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

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

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)

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

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

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

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

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.

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

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

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

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)

Infinitesimal Change in action will be

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

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

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

Eq. of motion: -------(21)

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

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

Total Hamiltonian -------(28)

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.

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.

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.

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.