By M. Y. Han
Provides the basic features of relativistic quantum box conception with minimum use of arithmetic. It covers the advance of quantum box idea from the unique quantization of electromagnetic box to the gauge box idea of interactions between quarks and leptons.
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Additional info for A story of light: a short introduction to quantum field theory of quarks and leptons
September 23, 2004 10:10 WSPC/SPI-B241: A Story of Light 52 chap09 A Story of Light brevity, we often write φ(x) = φ(+) (x) + φ(−) (x) with φ(+) = d3 ka(k)fk (x) and φ(−) (x) = d3 ka∗ (k)fk∗ (x). We now quantize the ﬁeld by imposing the canonical quantization rule, mentioned in Chapter 8, namely: [φ(x, t), φ(x , t)] = [π(x, t), π(x , t)] = 0 [φ(x, t), π(x , t)] = iδ 3 (x − x ) where π(x, t) ≡ ∂L/∂(∂φ/∂t) = ∂φ/∂t. These commutation rules become commutation relations among a(k)’s and a∗ (k)’s, thus: [a(k), a(k )] = [a∗ (k), a∗ (k )] = 0 [a(k), a∗ (k )] = δ 3 (k − k ).
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 September 23, 2004 56 10:12 WSPC/SPI-B241: A Story of Light chap10 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 Klein– Gordon equation is a second-order diﬀerential equation — second derivatives with respect to both space and time — and as a relativistic equation for single particle, it encounters some diﬃculties; the nature of second-order diﬀerential equations and the probability interpretation of quantum mechanics clash. ) Rather than a second-order equation, Dirac wanted a ﬁrst-order linear equation containing only the ﬁrst derivatives with respect to both space and time, that is, linear with respect to four-vector derivates.
A story of light: a short introduction to quantum field theory of quarks and leptons by M. Y. Han