Canonical Quantization Chapter I Canonical Quantization Lecture 4 Books Recommended: Advanced Quantum Mechanics by Schwabl Quantum Field Theory by Michio Kaku (2nd chapter for understanding Lorentz group)

Noether Theorem: Relate symmetry to conservation laws. Continuous transformations which leave the action unchanged lead to conservation laws. e.g. , Temporal invariance: conservation of energy Translational invariance : four momentum conservation Rotational invariance: Angular momentum

Consider infinitesimal Lorentz transformations ------(1) Corresponds to same point in space-time viewed from two Inertial frames. Anti symmetric tensor ----(2)

causes infinitesimal translation. Anti- symmetric in μ and ν . These depends upon the transformation properties of fields. For example, for spinors

Invariance under transformation (1) means --------(3) Variation in field will be ----(4) Total variation in field ----(5)

Eq. (4) and (5) are related as ------(6)

Total variation in the Lagrangian density will vanish as per Eq. (3) i.e., -------(7)

First term of Eq (7) will be written as -----(8) Where in 2nd line Euler Lagrange Eq is used. In last line Eq (6) is used.

Also, we can write ---------(9) Using (8) and (9) in (7), we get -----(10) Where -----(11)

Eq. (10) is continuity eq and is a statement of Noether theorem. is called Noether current. Corresponding Noether charge -----(12) We can write ----(13)

Eq. (13) will vanish i.e., charge will conserved if field vanish sufficiently rapidly at infinity. ---------(14)

Space-time translation ---(15) From (1), we get --------(16) And thus, -------(17)

Thus, Noether theorem (Eq. (11)) reduces to -----(18) And thus, From (10) and (18), ----(19) For ----(20) which is four momentum.

For --------(21) In above for zero compoent ie. , μ = 0, spatial momentum will be conserved and For spatial component stress tensor will be conserved i.e., .

Rotations Here, ---------(22) and ----(23) From (11), ---(24)

From (24), using antisymmetrization property --------(25) Where, -------(26)

From (25), we write using continuity Eq ------(27) From above, we write six quantities -----(28)

For spatial component, we obtain angular momentum operator -----(29) Here, 2nd and 3rd contain vector product of coordinate and space momentum density. First term is interpreted as intrinsic angular Momentum.

Gauge Transformation (Related to internal symmetry) Under internal symmetry fields transform But space time point do not. Consider Gauge transformations ----(30)

As space time point do not transform and therefore, from (1,4), ------(31) And also from (5), ----(32)

Four current density (From Noether Theorem) --------(33)

Noether Charge ----(34) Quantized form

We calculate