Summary
For two simple field theoretical models with infinite renormalization we show that the equations of motion can be formulated as differential equations for the renormalized fields involving only finite quantities. Thus the status of the initial value problem is not changed by the infinite renormalization. One has to observe that the field quantities at a sharp time, though not observables, have a well defined meaning as bilinear forms (infinite matrices). The equations involve certain limiting procedures which are closely related to Valatin’s work on Quantum Electrodynamics but more explicit.
Riassunto
Viene mostrato come, in due semplici modelli di teorie di campo con rinormalizzazione infinita, le equazioni del moto possano essere formulate quali equazioni differenziali per i campi rinormalizzati: tali equazioni contengono solo quantità finite. Di conseguenza, la rinormalizzazione infinita non altera il carattere del problema dei valori iniziali (problema di Cauchy). Gli operatori di campo per un fissato valore del tempo, benchè non siano osservabili, hanno un significato matematico preciso come forme bilineari (matrici). Le equazioni contengono alcuni processi di passaggio al limite che sono strettamente connessi con quelli usati da Valatin in Elettrodinamica quantistica: noi diamo di essi una formulazione più esplicita.
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References
T. D. Lee:Phys. Rev.,95, 1329 (1954).
Integral equations for the renormalized fields in the Lee model have been previously given byW. Heisenberg (5) and byM. Froissart (private communication).
J. G. Valatin:Proc. Roy. Soc., A226, 254 (1954).
G. Källén andW. Pauli:Dan. Mat. Fys. Medd.,30, no. 7 (1955).
W. Heinsenberg:Nucl. Phys.,4, 532 (1957).
Since we do not consider more than one heavy particle it is immaterial whether we treat these particles as bosons or as fermions.
K. O. Friedrichs, private communication.
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with the support of a grant of the U.S. Government (P.L. 402 and P.L. 584).
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Haag, R., Luzzatto, G. Differential equations for the renormalized fields in the point source Lee-model and scalar neutral meson theory. Nuovo Cim 13, 415–429 (1959). https://doi.org/10.1007/BF02732952
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DOI: https://doi.org/10.1007/BF02732952