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Large Gauge Symmetries and Asymptotic States in QED

Barak Gabai, Amit Sever

TL;DR

This work reframes Large Gauge Transformations (LGT) in QED as infinite global symmetries and analyzes their action on physical, Kulish-Faddeev dressed asymptotic states. It shows that LGT charges can be assigned to the vacuum and, for dressed states, depend only on total electric charge, not on particle momenta, with momentum-dependent pieces canceling between bare charges and their soft photon clouds. The soft-photon Ward identities familiar from undressed states arise only when one neglects the physical dressing, while the dressed S-matrix exhibits IR finiteness and a trivial leading soft behavior. A Wilson line picture provides an intuitive interpretation: the dressing distributes LGT charge into a universal, momentum-independent vacuum pattern. These results point toward similar structures for gravity and BMS symmetries and illuminate the role of memory effects and vacuum structure in infrared-safe scattering.

Abstract

Large Gauge Transformations (LGT) are gauge transformations that do not vanish at infinity. Instead, they asymptotically approach arbitrary functions on the conformal sphere at infinity. Recently, it was argued that the LGT should be treated as an infinite set of global symmetries which are spontaneously broken by the vacuum. It was established that in QED, the Ward identities of their induced symmetries are equivalent to the Soft Photon Theorem. In this paper we study the implications of LGT on the S-matrix between physical asymptotic states in massive QED. In appose to the naively free scattering states, physical asymptotic states incorporate the long range electric field between asymptotic charged particles and were already constructed in 1970 by Kulish and Faddeev. We find that the LGT charge is independent of the particles' momenta and may be associated to the vacuum. The soft theorem's manifestation as a Ward identity turns out to be an outcome of not working with the physical asymptotic states.

Large Gauge Symmetries and Asymptotic States in QED

TL;DR

This work reframes Large Gauge Transformations (LGT) in QED as infinite global symmetries and analyzes their action on physical, Kulish-Faddeev dressed asymptotic states. It shows that LGT charges can be assigned to the vacuum and, for dressed states, depend only on total electric charge, not on particle momenta, with momentum-dependent pieces canceling between bare charges and their soft photon clouds. The soft-photon Ward identities familiar from undressed states arise only when one neglects the physical dressing, while the dressed S-matrix exhibits IR finiteness and a trivial leading soft behavior. A Wilson line picture provides an intuitive interpretation: the dressing distributes LGT charge into a universal, momentum-independent vacuum pattern. These results point toward similar structures for gravity and BMS symmetries and illuminate the role of memory effects and vacuum structure in infrared-safe scattering.

Abstract

Large Gauge Transformations (LGT) are gauge transformations that do not vanish at infinity. Instead, they asymptotically approach arbitrary functions on the conformal sphere at infinity. Recently, it was argued that the LGT should be treated as an infinite set of global symmetries which are spontaneously broken by the vacuum. It was established that in QED, the Ward identities of their induced symmetries are equivalent to the Soft Photon Theorem. In this paper we study the implications of LGT on the S-matrix between physical asymptotic states in massive QED. In appose to the naively free scattering states, physical asymptotic states incorporate the long range electric field between asymptotic charged particles and were already constructed in 1970 by Kulish and Faddeev. We find that the LGT charge is independent of the particles' momenta and may be associated to the vacuum. The soft theorem's manifestation as a Ward identity turns out to be an outcome of not working with the physical asymptotic states.

Paper Structure

This paper contains 11 sections, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review
  3. LGT of a Photon
  4. LGT of an Undressed Particle
  5. The LGT Generator
  6. Physical Asymptotic States
  7. LGT of the Vacuum
  8. LGT of Dressed Particles
  9. Wilson Lines Interpretation
  10. Asymptotic Field Expression
  11. Memory Effect of Physical Asymptotic States

Figures (3)

  • Figure 2.1: a) A Penrose diagram of flat space where $r'$ and $t$ are the radial and time coordinates. In blue are lines of constant retarded time $u$, and in red are lines of constant retarded $r$. b) The Rescaled Radial coordinates $(\rho,\tau)$ cover the patch $t>r$. In blue are lines of constant $\tau$ and in red are lines of constant $\rho$.
  • Figure 3.1: All the ways to connect an external soft photon to a Feynman diagram. In (a)-(c) the soft photon connects to an external or internal leg. These are the traditional soft contributions. In (d) and (e) the soft photon is connected to the "clouds" dressing the asymptotic states.
  • Figure 4.1: On the conformal sphere, $W_L$w$$ emerges from the point parametrized by $$w,w̅$$ and reaches the south pole. For example, in this plot $w=e^{-i\pi/6}$.