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Il Nuovo Cimento (1955-1965)

, Volume 13, Issue 5, pp 988–1006 | Cite as

A generally relativistic field theory

  • C. Fronsdal
Article

Summary

In an effort to make contact between field theory and general relativity it is suggested to study space-times embeddable in flat spaces of higher dimensions, as intermediate steps between special and general relativity. The Lorentz invariant action principle formulation of single particle field theory is found to possess a very natural and simple extension to space-times embeddable in a flat six-space. Viewing space-time as a 4-surface in six-space, the metric properties of the former are determined by requiring the action to be stationary with respect to variations of the 4-surface. A more definite theory is obtained by imposing some rather severe restrictions on the form of the Lagrangian. A simple example is studied in detail and is found to describe a pair of Dirac fields interacting through the gravitational field. The main virtue of the theory is that it takes into account the mutual interaction between matter and metric in a consistent manner. It seems plausible that Mach’s principle is contained. Second quantization is not carried out, but may be fairly simple by Schwinger’s method.

Riassunto

Nel tentativo di creare un punto di contatto fra la teoria del campo e la relatività generale si propone di studiare gli spazi-tempi contenibili negli spazi piani a maggior numero di dimensioni come passi intermedi fra la relatività speciale e generale. Si trova che la teoria del campo di una singola particella formulata per mezzo del principio dell’azione invariante di Lorentz è suscettibile di una estensione molto naturale e semplice agli spazi-tempo contenibili in uno spazio piano a sei dimensioni. Considerando lo spazio-tempo come una quadrisuperficie nello spazio a sei dimensioni, si determinano le proprietà metriche del primo richiedendo che l’azione sia stazionaria rispetto alle variazioni della quadrisuperficie. Si ottiene una teoria più esatta imponendo alcune restrizioni piuttosto severe alla forma del lagrangiano. Si studia dettagliatamente un esempio semplice e si trova che esso descrive una coppia di campi di Dirac interagenti attraverso il campo gravitazionale. Il principale pregio della teoria è che essa prende in considerazione in maniera coerente le mutue interazioni fra materia e metrica. Appare plausibile che sia contenuto in essa il principio di Mach. Non si esegue la seconda quantizzazione, tuttavia essa può essere eseguita abbastanza semplicemente ricorrendo al metodo di Schwinger.

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References

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Copyright information

© Società Italiana di Fisica 1959

Authors and Affiliations

  • C. Fronsdal
    • 1
  1. 1.CERNGeneva

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