Abstract
We use the interpolating coordinates studied by Hornbostel to investigate a transition from equal-time quantization to light-front quantization, in the context of two-dimensional $\phi^4$ theory. A consistent treatment is found to require careful consideration of vacuum bubbles, in a nonperturbative extension of the analysis by Collins. Numerical calculations of the spectrum at fixed box size are shown to yield results equivalent to those of equal-time quantization, except when the interpolating coordinates are pressed toward the light-front limit. In that regime, a fixed box size is inconsistent with an accurate representation of vacuum-bubble contributions and causes a spurious divergence in the spectrum. The light-front limit instead requires the continuum momentum-space limit of infinite box size. The calculation of the vacuum energy density is then shown to be independent of the interpolation parameter, which implies that the light-front limit yields the same spectrum as an equal-time calculation. This emphasizes the importance of zero modes and near-zero modes in a light-front analysis of any theory with nontrivial vacuum structure.
Highlights
There has been a resurgence of interest in the spectrum of two-dimensional φ4 theory [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],1 partly because of what appeared to be an inconsistency between results from equal-time quantization and light-front quantization
The apparent inconsistency has been resolved, as a difference in mass renormalizations [7,9, 11,16], there remain various issues related to the structure of the vacuum
The mass spectrum that was used in [7] lacked the correct behavior near the critical coupling
Summary
There has been a resurgence of interest in the spectrum of two-dimensional φ4 theory [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],1 partly because of what appeared to be an inconsistency between results from equal-time quantization and light-front quantization. The mass spectrum that was used in [7] lacked the correct behavior near the critical coupling This caused the incorrect behavior for the computed value of hφ2i and created the need for extrapolation. The zero modes (p− 1⁄4 0) and negative p− states have infinite light-front energy and are removed from the spectrum, as c → 0. These modes can contribute to light-front computations and, in particular, to vacuum expectation values [23,31]. The calculations are done numerically, in a Fock basis of discrete momentum states in an x− box These lead to a much better understanding of the c → 0 limit.
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