Lorentz invariance in non-critical string theory

<h2 id="see-also">See also</h2>
The following textbooks on string theory mention a possibility
of anomaly-free quantization of the string outside critical dimension:

<ul><li>L. Brink, M. Henneaux, Principles of String Theory, Plenum Press, New York and London, (1988), p. 157</li></ul>
Should one not try to use a different representation
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.

<ul><li>B.M. Barbashov, V.V. Nesterenko, Introduction to the Relativistic String Theory, Singapore, World Scientific, (1990), p. 64:</li></ul>
It is difficult to understand that such a thoroughly studied
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.

<ul><li>M. Green, J. Schwarz, E. Witten, Superstring Theory Vol. 1, Cambridge Univ. Press, (1987), p. 124:</li></ul>
considering contribution of conformal factor φ 
in the path integral, it is noticed:

It is conceivable that the integral (3.1.13) is physically
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.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3). Elsevier BV: 207–210. doi:10.1016/0370-2693(81)90743-7. ISSN 0370-2693.:
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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Rohrlich, F. (1975-03-31). "Quantum Dynamics of the Relativistic String". Physical Review Letters. 34 (13). American Physical Society (APS): 842–845. doi:10.1103/physrevlett.34.842. ISSN 0031-9007. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Pron'ko, G.P. (1990). "Hamiltonian theory of the relativistic string". Reviews in Mathematical Physics. 02 (3). World Scientific Pub Co Pte Lt: 355–398. doi:10.1142/s0129055x90000119. ISSN 0129-055X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Pron'ko, G.P. (1990). "Hamiltonian theory of the relativistic string". Reviews in Mathematical Physics. 02 (3). World Scientific Pub Co Pte Lt: 355–398. doi:10.1142/s0129055x90000119. ISSN 0129-055X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Berdnikov, E. B.; Nanobashvili, G. G.; Pron'Ko, G. P. (1993-06-10). "The relativistic theory for principal trajectories and electromagnetic transitions of light mesons part I". International Journal of Modern Physics A. 08 (14). World Scientific Pub Co Pte Lt: 2447–2464. doi:10.1142/s0217751x93000965. ISSN 0217-751X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Berdnikov, E. B.; Nanobashvili, G. G.; Pron'Ko, G. P. (1993-06-20). "The relativistic theory for principal trajectories and electromagnetic transitions of light mesons part II". International Journal of Modern Physics A. 08 (15). World Scientific Pub Co Pte Lt: 2551–2567. doi:10.1142/s0217751x93001016. ISSN 0217-751X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Lorentz invariance in non-critical string theory open-in-new

Lorentz invariance in non-critical string theory