Standard Model Requires Treatment of Particles as Fields Hamiltonian, H=E, is not Lorentz invariant. QM not a relativistic theory. Lagrangian, T-V, used in particle physics. Creation and annihilation must be described. Relativistic Quantum Field theory!

Energy is not an invariant: consider four vector momentum

4-momentum transformation rest frame fixed frame

p = mv  rest frame fixed frame p 0 = p x / 

4-vector dot product:  µ=0,1,2,3 p µ p µ Covariant components rest frame fixed frame

Motivation for Lagrangians and the Law of Least Action

The Euler-Lagrange Equations give F = ma

Lagrangians are used in Field Theory

What is a Particle Field? A good example of a particle field is the electromagnetic field. It can be represented by the field function, A  = (  (r,t), A(r,t)). Classically  and A are related to E and B. The zero mass particles, photons, can be created and destroyed and represent the “quantization” of the field.