Magnetic Catalysis of Chiral Symmetry Breaking and the Pauli Problem

The non-perturbative Schwinger-Dyson equation is used to show that chiral symmetry is dynamically broken in QED at weak gauge couplings when an external uniform magnetic field is present. A complete analysis of this phenomenon may shed light on the Pauli problem, namely, why α = 1/137. Let me begin with a joke which some of you may have heard before. One of Wolfgang Pauli's lifelong dreams was to understand why the fine structure constant in electrodynamics is 1/137 (in the infrared regime). Pauli was also known to be a difficult person, very hard to please. As the joke goes, the first thing Pauli asked God after his death was to explain why α = 1/137. As God went on with His explanation, Pauli grew more and more dissatisfied. After five minutes, Pauli was seen storming out of Heaven's Gates mumbling, " Ridiculous! " Like Pauli, I also would like to understand why α = 1/137. To dignify this problem, I will call it the Pauli problem. It is possible that chiral symmetry breaking by an external field in QED may provide some insight on this old problem by giving a critical value of α close to 1/137. 1 Admittedly, nothing close to that magic value has arisen in the results we have obtained so far 2 , but our analysis is not yet complete. My interest in chiral symmetry breaking by an external field dates back a dozen years ago, to the time when a multiple correlated and narrow-peak structures in electron and positron spectra 3 was observed in heavy-ion experiments at GSI. Kikuchi and I 4 interpreted the e + e − peaks as decay products of a new type of positronium, which is formed in a new QED phase induced by the electromagnetic fields of the colliding heavy ions. The theoretical underpinning of this scenario was provided by earlier works 5 which indicated that QED might have a non-perturbative strong-coupling phase, characterized by spontaneous chiral symmetry breaking, in addition to the familar weak-coupling phase. The negative results in recent heavy-ion collision experiments at Argonne 6 have rendered our interpretation moot. Nevertheless, the problem of chiral symmetry breaking by an external field is still interesting as it may shed light on the Pauli problem, and as it provides an example of vacuum