Jacques Distler, our blogging string-theoretical friend, just spoke about

- The fun with random polynomials

which is a project he works on with Uday Varadarajan (Austin). The main point of the project is to generalize

to the case of arbitrary supersymmetric renormalizable (quartic) potentials for the "N" fields that play the role of the redundant anthropic superstructure. Recall that in the picture of Nima, Savas, and Shamit, there are "2^N" vacua because each quartic potential for one of the "N" scalar fields has two minima. For "N=400" or so, this is a large number of vacua. Some of them will have a realistic (i.e. very small) value of the cosmological constant and the Higgs mass, but the observation of Nima et al. is that under certain assumptions, all other parameters may have nearly constant values across the "friendly neighborhood".

Jacques considered generic cubic superpotentials of "N" chiral superfields that are constrained to preserve a "Z_4" R-symmetry under which the fields are odd, and therefore the superpotential must be an odd function. Renormalizability implies that there are linear and cubic terms only. He fixed the "GL(N,C)" symmetry in such a way that his model only differs from the superpotential of Nima et al. by an extra trilinear term

- sum_{i smaller j smaller k} b_{ijk} phi^i phi^j phi^k

with a huge number (N choose 3) of new coefficients b_{ijk}. Generically - except for sets of measure zero in the (a,b)-parameter space (where "a_i" are the coefficients of the linear terms in the superpotential) - one still obtains "2^N" different vacua. It is straightforward but tedious to calculate the average values of the superpotential, the squared potential (second moment), as well as all other conceivable polynomials of the roots (vacua) - sums over all roots (vacua); all the average values are polynomials in the variables "a,b" themselves. The vacuum energy and perhaps other physical parameters are calculated in this way.

You essentially know how to do it if you know that the coefficients in a quadratic equation are the sum and the product of the roots, respectively, and similar rules apply for higher degree polynomials, even if they are polynomials of many variables. It requires some neat linear algebra and characteristic polynomials.

This was the less controversial - the mathematical - part of the talk. The more controversial part of the talk was the physical interpretation. Jacques intended to avoid the anthropic principle, but he could not. He did not avoid it simply because he was talking about "generic values of b_{ijk}" for which some conclusions apply, and so forth. This very fact has essentially separated the room into two political parties of theoretical physics defined - more or less - by their relation to the anthropic principle:

- Jacques and Nima defended the straightforward and somewhat ad hoc procedures.
- The rest of us who participated in the discussions - which means especially Cumrun Vafa and Nati Seiberg who is visiting us because of the Sidneyfest on Friday and Saturday - were dissatisfied with the vague anthropic rules of the game.

Nati Seiberg liked the talk, as he explained us now at the dinner - and he proved it by the fact that he did not sleep. Nati Seiberg's main criticism was the apparent two-stage process in which the vacua were chosen:

- First, one talks about various distributions in the "b" space - where "b" are the coefficients of the trilinear terms in the superpotential.
- Second, one decides about some values of "b", and he talks about the "landscape" of different values of "phi's" and the distribution in this landscape assuming fixed values of "b".

Nati did not like it because in reality (i.e. in the full string theory), one should consider "b's" to be just other examples of the fields "phi" and consider all of them simultaneously and look for the allowed solutions involving all of these fields.

Cumrun complained about the vague and arbitrary sense in which the word "generic" was used in the sentence "generically, the qualitative conclusions of Nima et al. are not changed". He argued, and I agreed with him, that there are infinitely many "measures" that can decide what choice of the couplings "b" is generic and what choice is not. For example, Cumrun's "generic" choice is to choose "b" in such a way that the resulting superpotentials "W" are distributed according to any distribution we like. We believe that both of these choices are equally (un)justified.

Moreover, I was trying to convince others that generically, for a "naturally generic choice of b" one will be very *close* to one of the bad regions where the approximation breaks down and "b" imply a very different picture from the case of Nima et al. (where all "b" equal zero). More concretely, the statement that the other couplings have a small variance will fail. It's simply because there are roughly 1 million different coefficients "b" that one works with in Jacques' superpotential, and it is more or less guaranteed that at least one of them will be in some very special interval whose width is 1/100,000 of the allowed interval for values of any "b". I think that Nati was trying to convey a similar point, in a sense.

The simplified point of the anti-anthropic people is that there exists a huge number of ad hoc procedures with different outcomes, and one should use physical, not sociological, principles to select the right rules. Of course, according to Cumrun, me, and probably others, the preferred physical principles that should decide which choices are generic and natural should be based on the actual dynamics of our theory itself, as exemplified by the Hartle-Hawking wave function in Cumrun+Hiroshi+Erik's paper about the

entropic principle. They should not be based on some extra external (or political correct) assumptions about "uniformity" or "democracy" of something.

Once again, the viewpoint that I share with Cumrun is, once again, that no arbitrary assumptions about the selection etc. should be added to our theory. If we know what dynamics of our theory is - something equivalent to its action - all physically meaningful predictions should be obtained simply by an appropriate mathematical manipulation with this action (or whatever it is replaced by). Dividing fields and constants to fields and constants of 1st category and 2nd category and assuming various randomly chosen and unjustified probability distributions for these variables is simply not a scientifically satisfactory approach.