Longitudinal, Massive Photon in an External Magnetic Field

Abstract

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 -

The polarization tensor for a magnetized vacuum

“…In the long-wavelength limit, kll= 0 in (A.16) gives with nl and n2 such that { w 2 [ l -( 2 e B / 0 2 ) ] 2 -4 m 2 } / 8 e B and ( w 2 -4 m 2 ) / 8 e B are greater than or equal to nl and n2, respectively, and less than n l + l and n 2 + l respectively. Cover and Kalman (1974) obtained the opposite sign for a33'a)(0, w ) , but this is probably only a difference of notation.…”

Extending the Euler-Heisenberg Lagrangian to some nonperturbative and inhomogeneous field configurations

“…The effectiveness of the above-mentioned mechanism was discussed in Refs. [24,29] for homogeneous background configurations (see also [8,30]). Equations (34) confirm and generalize the results obtained in those works.…”

Pair-creation collective modes in an electron gas

“…Normally, the virtual electrons and positrons emerging from the vacuum do not bind, despite their electrostatic attraction. One case where they do is in the presence of an "ultrastrong" magnetic field, B & 1.6 X 10' 6, where a boson excitation in the form of a massive longitudinal photon has been reported [2]. In the case described here, the high-density degenerate electron background enhances the binding of the virtual pair, making binding possible for any value of the magnetic field, although the strength of the resulting excitation is still a sensitive function of the magnetic-field strength.…”