Exploring the Multidimensional Virasoro Algebra and its Significance for Physics

[Thomas Larsson]

Originally Posted by Thomas Larsson
"This is the reason why I desided to discover the multidimensional Virasoro algebra and its representation theory, many years ago, cf. my 1999 manifesto in http://www.arxiv.org/abs/gr-qc/9909039 ."

This seems nice. And what is the significance for physics - is there any model you can solve with it, or is there otherwise a physical quantity you can compute with it?

There is place one where it does not arise: QFT.

The Virasoro extension is a diff anomaly, defined in any dimension, and the analogous Kac-Moody-like extension is a gauge anomaly, proportional to the second Casimir. However, diff and gauge anomalies in QFT were classified many years ago; there are no diff anomalies in 4D, and gauge anomalies are proportional to the third Casimir. Hence these extensions can not arise in QFT proper.

However, they do arise in a minor modification of QFT, QJT (J for jet). To build the representations, all fields have to be replaced by their Taylor series. In doing so, a new datum is introduced: the expansion point (or rather curve). This is an important modification, because the relevant cocycles depend on it. The physical interpretation is that one must explictly consider the observer's trajectory in spacetime together with the quantum fields.

In view of this lesson from representation theory, I have been working on a reformulation of physics in terms of Taylor data. What is available from 2004, http://www.arxiv.org/abs/hep-th/0411028 , is still seriously flawed, although there was considerable progress compared to what I wrote in 1999, and a better version will be completed early next year. So it is not a new model, but rather a somewhat different framework for quantization.

There is easy to see that if you consider the Laurent polynomial version of the gauge algebra, anomalies are necessary for nonzero charge, cf. http://www.arxiv.org/abs/math-ph/0603024 . It does not really matter whether you consider diffeomorphisms, Yang-Mills or conformal transformations - the important difference is between Laurent and ordinary polynomials, or compact support. In particular, there is no conformal anomaly for polynomials, i.e. L_m with m >= -1.

Since I believe that Laurent polynomial gauge transformations should not be outlawed, and I certainly believe in nonzero charge, it follows that the new gauge anomalies exist and QFT must be replaced QJT.

 

[AndyW]

Dear Thomas Larsson,

Thank you for sharing your work and insights on the multidimensional Virasoro algebra and its representation theory. It is clear that your research has significant implications for physics, particularly in the area of quantization. The fact that your extensions do not arise in traditional QFT but are necessary in QJT highlights the importance of considering the observer's trajectory in spacetime when dealing with quantum fields.

Your reformulation of physics in terms of Taylor data is also intriguing. It seems to offer a different perspective on the quantization process and could potentially lead to new insights and approaches in the field.

I am particularly interested in your mention of a conformal anomaly for polynomials, and how it relates to the existence of gauge anomalies in QJT. This raises questions about the limitations of traditional QFT and the need for a modified framework to fully understand the behavior of quantum fields.

I look forward to reading your updated work and learning more about the implications of your research for the field of physics. Thank you for your contribution to the scientific community.

 

[Entropia]

I appreciate your curiosity about the significance of my work on the multidimensional Virasoro algebra and its representation theory. My research in this area has led to a better understanding of diff and gauge anomalies in QFT, as well as the development of a new framework for quantization, known as QJT (quantum jet theory). This framework considers the observer's trajectory in spacetime together with the quantum fields, and has implications for the existence of gauge anomalies and the need to replace QFT with QJT.

Furthermore, my work has also shed light on the importance of considering Laurent polynomial gauge transformations, which can lead to nonzero charge and the existence of new gauge anomalies. This has significant implications for our understanding of fundamental physical theories and the need for a more comprehensive approach to quantization.

In summary, the multidimensional Virasoro algebra and its representation theory have greatly contributed to our understanding of diff and gauge anomalies in QFT and have opened up new avenues for research in the field of quantum mechanics. I am excited to continue exploring these concepts and their implications for physics.