Abstract
The history of quantum electrodynamics (QED) has been one of unblemished triumph. Given the apparent inevitability of its success, why should we continue studying QED?
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References
R. Brandelik et al. (TASSO Collaboration), Deutsches Elektronen-Synchrotron preprint DESY 80/77 (April 1980).
C. Itzykson and J.-B. Zuber,Quantum Field Theory (McGraw-Hill, 1980), Section 9–4.
Standard references for field theory and QED include J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields (McGraw-Hill, 1965)
a more elementary treatment is given in J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967); see also Ref. 2.
Earlier and more comprehensive reviews include T. Kinoshita, in the Proceedings of the XIX International Conference on High Energy Physics, Tokyo, Japan (1978)
E. Lautrup et al, Phys. Rep. 3, 193 (1972).
Advances in experimental QED are discussed in the paper by D. W. Gidley and A. Rich presented at this conference.
S. J. Brodsky and P. J. Mohr, in Heavy Ion Atomic Physics, I. A. Sellin, ed. (Springer-Verlag, 1977).
See, for example, P. 0. Egan’s talk given at this conference.
T. Kinoshita, Cornell preprint CLNS 79/437 (October 1979).
H. Dehmelt, talk presented at this conference.
R. S. Van Dyck, Jr., Bull. Am. Phys. Soc. 24, 758 (1979).
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S. J. Brodsky and S. D. Drell, Stanford Linear Accelerator Center preprint SLAC-PUB-2534 (June 1980).
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The earliest application of these ideas was by E. Salpeter and H. Bethe, Phys. Rev. 84, 1232 (1951).
A thorough review of these ideas and of earlier references is given by G. P. Lepage, “Two-Body Bound States in Quantum Electrodynamics ”, Stanford Linear Accelerator Center report SLAC-212 (July 1978).
See also G. P. Lepage, Phys. Rev. A16, 863 (1977).
W. E. Caswell and G. P. Lepage, Phys. Rev.A18, 810 (1978).
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We are working in the atom’s rest frame. Also Gt is in general weakly dependent upon the relative energies k° and p°. Here we choose k° = p° = 0 for convenience; fixing the relative energies does not shift the poles in the total energy E. Finally, notice that Gt depends (implcitly in Section IV) upon the spins of the atom’s constituents. We can define \(G_T \left( {\vec p,\vec k,E} \right)_{\lambda \lambda ',\mu \mu '} = u^{(1)} \left( {\vec k\mu } \right)^\dag u^{(2)} \left( { - \vec k\mu '} \right)^\dag \bar G_T \gamma _O^{(1)} \gamma _O^{(2)} u^{(1)} \left( {\vec p\lambda } \right)\) \({\text{x u}}^{{\text{(2)}}} ( - \vec p\lambda '{\text{)}}\) without loss of generality.
This follows since each GT(n)in Eq. (1) is finite at the bound state energies. It is also evident if one realizes that poles in E, as in Eq. (2), are also poles in á since -1 ot m ~I (Å-E ) - (E +-y) . Thus these poles cannot appear in any n 2n2 finite order polynomial of a, such as would result from truncating Eq. (1).
Eqs. (1)–(3) have close analogues in non-relativistic quantum mechanics. GT is analogous to the T-matrix used in describing P2 1 non-relativistic scattering. Formally T(E) = (E - %-) -*---V ^2 2m Å-Ç+ßå Ë ç where Ç = ^- + V is the complete Hamiltonian. Expanding in powers of V gives the Born series for Ô (= V + VSV + ...), which is analogous to Eq. (1). Note also that since a' .- = J * * , T(E) has poles at bound state energies En with residues related to the corresponding wave functions. Finally Ô satisfies the Lippmann-Schwinger equation T=V + VST, just as in Eq. (3).
Potential V can be derived by rewriting Eq. (3) as V = GT - VSGT = GT - GTSGT + GTSGTSGT - ... Substituting the perturbative expansion (1) for GT, we obtain recursion relations for the V(n): V(1)=GT(1), V(n)=G(n) - Õ V(n-m)SGW. m=l
This statement must be qualified. The expansion for V is convergent except for certain radiative corrections, such as contribute to the Lamb shift for example. These are always high order corrections to the spectrum, and as such are readily computed. What remains once these terms have been removed can be shown to converge using general power counting arguments. An excellent pedagogical review of the problems associated with Lamb shift-like corrections is in D. R. Yennie, in Lectures on Strong and Electromagnetic Interactions, Brandeis Summer Institute, Vol. I (1963).
Notice that δV is in general energy dependent. This results in additional terms in the perturbation series (see Ref. 14).
V. W. Hughes, “Muonium”, in Exotic Atoms 79, K. Crowe et al., eds. (Plenum Press, 1980).
This number combines that given in Ref. 23 with a new result quoted in E. Borie, preprint to be published in the proceedings of the Symposium uber Stand und Ziele der Quantenelektrodynamik, Mainz, W. Germany (1980).
These terms are reviewed in S. J. Brodsky and G. W. Erickson, Phys. Rev. 148, 26 (1966).
These terms are reviewed in G. P. Lepage, Ref. 14, and in Ref. 15. See also G. T. Bodwin et al., Phys. Rev. Lett. 41, 1088 (1978).
W. E. Caswell and G. P. Lepage, Phys. Rev. Lett.41, 1092 (1978).
Terms through O(á3Ry) are explained in Ref. 2, Section 10-3-2. Higher order terms are derived in Refs. 14 and 31.
P. O. Egan et al., Phys. Rev. A15, 251 (1977).
A. P. Mills, Jr. and G. H. Bearman, Phys. Rev. Lett. 34, 246 (1975).
Some progress has already been made. See V. K. Cung et al., Phys. Lett. 68B, 474 (1977),
W. Buchmuller E. Remiddi, Nucl. Phys. B162, 250 (1980).
Theory for these rates is reviewed in W. E. Caswell and G. P. Lepage, Phys. Rev. A20, 36 (1979).
D. W. Gidley et al., Ref. 4.
S. Berko et al., Proc. 5th Int. Conf. on Positron Annihilation, Japan (1979).
T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).
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Lepage, G.P. (1981). Theoretical Advances in Quantum Electrodynamics. In: Kleppner, D., Pipkin, F.M. (eds) Atomic Physics 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-9206-8_12
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