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Table 1: Detail of the equivalent linear QBM system for a $\mathit{\lambda }{\mathit{\varphi }}^{\mathbf{4}}$ quantum field theory. The $\mathit{\lambda }{\mathit{\varphi }}^{\mathbf{4}}$ model has been chosen for concreteness, but the correspondence would be analogous for any other field theory model. The symbol $(=)$ indicates that the original and equivalent quantities are indeed identical despite the name change.

Original system Equivalent linear QBM System Two field modes Two identical oscillators System d.o.f. ${\varphi }_{\mathbf{p}}\text{,}$${\varphi }_{-\mathbf{p}}$ 2 copies of $q$ Environment Other modes 3-d field 2 1-d field Env. d.o.f. ${\varphi }_{\mathbf{q}}\text{,}$$\mathbf{q}\ne \pm \mathbf{p}$ ${\phi }_p$ Frequency $E_{\mathbf{p}}$ $\mathrm{\Omega }$ ($=$) Coupling $(\lambda /3!){\sum }_{{\mathbf{q}\mathbf{q}}^{\mathbf{\text{'}}}}{\varphi }_{\pm \mathbf{p}}{\varphi }_{\mathbf{q}}{\varphi }_{{\mathbf{q}}^{\mathbf{\text{'}}}}{\varphi }_{\mp \mathbf{p}-\mathbf{q}-{\mathbf{q}}^{\mathbf{\text{'}}}}$ $g^2{\sum }_pℐ(p)\dot{q}{\phi }_p$ 2-point function $G_{ab}(t,t^{\prime };\mathbf{p})$ $G_{ab}(t,t^{\prime })$$(=)$