Michio Kaku

What the nano-scale column stands on

chGravity-scales.html says the operator chain is “not metaphorical across scales”, and its table gives the compression operator C a technical identity at each one. At the nano scale the identity is level truncation in Witten's cubic string field theory — a projection to a finite-dimensional subspace of the string Hilbert space — and the scale's description says the theory has a T-duality self-dual point at R = 1. Those are two claims about string theory, and both rest on work that other people did first.

The script beside this page read all 873 chapters of the corpus as they stood at commit aaba379, before this chapter existed. No page names Michio Kaku. No page names Keiji Kikkawa. No page mentions the Kardashev scale. The three phrases — string field theory, T-duality, level truncation — appear on exactly one page, and it is this one. (One other chapter, book6/ch07-microtubule-fibonacci, cites a Kikkawa — M. Kikkawa, on the microtubule seam. A namesake, a different field. The script scopes its claim to Keiji Kikkawa so nobody merges the two.)

His account, and what it is worth

In Parallel Worlds (PDF pp. 209–210, book pp. 190–191) he says that in 1974, with Keiji Kikkawa of Osaka University, he extracted the field theory of strings, having set out to write the string as a field in the way Faraday's lines of force write electromagnetism. In Physics of the Impossible (p. 6) he says he wrote the theory of strings in terms of Faraday's force fields and so founded string field theory, and the jacket of the same scan calls him its cofounder. Later in the same book (PDF p. 256, book p. 237) he credits the R ↔ 1/R duality to Kikkawa and his student Masami Yamasaki, in 1984.

What the row needs, computed

C is a projection to a finite-dimensional subspace, so the first thing to know is how big the subspace is. In light-cone gauge, for the open bosonic string in 26 dimensions, there are 24 transverse oscillators and the number of states at level $N$ is the coefficient of $q^N$ in $\prod_{n\ge1}(1-q^n)^{-24}$. The script computes it exactly:

Three facts follow and the script checks each. Level 1 has 24 states, the 24 polarizations of the massless vector. Level 2 has $324 = 24 + 300$, which is also the dimension of a symmetric traceless tensor of $SO(25)$ — arithmetic, not a claim about representations. And the series is the coefficient list of $1/\Delta$: the product $\prod(1-q^n)^{24}$ expands to $1 - 24q + 252q^2 - 1472q^3 + 4830q^4 - 6048q^5 \cdots$, which is Ramanujan's function $\tau(1),\dots,\tau(6)$, so the counts are the reciprocal of the series with those coefficients. (This $\tau(n)$ is Ramanujan's, not the corpus's $\tau = 2$. The neighbouring chapter, Ramanujan, is about partitions and modular objects and does not develop $\tau(n)$; it is the neighbour, not the source.) The growth is exact to better than one part in $10^{9}$ from $N=10$ to $N=1000$:

So the cost of C at level $N$ is sub-exponential in $N$ and super-polynomial: a real number the row can be held to, and one the chapter it sits in never states.

The duality half of the row is a smaller check. For a closed string on a circle of radius $R$ (with $\alpha'=1$) the zero-mode part of $M^2$ is $(n/R)^2+(wR)^2$; the script confirms the spectrum at $R$ equals the spectrum at $1/R$ with $n$ and $w$ swapped, for 49 ratios, and that $R = 1/R$ has the single positive solution $R=1$. So “self-dual point at $R=1$” is right in units of the string length.

Three books, two ladders

The Type I / II / III scale is Kardashev's, as the books themselves say. All three books held here print it with the same step — each type ten billion times the last — and they do not print the same rungs. The paper itself is now held and prints a third set of rungs, different from both books'; the Kardashev chapter reads it in full.

Every entry above is on the page the script names. Then the arithmetic, all in the script:

Read plainly: Physics of the Future is internally consistent — its rungs sit on the physical quantities they name, and its years follow from its rates. Parallel Worlds is not: its Type I is a decade low, and its time to Type III does not follow from the growth rate it states. The two books are consistent with each other on the step and on nothing else. Which is the later, and so which corrects which, is not asserted here: Parallel Worlds' page footers carry the date 27 October 2004, and Physics of the Future refers to the recession of 2008, so it is the later of the two; the third book's date is not checked.

Why a corpus about scales should care

chGravity-scales says the same operator chain acts at every scale, and gives it three: 10⁻³⁵ m, 10⁻¹⁵ m, and 10⁻³ to 10¹ m. A ladder is a claim only when each rung has a number and there is a rate that takes you between them. Kaku's ladder is the case where that was done in public, more than once, and the numbers moved. The corpus's ladder has the rungs and no rate. What the rate would be — how the compression cost computed above changes from rung to rung — is OPEN, and it is the question this chapter leaves the nano-scale row with.

Physics of the Future (PDF pp. 187–188) adds an information scale and an entropy scale to the energy one. This chapter reads those pages and does not use them. Whether the three axes are independent, and whether the corpus's evidence tags map onto any of them, is not tested (OPEN).