page_title: Lorentz invariance in non-critical string theory
hierarchical_section_title: Lorentz invariance in non-critical string theory
caption_attribution_description: English: Spin-mass spectrum. Lorentz-invariant light cone gauge is related with singularities of DDF vector fields.
context_page_description: Usually non-critical string theory is considered in frames of the approach proposed by Polyakov. The other approach has been developed in. It represents a universal method to maintain explicit Lorentz invariance in any quantum relativistic theory. On an example of Nambu-Goto string theory in 4-dimensional Minkowski space-time the idea can be demonstrated as follows:
  Geometrically the world sheet of string is sliced by a system of
  parallel planes to fix a specific
  parametrization, or
  gauge on it.
  The planes are defined by a normal vector nμ, the gauge axis.
  If this vector belongs to light cone, the parametrization corresponds
  to light cone gauge, if it is directed along world sheet's
  period Pμ,
  it is time-like Rohrlich's gauge.
  The problem of the standard light
  cone gauge is that the vector nμ is constant, e.g.
  nμ =,
  and the system of planes is "frozen" in Minkowski
  space-time. Lorentz transformations change the position of the
  world sheet with respect to these fixed planes, and they are followed
  by reparametrizations of the world sheet. On the quantum level the
  reparametrization group has anomaly,
  which appears also in Lorentz group
  and violates Lorentz invariance of the theory.
context_section_description: Usually non-critical string theory is considered in frames of the approach proposed by Polyakov. The other approach has been developed in. It represents a universal method to maintain explicit Lorentz invariance in any quantum relativistic theory. On an example of Nambu-Goto string theory in 4-dimensional Minkowski space-time the idea can be demonstrated as follows:
  Geometrically the world sheet of string is sliced by a system of 
  parallel planes to fix a specific 
  parametrization, or 
  gauge on it.
  The planes are defined by a normal vector nμ, the gauge axis.
  If this vector belongs to light cone, the parametrization corresponds
  to light cone gauge, if it is directed along world sheet's 
  period Pμ,
  it is time-like Rohrlich's gauge.
  The problem of the standard light 
  cone gauge is that the vector nμ is constant, e.g. 
  nμ = (1, 1, 0, 0), 
  and the system of planes is "frozen" in Minkowski
  space-time. Lorentz transformations change the position of the 
  world sheet with respect to these fixed planes, and they are followed 
  by reparametrizations of the world sheet. On the quantum level the 
  reparametrization group has anomaly, 
  which appears also in Lorentz group 
  and violates Lorentz invariance of the theory. On the other hand, 
  the Rohrlich's gauge relates nμ with the world sheet itself. 
  As a result, the Lorentz generators transform nμ 
  and the world sheet 
  simultaneously, without reparametrizations. The same property holds 
  if one relates light-like axis nμ with the world sheet, using in 
  addition to Pμ other dynamical vectors available 
  in string theory. 
  In this way one constructs Lorentz-invariant parametrization of 
  the world sheet, where the Lorentz group acts trivially and does not
  have quantum anomalies.
  Algebraically this corresponds to a canonical transformation aᵢ -> bᵢ in the classical mechanics to a new set of variables, explicitly containing all necessary generators of symmetries. For the standard light cone gauge the Lorentz generators Mμν are cubic in terms of oscillator variables aᵢ, and their quantization acquires well known anomaly. Consider a set bᵢ = (Mμν,ξᵢ) which contains the Lorentz group generators and internal variables ξᵢ, complementing Mμν 
  to the full phase space. In selection of such a set, 
  one needs to take care that ξᵢ will have simple Poisson brackets with Mμν and among themselves. Local existence of such variables is provided by Darboux's theorem. Quantization in the new set of variables eliminates anomaly from the Lorentz group. Canonically equivalent classical theories do not necessarily correspond to unitary equivalent quantum theories, that's why quantum anomalies could be present in one approach and absent in the other one.
  Group-theoretically 
  string theory has a gauge symmetry Diff S¹, 
  reparametrizations of a circle. The symmetry is generated by 
  Virasoro algebra Lₙ. 
  Standard light cone gauge fixes the most of gauge degrees
  of freedom leaving only trivial phase rotations U(1) ~ S¹. 
  They correspond
  to periodical string evolution, generated by 
  Hamiltonian L₀.
  Let's introduce an additional layer on this diagram:
  a group G = U(1) x SO(3) of gauge transformations of the world sheet, 
  including the trivial evolution factor and rotations of the gauge axis
  in center-of-mass frame, with respect to the fixed world sheet. 
  Standard light cone gauge 
  corresponds to a selection of one point in SO(3) factor, leading to 
  Lorentz non-invariant parametrization. Therefore, one must select
  a different representative on the gauge orbit of G, this time 
  related with the world sheet in Lorentz invariant way. 
  After reduction of the mechanics to this representative
  anomalous gauge degrees of freedom are removed from the theory.
  The trivial gauge symmetry U(1) x U(1) remains, including evolution 
  and those rotations which preserve the direction of gauge axis.
  Successful implementation of this program has been done in 
  
  .
  These are several unitary non-equivalent versions of
  the quantum open Nambu-Goto string theory, where the gauge axis 
  is attached to different geometrical features of the world sheet.
  T
caption: Spin-mass spectrum.[4]  Lorentz-invariant light cone gauge  is related with singularities of DDF vector fields.
exif: {}