The Canonical Transformation and Duality in the 1+1 dimensional $\phi^4$ and $\phi^6$ theory

We investigate the self-organizing nature of relativistic quantum field theory in terms of canonical transformation and duality presenting simple but explicit examples of $(\phi^4)_{1+1}$ and $(\phi^6)_{1+1}$ theories. Our purpose is fulfilled by applying the oscillator representation (OR) method which allows us to convert the original strong interaction theory into a weekly interacting quasiparticle theory that is equivalent to the original theory. We discuss advantages of the OR method and compare the results with what was already obtained by the method of Gaussian effective potential (GEP) and the Hartree approximation (HA). While we confirm that the GEP results are identical to the Hartree results for the ground state energy, we found that the OR method gives the quasiparticle mass different from the GEP and HA results. In our examples, the self-organizing nature is revealed by the vacuum energy density that gets lowered when the quasiparticles are formed. In the $(\phi^6)_{1+1}$ theory, we found two physically meaningful duality-related quasiparticle solutions which have different symmetry properties under the transition of quasiparticle field $\Phi \to -\Phi$. However, these two quasiparticle solutions yield the identical effective potential in the strong coupling limit of the original theory.