12. The Interaction Picture and Dyson's Formula | QFT | Dr. Jamil Aslam

Описание: October 30, 2025

Course Instructor:
Dr. Muhammad Jamil Aslam
(Professor, Department of Physics, Quaid-i-Azam University, Islamabad)

✼ 𝘪𝘧 𝘺𝘰𝘶 𝘩𝘢𝘷𝘦 𝘢𝘯𝘺 𝘲𝘶𝘦𝘴𝘵𝘪𝘰𝘯𝘴 𝘳𝘦𝘭𝘢𝘵𝘦𝘥 𝘵𝘰 𝘵𝘩𝘦 𝘭𝘦𝘤𝘵𝘶𝘳𝘦, 𝘱𝘭𝘦𝘢𝘴𝘦 𝘧𝘦𝘦𝘭 𝘧𝘳𝘦𝘦 𝘵𝘰 𝘭𝘦𝘢𝘷𝘦 𝘵𝘩𝘦𝘮 𝘪𝘯 𝘵𝘩𝘦 𝘤𝘰𝘮𝘮𝘦𝘯𝘵𝘴 𝘣𝘦𝘭𝘰𝘸. 𝘢𝘴 𝘦𝘯𝘰𝘶𝘨𝘩 𝘲𝘶𝘦𝘴𝘵𝘪𝘰𝘯𝘴 𝘤𝘰𝘮𝘦 𝘪𝘯, 𝘸𝘦’𝘭𝘭 𝘳𝘦𝘤𝘰𝘳𝘥 𝘢𝘯 𝘦𝘹𝘤𝘭𝘶𝘴𝘪𝘷𝘦 𝘭𝘦𝘤𝘵𝘶𝘳𝘦 𝘸𝘩𝘦𝘳𝘦 𝘴𝘪𝘳 𝘸𝘪𝘭𝘭 𝘢𝘯𝘴𝘸𝘦𝘳 𝘺𝘰𝘶𝘳 𝘲𝘶𝘦𝘴𝘵𝘪𝘰𝘯𝘴! 𝘩𝘢𝘱𝘱𝘺 𝘭𝘦𝘢𝘳𝘯𝘪𝘯𝘨! ✼

Recommended Video Lecture:
Sidney Coleman Lecture 8:    • Sidney Coleman lecture 08  
Accompanying Notes: https://arxiv.org/pdf/1110.5013v1 (Pg 74-80)

Timestamps:

Q&As
00:00 Q: Poles and contours in contour integration
00:35 Q: Why we use the same Euler–Lagrange formalism in classical mechanics and quantum field theory
02:10 The endpoints do not necessarily have to be fixed while deriving the Euler–Lagrange equation
03:03 Constraints, sources, and extra terms in the Euler–Lagrange equation

The Interaction Picture
04:58 Recap: Schrödinger and Heisenberg picture
06:03 Both the state and the operators can be time dependent; take synchrotrons for example
06:59 The Free part and the Interaction part of the Hamiltonian
08:59 Relating the Interaction and Schrödinger picture with a Free Hamiltonian phase factor
17:20 Identity Operator
19:42 Interaction Hamiltonian in the Interaction Picture
21:16 Free Hamiltonian in the Interaction Picture
21:35 Violation of energy conservation at the time of interaction
22:43 Free Hamiltonian is time-independent
23:43 Q: How is energy conservation violated at the time of interaction?
26:43 Total Hamiltonian in the Interaction Picture
28:08 A General Operator in the Interaction Picture
29:27 Time Rate of Change
32:13 [H₀, f(p)]: Commutation of the Free Hamiltonian with Momentum-Dependent Operators
34:00 Summary of All Dynamical Pictures
35:25 Old ≠ Obsolete

Dyson’s Formula
36:17 Breaking: Turns Out a Theoretical Physicist’s Practical Isn’t Very Practical
36:58 Recommended resources for this part of the lecture
37:38 Notation in the Interaction Picture
38:45 Time Evolution in the Interaction Picture
40:15 Always Remember: Later is to the Left!
41:33 Unitary Group
44:11 Time Evolution Continued
48:13 Solving the differential equation using an integrating factor
50:58 Series expansion of the time-evolution ansatz
52:28 Verifying the Ansatz
55:55 Adding a perturbative parameter
56:31 Time derivative of the squared term
59:20 The interaction Hamiltonian may not commute with itself at different times
1:09:33 Time-ordering to the rescue!
1:11:27 Operators at two different times commute when time-ordered
1:19:52 Task: Show that the Dyson time-ordered exponential is a solution to the Schrödinger equation
1:21:08 Summary: The interaction Picture and Dyson’s formula
1:23:13 Up next: The S-matrix

✦ Behind the Scenes ✦
   • sir injures his hand during sync clap: lec...  

© Lectures are uploaded here with the express consent of Dr. Muhammad Jamil Aslam. All content remains his intellectual property.