that makes them look different. However for fermions there is a caveat, as the light-front anticommutation relations involve projection operators acting on the fermion fields. Nonetheless, not only can one still derive fermion unequal light-front time anticommutators starting from unequal instant-time ones, one can even derive unequal instant-time fermion anticommutators starting from unequal light-front time anticommutators even though the fermion projection operators that are relevant in the light-front case are not invertible. To establish the equivalence for gauge fields we present a quantization procedure that does not involve zero-mode singularities that are commonly encountered in light-front gauge field studies. We also study time-ordered products of fields, and again show the equivalence despite the fact that there are additional terms in the fermion light-front case. We establish our results first for free theories, and then to all orders in interacting theories though comparison of the instant-time and light-front Lehmann representations. Finally, we compare instant-time Hamiltonians and light-front Hamiltonians, and show that in the instant-time rest frame they give identical results.