The Standard Model of particle physics is typically expressed using a Lagrangian formulation rather than a Hamiltonian formulation for several reasons:
Relativistic Invariance: The Lagrangian formulation of the Standard Model is inherently covariant under Lorentz transformations, which maintain the principles of special relativity. The Lagrangian allows for the formulation of field theories, where the fields and their interactions can be described in a covariant manner. This is important for correctly describing the behavior of elementary particles at high energies and speeds.
Path Integral Formulation: The Lagrangian formulation is well-suited for the path integral formulation of quantum field theory. In this formulation, the quantum amplitudes associated with particle interactions are expressed as a sum over all possible paths taken by the particles in spacetime. The Lagrangian provides a natural starting point for constructing these path integrals, enabling calculations of scattering amplitudes and other quantum effects.
Symmetry and Gauge Invariance: The Lagrangian formulation allows for a clear treatment of symmetries and gauge invariance, which are fundamental concepts in the Standard Model. The Lagrangian can be constructed to be invariant under certain symmetries, such as the gauge symmetries associated with the electroweak and strong interactions. These symmetries are crucial for understanding the behavior of particles and the conservation laws that govern their interactions.
Field Equations and Equations of Motion: The Lagrangian formulation provides a concise way of deriving the field equations and equations of motion for the particles and fields in the theory. By varying the action (integral of the Lagrangian over spacetime) with respect to the fields, the Euler-Lagrange equations can be obtained, which describe how the fields evolve and interact with each other.
While the Lagrangian formulation is more commonly used in the theoretical development of the Standard Model, it is worth noting that the Lagrangian and Hamiltonian formulations are mathematically equivalent. The Hamiltonian can be derived from the Lagrangian through a Legendre transformation, and the two formulations provide equivalent descriptions of the dynamics of a physical system. However, the Lagrangian formulation is often more convenient and natural for the types of calculations and symmetries encountered in particle physics.