Revisiting Quantization of Gauge Field Theories: Sandwich Quantization Scheme

Quantization of field theories with gauge symmetry is an extensively discussed and well-established topic. In this short note, we revisit this old problem. While we confirm all details of the existing literature, we highlight a potentially important point which may provide a better understanding of and insights on the quantization of gauged systems. The gauge degrees of freedom have vanishing momenta, and hence their equations of motion appear as constraints on the system. We argue that to ensure consistency of quantization one can impose these constraints as ``sandwich conditions’’: The physical Hilbert space of the theory consists of all states for which the constraints sandwiched between any two physical states vanish. We solve the sandwich constraints and show they have solutions not discussed in the gauge field theory literature. We briefly discuss the physical meaning of these solutions and the implications of the “sandwich quantization scheme”.

💡 Research Summary

The paper revisits the quantization of gauge field theories, focusing on the treatment of constraints that arise because gauge degrees of freedom have identically vanishing canonical momenta. In the standard Dirac–BRST framework, these constraints are imposed as operator equations that annihilate physical states from the right (the “right‑action” or positive‑frequency condition). The authors argue that this requirement is stronger than necessary and may unnecessarily restrict the physical Hilbert space.

To address this, they propose a “sandwich quantization scheme”. Instead of demanding that the constraint operators vanish on every physical state, they require that the matrix elements of the constraints between any two physical states vanish: \