Against Symmetry II
Carl might not like the term background independence, but I have fond memories of independently lurking in the dark corners of a Sydney library in the early '90s, trying to figure out how quantum groups might allow us to throw out classical spaces in our construction of operators. I gave some seminars on applications of quantum groups to high temperature superconductivity, but these were no doubt naive and no one in the audience paid the least attention.
Not much has changed, although with the net I can now lurk from the comfort of home. Sometime in 1995, I decided that the early quantum group papers were trying to put spaces underneath the algebras, just like in old fashioned gauge theory. And even today, some people insist on pointing out that operads must act on something. Fortunately, at this level of abstraction, we can say that the spaces concerned are models or ideas imposed as experimental constraints, and they are not supposed to represent an actual aether, appearing universally in any question we might think to ask.
Not much has changed, although with the net I can now lurk from the comfort of home. Sometime in 1995, I decided that the early quantum group papers were trying to put spaces underneath the algebras, just like in old fashioned gauge theory. And even today, some people insist on pointing out that operads must act on something. Fortunately, at this level of abstraction, we can say that the spaces concerned are models or ideas imposed as experimental constraints, and they are not supposed to represent an actual aether, appearing universally in any question we might think to ask.
posted by Kea | 9:10 AM
8 Comments:
I have also pondered the role of quantum spaces. The motion of quantum spinors seems to make sense and would in my own approach provide a manner to characterize finite measurement resolution.
The non-commutativity of spinor components with different spin would mean that quantum measurement reducing the state to spin 1/2 or -1/2 is not possible. The proper observables would be now probabilities, the predictions of ordinary quantum theory. The probability of say 1/2 state would have quantized eigenvalue spectrum. Fuzzy qubit would be in question.
The idea of quantum space does not look attractive to me in general and non-commutative quantum field theories look horribly ugly to me. Quantum geodesic sphere (of CP_2) could be however useful notion allowing alternative formulation for dicretization associated with finite measurement resolution. At points of number theoretic braid one would have commutativity so that they would define the analog of reduced 2-sphere of causally independent points. For arbitrary two points there would be no commutativity and the fields at these points would not be causally independent so that discretization would emerge automatically from finite measurement resolution.
Quite generally non-commutativity would correspond to correlations and quantum dimension - which is smaller than the topological one- would express the presence of these correlations.
To sum up, non-commutativity would be an effective description summarizing finite measurement resolution. The Platonia behind the non-commutative veil would correspond to good old commutative manifolds. Good old Platonian symmetries would become quantum symmetries in finite measurement resolution.
An off topic comment,
Lubos summarized the ranking list of New Einsteins in Discoverer magazine. Lisi is the candidate number one. Witten is still hanging at the bottom of the list but might be replaced by Britney Spears any time.
Perhaps string model hypists should have thought twice: journalists are intelligent fellows and learned to make hype without the guidance of string theorists.
Matti, LOL! I have to admit, I also laughed a lot when I saw those photos. I don't think it is a fine example of intelligent journalism.
Wikipedia has a page on new einsteins
This is backward. Attempts at background independence focus on ridding ourselves of space - but the real solution is to rid ourselves of matter, by making matter emerge from vacuum geometry. This has been done. Ask for details. Open to quantization, should anyone care. General Yang-Mills theories included at no extra "charge". As it were.
-drl
Hello, drl. I am afraid my point of view is a bit more abstract and, by the way, the readers of this blog are familiar with your work. Whereas you talk about vacuum geometry, I like to think about diagrammatic reasoning using higher category theory. Either way, matter disappears, or rather, matter becomes less ontological.
Hi Kea,
I applaud such efforts, if only for their abstract beauty, but history shows that progress always occurs at the boundary between physics and metaphysics, and amounts to pushing back the vacuum, making it more "all inclusive" so to speak. The best example is the introduction of 4-d space of indefinite metric. This caused the idea of propagation to shed its mechanical weight, as it were, and made it simply an aspect of geometry - in the sense that metric geometry emerges from projective geometry with the postulation of a fundamental quadric - for Minkowski space, the light cone. Likewise, in quantum theory the substrate of reality is no longer tied to the material objects in themselves, rather their mutual interconnection as expressed in the Hilbert space of state functions.
So the essential question in physics should always be how to make the vacuum more inclusive.
-drl
Hi drl. Mutual interconnection is the reason for category theory, in my view. Your statement has a lot in common with twistor ideas, which are a large part of the inspiration for ideas discussed here. And yes, I am very tired of mainstream attempts to quantize fixed and limited geometrical structures, as if QG says nothing new for physics.
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