Transcript number operator

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
----(27)
The state is normalized
such that
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.