Extended Hamiltonian Formalism of the Pure Space-Like Axial Gauge Schwinger Model II

Canonical methods are not sufficient to properly quantize space-like axial gauges. In this paper, we obtain guiding principles which allow the construction of an extended Hamiltonian formalism for pure space-like axial gauge fields. To do so, we clarify the general role residual gauge fields play in the space-like axial gauge Schwinger model. In all the calculations we fix the gauge using a rule, $n{\cdot}A=0$, where $n$ is a space-like constant vector and we refer to its direction as $x_-$. Then, to begin with, we construct a formulation in which the quantization surface is space-like but not parallel to the direction of $n$. The quantization surface has a parameter which allows us to rotate it, but when we do so we keep the direction of the gauge field fixed. In that formulation we can use canonical methods. We bosonize the model to simplify the investigation. We find that the antiderivative, $({\partial}_-)^{-1}$, is ill-defined whatever quantization coordinates we use as long as the direction of $n$ is space-like. We find that the physical part of the dipole ghost field includes infrared divergences. However, we also find that if we introduce residual gauge fields in such a way that the dipole ghost field satisfies the canonical commutation relations, then the residual gauge fields are determined so as to regularize the infrared divergences contained in the physical part. The propagators then take the form prescribed by Mandelstam and Leibbrandt. We make use of these properties to develop guiding principles which allow us to construct consistent operator solutions in the pure space-like case where the quantization surface is parallel to the direction of $n$ and canonical methods do not suffice.Comment: 19 page

Light-Cone Quantization of the Schwinger Model

We consider constructing a canonical quantum theory of the light-cone gauge ($A_-$=0) Schwinger model in the light-cone representation. Quantization conditions are obtained by requiring that translational generators $P_+$ and $P_-$ give rise to Heisenberg equations which, in a physical subspace, are consistant with the field equations. A consistent operator solution with residual gauge degrees of freedom is obtained by solving initial value problems on the light-cones. The construction allows a parton picture although we have a physical vacuum with nontrivial degeneracies in the theory.Comment: 19 pages, two ps figures, uses ptptex.sty and psfi

Perturbative Formulation of Pure Space-Like Axial Gauge QED with Infrared Divergences Regularized by Residual Gauge Fields

We construct a new perturbative formulation of pure space-like axial gauge QED in which the inherent infrared divergences are regularized by residual gauge fields. For that purpose we perform our calculations in coordinates $x^{\mu}=(x^+,x^-,x^1,x^2)$, where $x^+=x^0\sin{\theta}+x^3\cos {\theta}$ and $x^-=x^0\cos{\theta}-x^3\sin{\theta}$. $A_-=A^0\cos{\theta}+A^3 \sin{\theta}=n{\cdot}A=0$ is taken as the gauge fixing condition. We show in detail that, in perturbation theory, infrared divergences resulting from the residual gauge fields cancel infrared divergences resulting from the physical parts of the gauge field. As a result we obtain the gauge field propagator prescribed by Mandelstam and Leibbrandt. By taking the limit $\theta {\to} \frac{\pi}{4}$ we can construct the light-cone formulation which is free from infrared difficulty. With that analysis complete, we perform a successful calculation of the one loop electron self energy, something not previously done in light-cone quantization and light-cone gauge.Comment: 29 pages; 1 figur

The Indispensability of Ghost Fields in the Light-Cone Gauge Quantization of Gauge Fields

We continue McCartor and Robertson's recent demonstration of the indispensability of ghost fields in the light-cone gauge quantization of gauge fields. It is shown that the ghost fields are indispensable in deriving well-defined antiderivatives and in regularizing the most singular component of gauge field propagator. To this end it is sufficient to confine ourselves to noninteracting abelian fields. Furthermore to circumvent dealing with constrained systems, we construct the temporal gauge canonical formulation of the free electromagnetic field in auxiliary coordinates $x^{\mu}=(x^-,x^+,x^1,x^2)$ where $x^-=x^0 cos{\theta}-x^3 sin{\theta}, x^+=x^0 sin{\theta}+x^3 cos{\theta}$ and $x^-$ plays the role of time. In so doing we can quantize the fields canonically without any constraints, unambiguously introduce "static ghost fields" as residual gauge degrees of freedom and construct the light-cone gauge solution in the light-cone representation by simply taking the light-cone limit (${\theta}\to \pi/4$). As a by product we find that, with a suitable choice of vacuum the Mandelstam-Leibbrandt form of the propagator can be derived in the ${\theta}=0$ case (the temporal gauge formulation in the equal-time representation).Comment: 21 pages, uses ptptex.st

1+1 Gauge Theories in the Light-Cone Representation

We present a representation independent solution to the continuum Schwinger model in light-cone (A + = 0) gauge. We then discuss the problem of finding that solution using various quantization schemes. In particular we shall consider equal-time quantization and quantization on either characteristic surface, x + = 0 or x − = 0

A High Body Mass Index and the Vacuum Phenomenon Upregulate Pain-Related Molecules in Human Degenerated Intervertebral Discs

Animal studies suggest that pain-related-molecule upregulation in degenerated intervertebral discs (IVDs) potentially leads to low back pain (LBP). We hypothesized that IVD mechanical stress and axial loading contribute to discogenic LBP’s pathomechanism. This study aimed to elucidate the relationships among the clinical findings, radiographical findings, and pain-related-molecule expression in human degenerated IVDs. We harvested degenerated-IVD samples from 35 patients during spinal interbody fusion surgery. Pain-related molecules including tumor necrosis factor alpha (TNF-alpha), interleukin (IL)-6, calcitonin gene-related peptide (CGRP), microsomal prostaglandin E synthase-1 (mPGES1), and nerve growth factor (NGF) were determined. We also recorded preoperative clinical findings including body mass index (BMI), Oswestry Disability Index (ODI), and radiographical findings including the vacuum phenomenon (VP) and spinal instability. Furthermore, we compared pain-related-molecule expression between the VP (−) and (+) groups. BMI was significantly correlated with the ODI, CGRP, and mPGES-1 levels. In the VP (+) group, mPGES-1 levels were significantly higher than in the VP (−) group. Additionally, CGRP and mPGES-1 were significantly correlated. Axial loading and mechanical stress correlated with CGRP and mPGES-1 expression and not with inflammatory cytokine or NGF expression. Therefore, axial loading and mechanical stress upregulate CGRP and mPGES-1 in human degenerated IVDs, potentially leading to chronic discogenic LBP