You have to ensure that any of the terms entering the Lagrangian is in the trivial representation of the SM gauge groups. If they are, then the term is gauge invariant. If not it is not.zaman786 said:TL;DR Summary: how can we check symmetry of SM in terms of Lagrangian
Hi, as we know SM is symmetric under SU(3) X SU(2) X U(1) , But my question is , how can we check the invariance of terms in Lagrangian under these symmetries - thanks
ok- got it - thanksOrodruin said:You have to ensure that any of the terms entering the Lagrangian is in the trivial representation of the SM gauge groups. If they are, then the term is gauge invariant. If not it is not.
Symmetry in terms of Lagrangian refers to the invariance of the Lagrangian under certain transformations. This means that the Lagrangian remains unchanged when the system undergoes a specific transformation, such as translation, rotation, or time reversal.
Symmetry is important in Lagrangian mechanics because it leads to conservation laws. Noether's theorem states that for every continuous symmetry of the Lagrangian, there exists a corresponding conserved quantity. These conserved quantities provide valuable insights into the dynamics of the system.
By exploiting symmetries in the Lagrangian, one can derive conserved quantities that simplify the equations of motion. These conserved quantities allow us to reduce the number of independent variables in the system, making it easier to analyze and solve the equations of motion.
Some common examples of symmetries in Lagrangian mechanics include translational symmetry (invariance under spatial translations), rotational symmetry (invariance under rotations), and time symmetry (invariance under time translations). These symmetries often lead to conservation of linear momentum, angular momentum, and energy, respectively.
Symmetries in Lagrangian mechanics can be broken if the system is subjected to external forces or constraints that violate the symmetry. In such cases, the conserved quantities associated with the broken symmetry may no longer hold, leading to a more complex dynamics of the system.