By M. Y. Han
This publication provides the fundamental elements of relativistic quantum box conception, with minimum use of arithmetic. It covers the advance of quantum box thought from the unique quantization of electromagnetic box to the gauge box thought of interactions between quarks and leptons.Aimed at either scientists and non-specialists, it calls for just some rudimentary wisdom of the Lagrangian and Hamiltonian formula of Newtonian mechanics and a simple figuring out of the specific conception of relativity and quantum mechanics.
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Extra info for A Story of Light. A Short Introduction to Quantum Field Theory of Quarks and Leptons
The canonical formalism for ﬁelds requires the canonically conjugate momenta that are to be paired with ﬁeld variables, and the momenta that canonically conjugate to ﬁelds can be deﬁned in terms of the Lagrangian that yields correct equations of motion via Lagrange’s equations of motion. In Chapter 2, we took the simplest approach to obtaining Lagrange’s equation, starting from Newton’s equations of motion. There is another way of obtaining Lagrange’s equations that is more formal than the direct approach we took in Chapter 2, and that is to derive Lagrange’s equation from what is called Hamilton’s principle of least action for particle mechanics.
The question then is what is the clue and prescription by which we can introduce interactions into the Lagrangian densities. There are very few clues. In fact, there is only one known prescription to introduce electromagnetic interactions and it comes from the Hamiltonian formalism of classical physics, as discussed in Chapter 2. Comparing the classical Hamiltonian (total energy) for a free particle with that of the particle interacting with 55 56 A Story of Light the electromagnetic ﬁeld, the recipe for introducing the electromagnetic interaction is the substitution rule (sometimes referred to as the “minimal” substitution rule) pµ ⇒ pµ − eAµ .
The source-free = 0) inhomogeneous Maxwell’s equations are (J µ ∂µ F µν = ∂µ ∂ µ Aν − ∂ ν ∂µ Aµ = 0. Using the freedom of gauge transformation, we can set ∂µ Aµ = 0. The choice of the arbitrary function Λ(x) to render ∂µ Aµ as always being zero is referred to as the Lorentz gauge. With such an option, Maxwell’s equations reduce to ∂µ ∂ µ Aν = 0, which, as mentioned in Chapter 1, is exactly the zero-mass case of Klein–Gordon equation. At the risk of being repetitive, let us emphasize this remarkable point that Maxwell’s equations are classical wave equations for the four-potential, and they predate both relativity and quantum mechanics.