P-adic Numbers and Cosmology

The next math department colloquium at Stanford will feature Lenny Susskind lecturing on p-adic numbers and cosmology, here’s the abstract:

The biggest conceptual problem of cosmology is called the measure problem. It has to do with the assignment of probabilities in an exponentially inflating universe, which falls apart into separate causally-disconnected regions. Neither I nor my friends had ever intended to learn about p-adic numbers until we realized how similar such a universe is to an endlessly growing tree-graph. The result has been some new insights from p-adic number theory into the measure problem and other puzzles of eternal inflation. Within the constraints of a one-hour lecture, I will explain as much of this as I can.

I’ve no idea what this is about, but I’m guessing that Susskind is somehow drawing inspiration from two facts:

  • p-adic integers can be represented using a “tree” diagram vaguely remniscent of the logo for the Stanford theoretical physics group on their web-site.
  • The p-adic integers, unlike the usual integers, are compact, so you can put a finite measure on them.
  • It’s hard to believe that any of the special features of these mathematical structures will make the problems of eternal inflation go away, but who knows…

    Coincidentally, I’ve spent a lot of time recently learning about the p-adics, with a very different motivation. The way these things come up in mathematics is that you can think of number theory as being about a space, the space of prime numbers. The p-adics appear naturally when you decide to ask what happens locally near one point (i.e. at one prime). P-adic integers correspond to power series expansions, p-adic numbers to Laurent series. Various people have thought about analogies between conformal field theories on a Riemann surface, where one also wants to focus on what happens at a point and use representation theory methods, and the Langlands program which does something similar in number theory. This is part of the geometric Langlands story, and has roots in a remarkable paper of Witten’s from 1988 entitled Quantum field theory, Grassmannians, and algebraic curves.

    As I’ve mentioned before, this semester here at Columbia we have Harvard’s Dick Gross as Eilenberg lecturer, and he’s giving a wonderful series of lectures starting with local Langlands. I’m hoping at some point to put together what I’ve been learning about this and possible connections to QFT in some readable form, but at the moment things are still too speculative and hazy. In any case, no sign that these ideas are going to solve the problems of cosmology…

    Update: The Susskind et al. paper on this topic is now out at the arXiv. A p-adic model is studied, but no reason is given to believe that it has anything to do with eternal inflation and cosmology.

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    18 Responses to P-adic Numbers and Cosmology

    1. Octoploid says:

      The link to the Witten paper is wrong (missing w in www):

      Why not link to the free version instead?

    2. Peter Woit says:


      Thanks, will fix and point to the Euclid version.

    3. Shantanu says:

      Peter when is the last time you met (or corresponded with) Lenny?
      Or let us know the talk and your meeting with him goes

    4. Chris Austin says:

      I would guess Lenny might be trying to deal with the problem that if eternal inflation is past eternal as well as future eternal, and the 3 extended spatial dimensions are infinite flat space, and the bubble nucleation rate \Gamma_{tot} is always less than R_{de Sitter}^{-4}, so that bubbles don’t collide, then every bubble is contained in an infinite tower of bubbles extending into the past, and every bubble has an infinite nest of descendant bubbles, which does indeed look like an infinitely branching tree.

      Using spatially flat FRW coordinates, the worldsheet of each bubble stops expanding relative to the spatial coordinates after about the de Sitter time and tends to become a tube of fixed coordinate radius, inside which the worldsheets of its descendant bubbles develop into tubes of ever smaller coordinate radius.

      The vacuum energy density of each child bubble has to be less than the vacuum energy density of its parent by a non-zero amount, so if you want eternal inflation to be some sort of “steady state” model for which you can define time-independent probabilities for finding yourself inside a bubble with given characteristics, you face the problem that since the branching tree containing our bubble is not a steady state because the vacuum energies are decreasing, your “steady state” model also has to contain an infinite number of other infinite branching trees of bubbles, shifted backwards and forwards in time relative to “our” infinite branching tree, by all possible amounts.

      But since the infinite tower of bubbles in the past of “our” infinite branching tree contains bubbles whose coordinate radii presumably get ever larger without limit as we go to earlier times, there is no “room” in the usual infinite flat extended spatial dimensions to put those other infinite branching trees of bubbles. So perhaps Lenny is trying to use the compactness property of p-adic numbers to get around this. I would guess that the resulting “enlarged” spatial dimensions might look something like infinite numbers from Non-standard analysis.

      In my opinion, a simpler resolution of the problem would be to give up the idea of having a “steady state” picture of eternal inflation, with time-independent probabilities, and the corresponding requirement that eternal inflation be past infinite.

    5. Bernhard says:

      I just hope Susskind will stay with cosmology problems and not try to use p-adic numbers to make nonsensical claims about multiverses.

    6. Peter Woit says:


      My only interaction directly with Susskind was asking him a question when questions were called for after a colloquium talk he gave here many years ago promoting the anthropic multiverse. He basically politely refused to answer. While we’ve never met, he does seem to have various opinions about me, positive and negative, which he made clear publicly during the “string wars” period, see for instance




      I have no idea what he’s trying to do with p-adics nowadays, but don’t see how they could help him deal with the measure problem. The general problem of most of the multiverse people seems to me that they are trying to sum over “all universes” without any non-trivial input as to what the set of “all universes” looks like. Unless you start with non-trivial structure, I don’t see how you get anything non-trivial out. But, who knows… I’ll be curious to hear what this is about from anyone who attends the talk at Stanford.

    7. plm says:

      Perhaps you can explain a little more what you thought when writing:

      “The p-adic integers, unlike the usual integers, are compact, so you can put a finite measure on them.”

      You can put finite measures on any measurable space. Any absolutely convergent series gives one on Z, any absolutely convergent integral on R, etc.


    8. Peter Woit says:


      To be more accurate I should have specified something like “translation invariant”. The point is that on the p-adic integers the Haar measure you get from thinking of them as an additive group can be normalized to get a finite result, something you can’t do for the usual integers.

    9. jpd says:

      so its bubbles all the way down?

    10. Chris Austin says:

      Correction to my comment above: the infinite number of branching trees of bubbles, shifted backwards and forwards in time by all possible amounts, that you need for a true “steady state” model of eternal inflation, actually can live side by side in the spatially flat FRW coordinates of what Harlow, Shenker, Stanford, and Susskind (HSSS) call the noncompact case, (page 4). The point being that the physical 4-volume of a region bounded by a fixed spatial coordinate sphere and extending from a fixed FRW time into the infinite past is finite, due to the exponential time dependence of the FRW R(t). So in the noncompact case, just like in the compact case, there can reasonably be a first, or outermost, bubble in the history of any bubble.

      It seems to me that the numbering of the descendant bubbles of a given outermost bubble by the p-adic integers, (Fig 6 on page 16), is simply the natural base p decimal numbering of the reals >= 0 and less than 1, reflected in the decimal point. And the p-adic numbers used for distinguishing descendant bubbles of different outermost bubbles, (eqn (3.7) on page 17), are the reals written in base p and reflected in the decimal point.

      The HSSS scheme usefully answers the following question: in the noncompact case, how should we label the outermost bubbles that are present at a fixed FRW time t? The answer: number the outermost bubbles such that their number roughly increases with their distance from the origin of the spatial coordinates, but such that the larger the coordinate radius of an outer bubble, the more powers of p its number is divisible by.

      The scheme then reproduces itself: one de Sitter time later, the outermost bubbles that were present at the earlier time all have almost the same coordinate radii as before, and for each of them, approximately p – 1 new outermost bubbles have appeared between them, all with approximately the same coordinate radius, 1/e times smaller than the previous minimum coordinate radius. Multiply the numbers assigned to the outermost bubbles present at the earlier time by p; these are then those outermost bubbles’ new numbers. And label each new outermost bubble by the new number of its nearest neighbour old bubble, plus a unique number from 1 to p – 1.

      So I think the HSSS scheme certainly helps with clarification and visualization. I am not able to judge, from a quick look through the article, whether the use of p-adic numbers gives you anything beyond what you would get by working with the real numbers obtained by reflecting the p-adic numbers in the decimal point.

    11. plm says:

      “the Haar measure you get from thinking of them as an additive group can be normalized to get a finite result”

      First, I do not see the importance of having a (multi)universe with finite measure. Or at least I do not really understand you mean.

      Second, the integers are amenable, they have a finitely additive translation-invariant probability measure (though not countably additive, and not the Haar measure).

      Third, Haar measure is unique up to multiplication by a nonzero real number. We “get a finite result” from all Haar measures on p-adic integers. Also, if you had an nonfinite Haar measure on your space (say Z) normalization would not help.

    12. Peter Woit says:


      The basic problem is that the multiverse people want to make statistical predictions, but have no real theory of what the space of universes should be or how they should be statistically weighted. So, they decide to weight everything equally. Even for a toy model where the integers parametrize your universes, this doesn’t work (since you can’t normalize such a weighting). If you take as your toy model the p-adic integers, then you can make this work. Again, I haven’t carefully looked at the paper, but I’m guessing this is what is going on.

    13. plm says:

      Thanks, I looked at the paper.

      Actually the p-adic integers do not seem to play any particular role. The p-adic numbers (rationals) are used in places in the paper, and infinite branches of the p-adic tree, i.e. p-adic integers which are not standard integers, are not really necessary for much of the discussion.

      What matters seems to be the p-adic distance itself, between vertices of the causal tree/multiverse cells/universes in the multiverse, arising from the assumption that all edges of the causal tree with p branches at each vertex have the same length (p^{-u}/2 at time u),.

      So compactness of the p-adic integers does not seem to be basic, neither translation invariance, rather the “homogeneous distance”/ultrametric assumption (which can be restricted to Z).

      In any case thanks for the blog post and replies to my comments. It was all interesting to think about.

    14. plm says:

      Correction to my comment: the assumption is that all vertices causally separated at time u are at distance p^{-u}.

    15. Shantanu says:

      how was the colloquium and did you get a chance to interact?

    16. Peter Woit says:


      The talk was at Stanford on the West Coast, I’m in New York….

    17. Shantanu says:

      oops, I thought its at columbia

    18. Jonathan Langdale says:

      Was this recorded and put up online?

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