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 the Lagrangian refers to a transformation that leaves the Lagrangian invariant. This means that if you apply a certain transformation to the coordinates and fields in the system, the form of the Lagrangian remains unchanged. Such symmetries are crucial because they often lead to conserved quantities due to Noether's theorem.
Noether's theorem states that for every continuous symmetry of the Lagrangian, there corresponds a conserved quantity. For example, if a system's Lagrangian is invariant under time translation, the conserved quantity is energy. If it is invariant under spatial translations, the conserved quantity is momentum.
An example of a symmetry is rotational symmetry. If the Lagrangian of a system does not change under rotations, it implies that the system has rotational symmetry. According to Noether's theorem, this leads to the conservation of angular momentum.
Symmetries are important because they simplify the analysis of physical systems and often lead to conserved quantities, which are useful in solving equations of motion. They also provide deep insights into the fundamental nature of physical laws and help in the formulation of unified theories.
Yes, discrete symmetries, such as parity (spatial inversion) or time-reversal symmetry, can also be described in terms of the Lagrangian. While Noether's theorem specifically applies to continuous symmetries, discrete symmetries still play a crucial role in determining the properties and behavior of physical systems.