Lecture 4 | New Revolutions in Particle Physics: Basic Concepts

Sanjay introduces a project to compile all old video lectures from YouTube into a single collection for easy navigation by students taking classes both online and offline. A temporary web address is provided for this collection, with the promise of a permanent link in the future.

The concept of the direct Delta function is explained, which is a function that is zero everywhere except at a particular point where it is infinitely sharp. It is defined by the property that its integral over its argument is one. The function is also shown to emerge from the integral of a periodic function, e^(iKX), over a cyclic dimension, where K is the wave number.

A discussion on Quantum Field Theory (QFT) is initiated with a focus on simplifying a two-year course into a few lectures. The use of QFT to describe particle processes such as scattering and annihilation is introduced. The need for a bit of mathematics, specifically the direct Delta function, is emphasized for understanding these complex theories.

The use of bra-ket notation in quantum mechanics is explained, with the bra vector representing initial states and the ket vector representing final states. The inner product of these vectors yields numerical results that can be interpreted experimentally. Creation and annihilation operators are also discussed in the context of their action on bra and ket vectors.

The rules for how creation and annihilation operators act on quantum states are defined. It is shown that these operators, when acting on bra vectors, follow a specific rule that results in another bra vector. The implications of these actions for the calculation of quantum mechanical operators are explored.

Quantum fields are introduced as mathematical objects that can create or annihilate particles. The time-dependence of these fields is incorporated, leading to a wave equation for the quantum field. The connection between the wave vector k and the frequency of oscillation is discussed, along with the implications for the quantum field's behavior in space and time.

The wave equation for the quantum field is derived, starting from the time and space derivatives of the field. The relationship between the frequency and the wave vector is used to find the form of the wave equation, which is identified as a form of the Schrodinger equation, but for operators rather than simple wave functions.

The application of quantum fields to describe scattering processes is discussed. A simple model is presented where a particle scatters off a heavy target, and the probability amplitude for the process is calculated using the quantum fields. The importance of energy and momentum conservation in these processes is highlighted.

The concept of the coupling constant is introduced as a measure of the strength of interaction between particles. Its role in calculating the probability of scattering processes is explained. The impact of the coupling constant on the likelihood of particle scattering is discussed, with examples provided to illustrate its significance.

The principle that phase invariance corresponds to charge conservation is explored. The conditions under which certain particle interactions are allowed or disallowed based on the number of field operators and their conjugates are discussed. The transformation properties of these operators under a change of phase are also examined.

The conversation touches on the possibility of turning the model into a relativistic one and the implications of complex coupling constants, which may indicate a violation of time reversal invariance. The distinction between the conservation of charge and rest mass in relativistic versus non-relativistic processes is clarified.

The discussion concludes with a look ahead to future topics, including the treatment of multiple particle species and the exploration of more complex scattering processes. The importance of understanding quantum fields in describing particle physics processes is emphasized.