Light-front vacuum

Abstract

Background: The vacuum in the light-front representation of quantum field theory is trivial while vacuum in the equivalent canonical representation of the same theory is non-trivial. Purpose: Understand the relation between the vacuum in light-front and canonical representations of quantum field theory and the role of zero-modes in this relation. Method: Vacuua are defined as linear functionals on an algebra of field operators. The role of the algebra in the definition of the vacuum is exploited to understand this relation. Results: The vacuum functional can be extended from the light-front Fock algebra to an algebra of local observables. The extension to the algebra of local observables is responsible for the inequivalence. The extension defines a unitary mapping between the physical representation of the local algebra and a sub-algebra of the light-front Fock algebra. Conclusion: There is a unitary mapping from the physical representation of the algebra of local observables to a sub-algebra of the light-front Fock algebra with the free light-front Fock vacuum. The dynamics appears in the mapping and the structure of the sub-algebra. This correspondence provides a formulation of locality and Poincaré invariance on the light-front Fock space.

Nonperturbative light-front Hamiltonian methods

“…The latter aspect makes the Fock-state wave functions unambiguous as probability amplitudes for constituent momentum distributions. This does leave open the question of the connection with the equal-time vacuum and properties such as symmetry breaking that are usually associated with the vacuum [38,39]; however, these considerations are beyond the scope of this review.…”

Nonperturbative matching between equal-time and lightcone quantization

“…This agreement in the observable Z indicates that at a fixed value of µ 2 gap /λ both ET and LC truncation are describing the same theory, confirming our procedure for matching bare parameters. 13…”

“…The main goal of this paper has been to obtain a deeper understanding of the relation between ET and LC quantization, focusing on the special case of λφ 4 theory in 2d. In 13 In this case, we have not literally used our mapping of bare parameters, but rather went directly to Z as a function of µ 2 gap /λ in both quantizations. We have already seen that our map matches µ 2 gap /λ in terms of the bare parameters to reasonably high accuracy, so in principle there is not much difference between first writing Z in terms ofλET andλLC and then mapping, versus directly expressing Z in terms of µ 2 gap /λ in each quantization.…”

“…One pays a conceptual price for this simplification, however. Important physics effects, such as spontaneous symmetry breaking and renormalization of the vacuum energy, are subtle to uncover in LC quantization [6][7][8][9][10][11][12][13][14][15]. Many of the difficult subtleties of LC can be traced to the fact that energy p + = µ 2 2p − is inversely proportional to momentum p − in terms of the Lorentz-invariant µ 2 = p 2 , and consequently "zero modes" with vanishing p − have infinite LC energy and are lifted out of the spectrum.…”

The Light-Front Vacuum

“…The resolution of this apparent inconsistency is that the annihilation operator does not completely characterize the vacuum [5]. Another characterization of the vacuum is as an invariant linear functional on an algebra of operators.…”