Il Nuovo Cimento (1943-1954)

, Volume 8, Issue 2, pp 108–131

# Sulla diagonalizzazione della hamiltoniana nella teoria dei campi d’onda e sulla teoria dei sistemi chiusi

Nota I
• B. Ferretti
Article

## Riassunto

Nella prima parte di questo lavoro si discutono alcune questioni preliminari relative al problema della generalizzazione agli stati legati dei metodi con cui si possono isolare le divergenze della teoria dei campi.

## Summary

Following Dyson, when the perturbation theory can be applied, the divergences of electrodynamics may be circumvented to any desired order of perturbation theory i.e. in the fine structure constant. However Dyson’s treatment cannot be immediately extended to bound state phenomena, because in this case the perturbation theory is not sufficient. In the first part of this paper some preliminary questions related to the problem of extending Dyson’s method to the calculation of the energy eigenvalues are considered. For this purpose a Schrödinger equation
$$i\hbar \frac{{\partial \varphi }}{{\partial t}} = (H_0 + \lambda V)\varphi$$
which in the <2 interaction representation >2 is of the form
$$i\hbar \frac{{\partial \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{\varphi } }}{{\partial t}} = \lambda \underline {V\varphi }$$
is discussed. An operator
$$\Pi = lim \Pi _i \left| {1 - \frac{i}{\hbar }\int\limits_{t_i }^{t_{i + 1} } {\lambda \underline {V dt} } } \right.$$
is defined. This operator is similar to the operator defined by the equation (10) of Dyson I (F. J. Dyson: Phys. Rev., 75, 486 (1949)). It is then shown that this unitary operator transforms by a canonical transformation the total Hamiltonian H0 + λV into an operator which is commutable with H0, provided that the <2 charge >2 λ is adiabatically switched on. It is further shown that this adiabatic switching on of the charge must be explicitly taken into account in the calculation of the transformed Hamiltonian operator, in order to get sensible results. Use is made in fact of the relation :
$$[\lambda \underline V ,H_0 ]_ - = i\hbar \frac{{d(\lambda \underline V )}}{{dt}} - i\hbar \frac{{d\lambda }}{{dt}} \cdot \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{V}$$
and it is shown that from −iħ(dλ/dt). V there arises an essential contribution to the energy eigenvalues, even in the limit dλ/dt=0. Furthermore it is shown that IT cannot be in general expanded in powers of λ even when the perturbation theory is valid, if there are discrete eigenvalues of the energy. The reasons for this fact are discussed. A consequence of this result is that the equation (13) of Dyson I will be no longer valid without restrictions, even in the case of the validity of the perturbation theory. It is then necessary to give a proof of the existence of the operator II and this will be performed in the second part of this paper, together with the more general treatment for the bound state problems. At the end of the first part, relativistic invariance questions related to the present problems are discussed. For the purpose of achieving the relativistic invariance, a world-scalar function λ(x) is introduced in place of the charge. λ(x) is taken to be zero at infinity of the space-time in all directions but to be practically a constant in any finite region. This allows an invariant generalization of the switching on of the charge.

## References

1. (1).
Sin-itiro Tomonaga:Prog. Theoret. Phys.,1 27 (1946);Koba, Tati e S. Tomonaga:Prog. Theoret. Phys.,2, 101 e 198 (1947);S. Tomonaga:Phys. Rev.,74, 224 (1948).
2. (2).
J. Schwinger:Phys. Rev.,73, 416 (1948);74, 1439 (1948);R. P. Feynmah:Rev. Mod. Phys.,20, 367 (1948);F. J. Dyson:Phys. Rev.,75, 486 (1949) (questo lavoro sarà indicate nel seguito con Byson I) ;75, 1736 (1949) (indicato in seguito con Dyson II) ; W. Pauli e F. Villars :Rev. Mod. Phys.,21, 434 (1949).
3. (5).
E. Fermi eC. N. Yang:Phys. Rev.,76, 1739 (1949);W. Heisenberg: in corso di stampa.
4. (6).
Dyson II; per le considerazioni di questo e dei prossimi numeri confronta ancheB. Ferretti:Nuovo Cimento,7, 79 (1950).
5. (8).
Dyson I — formula 10.Google Scholar
6. (13).
Vedi, per esempio,G. Wentezl:Quantum Theory of Fields, p. 17.Google Scholar
7. (15).
La verifica è identica a quella. relativa alla formula (31) di Dyson I.Google Scholar
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Vedi anche B. Ferretti:Nuovo Cimento,7, 783 (1950).
9. (19).
Dyson I, paragrafo III. 9 —II Nuovo Cimento.Google Scholar
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J. Kanesawa eS. Tomonaga:Prog. Theoret. Phys.,3, 1 (1948); vedi anche B. Ferretti:Nuovo Cimento,7, 375 (1950).