1. Canonical Quantization
Chapter I
Canonical Quantization
Lecture 2
Books Recommended:
Lectures on Quantum Field Theory by Ashok Das
Advanced Quantum Mechanics by Schwabl

2. Euler Lagrange Equation of motion in point
Particle mechanics:
Lagrangian for a system with n degrees of
Freedom
-----(4)
Action will be
(5)

3. To find equation of motion, we use
Hamilton principle.
Actual trajectory of particle will
be along which action is stationary
(6)
Restriction on variation of trajectories
------(7)

4. Eq. (6) implies
(8)
2nd term in the above equation vanished because of
Eq (7)

5. Eq.(8) will be true if
(9)
Which is Euler Lagrange equation of motion.

6. Also, from Lagrangian,
-----(10)
We define conjugate momentum
(11)
and Hamiltonian using Legendre transformations
---(12)

7. Poisson brackets
---(13)
Dynamical eq in terms of Hamiltonian
-----(14)
Now, we repeat above procedure for classical
Field theory.

8. Classical Field Theory:
Field is a set of numbers at each point in
space-time.
We want to use action again as time integral of
Lagrangian .
But also need to incorporate the information
from each point in space.

9. For this Lagrangian is written as spatial integral
of some function of field
Where is called
Lagrangian density.
Action:

10. Definitions:
Lagrangian density: depend upon field variables and
their first derivatives
-----(1)
Total Lagrangian:
(2)

11. Action:
-----(3)
Units:
S = Energy * time
L = Energy
= Energy/vol
What about natural units ?

12. Consider the infinitesimal change in the field i.e., -----(15)
Euler Lagrange Equation of motion in field theory:
Consider the infinitesimal change in the field i.e.,
-----(15)
With restriction
-----(16)

13. Infinitesimal Change in action will be

14. ----(17)
Where we used Gauss surface theorem and
also used relation
----(18)
proof

15. Using in (17), we get
-----(19)

16. Minimizing action in Eq (19), we get
-----(20)
which is Euler Lagrange Eq in classical field theory.
E.g.: Lagrangian density for Kliein Gordan field

17. Eq. of motion:
(21)

18. Also, from Lagrangian density
----(22)
We write, conjugate momentum as
---(23)
Hamiltonian density
----(24)
Total Hamitonian
---(25)

19. For Klein Gordan field,
Conjugate momentum
----(26)
Hamiltonian density
-----(27)

20. Total Hamiltonian
(28)

21. Remarks:
To the Lagrangian density we can always add a
total divergence or constant or can multiply it.
This will not change equation o
f motion.
Integral of total divergence can be added to action
This will again not change the EoM.

22. Change in action due to two Lagrangian density
will be
Using Gauss divergence theorem abvove
Eq can be rewritten as integral over the surface.
Since vanishes at the boundary and
Thus right side will be zero and hence, EoM will
Be same from two Lagrangian densities which
Are differ by four divergence some function.

23. Lagrangian density we write should be Lorentz invariant.
This is required for the action to be invariant
In above, is scalar i.e. invariant under Lorentz
transformations and thus should also be scalar so that
action is invariant.
Lagrangian density cannot be explicit function of
space-time coordinates. It depends upon space-time
coordinates through fields and their derivatives. Explicit
dependence on will destroy Lorentz invariance.

24. Higher order derivatives are not written in
the argument of Lagrangian density. These
higher order derivative will result in differential
equation of order more than two i.e. Higher
than second order.
Lagrangian density must be local.