My impression was that John and others regard as a problem that the representations for 2-counterparts of Lie groups seem to reduce to representations of permutation groups for a discrete set of objects. The reason is basically that at the level of abstraction they are working the points of n-dimensional space are replaced with n-tuples of linear spaces of varying dimensions. Vector space replaces the point of the vector space. For ordinary vector spaces one has a continuum of different choices of basis vectors transformed to each other by matrices representing group elements. One cannot however superpose linearly vector spaces and representation matrices can just permute the vector spaces forming the components of the vector. For instance, for Poincare group the representations would be induced from the representations of discrete subgroups of Lorentz group known as Fuschian groups and realized in terms of discrete Möbius transformations of complex plane (Fuschian groups).
I am not sure whether I would like to see this as a problem. Categories, quantum groups, n-groups, hyper-finite factors and their inclusions, etc.. arise in TGD in a close association with the notion of finite measurement resolution which among other things led to a stringy formulation of quantum TGD and to a precise formulation of QFT limit of TGD allowing to predict the values of gauge coupling strengths.
At space-time and imbedding space level finite measurement resolution has discreteness as a space-time correlate. Discrete groups are obviously a natural correlate for the finite measurement resolution when one speaks about symmetries. If there is a connection, this discretization - something very concrete- could have also interpretation in terms of an abstraction process in which one replaces points with vector spaces. Could it really be that these mathematicians are becoming conscious of the mathematics needed to realize elegantly the basic physical picture of quantum TGD? Maybe we indeed are pieces of a Very Great Mind and our feeling about working independently is only an illusion. In Great Experiences, which we may have once or twice during lifetime, we can experience directly a contact with this Great Mind and may have the mysterious and paradoxical Brahman=Atman experience of actually being the Great Mind ourselves. Just asking;-).
2 comments:
I just love this one: "Could it really be that these mathematicians are becoming conscious of the mathematics needed to realize elegantly the basic physical picture of quantum TGD? Maybe we indeed are pieces of a Very Great Mind and our feeling about working independently is only an illusion. In Great Experiences, which we may have once or twice during lifetime, we can experience directly a contact with this Great Mind and may have the mysterious and paradoxical Brahman=Atman experience of actually being the Great Mind ourselves."
Brahman=Atman should be the same as soul and holy spirit, or nous and Sophia; that is the holy trinity, or the same thing as in your theory TGD:-)
Almost every big scientist is sometimes in contact with this wisdom. The get inspiration from it. In this collective mind is all the knowledge and all the wisdom in the world. You don't have to read and study, you just tap in and get it.
This is in theory. In practise it isn't that simple. Oracles and prophets may know a lot, but their knowledge may also easily be distorted and wrong. To be sure to get it right there are no other way than studying. So there are no easy way,remember, in the Bible they talk about that narrow way:-)
Musicians and artists use this all the time. Also writers and healers. Great scientists like Einstein too, and you yourself. Often they describe how they got the idea "in a sudden glimpst", and they can't describe how. But when they do their work and unfold the idea, they often see that it is the truth.
Can somone understand this:-)
I just have to add to my comment some reflections.
Firstly: In ancient times they had no sciense in our sense. They had no tools, so their most appreciated "sciense" was that of magic. They just tapped in in the collective unconsciousness, as Jung called it. As we can see it today with our very limited understandings of universe their picture is at most very interesting. Those who don't understand so much think it's rubbish:-)
Secondly: Today we have maybe tools to understand, at least something, of what universe is. But very few scientists use these tools, because it is very hard work of the mind. It's much easier to climb to reductionistic tools. And maybe reductionistic worldwiev is a must too, otherwise we can't grasp anything. But as I see it, it is also very, very important to have some kind of holistic wiev of what universe is, otherwise our sciense work in blindness.
This is perhaps the most exciting thing in sciense today, and that's why I see TGD as such an interesting object for my studies. Although I can't understand much at all, but something:-). At least I hope so.
Ulla.
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