Revisiting Quantization of Gauge Field Theories: Sandwich Quantization Scheme

TL;DR

This work revisits canonical quantization of gauge theories by promoting the gauge-constraint equations to sandwich conditions, i.e., $\langle\phi| (\boldsymbol{\varphi}_i-\boldsymbol{\varphi}_i^0)|\psi\rangle=0$ and $\langle\phi|\boldsymbol{\cal C}^i|\psi\rangle=0$, offering a mixed approach that fixes gauge classically while enforcing EoM constraints quantum-mechanically. The authors derive the scheme from path integrals, showing gauge-invariant n-point functions must satisfy the constraints, and classify solutions into Class 1 (vanishing $ (\boldsymbol{\cal C})^+$) and Class 2 (states mapped into a complement Hilbert space $\mathcal{H}_{\text{c}}$). In Maxwell theory, Class 1 corresponds to transverse photons, while Class 2 yields new vacuum sectors labeled by background charge density $\mathcal{Q_B}(\mathbf{x})$, forming super-selection sectors and enriching the physical Hilbert space. The paper discusses consistency with BRST symmetry and proposes a sandwich equivalence principle, with potential extensions to gravity and implications for the arrow of time. Overall, the sandwich quantization framework offers a broader perspective on gauge quantization that could reveal new quantum sectors and connections to observer-dependent backgrounds.

Abstract

Quantization of field theories with gauge symmetry is an extensively discussed and well-established topic. In this short note, we revisit this old problem. 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 implications of the "sandwich quantization scheme".