Chapter II Klein Gordan Field Lecture 5

Charge operator We consider the internal symmetry Transformations ----(1) Above tranformation will not change space- Time coordinates.

Infinitesimally, we can write --(2) Infinitesimal constant parameter

Lagrangian does not change under above symmetry transformations i.e., -----(3)

Recall the following from Noether theorem Whenever the Lagrangian density is invariant It leads to conserved current Where For internal symmetry

We can write for complex Klein Gordon field = ----(5)

Parameter independent current ----(7) Which is a vector current and is conserved ----(8)

Conserved charge -----(9) Normal ordered charge operator ---(10)

Charge operator is associated with the global phase transformation similar to electromagnetic Interactions (U(1) phase transformation). Above charge operator is identified as electric Charge. Note, using (10), ----(12) Vacuum does not carry any charge.

Acting on 1st of the one particle state, we get ----(13) Thus is eigenstate with eigenvalue +1

Acting on 2nd of the one particle state, we get ----(14) Thus is eigenstate with eigenvalue -1

Complex Klein Gordon field describe the particles carrying charge and have spin 0. Real Klein Gordon field is for neutral spin 0 Particles. Hamiltonian Is invariant under This is called charge conjugation property