By Luis Alvarez-Gaumé

Why will we desire 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 traditional 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.- unique Topics.- Notation, Conventions and Units.- A Crash path in crew Theory.- Index

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57) would give the number operator d † (k, s)d(k, s) with a minus sign in front. As a consequence, the Hamiltonian would be unbounded from below and we would be facing again the instability of the theory already noticed in the context of relativistic quantum mechanics. 57) takes the form Hˆ = s = ± 12 −2 d 3k 1 E k b† (k, s)b(k, s) + E k d † (k, s)d(k, s) (2π )3 2E k d 3 k E k δ(0). 58) As with the scalar field, we find a divergent vacuum energy contribution due to the zero-point energy of an infinite number of harmonic oscillators.

Hence, the wave function for both positive- and negative-energy solutions is labelled by the three-momentum k of the particle, u(k, s), v(k, s). Before proceeding any further we consider the case of a massless Dirac fermion. 48) and similarly for v(k, s). , the projection of the spin along the direction of motion) and chirality (the eigenvalue of the γ5 matrix) are equivalent concepts. In this case the helicity of the spinor is a relativistic invariant. This is no longer true when m = 0 because when the particle moves with a speed smaller than the speed of light the sign of λ can be changed by a boost reversing the direction of k.

The plates fix the boundary condition of the vacuum modes of the electromagnetic field. These modes are discrete in between the plates (region II), while outside them they have a continuous spectrum (regions I and III). The vacuum energy of the electromagnetic field is equal to that of two massless scalar fields, corresponding to the two physical polarizations of the photon (see Sect. 2). Hence we can apply the formulae derived above. A naive calculation of the vacuum energy in this system gives a divergent result.