We will also need transformation properties of fermion bilinears:

Transcription

1 We will also need transformation properties of fermion bilinears: Parity: some product of gamma matrices, such that so that is hermitian. we easily find: 88

2 And so the corresponding bilinears transform as: scalar pseudoscalar vector axial vector (pseudovector) even under a parity transformation odd 89

3 Time reversal: we easily find: 90

4 And so the corresponding bilinears transform as: even under time reversal odd 91

5 Charge conjugation: taking the transpose of the RHS it can be also written as: we easily find: For a Majorana field we have: 92

6 And so the corresponding bilinears transform as: even under charge conjugation odd For a Majorana field: 93

7 Combined C, P and T transformation: 94

8 CPT theorem: even under CPT odd General rule: a fermion bilinear with n vector indices is even (odd) under CPT if n is even (odd); this also applies to derivatives. Thus any hermitian combination of fields and derivatives that is Lorentz invariant (has no uncontracted Lorentz indices) is even under CPT! Lagrangian is formed from such terms, under CPT, and so the action is invariant under CPT. Lorentz invariance CPT 95

9 LSZ reduction for spin-1/2 particles In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory: one particle states: based on S-41 with Lorentz invariant normalization: 96

10 Recall, for a scalar field theory we defined an operator: that creates a particle localized in the momentum space near wave packet with width! and localized in the position space near the origin. (go back to position space by Fourier transformation) is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at. for in the past. In the interacting theory is a state describing two particles widely separated is not time independent 97

11 A guess for a suitable initial state: Similarly, let s consider a final state: we can normalize the wave packets so that where again and The scattering amplitude is then: 98

12 Similarly, let s define an operator: that creates a particle localized in the momentum space near wave packet with width! and localized in the position space near the origin. (go back to position space by Fourier transformation) is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at. for separated in the past. In the interacting theory is a state describing two particles widely is not time independent 99

13 A guess for a suitable initial state: Similarly, let s consider a final state: we can normalize the wave packets so that where again and The scattering amplitude is then: and similarly for d-type particles 100

14 A useful formula: Integration by parts, surface term = 0, particle is localized, (wave packet needed). is 0 in free theory, but not in an interacting one! 101

15 Thus we have: or for its hermitian conjugate: Similarly for d-type states: The scattering amplitude: we put in time-ordering symbol (without changing anything) extra minus sign for each exchange of operators! 102

16 The scattering amplitude can be written as: Lehmann-Symanzik-Zimmermann formula (LSZ) 103

17 The scattering amplitude for any process can be obtained from the time ordered product of creation and annihilation operators representing the initial and final states with the following replacements: For a Majorana field,, everything we derived holds. we can use expressions for b or d, whichever is more convenient 104