## Michael FrySenior Lecturer in Mathematics |

Every physical process calculable within the standard model ultimately depends on the model's fermion determinants. These are part of the effective functional measure for the gauge fields after integrating out the fermion fields. Without them, charge and color screening, quark fragmentation into hadrons, unitarity and much more would be lost. Accordingly, they are fundamental, and the nonperturbative structure of the standard model requires corresponding information about its determinants.

They are also hard to calculate, and physicists by and large lost interest in them during the early 1980s. With the advent of large machines lattice QCD physicists now include the determinant in their calculations. But analytic results remain very scarce.

As with any problem in physics one begins with the simplest cases first, namely, QED determinants in 2, 3, and 4 dimensions. To be useful these determinants need to be known for a general background gauge field. Because of this the best one can hope for are restrictive upper and lower bounds on a determinant and its asymptotic behavior for small fermion mass and/or strong coupling. This is the subject of the papers listed below.

These lead to the 2015 paper dealing with the electroweak model. It asks if the model can be nonperturbatively quantized. Stated differently, is perturbation theory the only means by which it can be understood?

This question depends on whether QED itself can be nonperturbatively quantized. That is, does the functional integral--path integral--representation of any physical process in QED converge? The QED fermion determinant is central to this question. This paper derives a new representation of the determinant that allows its large ampltitude Maxwell field variation to be bounded for a class of random fields. Whether the functional integral over the Maxwell potential converges is reduced to the question of what constitutes a set of random potentials of measure 1. This paper gives *prima facie* evidence that Maxwellian zero modes are necessary for the nonperturbative quantization of QED and, by implication, for the nonperturbative quantization of the electroweak model.

For the 2015 paper to be relevant to the complete electroweak model it has to be shown that QED decouples from the remainder of the model for large amplitude variations of the Maxwell field. The 2018 paper establishes that a necessary condition for this is satisfied for a broad class of random Maxwell fields. The main result of this paper--that the Euclidean Dirac propagator in four dimensions vanishes for large amplitude variations of the Maxwell field--resolves the long-standing open question of whether this happens.

- M. P. Fry,
*Massive Schwinger model and four-dimensional QED: The connection,*Physical Review D**45**(1992) 682-686. - M. P. Fry,
*Fermionic Determinant of the Massive Schwinger Model*, Physical Review D**47**(1993) 2629-2632 hep-th/9301013 - M. P. Fry,
*Fermion determinants in static, inhomogeneous magnetic fields*, Physical Review D**51**(1995) 810. hep-th/9410126 - M. P. Fry,
*QED in strong, finite-flux magnetic fields*, Physical Review D**53**(1996) 980. hep-th/9510203 - M. P. Fry,
*QED in inhomogeneous magnetic fields*, Physical Review D**54**(1996) 6444. hep-th/9606037 - M. P. Fry,
*Paramagnetism, zero modes and mass singularities in QED in 1+1, 2+1 and 3+1 dimensions*, Physical Review D**55**(1997) 968. hep-th/9611055 - M. P. Fry,
*Effective actions in QED in 1+1, 2+1 and 3+1 dimensions*, Int. J. Mod. Phys. A**12**(1997) 1153. - M. P. Fry,
*Chiral limit of the two-dimensional fermionic determinant in a general magnetic field,*J. Math. Phys.**41**(2000) 1691-1710. hep-th/9911131 - M. P. Fry,
*Mass zeros in the one-loop effective actions of QED in 1 + 1 and 1 + 3 dimensions*, Physical Review D**62**(2000) 125007. hep-th/0010008 - M. P. Fry,
*Fermion determinants*, Int. J. Mod. Phys. A**17**(2002) 936. hep-th/0111244 - M. P. Fry,
*Fermion determinant for general background gauge fields*, Physical Review D**67**(2003) 065017. hep-th/0301097 - M. P. Fry,
*Fermion Determinants 2003,*Proceedings of the Sixth Workshop on Quantum Field Theory under the Influence of External Conditions, U. Oklahoma, Sept. 15-19, 2003, edited by Kimball A. Milton (Rinton Press, Princeton, NJ, 2004) p. 271 hep-th/0311145 - M. P. Fry,
*Fermion Determinants: Some Recent Analytic Results*. Int. J. Mod. Phys. A**20**(2005) 4492. hep-th/0411011 - M. P. Fry,
*Evidence for mass zeros of the fermionic determinant in four-dimensional quantum electrodynamics*, hep-th/0612218 - M. P. Fry,
*Mass zeros of the fermionic determinant in four-dimensional quantum electrodynamics*, Physical Review D**75**(2007) 065002. [This is a reduced version of hep-th/0612218] - M. P. Fry,
*Nonperturbative results for the mass dependence of the QED fermion determinant*, Physical Review D**81**(2010) 107701. An extended version of this paper appears in hep-th/1005.4849 - M. P. Fry,
*Stability of QED*, Physical Review D**84**(2011) 065021, arXiv: 1108.4345. - M. P. Fry,
*Nonperturbative quantization of the electroweak model's electrodynamic sector,*, Physical Review D**91**(2015) - M. P. Fry,
*Strong, random field behavior of the Euclidean Dirac propagator,*, Physical Review D**98**(2018) 033002. arXiv:1807.11741 085026. arXiv:1504.03117

**Postal address:**

School of Mathematics

Trinity College

Dublin 2

Ireland

Departmental fax number: +353-1-896 2282.

Updated 25th August 2015