DDF=Pohlmeyer known in 1981
Posted by Urs Schreiber
Readers of this blog will recall that I was involved in some thinking about Pohlmeyer invariants.
It all started with Th. Thiemann’s paper ‘The LQG-String’ in which the Pohlmeyer invariants were used in an attempt to find an alternative quantization of the relativistic string (which had some relevance to approaches of higher dimensional quantum gravity). This requires to find a consistent quantization of their algebra.
I thought that such a consistent quantization should be obtainable by expressing them in terms of the standard DDF invariants, which have a well known standard quantization and form a closed algebra.
When asking (I, II) K. Pohlmeyer, K.-H. Rehren and D. Bahns if this has been considered I received negative answers of various kinds, most of them saying that it cannot work in principle. At the same time Luboš Motl reported a conversation with Edward Witten about the suspicion that maybe the Pohlmeyer invariants contain no real worldsheet information at all. Related doubts were voiced by others.
Due to this I finally decided to write up how I thought the Pohlmeyer invariants could be re-expressed in terms of DDF invariants, and I had intensive discussion about these ideas with K.-H. Rehren at the DPG meeting in Ulm. The insight finally published as hep-th/0403260 was among the reasons for H. Nicolai to invite me to the AEI and for R. Flume to invite me to speak in the theory seminar in Bonn.
This may explain my surprise when a couple of days ago I received an email by A. P. Isaev, telling me that he had published precisely this result already in the early 1980s in
A. P. Isaev: Integrals of the motion of a closed relativistic string, Theor. Math. Phys. 54 (1983) 134 (original Russian version published in 1981)
and
V. I. Borodulin & A. P. Isaev: On the infinite set of integrals of motion for a closed supersymmetric string, Phys. Lett. B 117 (1982) 69.
Of course I apologize to A. P. Isaev for not having been aware of this old work - I was apparently not the only one. I will include a respective reference in my paper on the arXiv.