Summary
Certain nonlinear field equations are invariant with respect to the scale (dilatation) transformation. A quantum theory based on such an equation can be invariant only if it does not contain states with finite rest mass. The invariance could however show up at high energies and momentum transfers (short distances), where rest masses are negligible. The consequences of this asymptotic invariance for then-point functions are studied. The functions are in general less singular than in perturbation theory. As a consequence the metric in the space of states is indefinite. The problem of compatibility of the invariance with locality and the equations of motion is treated in a very rough approximation only.
Riassunto
Alcune equazioni di campo non lineari sono invarianti rispetto a trasformazioni di scala (dilatazioni). Una teoria quantistica basata su una tale equazione può essere invariante solo se non contiene stati con massa di quiete finita. L’invarianza potrebbe rivelarsi ad alte energie e alti momenti trasferiti (piccole distanze), quando le masse di quiete sono trascurabili. Si studiano le conseguenze di questa invarianza asintotica per le funzioni din punti. Le funzioni sono in generale meno singolari che nella teoria delle perturbazioni. Come conseguenza la matrice nello spazio degli stati è indefinita. Si discute con una grossolana approssimazione il problema delle compatibilità dell’invarianza con la località e le equazioni del moto.
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This work was supported in part through AEC Contract AT(30-1)-2098, by founds provided by the U. S. Atomic Energy Commission, the Office of Naval Research, and the Air Office of Scientific Research.
He also has benefited from discussions with Drs.H. A. Kastrup, H. Reeh andS. Schlieder.
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Mitter, H. Quantization of nonlinear field theories and scale transformation. Nuovo Cim 32, 1789–1808 (1964). https://doi.org/10.1007/BF02732811
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DOI: https://doi.org/10.1007/BF02732811