1. Chapter II
Klein Gordan Field
Lecture 5

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

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

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

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

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

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

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

9. 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.

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

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

12. 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