Chapter V Interacting Fields Lecture 1 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl

Interaction Lagrangian Density Example: Consider the Lagrangian density for free Dirac field ---(1) Above Lagrangian is invariant under global Phase transformations ---(2)

Invariance of (1) under (2) lead to conserved current ----(3) Now, consider the invariance of Eq (1) under Transformations in which phase angle Function of space-time coordinates i.e. --(4)

Under transformation (4), Free Dirac Lagrangian given by (1) will not be invariant ----(5) We can make the Lagrangian invariant by Introducing new field

Consider the Lagrangian -----(6) Consider the transformation -----(7) Transformed Lagrangian -----(8)

Comparing (6) and (8) ----(9) Recall that the above transformation leave the free Lagrangian of e.m. field invariant. We identify as photon field and in (6) Introduce corresponding free term -----(10)

In (10), we identify ----(11) Eq (10) describe QED. We write Eq (10) as ----(12) Where, covariant derivative ----(13)

Example 2: Phi-four theory λ dimensionless coupling constant. Self-interactions of Higgs field in electroweak theory involve above interaction. Eq of motion:

Example 3: Yukawa Theory g is dimensionless coupling constant. Use: Interaction of nucleons with pions In standard model, scalar Higgs field couple Quarks and leptons.

Remark: Any QFT should be renormalizble. In perturbation theory higher order terms will Involve integrals over 4-momenta of virtual Particles. To avoid divergence cut off parameter Λ is used At end one again takes and if physical quantity remain independent of Λ, we say theory is renormalizable.

Suppose theory have coupling constants having dimensions mass raise to power negative dimensions. Coupling constant is multiplied by factor having dimension mass raise to power positive dimensions so that scattering amplitude is dimensionless. This quatitiy will be and When we take theory will Not remain renormalizble. Theory should not have coupling constant with –Ve mass dimensions.

Action is dimensionless Lagrangian density should have dimension mass4 or simple 4 . Scalar filed d = 1, Fermion , d = 3/2 E.M field, d = 1 . Now consider in scalar theory d =1, d = 0

For n>4 i.e. . No terms will be allowed as coupling constant then should have dimension 4-n, which is –ve and theory will diverge. Spinor self interactions are not allowed e.g. , d = 9/2. Not Lorentz invariant. allowed.

Vector-spinor interaction QED Scalar QED Lagrangian Vector self interactions

Interaction Representation The Lagrangian or Hamiltonian for an interacting system can be divided into Free part and interacting part ----(14) If interaction Lagrangian do not have derivative Term, then the corresponding interacting part of Hamiltonian will be ----(15)

In Schrodinger picture -----(16) In interaction picture --(17) Eq. of motion ---(18)

Notations in interaction picture in subsequent Discussion -----(19) Eq. Of motion for field operators ---(20)

Since interacting part of Lagrangian does not contain derivative terms and therefore, the canonical conjugate fields remain same -----(21) Equal time commutation relations of Interacting fields are same as for free fields.

The interacting fields obey the same commutation relations as the free fields. The plane waves (spinor solutions, free photons, and free mesons) are still solutions of the equations of motion and lead to the same expansion of the field operators as in the free case. The Feynman propagators are still same.

Schrodinger and Heisenberg operators are related through ----(22) In interaction representation ---(23)

Interacting part of Hamiltonian in Schrodinger Picture ---(24) In interaction picture ----(25) Where, x = (x,t)

Time evolution operator in interaction picture In Schrodinger picture: ---(26) In interaction picture: ---(27)

From (27), the time evolution operator in interaction picture will be ----(28) It obey ----(29) And also unitary relation ---(30) Require H and H0 to be hermitian

Eq. of motionon for time evolution operator ----(31)