If the function / is such that arbitrarily large I3; I are admissible in our coordinate system, then we fix the multiplicative factor in/ (and, therefore, in A, £, and expx) by specifying that the asymptotic values be |A;yl 3/4 or |Aj;| l/4 . Then, if the asymptotic form is I A3; l 3/4 , the limiting case A=0 corresponds to a Minkowski space with Cartesian coordiantes x a which are related to ours by2u = x 3 -x 4 , np=x 19 ua = x 2 , and 2v = 2(x 3 +x 4 ) + u(p 2 + cP), where v = 4^/ 4 /l.Similar relations hold when the asymptotic form is |Ay| l/4 . I thank Dr. Fred Ernst for verifying the solution, for the numerical integration of Eq. (15), and for the relations of the coordinates to Cartesian coordinates when A =0.The existence of strong magnetic fields in the vicinity of collapsed stars has brought into focus the question of the dispersive properties of the vacuum in the presence of strong magnetic fields. 1 " 7 Such dispersive features become significant when the magnetic field reaches and exceeds the critical field value B^=m 2 c 3 /fie =4.41 xlO 13 G. In this Letter a novel feature of the electromagnetic vacuum is pointed out: When the magnetic field strength exceeds a second critical value B 0 of the order of {Hc/e 2 )B^, a massive, longitudinal photonlike resonance appears whose mass value roughly coincides with the energy of an electron-positron pair occupying the lowest Landau level. For magnetic field values in the vicinity of the critical field the photon is heavily damped, but for increasing field strengths the lifetime rapidly increases: Typically for B ^10B 01 T^27rxl5xco" 1^7 xl0" 18 sec.The existence of the massive photon is inferred from the study of the longitudinal dispersion rela-with e(k,co), the longitudinal "dielectric function," and a(k, co), the longitudinal polarizability of the vacuum, being related to the regularized longitudinal vacuum polarization ff 33 (k,co) by e(k, co) = 1 + «(k, w) = 1 -ar 2 n 33 (k, w).(We calculated the vacuum polarization in the presence of an arbitrarily strong, uniform magnetic field, to lowest order in e 2 . The photon momentum is restricted to be parallel to B. For small momenta the k dependence of n 33 is not significant and in this Letter we consider the fc -0 limit only. n 33 can be obtained by standard methods, using the appropriate Green's functions for electrons in a magnetic field. 8 The unrenormalized value of II 33 is given by rr /A \ o e * B ^ f°° m 2 + 2neB n 33 (o,w) = 8-XX rrrir^^ (3) The study of the vacuum polarization for strong magnetic fields reveals the existence of a massive, longitudinal photonlike resonance for magnetic fields exceeding B =*ra 2 c 4 /# 3 -

A unified formalism appropriate for calculating the electromagnetic response of both an electron gas and the vacuum in an arbitrarily strong magnetic field is developed. The particular case of propagation parallel to the magnetic field in vacuum is discussed in detail in rhis paper and the full expression for the corresponding . . II,,,(~, w ) is obtained. The pole structures of the dynarnical polarizabilities related to n,,,(O,w) are examined and the existence of a singlet longitudinal massive photon ~nferred. The static, k-dependent. electr~c and magnetic polarizabilities of the vacuum are evaluated from n,,,(k?,0).

The particle part of the polarization matrix for a relativistically dense electron gas in an external magnetic field is explicitly calculated for the case of propagation vector parallel to the magnetic field. The results are combined with the corresponding renormalized vacuum elements to give the complete o -and kdependent dielectric response function eu(E,w) for the system. The static longitudinal element r,,(Z,0) is used to determine the density and field dependence of the screening along the field. The static transverse element el(C,0) determines the magnetic properties of the system. The possibility of magnetic phase transitions is determined for densities greater than a certain B-dependent value. The longitudinal k = 0 element e,,(O,o) is used to investigate the longitudinal plasma mode and the associated plasma frequency of the system; its remarkable magnetic field dependence is displayed. The Landau-level structure of the plasma and the vacuum contribution have important effects on the structure of the mode. The transverse k = 0 lefthanded e+(O,w) and right-handed e-(0,w) elements are employed to analyze the transverse modes and to investigate o,, the associated cyclotron resonance. The latter acquires a complex fine structure owing to the relativistic Landau energy states of the system.

No abstract

The quality of a wiggler field is often characterized by its rms deviation from the ideal. Ths deviation, however, is a poor predictor of actual system performance, particularly in light of the possibilities for optimized error ordering. An alternative field-quality characterization in terms of trajectory wander and cumulative phase shake shows greatly improved correlation with performance.