1. References

2. See also

Usually non-critical string theory is considered in frames of the approach proposed by Polyakov.^[1] The other approach has been developed in^[2][3].^[4] 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 n_μ, 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 P_μ,

it is time-like Rohrlich's gauge.

The problem of the standard light

cone gauge is that the vector n_μ is constant, e.g.

n_μ = (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 n_μ with the world sheet itself.

As a result, the Lorentz generators transform n_μ

and the world sheet

simultaneously, without reparametrizations. The same property holds

if one relates light-like axis n_μ with the world sheet, using in

addition to P_μ 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.

a_i -> b_i

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 M_μν

are cubic in terms of

a_i, and their quantization

acquires well known anomaly. Let's consider a set

b_i = (M_μν,ξ_i)

which contains the Lorentz group generators and internal variables

ξ_i,

complementing M_μν

to the full phase space. In selection of such a set,

one needs to take care that ξ_i

will have simple Poisson brackets with

M_μν 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.

string theory has a gauge symmetry Diff S¹,

reparametrizations of a circle. The symmetry is generated by

Virasoro algebra L_n.

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

.^[5]

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

.^[6]

References

See also

The following textbooks on string theory mention a possibility

of anomaly-free quantization of the string outside critical dimension:

• L. Brink, M. Henneaux, Principles of String Theory, Plenum Press, New York and London, (1988), p. 157

of the string operators so as to avoid the central charge?

Again, it might very well be possible to construct such

a representation and, if so, it is very likely that the

resulting quantum theory would be very different from the one

explained here. It could be that this yet-to-be-constructed

theory would possess an intrinsic interest of its own

(e.g., through the occurrence of infinite-dimensional

representations of the Lorentz algebra). Moreover,

because this theory would not be based on the use of

oscillator variables, it might be more easily extendable

to higher-dimensional objects, such as the membrane.

Further, on pp. 157–159, the quantum solutions of closed string theory

in the class of non-oscillator representations possessing no anomaly

in Virasoro algebra at arbitrary even value of dimension are

explicitly presented.

• B.M. Barbashov, V.V. Nesterenko, Introduction to the Relativistic String Theory, Singapore, World Scientific, (1990), p. 64:

object in classical theory and nonrelativistic quantum mechanics

as the string cannot consistently be analyzed at the quantum level

in a realistic 4-dimensional space-time. Attempts were undertaken

to find other quantum solutions for the relativistic string problem

which would not encounter the above difficulties.

Further, in Sec.11 and Sec.30 quantization of non-critical

string theory in frames of the approaches by Rohrlich and Polyakov

is described.

• M. Green, J. Schwarz, E. Witten, Superstring Theory Vol. 1, Cambridge Univ. Press, (1987), p. 124:

considering contribution of conformal factor φ

in the path integral, it is noticed:

sensible even if the φ dependence does not cancel.

This possibility has motivated some very ingenious suggestions,

but remains uncertain. In any case, for superstring unification

the critical dimension in which the φ dependence cancels

is preferred, since in most suggestions about how to get outside

of the critical dimension, one expects to lose the massless

particles that are present in the critical dimension.

Note: this does not exclude usage of non-critical string theory

in the physics of hadrons, where all coupled states are massive.

Here only self-consistence of the theory, particularly its

Lorentz invariance, is required.

• Qal'eh Sheykhan
• Qaleh Sheykhan
• Qaleh Sheykh, Markazi
• Qaleh Sheykh, Tehran
• Qal'eh Shir
• Qaleh Shir
• Qaleh Shira
• Qal'eh Shira
• Qaleh Shir Khan
• Qal'eh Shir Khan
• Qal'eh Shisheh
• Qaleh Shisheh
• Qaleh Shorkh
• Qal'eh Shorkh
• Qal'eh Shur
• Qaleh Shur
• Qal'eh Shu Shur
• Qaleh Shu Shur
• Qaleh Siah
• Qal'eh Siah
• Qaleh Sin
• Qal'eh Sin
• Qal'eh Siun
• Qaleh Siun
• Qal'eh Sohrab