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The Third Road

  1. Feb 9, 2005 #1
    The "Third Road" to Quantum Gravity

    I have been thinking lately about where the search for Quantum Gravity may be headed in the near future and it has struck me that the LHC is going to be a major pivot point in the future of research- if they are able to observe micro black-holes with the LHC- it seems to me that such a tremendous achievement will capture the imaginations of everyone- at that point it would seem that the so-called “third road” to QG- namely Black Hole Thermodynamics will surge ahead of strings and LQG/quantum geometry- depending on the particulars the case for strings and/or LQG may be strengthened on some front but would still be absorbed by the momentum of black-hole physics- after all we will HAVE real black-holes to study!

    when one considers the direction of science and technology- it seems even more inevitable- in Computing people like Seth Lloyd have shown us that the ultimate end of our advances in computation would lead to black-holes which are by definition ultimate quantum computers- and Lee Smolin has suggested that black-holes may be the very source of universes themselves with his CNS idea-

    a positive result at the LHC could only explode all these black-hole ideas- and begin to dominate all research in physics-[which is already pretty black-hole saturated as it is!]
    Last edited: Feb 9, 2005
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  3. Feb 9, 2005 #2


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    the third road

    Hi setAI

    As a category theorist, I like where you're coming from. A paper that you might want to reference is

    The Computational Universe: Quantum gravity from quantum computation
    Seth Lloyd

    By the way, Smolin's ideas should probably be traced back to Bekenstein in this context. And Lloyd's ideas are, I believe, already well appreciated by a certain sector of the spin foam/computation community.

    On this note: a request to Integral - isn't it time we had a new master thread entitled (just a suggestion) "Categories, Gravity and Logic" ?


  4. Feb 9, 2005 #3


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    You mean a sticky? Or a new subforum? The first is easy and could hold useful links on the subject. The second is like pulling teeth.
  5. Feb 9, 2005 #4


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    Well, I meant a new subforum. I guess I shouldn't be so lazy, and write a sticky...but I think the subject deserves its own subforum.
  6. Feb 11, 2005 #5
    I've been meaning to thank you for the link to Seth's paper- Kea- but I've been so engrossed in it I havent had a chance!
  7. Feb 14, 2005 #6


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    For setAI and selfAdjoint

    That poll of yours is moving slowly, selfAdjoint. Just in case the informational option wins, I thought I would reintroduce it.


    Recent interest in category theory amongst the String theorists, and the growing interest in the intersection between LQG and Strings, suggests that perhaps it is time to recognise the existence of QG ideas outside the scope of Strings, Branes and LQG.

    The ideas to which I refer do have an intersection with both Strings and LQG. In the first case, the notion of a gerbe, as discussed in

    Higher Gauge Theory: 2-Connections on 2-Bundles John Baez,
    Urs Schreiber, http://www.arxiv.org/abs/hep-th/0412325

    is a category theoretic one. In the second case, the spin foam (for a review see Spin Foam Models for Quantum Gravity Alejandro Perez, http://arxiv.org/abs/gr-qc/0301113 ) approach to QG uses the functorial aspect of topological field theories, and has its origins in Penrose's spin networks, which in turn arose from the study of twistors, about which more is said below.

    This is a very short introduction to this subject. A few useful web references are collected. It is worth noting here John Baez's homepage http://math.ucr.edu/home/baez/README.html

    The Third Road to Quantum Gravity

    The third road is not about the application of a few category theoretic concepts, such as gerbes or functors, to physics modelled entirely on existing principles. It is about trying to understand what we mean by observation and quantum geometry at a fundamental level. The idea of a path integral summation over preselected geometries is dismissed outright.

    Only category theory can discuss logic, geometry, algebra and number theory in the same language. The third road says "get the logic right, and you'll see how computational the universe is".

    Now this might all be pie-in-the-sky philosophy, but actually it is a well-developed approach to Quantum Gravity. The reason that it remains unrecognised as such is partly due to its interdisciplinary nature. Experts in logic tend to reside in Philosophy departments, experts in computation in Computer Science departments and so on.

    For a real philosopher's introduction to these ideas see

    Loop and Knots as topoi of substance R.E. Zimmermann

    or maybe look at some of my previous posts at

    In the next section, I would like to point out that General Relativity itself is category theoretic in nature.

    General Relativity and Categories

    Background independence is about more than coordinate invariance. I shouldn't have to say this, but String theorists don't seem to know this. If you take all the matter out of the universe then there isn't any spacetime. Penrose understood this well. That is why he started using sheaves - to do twistor theory.

    The question is: how can we describe a point in spacetime? Well, a point in spacetime isn't of any physical importance. In fact it was only by realising this that Einstein came to accept general covariance in the first place (see the book by J. Stachel, Einstein from B to Z Birkhauser 2002). What is physical are the (equivalence classes of) gravitational fields.

    If we work with sheaves over a space [itex]M[/itex] then a point is indeed a highly derived concept. So the physics is telling us we should use sheaves to do GR. But sheaves are examples of functors - maps between categories.

    Carrying this much further, one can model a differential manifold on something called a local ringed topos, namely the topos of sheaves on some subset of [itex]\mathbb{R}^{n}[/itex] which contains the distinguished sheaf of differentiable [itex]\mathbb{R}[/itex] valued functions on the subset.

    But why do we need manifolds at all? Some people take this question very seriously. See, for instance, the recent 400+ page tour-de-force

    [itex]C^{\infty}[/itex]-smooth singularities exposed: Chimeras of the differential spacetime manifold; A. Mallios, I. Raptis, http://arxiv.org/abs/gr-qc/0411121

    on the use of Abstract Differential Geometry in classical and quantum gravity, with its extensive bibliography.

    Anyone still reading this will at least grudgingly admit that maybe a physicist needs to know a little bit about what a category is.....

    Quick Introduction to Categories

    Whereas a set has elements, and a map between sets takes elements to elements, a category has both elements, called objects, and relationships between elements, called arrows. Every object [itex]A[/itex] is equipped with at least an identity arrow [itex]1_{A}[/itex] from [itex]A[/itex] to [itex]A[/itex]. Maps between categories, called functors, take objects to objects and arrows to arrows. Arrows may be composed [itex]f \circ g[/itex] if their ends match appropriately. An arrow is monic if for any [itex]g: A \rightarrow B[/itex] and [itex]h: A \rightarrow B[/itex], [itex]f \circ g = f \circ h[/itex] implies [itex]g = h[/itex].

    For example, there is a category [itex]\mathbf{Set}[/itex] whose objects are sets and whose arrows are functions between sets. In [itex]\mathbf{Set}[/itex] there is an object [itex]\{ 0,1 \}[/itex]. There are also many arrows of the form [itex]f: S \rightarrow \{ 0,1 \}[/itex] for a set [itex]S[/itex]. Such arrows may be thought of as the selection of a subset of
    [itex]S[/itex], namely those elements that are mapped to [itex]1[/itex]. A one element set, [itex]\{ \ast \}[/itex], has precisely one arrow into it from any other set, making it an example of a terminal object in [itex]\mathbf{Set}[/itex].

    Functors are contravariant if they actually act on the category with all arrows reversed. Contravariant functors from a (small) category [itex]C[/itex] into [itex]\mathbf{Set}[/itex] are known as presheaves, providing a preliminary example of a topos. When [itex]C[/itex] comes equipped with a topology (definition omitted) one restricts to a subcategory of sheaves.

    The intended interpretation of pieces of categories is that they are geometric entities. Objects are zero dimensional and arrows are one dimensional. In a category there is no equality between objects, but we consider objects isomorphic if there exists two arrows [itex]f[/itex] and [itex]g[/itex] such that [itex]f \circ g = 1_{A}[/itex] and [itex]g \circ f = 1_{B}[/itex].

    Now one may also consider the category [itex]\mathbf{Cat}[/itex], with categories as objects (which are small enough in a suitable sense) and arrows functors between them. One may naturally include in this category the natural transformations [itex]\tau[/itex] between functors, as another level of arrows, as some commuting squares, which I would like to draw but I need xypic....These squares may be composed, both vertically and horizontally, in the obvious way. Thus [itex]\mathbf{Cat}[/itex] is an example of a 2-category: an inherently two dimensional structure. In a 2-category, all arrows between two objects [itex]A[/itex] and [itex]B[/itex], denoted Hom[itex](A,B)[/itex], form a category.

    Another example of a 2-category is the category of topological spaces, with homeomorphisms for 1-arrows and homotopy maps as 2-arrows.

    Given a subset [itex]S[/itex] of the arrows of a category [itex]C[/itex] one defines the localisation category [itex]S^{-1} (C)[/itex] by sending all arrows in [itex]S[/itex] to isomorphisms under a functor [itex]C \rightarrow S^{-1}(C)[/itex] which has a nice universal property.

    A category representing the ordinal [itex]\mathbf{4}[/itex] is visualised as a 3-simplex equipped with oriented edges and faces.

    Recall that in three dimensions gravity is a topological theory because it has no local degrees of freedom. If one is interested in (physical) spaces that are topological (ie. there is an equivalence up to continuous deformation) and oriented it is sufficient to describe them by a space made out of simplices,
    suitably glued together. A TFT is, axiomatically, a functor from such spaces, thought of as arrows between boundary components, into an algebraic category.

    However, this isn't category-theoretic enough for the third road.
  8. Feb 14, 2005 #7


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    Topos Theory and Twistors

    "Indeed, that the quantum nature of reality should affect the very structure of space-time at some scale is now a more-or-less accepted viewpoint among those physicists who have examined this question in some depth (cf. Schrödinger 1952, Wheeler 1962). But I think that most physicists would believe that such effects should be relevant only at the absurdly small quantum gravity scale of 10-33 cm. (or smaller). My own attitude was somewhat different from this. While it might be that only at 10-33 cm is it necessary to invoke a description of space-time radically removed from that of a manifold, my view was (and still is) that even at the much larger levels of elementary particles, or perhaps atoms, where quantum behaviour holds sway, the standard space-time descriptions have ceased to be the most physically appropriate ones, and some other picture of reality, though at that level equivalent to the space-time one, should prove to be the more fruitful."

    quoted from

    Roger Penrose On the Origins of Twistor Theory

    The massless free field equations for particles of spin [itex]\frac{n}{2}[/itex] in terms of [itex]n[/itex] indices are

    [tex]\nabla^{AA'} \phi_{AB....} = 0 \hspace{20mm} \nabla^{AA'}
    \widetilde{\phi}_{A'B'....} = 0 [/tex]

    where in spinor terms, for the photon, the Maxwell curvature is

    [tex]F_{AA'BB'} = \phi_{AB} \varepsilon_{A'B'} + \varepsilon_{AB}

    for [itex]\varepsilon_{AB}[/itex] the skew symmetric spinor. It was noted
    that these equations are actually invariant under a conformal group. One needs to compactify Minkowski space so this works properly. The double cover of the Lorentz group is then replaced by the twistor group [itex]SU(2,2)[/itex]. It acts on twistor space [itex]\mathbb{T}[/itex], with coordinates given by a spinor pair. For details see the two volume

    Spinors and Spacetime Penrose and Rindler (Cambridge 1986)

    The twistor correspondence looks at flag manifolds such as

    [tex]\mathbb{F}_{12} \equiv \{ V_{1} \subset V_{2} \subset
    \mathbb{T} \}[/tex]

    where [itex]V_{1} \simeq \mathbb{C}[/itex] and [itex]V_{2} \simeq \mathbb{C}^{2}[/itex]. Points of Minkowski space correspond to spheres in a projective twistor space under the correspondence

    [tex]\mathbb{P}\mathbb{T} \leftarrow \mathbb{F}_{12} \rightarrow

    and the beauty of this is that solutions to the (primed) spin [itex]s[/itex]
    equations correspond (one to one) to elements of a sheaf cohomology


    and similarly for the unprimed case. Don't worry too much about this if you don't know anything about it. The point is that this cohomology is Abelian. Now one can do non-Abelian cohomology in 1D, but trying to do it in 2D is another matter altogether. Why would we want this?

    The first interesting step towards a modern category theoretic understanding of mass, IMHO, is the study of the Klein-Gordon equation in

    L.P. Hughston T.R. Hurd A cohomological description of massive
    Proc. Roy. Soc. Lond. A378 (1981) 141-154

    In this paper, Hughston and Hurd combine two solutions to the massless equations for spin [itex]s[/itex] particles thought of as elements of the sheaf cohomology group [itex]H^{1}(\mathbb{P}\mathbb{T}^{+} , S(-2 s - 2))[/itex] on twistor space. The Klein-Gordon equation solutions then belong to a second cohomology group [itex]H^{2}(\mathbb{P}\mathbb{T}^{+} \times \mathbb{P}\mathbb{T}^{+} , S_{m,s}(- \mu - 2 , - \eta - 2))[/itex] for [itex]s - \frac{1}{2} | \mu - \eta | \in \{ 0,1,2,3 \cdots \}[/itex].

    Assuming we believe the need to understand this in category theoretic terms, it is simply a fact that categories of sheaves are toposes, which are categories with certain nice properties.

    Some notable web references on toposes and spacetime are

    Toposes, Triples and Theories Michael Barr and Charles
    Wells, http://www.cwru.edu/artsci/math/wells/pub/ttt.html

    A New Approach to Quantising Spacetime C.J. Isham,

    I: Quantising on a General Category http://arxiv.org/abs/gr-qc/0303060
    II: http://arxiv.org/abs/gr-qc/0304077 III: http://arxiv.org/abs/gr-qc/0306064

    The internal description of a causal set: what the universe
    looks like from the inside
    F. Markopoulou

    Knots and Quantum Computation

    Everyone seems to agree these days that knots are wonderful. To category theorists, knots have a lot to do with

    A. Joyal R. Street, Braided tensor categories, Adv. Math.

    or other higher dimensional categorical structures of a similar kind. In

    A modular functor which is universal for quantum
    M. Freedman M. Larsen Z. Wang

    the authors show why the Jones polynomial for a certain root of unity is pretty good at modelling quantum computation. An even more thorough use of categories for computation appears in the seminal paper

    S. Abramsky B. Coecke A categorical semantics of quantum

    Cohomology: Descent Theory references

    Categorical and combinatorial aspects of descent theory
    Ross Street http://arxiv.org/abs/math/0303175

    Notes on Grothendieck topologies, fibered categories and
    descent theory
    Angelo Vistoli

    Notes on Motivic Cohomology C. Mazza V. Voevodsky C.
    Weibel http://www.math.uiuc.edu/K-theory/0486

    Best regards
    Kea :smile:
    Last edited: Feb 14, 2005
  9. Feb 15, 2005 #8


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    I'm a big fan of information theory cosmology. It does a pretty amazing job of modeling certain things - like entropy.
  10. Feb 15, 2005 #9


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    Do you have any favourite references on the cosmology?

  11. Feb 15, 2005 #10


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    As it happens, I do. This very obscure paper sparked my initial interest:

    The Nature of Information and the Information of Nature

    I later became intrigued by some work done by Davies on emergent phenomena - in particular life:

    How bio-friendly is the universe

    Emergent biological principles and the computational properties of the universe

    Application of information theory to physics has evolved considerably since the Shannon entropy days. Quantum information theory has become quite popular in recent years.
  12. Feb 16, 2005 #11


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    Thanks, Chronos

    I had a look through this. It's something that I'm sure I would have scoffed at 10 years ago, but of course now I've changed my mind! He states: Emergent laws of biology may be consistent with, but not reducible to, the normal laws of physics operating at the microscopic level. I disagree with this. I believe the post-quantum laws of physics are sufficiently radical to encompass emergent phenomena. The link between knots and DNA is already a trendy topic. Davies seems to think a bit too much like a classical cosmologist. But great stuff.

    While I think of it, I recently discovered a truly 21st century series of papers which I suspect very few people understand...and I'm not one of them!....but I'm going to recommend anyway, by Paul Taylor, whose homepage is
  13. Mar 14, 2005 #12
    an interesting paper on q/comp-ing the bosonic oscillator


    "By the early eighties, Fredkin, Feynman, Minsky and others were exploring the notion that the laws of physics could be simulated with computers. Feynman's particular contribution was to bring quantum mechanics into the discussion and his ideas played a key role in the development of quantum computation. It was shown in 1995 by Barenco et al that all quantum computation processes could be written in terms of local operations and CNOT gates. We show how one of the most important of all physical systems, the quantized bosonic oscillator, can be rewritten in precisely those terms and therefore described as a quantum computational process, exactly in line with Feynman's ideas. We discuss single particle excitations and coherent states. "
  14. Mar 30, 2005 #13
    "It is likely that the holographic principle will be a consequence of the would be theory of quantum gravity. Thus, it is interesting to try to go in the opposite direction: can the holographic principle fix the gravitational interaction? It is shown that the classical gravitational interaction is well inside the set of potentials allowed by the holographic principle. Computations clarify which role such a principle could have in lowering the value of the cosmological constant computed in QFT to the observed one."

    Fri, 18 Mar 2005 08:24:20 GMT
  15. Mar 31, 2005 #14


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    Let's for the moment consider the possibility that there is no cosmological constant in our observed classical cosmos, as per recent discussions. What would the various approaches to QG have to say to this?

    As Lubos has already pointed out, the String theorists might be very happy because they could then simply return to the unification paradigm without worrying about a positive [itex]\Lambda[/itex]. However, doing away with
    [itex]\Lambda[/itex] might mean doing away with varying fundamental constants, such as [itex]\alpha[/itex], and as far as I am aware current gravitational String theory requires this variability (please correct me if I'm wrong, Lubos).

    I don't really need to point out that an absence of [itex]\Lambda[/itex] is a serious problem for the naive application of 3D state sums to cosmology. Even my flatmate can see that.

    So as far as I know this leaves 'fundamental LQG' and it's relatives.

    It would be nice if people commented on this issue. Personally I cannot say anything at present.

  16. Apr 15, 2005 #15
    General Relativity and Quantum Cosmology- Computability at the Planck scale

    "We consider the issue of computability at the most fundamental level of physical reality: the Planck scale. To this aim, we consider the theoretical model of a quantum computer on a non commutative space background, which is a computational model for quantum gravity. In this domain, all computable functions are the laws of physics in their most primordial form, and non computable mathematics finds no room in the physical world. Moreover, we show that a theorem that classically was considered true but non computable, at the Planck scale becomes computable but non decidable. This fact is due to the change of logic for observers in a quantum-computing universe: from standard quantum logic and classical logic, to paraconsistent logic. "

  17. Apr 15, 2005 #16


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    Thanks, setAI. This is a nice, readable paper. The list of references in it is an excellent resource, too.

    From page 4:
    "If instead the observer focuses on his perceptions, he will make, in his mind, automatically the two limits [itex]N \rightarrow \infty[/itex] and [itex]A \rightarrow 0[/itex]"

    where [itex]A[/itex] is the area of a cell of the cosmological horizon and the limit on [itex]N[/itex] means that, classically, we should expect no cosmological constant, even though the number [itex]10^{120}[/itex] is significant for physics at the Planck scale.
    Last edited: Apr 15, 2005
  18. Apr 25, 2005 #17


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    some thoughts


    If Peter Bergmann has taught us one thing above most others, it is surely that if we remove the life from Einstein's beautiful theory by steam-rollering it first to flatness and linearity, then we shall learn nothing from attempting to wave the magic wand of quantum theory over the resulting corpse. Roger Penrose (1976)

    Perhaps it is time we reviewed the principle of QGC. Some people would prefer to think of this as a generalisation of Mach's principle perhaps, but Mach predates both GR and quantum physics, let alone quantum gravity. QGC probably isn't a great term either. Maybe we should use the term cosmic duality as inspired by Cartier's dream. At least when one says QGC it is clear that one is operating outside the framework of String unification, which
    contains the nonsensical notion that we can discuss objectively the nature of observations on a large range of scales in a classical background.

    It is important to distinguish QGC from String theory because at the end of the day, these days, the mathematics looks very simliar and it won't be long before the String theorists are saying it's all String theory anyway, which is a worrisome state of affairs because we then run the risk of obscuring the rich physical ideas coming from the informational perspective.

    Recall that the thought experiments behind GR are about visualising processes, which can only be given meaning via a partitioning of the cosmos. The simplest idea of a partition is inherently discrete. But what tools were available to Einstein? The primary objective was to construct non-trivial geometries that were locally Minkowskian. Minkowski space had been defined upon a long tradition of Cartesian geometry, and hence differential geometry arose as the only available option. The same is not true today.

    Mach's idea of inertial frames being defined by the universe at large could not be incorporated into GR, which is a purely local theory. Instead (as explained nicely in a seminar I went to recently) GR predicts a frame dragging effect for massive objects such as the earth.

    But Mach's principle was a great inspiration to Einstein. The possibility of respecting it in a theory of quantum gravity is an important physical consideration. Now we may have a viable classical cosmology, the Wiltshire cosmology, which relates Mach's principle of inertial frames to the existence of isotropic observers with respect to the CMBR.

    Let's take this classical picture seriously. It states that the homogeneous bulk universe exists on scales much larger than the observable cosmos. Is this a problem for the quantum picture? Not at all. As observers, we operate on modest energy scales in comparison to the Planck scale. Moreover, the classical inference of a universal observer is highly derived and does not exist at the fundamental level of quantum observation. In fact, the Third Road looks at the notion of a universal observer as a definition of classical limit.

    GR without Mach's principle works well on earth and very nearby. It may not work so well on the scale of the solar system. We observe that it does not work very well on the scale of the galaxy. Why then does the standard cosmology, based on GR, have anything at all to do with the CMBR? The assumption of homogeneity is a paradoxical property of the standard cosmology.

    What does it mean to observe large distances? We say it is the observation of large redshifts due to the expansion of the universe, but this presumes an objective classical reality that is unacceptable in the computational approach to quantum gravity. Large distances are therefore more about an enhanced probability of seeing less energetic photons. But this is almost what a particle physicist means by distance. The more energy
    available in the rest frame of the accelerator, the shorter the distances that may be probed. The more energy, the greater the number of possible observations in a universe of possibilities. What is the limit of this process? What does it mean, for instance, to say that the universe contains a finite amount of energy? In the informational approach to quantum gravity, this
    becomes a question of computability.

    The big question, of course, is: can we recover the classical picture mathematically from a higher categorical (informational) description of QGC? Or, in what general form does Mach's principle restrict cosmological solutions to Einstein's equations?


    At atomic scales, or in highly curved regions of spacetime, we know that the classical laws of physics break down. In this supposed fact is an assumption that a physical definition of energy scale is fundamental to any laws under consideration. But such a notion of scale is strictly classical. It appears in standard quantum theory only because one always works with a classical background.

    In quantum gravity we imagine different mass quantum numbers for the same type of particle (not a good word, 'particle'). In other words, the generations are not to be thought of as different particles but as different observations of the same thing, the probability depending in some way upon scale.

    So before we even begin to contemplate the structure of quantum gravity we need to ask ourselves: what (dimensionless) notion of scale can replace the naive one? What sort of measure of an observation characterises scale?

    Planck's constant [itex]\hbar[/itex], whatever its value, is necessary to compare the chosen units for [itex]x[/itex] and [itex]p[/itex]. It is necessary, therefore, to understand how these physical quantities are related on an abstract level, beyond the phase space duality of QM. This immediately means understanding non-conservation of particle number in a mathematically rigorous fashion.

    Leaving aside Gray tensor products and tricategories for now, consider the opening sentence of the recent paper

    Quantum Fields and Motives
    A. Connes, M. Marcolli

    The main idea of renormalization is to correct the original
    Lagrangian of a quantum field theory by an infinite series of
    counterterms, labelled by the Feynman graphs that encode the
    combinatorics of the perturbative expansion of the theory.

    This paper, and the references within, describe the rich structure of renormalisation in ordinary QFT. There are a number of category theoretic elements involved. For a different, and easier, introduction to a category theoretic understanding of Feynman diagrams, see

    From Finite Sets to Feynman Diagrams
    John C. Baez, James
    Dolan http://arxiv.org/abs/math/0004133

    It is important to realise that in the category theoretic SM quantization itself may be seen as a natural result of the requirement of higher dimensionality. Braided monoidal categories are examples of tricategories. Particle number becomes a measure of categorical dimension, and non-conservation of particle number is forced by dimension raising products.

    Well, I must have sent enough people to sleep for now...
    Kea :smile:
  19. Apr 26, 2005 #18
    Hi Kea

    You bring to light some deep, unsolved problems. Now the method for dealing with classical backgrounds and observations is still up for grabs. You and Urs and have mentioned categorification and NCG, but still the details remain elusive. This category-NCG line of attack seems promising, as it is related to Matrix theory.

    In the IKKT matrix model, for instance, the authors claim spacetime is dynamically generated by the eigenvalues of the hermitian scalar fields Phi^mu. This claim was supported by the series of papers on K-matrix theory, which is an attempt to re-express matrix models in terms of spectral triples. In the K-matrix papers the exact C*-algebra and Hilbert space are not mentioned, but the general framework is elucidated. The algebraic K-theory, K-homology, and KK-theory are all reinterpreted in D-brane terminology. The K-structures bring one into the realm of category theory, as (in K-matrix theory) K_0 (a Grothendieck Group) is a functor from the category C*-Alg to the category Ab of abelian groups.

    NCG specifically enters IKKT matrix theory through the idea of "dynamically generating spacetime." The scalar field operators Phi^mu are hermitian elements of a C*-algebra A, and spec(A) is giving the classical positions of D-branes. If A is finite dimensional (NxN matrices), the spectrum is a compact, Hausdorff, totally disconnected set of points. The functions over spec(A)=X are matrix elements of A (as by Gel'fand-Naimark A=C(X)), and the vector fields over X come from the derivations of A. Therefore, the "strings" are collectively encoded as entries of the matrix functions Phi^mu over X, and the derivations of A perturb the "strings", i.e., transform the matrices. For matrix algebras over C, the only interesting derivations are from the unitary transformations, hence the SU(N) symmetry of matrix models.

    We move beyond 1-NCG by noticing that in the BFSS paper, the authors refer to commuting diagonal block matrices, with hermitian matrices Phi^mu as entries. This cries out for a construction of a larger C*-algebra M_n(A), where the Hilbert space H must be specified. The commuting block matrices are then elements of diag(A), the diagonal matrices with entries from A. By unitary transformations, we can reduces such matrices to diagonal matrices of eigenvalues, where we then physically behold collective collections of "string networks" (D0-brane bound states), given by many copies of spec(A)=X. The construction depends on the use of observables Phi^mu, and makes no classical sense for non self-adjoint elements.

    Hence, if we construct space-time dynamically, we are in effect saying the universe emerges from a large quantum mechanical system. This allows us to speak of observers and classical spacetime in a sound fashion.

    I hope my review supports my claim that Matrix theory, NCG, and category theory coincide. Feel free to let me know if I'm completely off base.
  20. Apr 26, 2005 #19


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    Hi kneemo

    Very pleased to meet you! Yes, the details are elusive, indeed. I have not put any effort into understanding the Matrix theory approach, but the points you bring up sound very reasonable. Perhaps you could provide us with some web references?

    Best regards
  21. Apr 27, 2005 #20
    Hello again Kea

    The pleasure is all mine. Thank you for reminding me to provide references; here they are:

    M Theory As A Matrix Model: A Conjecture [BFSS model]

    IIB Matrix Model [IKKT model]

    D-branes, Matrix Theory and K-homology

    Exact description of D-branes in K-matrix theory

    D-Branes, Tachyons and K-Homology
    hep-th/0209210 .

    After reading these you will find that Matrix theory is merely a proposed formulation of M-theory using noncommutative geometrical methods based on matrix C*-algebras. Thus Matrix theory makes string theory much more palpable for mathematicians, such as you and I. For after the theoretical dust settles, there are no strings and space-time, but only algebras, their spectral topological spaces, and functors describing their large-scale quantum evolution.

    p.s. On page 46 of hep-th/0108085, the authors mention the relationship between K-homology and the derived category of coherent sheaves, which may be of interest to you. For further insight, they reference:

    D-Branes, Derived Categories, and Grothendieck Groups
    hep-th/9902116 .


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