Chapter 1 Lagrangian

L=L0+L1+LIZ2L0=12μφ0μφ0+12m20φ20+λ0φ40L1=12μφ1μφ1+12m21φ21+λ1φ41LIZ2=μφ20φ21LI=gφ1φ30 On the lattice we can discretise the derivative μφ(x)=1a(φ(x+μ)φ(x)). On the lattice the above lagrangian can be written in a more convenient way for simulations L0=x[2κ0μϕ0(x)ϕ0(x+μ)+λ0L(ϕ0(x)21)2+ϕ0(x)2],L1=x[2κ1μϕ1(x)ϕ1(x+μ)+λ1L(ϕ1(x)21)2+ϕ1(x)2],LIZ2=μLxϕ0(x)2ϕ1(x)2,LI=gLxϕ1(x)ϕ0(x)3. With m20=12λ0Lκ08,λ0=λ0L4κ20,φ0=2κ0ϕ0m21=12λ1Lκ18,λ1=λ1L4κ21,φ1=2κ1ϕ1, and μ=μL4κ0κ1,g=gL4κ0κ3/21

1.0.0.1 necessary to set up the rendering inside loops rendering