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THREE
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QUANTIZATION-FREE FIELDS
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Quantization is presented in the canonical way and applied to free fields. Starting from a mechanical analog we discuss in turn the neutral and charged scalar field. The Gupta-Bleuler indefinite metric is used in the electromagnetic case with emphasis on gauge invariance and the problems arising from the vanishing photon mass. The vacuum fluctuations are beautifully evidenced by the Casimir effect. For Dirac fields, the stability of the vacuum and the exclusion principle lead to quantization according to anticommutation rules. This implies the connection between spin and statistics and the peT theorem.
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3-1 CANONICAL QUANTIZATION
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In the nonre1ativistic case we obtain Schrodinger's equation by replacing classical observables by operators and Poisson brackets by commutators. As usual, conjugate momenta are derivatives of the Lagrange function with respect to velocities. We have seen in Chap. 1 that the classical procedure can be extended to infinite systems if discrete indices are replaced by continuous ones and the Kronecker symbol by a Dirac b function. We shall therefore boldly generalize quantization to this case without worrying too much at first. The only point which we should be cautious about is to preserve Lorentz invariance. In the specific case of electrodynamics the lagrangian is known, at least classically. The precise role of the lagrangian after quantization will have to be clarified. This will be done in Chap. 9, where we shall investigate an alternative and fruitful interpretation of quantization using the methods of path integrals.
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QUANTUM FIELD THEORY
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It is clear that the correct interpretation of a quantum theory depends on its dynamics (which can lead, for instance, to the appearance of bound states, etc.). Unfortunately, in most cases we cannot solve the equations of motion and we have to treat interactions using approximations. This is a major difficulty which sometimes prevents us from reaching a logical and coherent presentation. Part of this difficulty can be overcome by renormalization which hides some of the interactions by reexpressing the observable quantities in terms of measured masses and coupling constants. A more consistent but difficult axiomatic approach was pioneered by Wightman. One tries to develop a quantum field theory from a few well-defined axioms obtained from an idealization of physical requirements. To uncover the content of the theory requires sophisticated mathematical developments. Two main lines of attack have been pursued. According to one of them, one studies Green functions in detail: analyticity in momentum space, algebraic properties, discontinuities. At an ultimate stage one may reach a scattering theory after proper identification of the asymptotic states. A more ambitious program involves the explicit construction of the fundamental operators, such as the hamiltonian, in order to study their spectral properties. The latter should provide the particle interpretation, bound states, scattering states, etc. Our goal will be more modest and follows partly the historical path. It combines approximations, physical intuition, and mathematical deduction. One may hope that these various methods will at some point merge together and that some rigorous basis will be given to the above treatment. At first sight a field theory does not seem to have an interpretation in terms of microobjects identified as particles. The long story of the wave-particle duality in the theory of light is a testimony of the confusion which may arise and still remains in certain aspects up to this day. One of the triumphs of the quantum theory will be to give a better understanding of this phenomenon. Naive pictures derived from classical experiences have limitations, however, arising in particular from the concept of indistinguishable particles. The structure of the Hilbert space of states constructed for free fields-the Fock space-will reflect this aspect. We shall only deal here with local theories. The meaning of locality will become clearer as we go along. It assumes an idealization of space-time measurements in arbitrarily small regions. As required by Lorentz in variance and a weak form of causality, measurements separated by a space-like interval cannot influence one another. In other words, (1) local observables exist and (2) local observables relative to space-like separated regions commute. Experimental verifications of this postulate are only indirect since we do not possess any apparatus comparable to nuclear or subnuclear sizes. In this case, measurement necessarily involves objects of the same nature as those under study. Nevertheless, it is a fair statement to say that no violation of this principle has been observed down to distances hc/Js where s is the squared center-of-mass energy of particle collisions in presentday accelerators (Js ;:s 60 GeV at the CERN ISR). These lengths (~1O- 15 cm) are minute fractions of atomic sizes (10- 8 cm).
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