By Luis Alvarez-Gaumé
Why will we want Quantum box conception After All?.- From Classical to Quantum Fields.- Theories and Lagrangian I: subject Fields.- Theories and Lagrangian II: Introducing Gauge Fields.- Theories and Lagrangian II: the normal Model.- in the direction of Computational ideas: Feynman Diagrams.- Symmetries I: non-stop Symmetries.- Renormalization.- Anomalies.- The foundation of Mass.- Symmetries II: Discrete Symmetries.- powerful box Theories and Naturalness.- distinct Topics.- Notation, Conventions and Units.- A Crash path in team Theory.- Index
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The noncommuting field operators are multiplied in the order in which they occur in time. For example, for the time-ordered product of two scalar fields we have ˆ 1 )φ(x ˆ 2 )φ(x ˆ 2 ) = θ x 10 − x 20 φ(x ˆ 2 ) + θ x 20 − x 10 φ(x ˆ 1 ). 63) The generalization to monomials with more than two operators is straightforward: operators evaluated at earlier times always appear to the right. In the case of our free scalar field theory the only independent time-ordered correlation function is the Feynman propagator G 2 (x1 , x 2 ).
It is also present in other field theories and in particular in quantum electrodynamics. In 1948 Hendrik Casimir pointed out  that although a formally divergent vacuum energy would not be observable, any variation in this energy would be (see [2–4] for comprehensive reviews). To show this he devised the following experiment. Consider a couple of infinite, perfectly conducting plates placed parallel to each other at a distance d (see Fig. 1). The plates fix the boundary condition of the vacuum modes of the electromagnetic field.
58) can be removed by the normal order prescription ˆ :H:= s = ± 12 d 3k 1 E k b† (k, s)b(k, s) + E k d † (k, s)d(k, s) . 59) Finally, let us mention that using the Dirac equation it is easy to prove the conservation of the four-current j μ = ψγ μ ψ, ∂μ j μ = 0. 60) As we will explain further in Chap. 7, this current is associated to the invariance of the Dirac Lagrangian under the global phase shift ψ → eiθ ψ. 61) is identified with the electric charge, with q the charge of the particle created by b† (k, s) acting on the vacuum.