explore the origin of spontaneous symmetry breaking from adaptive perturbation method

Spontaneous symmetry breaking occurs when the underlying laws of aphysical system are symmetric, but the vacuum state chosen by the system is not.the (3+1)d ϕ4 theory is relatively simple compared to other more complex theories,making it a good starting point for investigating the origin of non-trivial vacua. theadaptive perturbation method is a technique used to handle strongly coupled systems.the study of strongly correlated systems is useful in testing holography. it has beensuccessful in strongly coupled qm and is being generalized to scalar field theory toanalyze the system in the strong-coupling regime. the unperturbed hamiltonian doesnot commute with the usual number operator. however, the quantized scalar field admitsa plane-wave expansion when acting on the vacuum. while quantizing the scalarfield theory, the field can be expanded into plane-wave modes, making the calculationsmore tractable. however, the lorentz symmetry, which describes how physicallaws remain the same under certain spacetime transformations, might not be manifestin this approach. the proposed elegant resummation of feynman diagrams aimsto restore the lorentz symmetry in the calculations. the results obtained using thismethod are compared with numerical solutions for specific values of the coupling constantλ = 1, 2, 4, 8, 16. finally, we find evidence for quantum triviality, where selfconsistencyof the theory in the uv requires λ = 0. this result implies that the ϕ4theory alone does not experience ssb, and the ⟨ϕ⟩ = 0 phase is protected under therg-flow by a boundary of gaussian fixed-points.