Friday, July 19, 2019

M is for Maggot, N is for Nonsense

wormy apple
Imagine you bite into an apple and find a beheaded maggot. Yuck! But it could have been worse. Had you found only half a maggot, you’d have eaten more of it. Worse still, you may have found only a quarter of a maggot, or a hundredth, or a thousandth. Indeed, if you take the limit maggot to zero, the worst possible case must be biting into an apple and not finding a maggot.

Wait, what? That doesn’t make sense. Certainly a maggot-free apple is not maximally yucky. Where did our math fail us?

It didn’t, really. The beheaded maggot is an example of a discontinuous or “singular” limit and originally due to Michael Berry*. You know you have a discontinuous limit if the function whose limit you are taking (that’s the increasing “yuck factor” of the maggot) does not approach the value of the function at the limit (unyucky).

A less fruity example is taking the y-th power of x and sending y to infinity. If x is any positive number smaller than 1, taking its exponent to infinity will give zero. If x is equal to one, all values of y will give back 1. If x is larger than one, the result of taking y to infinity will return infinity. If you plot the limit y to infinity as a function of x, it’s discontinuous.

Such singular limits are not just mathematical curiosities. We have them in physics too.

For example in thermodynamics, when we take the limit in which the number of constituents of a system becomes infinitely large, we see phase transitions where some quantities, such as the derivative of specific heat, become discontinuous. This is, of course, strictly speaking an unrealistic limit because the number of constituents may become very large, but never actually infinite. However, the limit isn’t always unrealistic.

Take the example of massive gravity. In general relativity, gravitational waves propagate with the speed of light and the particle associated with them – the graviton – is massless. You can modify general relativity so that the graviton has a mass. However, if you then let the graviton mass go to zero, you do not get back general relativity. The reason is that if the graviton mass is not zero, then it has additional polarizations and those are independent of the mass as long as the mass isn’t zero**.

The same issue appears if you have massless fields that can propagate in additional dimensions of space. This too gives rise to additional polarization which don’t necessarily disappear even if you take the size of the extra dimensions to zero.

Discontinuous limits are often a sign that you have forgotten to keep track of global, as opposed to local properties. If you for example take the radius of a sphere to infinity the curvature will go to zero, but the result is not an infinitely extended plane. For this reason, there are certain solutions in general relativity that will not approximate each other as you think they should. In a space with a negative cosmological constant, for example, black hole horizons can be infinitely extended planes. But these solutions no longer exist if the cosmological constant vanishes. In this case, black hole horizons have to be spherical.

Why am I telling you that? Because discontinuous limits should make you skeptical about any supposed insights gained into quantum gravity by using calculations in Anti de Sitter space.

Anti De Sitter (AdS) space, to remind you, is a space with a negative cosmological constant. It is popular among string theorists because they know how to make calculations in this space. Trouble is, the cosmological constant in our universe is positive. And there is no reason to think the limit of taking the cosmological constant from negative values to positive values is continuous. Indeed, it almost certainly is not because the very reason that string theorists prefer calculations in AdS is that this space provides additional structure that exists for any negative value of the cosmological constant, and suddenly vanishes if the value is zero.

String theorists usually justify working with a negative cosmological constant by arguing it can teach us something about quantum gravity in general. That may be so or it may not be so. The case with the negative cosmological constant resembles that of finding a piece of a maggot in your apple. I find it hard to swallow.

* ht Tim Palmer
** there are ways to fix this limiting behavior so that you do get back general relativity.


  1. Another example from physics is viscosity: The smaller it becomes, the more pronounced the turbulence (Reynolds-number increases). But With viscosity=0, turbulence is not infinite, but gone.

    1. Thanks for this wonderful example!

    2. “… when we take the limit in which the number of constituents of a system becomes infinitely large, we see phase transitions … ”
      A. Zee dubs this “The emergence of nonanalyticity”.
      (At a second order phase transition all kinds of responds functions [susceptibilities] jump.
      What I find really amazing is that the critical exponents on the other hand show a universality, i.e. seem to be the same for different substances.)

  2. This comment has been removed by the author.

  3. Is it true that effective field theories have the same behavior?

    1. RGroves,

      The same behavior... as what? Sorry, I can't follow.

    2. I suppose the question is - are there discontinuities in renormalization "group" flows?

    3. Arun,

      Oh, I see. This is a most excellent question. I speculated about it in this essay. The brief answer is there aren't any known examples but if there were it would basically mean the end of reductionism.

      (I realized after writing the essay that there are some examples of the behavior I am talking about for time-dependence rather than energy-dependence, so it's not as crazy as I originally thought.)

  4. How about the weak gravity conjecture implying that cosmic censorship holds in AdS space? That's such a remarkable convergence, I consider it evidence that the theoretical frameworks involved are on the right track.

    1. Mitchell,

      The question is on the right track... to what? Most certainly, a remarkably rich understanding of AdS. But what does this teach us about the universe we live in?

    2. It suggests that consistent quantum gravity requires certain relationships between gravitational and nongravitational forces. It would be surprising if this was only true in AdS space.

  5. "String theorists usually justify working with a negative cosmological constant by arguing it can teach us something about quantum gravity in general. "

    how well do QG rivals like LQG and AS deal with a positve de-sitter spacetime?

  6. Thank you! Once again you have pointed out something that should have been obvious but seduced me into accepting what I was told.

  7. Building a toy universe inside a computer means making do with finite quantities of... well, everything. Finite resources as a starting point relentlessly drives toy universe design away from filling up space with arbitrarily precise objects, and instead towards much fuzzier objects combined with precise convergent processes (equations) that provide additional historically consistent details in direct proportion to resources allocated.

    This process of virtualization, of replacing infinitely precise levels of detail with as-needed equations that only approximate more precisely, results in a toy universe in which many of the dangers of infinities and ambiguities that arise from pushing everything to some limit simply do not exist. The problem of vacuum density is for example a non-sequitur in such a toy universe, since the chaotic densities of Planck space literally never come into existence… and even if they did, they would require most of the available mass of the toy universe to create them. Uncertainty at the smallest scales is inherent in such a toy universe, since future paths remain literally undefined until energy is applied to create a specific history. Ambiguities of interpretation at limits (e.g. is it a sphere or plane?) also never come into existence, because no such "absolute" limits can exist. Noether's theorem also gets flipped upside down: differential symmetries emerge because quantities are conserved, not the other way around. Absolute conservation drives all dynamics.

    The Standard Model of this toy universe is not a set of objects and limits, but a set of convergence and elaboration rules enabled by finite resources. The creation and annihilation operators are among the most fundamental of these rules, but in a more generalized and hierarchical form. Hydrogen existence for example is enabled not by defining separate creation operators for electrons and protons, but by allowing hierarchies of creation operators to cross paths and get a bit mixed up. The positive and negative charges of hydrogen would love to annihilate, but cannot do so because the electron took a different fermionic path from the more complex and piecemeal quark composition path of the protons. Stalemate!

    All of that is toy universe thinking, though. For a real universe to behave like that would mean that infinitesimal based mathematics would need to be rephrased and augmented with a more purely operator-based approach to physics math. Such operators need to include hierarchies of creating and annihilation operators. Fermionic particles and space itself would emerge only as side-effects of various creation paths getting really badly messed up, thermodynamically messed up in fact, as the brilliant Boltzmann almost uniquely seemed to understand.

    That would be a lot of work.

  8. I seem to remember you arguing that what we observe is the sum of the Cosmological Constant and the vacuum energy, not the CC itself. And therefore mathematically you can have a very large vacuum energy and a very large, negative (!) CC.

    1. Dlb,

      If by cosmological constant you mean an additional term in the action and by vacuum energy you mean the qft contribution (and not the source of the field equations in vacuum), then yes. But I don't understand what this has to do with the current blogpost. Are you sure this comment is in the right thread?

    2. This is indeed what I mean. My comment was prompted by this section of your post: "Anti De Sitter (AdS) space, to remind you, is a space with a negative cosmological constant. It is popular among string theorists because they know how to make calculations in this space. Trouble is, the cosmological constant in our universe is positive."

      For the record I completely agree with your feeling about the whole story. My own summary would be: people are calculating in AdS space not because they have a reason to do it, but because they can.

  9. What about wormhole theorists?

    1. Marten,

      You mean wormholes in Anti De Sitter space? Let me just say that combining several speculative ideas doesn't give you a less speculative idea.

    2. A wormhole in an apple is not speculative, it is reality and we know what caused it and how.

    3. Haha, now I get it :p Sorry for being slow.

  10. This has been something I have pondered over a lot. It does appear that quantum gravity prefers the AdS spacetime with the negative curvature Λ < 0. This is certainly the case for AdS/CFT correspondence. Supersymmetry is broken with positive energy as well. The observable universe is de Sitter-like with Λ > 0 and putting string theory here and potentially much of quantum gravity is maybe pounding a square peg into a round hole.

    One can think of the dS spacetime as a single hyperboloid sheets bounding a cone, while the AdS is one of a pair of hyperboloids within the top and bottom part of the cone. These do meet at I^± (scri^±). The AdS_4 and dS_4 might then exist on these holographic screens have junction conditions that are positive or negative. From a quantum perspective these are two states determined in a way similar to the Haldane chain for quantum critical points.

    The observable dS spacetime is a hyperboloid of one sheet, a four dimensional “sheet” in five dimensions, while the AdS spacetime is one a single surface of a 4-dimensional hyperboloid. The two surfaces in effect meet at I^+. As a result the two spacetimes share related conformal structures. In the case of AdS a conformal field theory exists on the conformal boundary with degrees of freedom holographic equivalent to those of gravitation in the bulk of the AdS_n, for n = 4 or 5. Curiously Penrose's CCC posits conformal equivalency with boson fields at or near I^+, which is in this perspective dual to the holographic concept of field on the boundary of the AdS. This means the two have some correlation as methods connecting gravitation with quantum field theory.

    These two sheet hyperboloids are surfaces bounded above the origin by the cosmological constant in a way similar to the gap in the Dirac equation. The one sheet hyperboloid is similarly so bounded away from the origin in a spacelike direction. The one feature that is most salient here that must be kept in mind for later is these two are bounded by a cone. In momentum space this is a Dirac cone, and in condensed matter physics these cones occur at the intersection of conduction and valence bands. This then leads to a number of analogues with topological insulators, or the theory of topological numbers in graphene as laid down by Haldane. Dirac cones occur in band structure where two sets of state intersect. The analogue of the mass-gap is a breaking of a degenerate vacuum state for anyons in Kitaev's toric code. The two hyperboloid states, one corresponding to dS and the other AdS are separated by this mass-gap. This then gives some type of duality, which at this time I do not fully understand, between the AdS and dS physics.

    So the insight with the worm not there, the apple is clean, the partial worm it is dirty, and again no worm, but you ate it, is a decent analogue. The continuous variation in the worm size is a bit like having a positive vacuum energy on the dS that when adjusted to zero discontinuously demolishes everything. However, if you think about supersymmetry should be there. Carrol Johnson and Randall showed that a black hole made to become extremal discontinuously projects the spacetime between the inner and outer horizons into S^2x×AdS_2. Think also of bifurcation maps in chaos theory as well.

  11. "Take the example of massive gravity. In general relativity, gravitational waves propagate with the speed of light and the particle associated with them – the graviton – is massless. You can modify general relativity so that the graviton has a mass. However, if you then let the graviton mass go to zero, you do not get back general relativity. The reason is that if the graviton mass is not zero, then it has additional polarizations and those are independent of the mass as long as the mass isn’t zero**."

    does this analogy sort of explains why directly canonical quantizing GR via new variables does not necessarily lead back to GR?

    1. neo,

      I am not aware of a relation.

    2. I thought about commenting on this, but time and space were limits. The LIGO detection of gravitational radiation has placed bounds on graviton masses. The Compton wavelength λ = ħ/mc as 1.6×10^{16}m or about 1.6 light years or a mass m_g < 8×10^{-23}eV --- ref .

      The mass of a graviton corresponds to a longitudinal degree of freedom, and this is not removed so easily by taking m → 0 in a naive setting. I would say one of the central aspects of physics is conserving, counting and combining degrees of freedom. How might a graviton get a mass? It could be a Higgs or related mechanism where a scalar field is absorbed by a graviton. If so one would carefully take m → 0 as the restoration of this scalar or auxiliary field.

      The massive graviton has lurked in the wings since it was found in the early 1970s that they can form excitons of supergravity and the s = 2 states. The massive graviton can exist in a Chern-Simons Lagrangian, and this might be thought of as a manifestation of some boundary, maybe of this pocket world or holography or … . Maybe massive gravitons are manifestions of states that are then cocyles or topological fields. In this way if one takes the mass to zero this topological field does not converge to a standard graviton or gravitational field that has curvature determined by a covariant coboundary operation on connections. The massive graviton might then play some role in dark matter.

  12. Sabine,

    A small biological quibble for both you and Berry. The critter you would like to avoid biting is actually the vegetarian caterpillar of the codling moth that grazes within fruit on six continents. The maggot is the mostly carnivorous larva of the various fly species that are also well traveled. Folks making cider usually throw in the whole apple, caterpillar and all because – hey, the caterpillar it is basically apple, right?

    More to the point of your blog post, here are a couple of quotations from Berry’s paper that seem to suggest that singularities can lead to truly emergent physics:

    “However, there is a creative side to singular limits: They lead to new physics. For large N, where a central idea is symmetry-breaking, this creative side is concisely expressed in Philip Anderson’s celebrated phrase: More is different.”

    “Such postmodern quantum effects are emergent phenomena par excellence: The discrete states they describe are essentially nonclassical, but can be unambiguously identified only for highly excited states, that is, under near classical conditions.” — ‘Singular Limits,’ Michael Berry

    Is that the way one should read this?


    1. Don Foster: hey, the caterpillar it is basically apple, right?

      So I can call myself a Vegan if I restrict my diet to vegan animals? Good news, I don't have to give up Barbecue night!

      I suppose cannibals can claim they are vegetarians if they only consume Vegan people.

    2. This critter is usually called a worm in informal English, not a maggot.

  13. But in classical General Relativity, one can move continuously from a positive CC to a negative one, going through the value zero, without any singularity, if I'm not mistaken. Why is it different in quantum gravity?
    From you brief sentence on why string theorists prefer to work with AdS space, it is not obvious that there is a singularity at Lambda=0

    1. Opamanfred,

      I am not sure what makes you say that. What exactly do you mean with "continuously"? I have named above an example (planar black holes) this is not the case and you probably also know that the FRW solutions with positive and negative cosmological constant don't continuously deform into each other. What my (admittedly cryptic) sentence was referring to is that AdS has a conformal boundary and isn't globally hyperbolic. This has nothing to do with quantum gravity in particular.

    2. @opamanfred: This would be the case if there is no mass-gap separating quantum states of a de Sitter vacuum from an anti-de Sitter vacuum. One has to consider however there are topological differences between the dS and AdS. The dS in n+1 dimensions is basically R×M^n, for the reals R representing time and M^n either an n-dimensional sphere or Euclidean flat space. The anti-de Sitter spacetime is S^1×R^n, where S^1 is a circle and R^n is a Euclidean flat space of n dimensions. This means there are closed timelike curves, which is an oddity. Often one considers only a conformal patch in the AdS that avoids time loops. These topological differences are identified with obstructions such as a mass-gap.

  14. Hi Sabine, !
    Please feel free to disregard
    My comment from last night.
    All apologies, I was Extremely
    tired and (usually :) know better than to comment when I'm drifting in and out of sleep.
    I understand perfectly what You're saying and,tbh, I have
    no idea What I was talking about! lol
    Just wanted to say
    Welcome Back!
    - and should have waited
    'till the morning lol

    All the best,

    This thread is about limits and infinities
    Mathematically their interaction was explained
    by ''method of renormalization''
    Physical aspect is unknow.

    1. In many physical experiments the INFINITY appears.
      Sooner or later all Fields are finished by ''waves collapse'' and then
      needs ''method of renormalization'' to bring them again to existence.
      Sometimes Electron's parameters become INFINITE.
      What if INFINITY is a real concept ?
      Can INFINITY be understood by Physical Laws ?

  16. What about singularity of particles as perfect points? - what is a common belief e.g. for electron and would mean infinite energy of its electric field alone.
    I was searching in literature for experimental evidence of this belief for electron and it usually leads to 1988 Dehmelt's g-factor argument, where he extrapolates from proton and triton - literally by fitting parabola to two points (!).
    In contrast, looking at electron-positon scattering cross-section and extrapolating to resting energy to remove Lorenz contraction contribution, we get ~100mb corresponding to ~2fm radius.
    Gathered materials:

  17. Infinity appears in many (!) physical theories and experiments.
    We were taught how to escape from this problem
    But what if INFINITY is a real physical concept ?
    For example: there is only one physical INFINITE reality -
    the spacetime between billion and billion galaxies.
    All the rest is limited and finite.



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