![]() “I was not interested in considering any theory which would not fit in with my darling,” Dirac wrote, referring to unitarity. In the 1920s, for instance, this unitarity requirement guided the British physicist Paul Dirac to discover an equation that implied the existence of antimatter. In working out how a particle is allowed to evolve or interact, physicists use the fact that amplitudes never change in a way that disrupts the fixed sum of their squares. As a particle’s state changes (as it flies through a magnetic field, say, or collides with another particle), its amplitudes change too. It’s this twist-the squaring of hidden amplitudes to calculate the outcomes we actually see-that gives unitarity teeth. The extension of our model to higher dimensions and the renormalization of interacting (scalar) field theories are briefly discussed.Illustration: Merrill Sherman/Quanta Magazine +- infinity independently of any weak-field approximation. Finally, the validity of weak-field perturbation theory (in the appropriate range of parameters) is checked directly in the solvable model, and the trace anomaly computed in the asymptotic regions t.->. The particle production ( for t.->.+infinity) is computed explicitly. It is shown that the action previously considered leads, in this model, to a well-defined finite expectation value for the stress-energy tensor. ![]() We then study a specific solvable two-dimensional model of a massive scalar field in a Robertson-Walker asymptotically flat universe. An action is constructed which renormalizes the weak-field perturbation theory of a massive scalar field in two space-time dimensions, it is shown that the trace anomaly previously found in dimensional more » regularization and some point-separation calculations also arises in perturbation theory when the theory is Pauli-Villars regulated. The method avoids the conceptual difficulties of covariant point-separation approaches, by always starting from a manifestly generally covariant action, and the technical limitations of the dimensional regularization approach, which requires solution of the theory in arbitrary dimension in order to go beyond a weak-field expansion. It is proposed that field theories quantized in a curved space-time manifold can be conveniently regularized and renormalized with the aid of Pauli-Villars regulator fields. The extension of the model to higher dimensions and the renormalization of interacting (scalar) field theories are briefly discussed. +- infinity independently of any weak field approximation. The particle production (less than 0 in/vertical bar/theta/sup mu nu/(x,t)/vertical bar/0 in greater than for t. It is shown that the action previously considered leads, in this model, to a well defined finite expectation value for the stress-energy tensor. One then studies a specific solvable two-dimensional model of a massive scalar field in a Robertson-Walker asymptotically flat universe. An action is constructed which renormalizes the weak-field perturbation theory of a massive scalar field in two space-time dimensions-it is shown that the trace anomaly previously found in dimensional regularization more » and some point-separation calculations also arises in perturbation theory when the theory is Pauli-Villars regulated. The method avoids the conceptual difficulties of covariant point-separation approaches, by starting always from a manifestly generally covariant action, and the technical limitations of the dimensional reqularization approach, which requires solution of the theory in arbitrary dimension in order to go beyond a weak-field expansion.
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