the problem is on page 26 of "relativistic quantum mechanics and field theory" by Franz Gross. consider the lagrangian density: L=(1/2)[(∂ψ/∂t)^2 -(∂ψ/∂z)^2 -m^2ψ^2] a) find the momentum conjugate. b) find the equation of motion for the fields and the solution. use periodic boundary conditions. Attempt at solution: taking partial derivative with respect to (∂ψ/∂t) and (∂ψ/∂z) (a) the momentum conjugate is: π(z,t)= (∂ψ/∂t)-(∂ψ/∂z) (b) Using Euler-Lagrange for function with more than one variable we get: -m^2ψ=(d^2ψ/dt^2)- (d^2ψ/dz^2) using separation of variable ψ=Z(z)T(t) The part which is only a function of "t" equate it to square of constant "ω" then we get: T= Ae^(iωt) + Be^(-iωt) and equation becomes: -m^2=ω^2 - (1/Z)(d^2Z/dz^2) → (m^2 + ω^2)Z= (d^2Z/dz^2) call m^2 + ω^2 = κ^2 the Z=Ce^(ikz) + De^(-ikz) at this step I don't know how to apply the periodic boundary conditions. the periodic conditions are given as following given in the page 4 : ψ0=ψN dψ0/dt=dψN/dt where 0 and N indicate 0th and Nth oscillators. I would be really grateful if anyone could help me out with this. the rest of the question is related to this part and i can't do it without getting this part right.