Chapter II Klein Gordan Field Lecture 2 Books Recommended: Lectures on Quantum Field Theory by Ashok Das A First Book of QFT by A Lahiri and P B Pal

Creation and Annihilation operators Consider -----(1) Conjugate Momentum --(2)

We can also write -----(3) ----(4) Check that right side is independent of t.

We can write following commutation relations between the operators and , similar to harmonic oscillator: -------(5)

We now calculate following commutation relation and use (5) ----(6)

Also,

Similarly, ---(7)

We can also prove the commutators -----(8) By inserting the expressions of above Operators in terms of field operators i.e., Using(3) and (4). Perform above exercise.

Now we calculate Hamiltonian using field expansions (1) and (2) ---(9) Where, 1st term of (9)

----(10) Where we used -(11)

2nd term of (9)

-----(12) 3rd term of (9)

-----(13)

Using (10) , (12) and (13) in (9)

---(14) Which is H for KG field.

Note the following ---(15) ----(16)

Considering analogy between SHO and above discuss field theory, can be interpreted as annihilation operator and as creation operator. Positive energy component of the field Annihilate the quantum whereas the negative energy component create the quantum. This quantum is called a particle of positive energy.

Normal Ordering We can defined the ground state such that ----(27) The state is normalized

Consider Hamiltonian ----(28) Using commutator Eq (27) become --(29)

Using (27) and (29), ground state energy will be ----(30) Which is infinite. Infinite number of oscillators are contributing to energy. Need to redefine Hamiltonian

Energy differences are physical quantities not absolute energy. We can subtract the infinities and can Redefine the Hamiltonian. A consistent approach for this is known as normal ordering. When we have the expression involving the Product of annihilation and creation operators , we defined the normal ordered product by Moving all annihilation operators to the right Of all creation operators as if commutators Were zero

For example: ------(31)

Normal ordered Hamiltonian -----(32) Where number operator N(k) ---(33)

Total number operator ---(34) We can find -------(35)

Also ---------(36)

Exercise: (i) Find the linear momentum operator for the Klein Gordon field using the Noehter Theorem i.e. find (ii) Use the field expansions in above and show That (iii) What will normal ordered form for P? Do we need Normal ordering for momentum?

Note: For a gneral field as KG field is scalar i.e., And therefore, = 0 i.e., spin of particles Described by KG eq is zero.