Canonical Quantization Chapter I Canonical Quantization Lecture 3 Books Recommended: Field Quantization by W. Griener Lectures on Quantum Field Theory by Ashok Das

We start from classical mechanics: Quantization We start from classical mechanics: For two dynamical quantities A and B, Poission bracket ----(1) Time derivative of A -----(2) Where, we used Hamilton Eqs

Note that, for coordinates ----(3) And ------(4) In going from CM to QM, we use ---(5)

We now defined variation in functional -------(6) Functional derivative It tells us how the functional changes with the change in field at x.

Momenta canonical to field ---------(7) Note that L is total Lagrangian Also ------(8) Hamilton Eqs -----(9) H is total Hamiltonian.

We define the Poission bracket for functional as ---(10) Time dependence of Functional is defined as ----(11) Where (9) is used.

Considering function as functional, we define --(12) Functional derivative of above will be --------(13)

Eq. (13) is true can be verified using definition (6), ----(14) Using (13) in above we get left side. Also, we have ---(15) -----(16)

We can now write --(17) ---(18)

Poission bracket relations -------(19) And --(20) Above relations are equal time possion bracket Relations.

We may also use following notations later -----(21) In above we used space time coordinate, Whose 0th component is time.

Single particle classical mechanics is Quantized by replacing the Poisson bracket among the coordinates by corresponding Commutation relations and operators. This is called first quantization. In moving to the Quantum field theory, we quantize the classical fields and treat them as operators. Poisson bracket among the fields given inn Eq (21) Is replaced by corresponding Commutation Relations. This is called second quantization.

Thus, we write for QFT, commutation relations ---(22) Note that for Bose particles commutation Relations are used whereas as we will see Later for Dirac field anti-commutation relations Are used. Above procedure of quantization is Called Canonical quantization