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Non-critical string theory: Lorentz invariance : ウィキペディア英語版
Lorentz invariance in non-critical string theory

Usually non-critical string theory is considered
in frames of the approach proposed by Polyakov
.〔
A.M. Polyakov, Phys. Lett. B, V.103, p.207 (1981):
the paper shows in frames of path integral formulation,
that quantum Nambu-Goto string theory at d=26
is equivalent to collection of linear oscillators,
while at other values of dimension the theory exists as well,
and contains a non-linear field theory
associated with Liouville modes.
Papers cited below use for quantization
Dirac's operator formalism.〕
The other approach has been developed in
〔F. Rohrlich, Phys.Rev.Lett., V.34, p.842 (1975).〕

G.P. Pron'ko, Rev. Math. Phys., V.2, N.3, p.355 (1991).

.〔S.V. Klimenko, I.N. Nikitin,
Non-Critical String Theory:
classical and quantum aspects, Nova Science Pub.,
New York 2006, ISBN 1-59454-267-8.〕
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 = (Mμνi)
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


.〔
Concise Encyclopedia of Supersymmetry
and noncommutative structures in mathematics and physics,
entry "Anomaly-Free Subsets",
Kluwer Academic Publishers, Dordrecht 2003, ISBN 1-4020-1338-8.

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
.〔
E.B. Berdnikov, G.G. Nanobashvili, G.P. Pron'ko,
Int. J. Mod. Phys. A, V.8, N14, p.2447 (1993); N15, p.2551 (1993).

== References ==


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Lorentz invariance in non-critical string theory」の詳細全文を読む



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