Quantum Field Theory

An Example: A Real Scalar FieldFor the LagrangianL = 1 2 ˙φ 2 − 1 2 (∇φ)2 − V (φ) (1.71)the momentum is given by π = ˙φ, which gives us the Hamiltonian,∫H = d 3 x 1 2 π2 + 1 2 (∇φ)2 + V (φ) (1.72)Notice that the Hamiltonian agrees with the definition of the total energy (1.45) thatwe get from applying Noether’s theorem for time translation invariance.In the Lagrangian formalism, Lorentz invariance is clear for all to see since the actionis invariant under Lorentz transformations. In contrast, the Hamiltonian formalism isnot manifestly Lorentz invariant: we have picked a preferred time. For example, theequations of motion for φ(x) = φ(⃗x, t) arise from Hamilton’s equations,˙φ(⃗x, t) =∂H∂π(⃗x, t)and ˙π(⃗x, t) = − ∂H∂φ(⃗x, t)(1.73)which, unlike the Euler-Lagrange equations (1.6), do not look Lorentz invariant. Nevertheless,even though the Hamiltonian framework doesn’t look Lorentz invariant, thephysics must remain unchanged. If we start from a relativistic theory, all final answersmust be Lorentz invariant even if it’s not manifest at intermediate steps. We will pauseat several points along the quantum route to check that this is indeed the case.– 20 –