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. Let's 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. It is well known that 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. Their common properties are
- explicit Lorentz-invariance at d=4
- reparametrization degrees of freedom fixed by the gauge
- Regge-like spin-mass spectrum
The reader familiar with variety of branches co-existing in modern string theory will not wonder why many different quantum theories can be constructed for essentially the same physical system. The approach described here does not intend to produce a unique ultimate result, it just provides a set of tools suitable for construction of your own quantum string theory. Since any value of dimension can be used, and especially d=4, the applications could be more realistic. For example, the approach can be applied in physics of hadrons, to describe their spectra and electromagnetic interactions .
Famous quotes containing the word string:
“... looped with the creep of varying light,
Monkey-brown, fish-grey, a string of infected circles
Loitering like bullies, about to coagulate....”
—Philip Larkin (19221986)