I'm having trouble figuring out how a field and it's conjugate are independent quantities. How can they be, when they are related by conjugation? Suppose you have real fields x and y, and form fields: L=(x+iy)/sqrt2 R=(x-iy)/sqrt2 In a path integral, you'd have .5(∂x∂x+∂y∂y) in your Lagrangian. Changing variables to L and R you'd have ∂L∂R. How can L and R be independent in the path integral? At each spacetime point, the quantity in the Lagrangian is of the form ∂L∂R=number*(its conjugate). If L and R vary independently, then your Lagrangian might not even be real, since only a number times its conjugate is real.