We present a multimode Hamiltonian formulation for the problem of opto-acoustic interactions in optical waveguides. We develop a quantised Hamiltonian representation of the acoustic field and then introduce a full system with a simple opto-acoustic coupling that includes both photoelastic/ electrostrictive and radiation pressure/moving boundary effects in a particularly transparent manner. The interaction is applied to a Fermiʼs golden rule calculation of spontaneous Brillouin scattering in uniform waveguides. The Heisenberg equations of motion are then used to obtain coupled mode equations for quantised envelope operators for the optical and acoustic fields. We show that the coupling coefficients obtained coincide with those established earlier. Our formalism provides a new basis for future work involving quantum photon and phonon noise in the low intensity limit, phonon-phonon scattering and anharmonicity effects.Consequently, on-chip SBS was first achieved [21] in rib waveguides made from nonlinear chalcogenide glasses, which combine high refractive index and nonlinearity with relative mechanical softness. Subsequently, SBS in silicon waveguides has been observed in Si/SiN membranes [22] and elevated rails [23,24], which both exploit physical isolation of the waveguide to minimise acoustic losses. Considerable development will be needed to reach designs suitable for mass-fabrication, but a practical platform for on-chip SBS would enable numerous applications [25] in microwave photonics [10,12,[26][27][28][29], sensing, isolators [30, 31] and chip-based lasers [32,33].A key driver for developing SBS in sub-micron waveguides was the realisation by Rakichet al [34,35] that at small scales, there are new contributions to SBS associated with radiation pressure of light on the waveguide boundaries, and the back-action of 'moving boundaries' on the optical field [36]. Depending on the particular waveguide configuration and combination of optical modes, these contributions can either reinforce or counteract the more familiar bulk contributions from electrostriction (the mechanical stress induced by the optical field) and the complementary process of photoelasticity (the change in the dielectric response induced by the acoustic strain) [34]. As well, waveguides in which the acoustic fields are strongly confined can enhance the scattering efficiency of near-stationary quasi-transverse acoustic waves, a requirement for efficient forward SBS where the pump and Stokes wave co-propagate [16, 35] (see figure 1).Through this work and subsequent contributions [23, 37], the coupled mode equations describing SBS in compact waveguides have been established through several routes including treatments based on optical forces and the Maxwell stress tensor [22,34,35], and an approach that focuses on susceptibilities and the Maxwell boundary conditions [37,38]. As well as providing precise forms for the coupling constants for both the electrostrictive and radiation pressure effects, these works confirmed that the radiation pressure a...