• DOI:10.1088/0253-6102/38/4/429
• Corpus ID: 119406277

Boundary Poisson structure and quantization

@inproceedings{Zhao2001BoundaryPS,
  title={Boundary Poisson structure and quantization},
  author={Liu Zhao and Wenli He},
  year={2001}
}

• Liu Zhao, Wenli He
• Published 1 November 2001
• Physics

Quantization of classical field in the presence of various boundary conditions is an old problem for which a systematical solution is still missing. This problem is important because it is related to a vast range of physical problems including, e.g. surface effect in condensed matter physics, cavity QED, Casimir energy, two-dimensional integrability [9, 10, 11, 12], mass generation [13, 14] and especially conformal field theory and open string theory . Recently, there has been a renewed interests in this problem among string theorists since the discovery of D-branes, noncommutativity and other extended objects (like the so-called Horizontal branes [5]) in string theory. In most recent papers on this subject, people are tempted to use the Dirac procedure [6] for constrained systems to treat the inconsistency of Poisson structure with the boundary condition [1, 2, 3, 4]. However, for at least several reasons we think that this treatment of boundary condition is not quite appreciated. First, since the boundary condition is a constraint at a specific spacial point (or a spacial hypersurface if there are more than one spacial directions) which is of functional measure 0 in the space of field, the direct application of Dirac procedure necessarily fails because the standard definition of Dirac Poisson brackets in this case would involve the inverse of δ(0), which could hardly be given any practical sense. An alternative way is to put the spacial direction with the boundary into a lattice form and then implement the Dirac procedure. In this case the construction of Dirac brackets seems to make perfect sense [1, 2, 3, 4], but, unfortunately, there seems to be no simple continuum limit for such lattice regularized theories for which the Dirac brackets remain consistent. More over, the appearance of an explicit length scale – the lattice spacing – is another drawback of this formalism, which is especially unfavorable in scale invariant theories.