Background Independence
Let's get back to basics. When one starts thinking about the mathematical manifestation of background independence, whatever that means, it is very easy to get in a muddle. Maybe one picks a collection of nice principal bundle spaces and then tries deleting as much as possible. Spacetime points? OK, gone. Oops! Everything else seems to have gone out the window as well. Internal symmetries? Disappeared. Poof. Is there anything left? What about the spacetime of quantum mechanics? Well, we'd better figure out a way to do quantum mechanics without the usual wavefunction paraphenalia.
Are Feynman diagrams all right? Well, they might be, in the context of diagrammatic reasoning, but then that pesky Minkowski background is still lurking there somewhere... Maybe if we turned that into twistor geometry things would finally look up. Points look like spheres now, which in itself doesn't sound like a huge advance, but the beauty of it is that causality now has a language capable of stepping outside of set theory. Category theory isn't just there to put a whole lot of complicated geometry into a neat package. It allows foundational mathematics and real physical calculations from new axiomatics.
Sigh. If only it was all a little bit easier...
Are Feynman diagrams all right? Well, they might be, in the context of diagrammatic reasoning, but then that pesky Minkowski background is still lurking there somewhere... Maybe if we turned that into twistor geometry things would finally look up. Points look like spheres now, which in itself doesn't sound like a huge advance, but the beauty of it is that causality now has a language capable of stepping outside of set theory. Category theory isn't just there to put a whole lot of complicated geometry into a neat package. It allows foundational mathematics and real physical calculations from new axiomatics.
Sigh. If only it was all a little bit easier...
posted by Kea | 8:37 AM
15 Comments:
03 19 07
Yes Kea:
I wish things were a bit easier regarding background independence too! From my explorations with LQG, I can say that the polymer representation is way to restrictive...But that does not mean that this polymer picture is the only background independent form of physics (despite what some may say about irreducibility of Ashtekar, Lewandowski picture), so I am curious to see if the concept makes sense in terms of M theory. It seems like as soon as terms like measure or boundary get thrown about, background independence no longer exists. But this is terribly confusing.
I suppose I should admit that I believe that background independence is completely wrong, and that a primary reason for the impass in physics is the attempt to make a theory that possesses it.
I can't give a proof. Some things one must take on faith. There is one existing universe, it is unique. There is one perfect reference frame, it is also unique.
If the universe did not possess approximate background independence, life would depend on the absolute velocity of a planet and life would be impossible. So background independence can be derived as an anthropic principle just as so many other (approximate) principles. If the orbit of the earth was highly elliplitical it would be difficult for life to exist here, but that does not mean that planets move on perfect circles.
So from the way I see it, approximate background independence is something that a low energy theory must possess to be consist with observation. But it is not needed in a high energy theory, and the concept prevents one from using many useful analogies with condensed matter (which always has a background).
But it is OK to talk about boundaries, so long as you understand it as an experimental constraint formulated in terms of the diagrammatic reasoning that canonically represents the experimental question. For M theory, that means knotty pictures for working in higher topos cohomology.
Just saw your post, Carl. This really comes down to the difference between our approaches. You are happy without the fancy maths, because it doesn't seem to buy us anything. But I am talking about a theory that could do MUCH, MUCH more than just the standard model, or cosmology. And I am also thinking of that nagging issue of high Tc superconductivity, which I do not believe can be explained entirely within your framework. Of course, this remains to be seen. Cheers.
03 19 07
Yes Kea, and viewing boundaries as experimental contraints on a system INDUCES a background methinks. I used to think that background independence meant that we weren't concerned about the hows and whys of what was inside a shape, simply a characterization of its boundary because what is inside is dynamic. Yet this does not seem to be what main proponents of BI seem to mean. The last paper I read showed a quantization of strings using BI, except they had to construct a series of shadow states to approximate a background to perform the procedures. Philosophically I think BI is a bit ill defined.
But in a general sense, there is elegance to the thought. Carl, I figured you may not like BI because you don't seem to like so called formalism very much. Regarding aspects of condensced matter physics popping out, well we have yet to understand many things in that realm. Anyons are pretty cool and in fact, a lot of anyonic mathematics has direct tie in to M theory mathematically...
Dea Kea and others,
I propose a different philosophy inspired originally by geometrization of loop-spaces. Loop-spaces (maps from circle to a compact Lie group) allow a unique Kahler geometry having maximal possible symmetries (Kac-Moody group). This follows from the mere existence of Riemann connection.
This suggests that Einstein's geometrization program should be generalized to the infinite-dimensional context. For some reason this approach has been followed. Probably because, the first finding is that the curvature scalar of loop space (constant curvature space) is infinite and this does not look at all encouraging and is in strong contrast with finiteness belief of stringy community.
In TGD framework the natural conjecture is that the mere existence of Kahler geometry (from the geometrization of hermitian conjugation) implies that this geometry is unique and allows maximal symmetry group generalizing Kac-Moody symmetries. "Background" understood as counterpart of stringy target space would be unique and classically dynamics would be dynamics of space-time as a 4-surface. Physics would be unique from the uniqueness of infinite-D Kahler geometric existence.
Generalized symmetric space property would follow and the vanishing of curvature scalar would imply vacuum Einstein equations for the world of classical worlds.
As an additional bonus a geometrization of fermionic statistics and super-symmetries follows. Oscillator operators define basis for gamma matrices for the "world of classical worlds" and having an interpretation as super-generators.
Finite-D imbedding space, call it H, would in general case be product of flat space and a symmetric space, etc.... Super-conformal symmetries force lightlike 3-surfaces and 4-D space-time and 4-D Minkowski space as Cartesian factor of H=M^4xS, etc, etc.. S=CP_2 is the only possible choice if one wants standard model physics and has number theoretic interpretation.
Of course, one cannot avoid completely perturbation theory and selecting preferred space-time surfaces (or equivalently lightlike 3-surfaces representing generalized Feynman diagrams). These correspond to maxima of Kahler function for the world of classical worlds defining generalized Feynman diagrams and perturbation theory around them. Symmetric space property for the world of classical worlds however strongly suggests, and quantum criticality (RG invariance) and number theoretic constraints require that loops sum up to zero in the configuration space functional integral (infinite perturbation series in general does not give algebraic outcome) so that in this sense theory would be exactly solvable and behave like free field theory.
Coupling constant evolution would not be lost: it would have p-adic origin and realized at the level of "free field theory" as log(p) scaling of spectra of the modified Dirac operator.
Best Regards,
Matti
Formalism works best the farther you get away from simplicity. The idea of elementary particles is that when you get things hotter and hotter, they become simpler. As such, elementary particles should be done from a point of view of realism.
Condensed matter is as far from elementary particles as you can get and there is no reason to doubt that formalism will be very useful for it.
That formalism has been very successful in elementary particles so far is not proof that the universe is "formalism all the way down". Instead, it's evidence that the elementary particles we know are already formal objects. That is, the elementary are condensed matter; they are composites.
If you want to understand a crystal, you do not have to understand atoms. All you need are formal symmetries and all that. This is what I mean by the utility of formalism in condensed matter.
Still a comment loosely relating to back ground independence.
Kea mentioned high T_c super-conductivity which seems to be quantum critical phenomenon involving large number of length and time scales. TGD Universe is by definition quantum critical and the challenge is to understand what this really means quantitatively. Quantization of Planck constant is one aspect of it and I have applied it to high T_c superconductivity. The larger the value of Planck constant, the longer corresponding quantum length and time scale (say Compton length or some other length parameter or time scale of quantum dynamics). What this would mean in practice is that electrons of exotic Cooper pairs would have zoomed up sizes of order 10 nm, which happens to correspond to cell membrane thickness (not an accident, zoomed up dark Cooper pairs and ions would have key role in TGD inspired quantum biology).
Super-conductivity has also another aspect which hints for a dramatic modification of the existing views. Coherent states of Cooper pairs do not possess a well defined fermion number, em charge, or any other quantum numbers. The question is should we give up this routinely use notion or is there some way out.
In TGD framework zero energy ontology provides an alternative interpretation. Net quantum numbers of any physical state vanish and states consist of positive and negative energy parts having interpretation as initial and final states of particle reaction in the usual positive energy ontology. The detection of particle reaction is essentially a reduction of time-like entanglement like ordinary quantum measurement is a reduction of space-like entanglement. In this framework states can be superpositions of pairs of positive and negative energy parts with arbitrarily high fermion numbers/charges since always net quantum number vanish for components.
This of course raises objection. How unique S-matrix can be if it characterizes states rather than the unitary process U associated with quantum jump? [U is also present but seems to be most interesting from the point of view of intentional action transforming p-adic space-time sheet to real one in quantum jump.] Super-conformal symmetries, unitarity, and extremely constraining properties of hyper-finite factors of type II_1 suggest that the S-matrices could be in well defined sense fractal and highly unique.
Thanks for Kea for introducing very interesting topic into discussion.
A comment to Carl Brannen,
the work with a quantum theory of consciousness has forcec me to challenge the belief on single existing universe and has led to the view that universe understood as a quantum state is replacing itself with a new one in each quantum jump and the classical universe is geometric sense is 4-dimensional space-time assigned *uniquely* (this is new!) to 3-D lightlike surface by classical dynamics. The super-position of 4-D space-time surfaces would be replaced by a new one in each quantum jump so that also the geometric past changes in the process. This is quite different view from that of standard QM but the basic paradox of quantum measurement theory disappears.
Carl mentions also the preferred reference frame. In number theoretic approach this question becomes very relevant since the restriction of general coordinate transformations to say rational maps with rational coefficients seems to be unavoidable. In which coordinate frames things are rational?
The vision about the uniqueness of physics as being due to uniqueness of infinite-D Kahler geometric existence implies that imbedding space (H=M^4xCP_2) possesses so high symmetries that it is possible to choose coordinate system uniquely apart from the action of isometries.
There is however still the question about the choice of quantization axes of spin and color quantum numbers and also the choice of rest system. There cannot be any universal choice and the only reasonable conclusion is that the choice is an aspect of quantum state itself and by quantum classical correspondence an aspect of geometric correlates of quantum state too.
The quantization of Planck constant based on Jones inclusion indeed implies that the world of classical worlds decomposes to sectors corresponding to different choices of these quantization axes. The choice of the quantization axes would be an aspect of both geometry of the world of classical worlds, of the imbedding space in the generalized sense, and of space-time surface itself. Mathematically this would however bring in only a relatively minor complication.
Best,
Matti
03 20 07
Interesting thread:
Carl, we may be talking around one another. I am simply saying that formalism can be useful. Regarding condensed matter physics I disagree because anyons are quite important theoretically and in practical world and they have quite rigorous mathematical description. When I studied a bit on Lqg last year, I saw Classical and Quantum Chern Simons theory pop up in description of anyons, also one might describe them as a projective representation of a Lie Group. What is even neater is that braid statistics emerge in descriptions of topological description of univese, just as Matti mentions. All of this from studying the formalism of anyons!
This paper by ZHENGHAN WANG entitled: "TOPOLOGIZATION OF ELECTRON LIQUIDS WITH
CHERN-SIMONS THEORY AND QUANTUM
COMPUTATION" really seems to tie in the ideas that Kea has mentioned before about quantum computing and similarities to the universe, what Matti says about the Jones inclusions and braid statistics. A lot of richness in this discussion:)
03 20 07
Here is the link.
Interesting discussion. Yes, I have felt for a while that we are talking around each other a lot. Carl, Mahndisa's 'induced background' idea is right for M theory. So we actually agree that computationally there is always a 'background'. Matti seems to understand Carl's 'unique universe' point of view better than I do. I don't think in terms of 'quantum jumps', but rather in terms of interacting 'induced' universes. This is probably closer to the more conventional MWI picture, but naturally we are not doing ordinary QM.
Matti, as you know, loop spaces lead also to higher operads. I am really quite convinced (given my discussions with kneemo) that this is the right computational framework.
Dear All,
thanks for a nice discussion and to Mahndisa for the link.
Still a comment on non-unique universe and Jones inclusions just to demonstrate that I begin to understand something also about the mathematics side;-).
Non-unique universe point of view allows to understand evolution as the increasing of the algebraic extension of rationals and having as a counterpart as inclusion sequence at the level of hyperfinite factors of type II_1 (HFF briefly). Basic idea is that Galois group for algebraic closure of rationals is infinite symmetric group extending to infinite braid group: the group algebra of both is HFF.
The basic construction of inclusion hierarchy N subset M subset M1 subset... of Jones as analog of hierarchy of inclusions and analogous to a hierarchy of algebraic extensions of rationals relies on representation of these algebras via they left action in L^2(M) obtained by regarding M as Hilbert space. This space contains the entire hierarchy of inclusions and is in this sense analogous to reals contra rationals.
The idea of Jones's construction is somewhat similar to that in the extension of complex numbers to quaternions to octonions to...: take complex conjugation as operation and add it as a formal element to the previous structure. Now one takes the projection from M to N as something from outside and adds it to to M regarded as subspace of L^2(M) and repeats this again and again to get N subset M subset M1....
Evolution could correspond to an infinite sequence of extensions of rationals represented in terms of an infinite inclusion sequence about which Jones inclusion sequence is the simplest example. Thus also state space would be dynamical and evolving and extending quantum jump by quantum jump.
To Kea, I would like to understand better operads and their relationship to planar algebras and planar algebras themselves since the latter characterize multi-step inclusions. For a couple of years I wrote some comments about operads and infinite primes but my understanding remained very shallow. I remember only that the action of symmetric group appeared in "fractal" manner: permutations of indices, of index groups,...: this brings again in mind representations of S_infty: my candidate for the Galois group of algebraic closure of rationals and HFFs.
Any easy-to-read references telling the ideas at conceptual level without too many technicalities?
Hi Matti. Unfortunately, there are no physics style references on this. There are plenty of books on operads for CFTS, which I guess you've come across, but they are not particularly helpful. My intuition comes from struggling for many years with pure category theory, and in this context Batanin's operads fit into the search for a proper definition of weak n-category. Batanin's papers are really the only thing I would highly recommend reading.
'Background Independence' may be relative rather than absolute.
Neutrinos are relatively independent of the background atomic particles comprising the Earth, with some passing through apparently untouched.
However, other neutrinos can interact with
a - neutrons to form protons and electrons or
b - emit Cherenkov radiation to form detectable photons. http://en.wikipedia.org/wiki/Neutrino
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