We calculate the changes in the yields of primordial nucleosynthesis which result from small corrections to rates for weak processes that connect neutrons and protons. We correct the weak rates by improved treatment of Coulomb and radiative corrections, and by inclusion of plasma effects. Our calculations lead to a systematic decrease in the predicted 4He abundance of about AY=O.0025. The relative changes in other primordial abundances are also 1-2%.

Underwater sparks have long been used by the geophysical prospecting community as a source of intense low-frequency sound. While bubble hydrodynamic models are well developed, the mechanism of transferring energy from the thermal power input, through the various energy conversion channels, to the work done by the bubble has not been adequately studied. In this work an ab initio model of the bubble dynamics, including blackbody ablation, ionization, dissociation, and radiative transport, is developed. This model is a first step in enumerating the important physical mechanisms within bubbles generated by underwater sparks. The predictions of this model are compared with experimental results. Experimental work is still needed to validate the model, and to determine if and how model parameters related to actual physical parameters and measurable effects.

This paper describes the time evolution of bubbles generated by underwater electrical discharges. The oscillations of these high-temperature vapor and plasma bubbles generate acoustic signatures similar to the signatures generated by air guns, underwater explosions, and combustible sources. A set of model equations is developed that allows the time evolution of the bubble generated by a spark discharge to be calculated numerically from a given power input. The acoustic signatures produced by the model were compared to previously recorded experimental data, and the model was found to agree over wide ranges of energy and ambient pressure on several characteristic values of the acoustic signatures. The bubble period in particular matched very well between model and experiment, indicating that the total energy losses predicted by the model over the oscillation of the bubble were approximately correct, although no reliable information was gained about the relative magnitudes of the individual energy loss mechanisms examined. The bubble period and the minimum rarefaction pressure were found to depend on depth, while the peak pressures in the expansion and collapse pulses and the acoustic energy in the expansion pulse were not found to depend on depth over the parameter ranges investigated.

A. H. VGlkel, Phys. Rev. D 1, 3377 (19701. 'A. H . Volkel, Phys. Rev. D 3, 917 (1971). 3~o r the sake of fairness we should mention that this i s denied by some physicists (H. Kleinert and B. Hamprecht, private communication). 4~. Renner, Current ,llgebras and their Applications (Pergamon, New York, 1968). %. L. Adler and R. F. Dashen, Current Algebras and Applications t o Particle Physics (Benjamin, Amsterdam, 19fi8). 6~h e distribution and use of indices a t the currents throughout this paper a r e a s follows: (i) Upper Greek indices p , v , A . . . = 0 , 1, 2, 3 to the left of the argument(s) indicate tensor properties with respect to the Lorentz group, the corresponding Latin indices k , I , Y . . . -1 , 2 , 3 their restriction to the space parts. (ii) Louver Latin indices a , b , c = V , A to the right of the argument(s) differentiate between vectors ( V ) and axial vectors ( A ) . In the commutation relations we have the connection a # bc = A and a = bc = V . (iii) Lower Greek indices a , 8 , y to the left of the a rg u m e n t (~) r e f e r to the internal broken-symmetry group with structure constants E~, ,~~Y .The usual summation convention for double indices i s used. The hfinkowski metric i s (+I, -1, -1, -1).IFurther restrictions on the c l a s s of admitted 6 s equences like cp(xo) 0, etc., would not influence our r esults.8~. F. Streater and A. S. Wightman, P C T , Spin and Statzstics a n d A l l That (Benjamin, New York, 1964). 9~. Girding and A. S. Wightman, Arkiv Fysik 28, 129 (1964).'k. J o s t , The General Theory 01-Quantzaed Fields (American Mathematical Society, Providence, R . I., 1965). "B. Schroer and P. Stichel, Commun. Math. Phys. 3, 258 (1966). l2v. Volkel and A. H. Volkel, Nuovo Cimento @, 203 (1969). 1 3~. Schwartz, ThSorie des Distributions (Hermann. P a r i s , 1959), Vol. II. 14From the work of Schroer and Stichel (Ref. 11) i t follows that this i s very probably not a new and s e v e r e a ssumption beyond the existence of equal-time commutat o r s .l5watch that the form factors depend a l s o on the isospin quantum numbers of the nucleon s t a t e s , which we have not written out explicitly. Therefore, in the following formulas, a summation over s o m e of these quantum numb e r s i s to be understood. 1 6~. Schildknecht. DESY Report No. 69/41, 1969 (unpublished).'?we want to point out that the one-pion pole in the t channel does not contribute to the invariant function A1-The general properties of analyticity, covariance, and unitarity a r e studied in quantum field theories regularized by finite-mass, indefinite-norm states. After reviewing the genera l status of indefinite-metric theories, a relativistic s c a l a r model i s analyzed for covariance and analyticity. This model shows that a commonly accepted prescription for treating the negative-norm states i s not covariant, and more sophisticated methods a r e required. The technique of shadow states developed elsewhere i s reviewed a d applied to this problenl.

The extension of relativistic quantum fields to dense systems of fermions strongly interacting via meson exchange is discussed. The direct application of quantum field theory methods in this context fails. The failure of this approach leads to an effective Lagrangian based on an approximation scheme naturally suited to highdensity matter. The resulting model can be solved exactly for cold systems, and small temperature corrections may be added. The applications of this model to neutron stars and problems in elementary particle physics will be made in the following paper.