Things in Berkeley and later here in Jerusalem were very hectic so I did not blog much since mid October. Much have happened so let me give brief and scattered highlights review.

Two “real analysis” workshops at the Simons Institute – The first in early October was on Functional Inequalities in Discrete Spaces with Applications and the second in early December was on Neo-classical methods in discrete analysis. Many exciting lectures! The links lead to the videotaped  lectures. There were many other activities at the Simons Institute also in the parallel program on “big data” and also many interesting talks at the math department in Berkeley, the CS department and MSRI.

To celebrate the workshop on inequalities, there were special shows in local movie theaters

My course at Berkeley on analysis of Boolean functions – The course went very nicely. I stopped blogging about it at weak 7. Just before a lecture on MRRW upper bounds for binary codes, a general introductory lecture on social choice, and then several lectures by Guy Kindler (while I was visiting home) on the invariance principle and majority is stablest theorem.  The second half of the course covered sharp threshold theorems, applications for random graphs, noise sensitivity and stability, a little more on percolation and a discussion of some open problems.

Back to snowy Jerusalem, Midrasha, Natifest, and Archimedes. I landed in Israel on Friday toward the end of the heaviest  snow storm in Jerusalem. So I spent the weekend with my 90-years old father in law before reaching Jerusalem by train. While everything at HU was closed there were still three during-snow mathematics activities at HU. There was a very successful winter school (midrasha) on analytic number theory which took place in the heaviest storm days.  Natifest was a very successful conference and I plan to devote to it a special post, but meanwhile, here is a link to the videotaped lectures and a picture of Nati with Michal, Anna and Shafi. We also had a special cozy afternoon event joint between the mathematics department and the department for classic studies  where Reviel Nets talked about the Archimedes Palimpses.

The story behind Reviel’s name is quite amazing. When he was born, his older sister tried to read what was written in a pack of cigarettes. It should have been “royal” but she read “reviel” and Reviel’s parents adopted it for his name.

4 thoughts on “Many Short Updates”

1. John Sidles

Hi Gil! In regard to Joe Fitzsimons’ recent critique of your quantum postulates over on Shtetl Optimized, here is a defense of (what I take to be) those same postulates, that amounts to a preview of the planned mathematical topics for our Seattle-based Soldier Healing Seminar for 2014.

Please let me say too, that we in the medical/engineering community increasingly appreciate the practical value of the quantum/mathematical researches of you and your Jerusalem colleagues. Thank you, Gil Kalai!

(also, please let me apologize in advance for any typographic errors in the following post … without a preview, it’s mighty hard to get the LaTex right!)

Joe Fitzsimons says (in effect): “Gil, I think the linear increase in decoherence with the number of qubits is basically impossible on physical grounds.”

The exchange between Joe Fitzsimons and Gil Kalai exemplifies (as it seems to me) a cognitive phenomenon that the preface to Joseph Landsberg’s (wonderful!) text Tensors: Geometry and Applications aptly calls the Clash of Cultures

Section 0.3: Clash of Cultures

In the course of preparing this book I have been fortunate to have had many discussions with computer scientists, applied mathematicians, engineers, physicists, and chemists.

Often the beginnings of these conversations were very stressful to all involved. I have kept these difficulties in mind, attempting to write both to geometers and researchers in these various areas.

Tensor practitioners want practical results. To quote Rasmus Bro (personal communication): “Practical means that a user of a given chemical instrument in a hospital lab can push a button and right after get a result.”

My goal is to initiate enough communication between geometers and scientists that such practical results will be realized. While both groups are interested in communicating, there are languages and even philosophical barriers to be overcome.

The following passage attempts a Landsberg-style reconciliation of the Fitzsimons/Kalai “culture clash”

Quantum dynamics on a Hilbert space is characterized by Hamiltonian flows $X$ on a state-space manifold $\mathcal{M}$ that respect four physical principles:

$\begin{array}{rllr} \langle X,\cdot\rangle_\omega = \text{d}\rlap{h}&&\text{(Hamiltonian dynamics)}&(\text{eq.1})\\ \mathcal{L}_{X} \omega \vphantom{\overset{?}{=}} = 0 && \text{(thermodynamical laws)}&(\text{eq.2})\\ \mathcal{L}_{X} (f_i) \vphantom{\overset{?}{=}} = 0 && \text{(conservation laws)}&(\text{eq.3})\\ \mathcal{L}_{X} g \overset{?}{=} 0 && \text{(superposition principle)}&\quad(\text{eq.4}) \end{array}$

Here (as usual) $\mathcal{L}$ is the Lie derivative, $\omega$ is the symplectic structure satisfying $\text{d}\omega=0$, and $g$ is the metric on the tangent space $T\mathcal{M}$. The link to Dirac notation is via the bra-ket symbol function $h=\langle\psi|H|\psi\rangle$ associated to the Hamiltonian operator $H$, and similarly $(f_i) =(\langle\psi|F_i|\psi\rangle)$ is a basis set of bra-ket symbol functions associated to a basis set of conserved operators $F_i$.

Familiar quantum mechanical relations like commutators are natural under pullback onto $\mathcal{M}$:

$\langle|\psi|[A,B]|\psi\rangle = \{a,b\}$

where $\{\cdot,\cdot\}$ is the Poisson bracket, and the symbol functions $a,b$ are given by $a=\langle\psi|A|\psi\rangle$, and $b=\langle\psi|B|\psi\rangle$; thus the discussion of any QM textbook can be pulled-back from Hilbert state-space to varietal state-spaces.

In this coordinate-free idiom — which strictly respects the dictum of Landsberg’s preface: “Don’t use coordinates unless someone holds a pickle to your head” — the familiar $i$ of QM is identified with the complex structure $J$ satisfying for all tangent vector fields $X,Y$ on $\mathcal{M}$

$\forall X,Y \in T\mathcal{M}\quad\left\{\begin{array}{l}\omega(X,Y) = g(X,JY)\latex 0.5ex] g(JX,JY)=g(X,Y)\latex 0.5ex] J(J(X)) = -X\end{array}\right.\phantom{\}}$

Notice that even the (seemingly) innocuous number “$i$” is expunged from this formulation of quantum mechanics (and this is why one so seldom encounters “$i$” in mathematical discussions of complexity geometry and dynamics).

Tensor practitioners (like me) and mathematical postulators (like Gil) now find common ground as follows:

Tensor practitioners  generically simulate large-$n$ quantum dynamics on varietal manifolds, upon which eqs. 1-3 (Hamiltonian flow, conservation laws, and thermodynamics) hold identically, while eq. 4 (quantum superposition) holds only approximately (that is, superposition holds microscopically, but is invalid in the macroscopic thermodynamic limit).

Mathematical postulators  inquire into generic properties of varietal dynamical systems, in effect by admitting $\mathcal{L}_{X} g \ne 0$ as a starting postulate, from which new mathematical results follow (wonderful mathematical results, we may all reasonably hope).

Students especially — and experienced quantum researchers too — may worry that horribly unphysical consequences may result from relaxing the postulate of metric isomorphism (of eq. 4). After all, this implies that the complex structure $J$ evolves dynamically, such that (in effect) the number “$i$” is no longer a constant!

It is reasonable to worry that in consequence (per #72) “Energy levels may broaden linearly, which is basically impossible on physical grounds.” Relief from these worries may be found by reflecting upon the paradigmatic example of a varietal state-space, namely Segre varieties of Veronese varieties, which are known to physicists as coherent states evolving on Bloch-sphere product spaces. Here $\mathcal{L}_{X} g \ne 0$, and yet no untoward dynamical and/or thermodynamical consequences result.

In particular, varietal line-widths in Bloch-sphere dynamics can be narrowed to any desired width, in complete accord with (for example) analyses like Steven Weinberg’s “Precision tests of quantum mechanics” (1989); as reference that is oftimes cited — mistakenly as it seems to me — as evidence that relaxing the metric isomorphism postulate (of eq. 4) is inconsistent with spectroscopic tests of the superposition principle. Here the abstraction $i\to J$ concretely augments both our physical insight and our capacity to critically assess the literature, and even more importantly (as it seems to me) it excites our mathematical imagination.

The Analysis in a Nutshell  The existing evidence for the superposition principle (as ensured by eq. 4) is considerably weaker — both experimentally and mathematically — than evidence for the thermodynamical Four Laws (as ensured by eqs. 1-3).

Aram Harrow asked me: “Where are these ideas written up?” The answer is: “Everywhere, and nowhere.”

“Everywhere” in the sense that no single one of these ideas is new (indeed each of the above ideas appears in hundreds of references under dozens of guises). “Nowhere” in the sense that — regrettably for students especially — no single reference summarizes and unifies these ideas (or even establishes notational conventions for them).

Conclusions Relaxing the metric isomorphism postulate (of eq. 4) yields dynamical flows that concretely model Gil Kalai’s postulates, that moreover are exceedingly useful (in Landsberg’s phrase) to “tensor practitioners who want to push a button and right after get a result.” Students (especially) can enjoyably benefit from these ideas by reading the scientific literature of the 20th century with the fresh vision that 21st century mathematical naturality provides.

2. John Sidles

Well that worked pretty well! The only “glitched” equation concerns the complex structure $J$ … hopefully this will parse OK:

$\forall\,X,Y \in T\mathcal{M}\quad\left\{\begin{array}{l}\omega(X,Y) = g(X,JY)\\[0.5ex] g(JX,JY)=g(X,Y)\\[0.5ex] J(J(X)) = -X\end{array}\right.\phantom{\}}$

Thank you (again) Gil, for conceiving quantum postulates that (as it seems to me and many) unite mathematical beauty, with fresh physical insights, with engineering elegance … postulates that help so greatly to accelerate the 21st century’s great enterprises of healing.

1. Gil Kalai Post author

Dear John, just a short response for now. Many thanks for defending my postulates! Of course, I also thank Joe for criticizing my postulate! best Gil

3. John Sidles

Dear Gil, you’re welcome!

To say a little more about how-and-why your quantum postulates are of great interest to “tensor practitioners” in general, (and we medical researchers in particular), your quantum postulates, as practically modeled by broad classes of varietal quantum simulations, turn two fundamental identities of Hilbert-space dynamics (as mentioned above), namely

$\begin{array}{rl} \mathcal{L}_{X} g \overset{?}{=} \rlap{0}\hphantom{\{a,b\}} & \text{(the superposition principle),}\\ \langle\psi|\,[A,B]\,|\psi\rangle \overset{?}{=} \{a,b\}& \text{(Dirac's commutator principle)}\end{array}$

into mere(?) approximations. That is to say, for purely practical reasons we “tensor practioners” choose our state-space varieties $\mathcal{M}$ and our operators-of-interest $H,A,B,\ldots$, such that the set of flows $X_H,X_A,X_B,\ldots$ are (projective) $g$-isomorphisms of $\langle X_A,X_B\rangle_g$ (per Eq.4), and are Dirac-natural under pullback (per Eq.5), and moreover, the (inevitable?) algebraic singularities of $\mathcal{M}$ are occult to Lindblad/Carmichael trajectory-unravellings that are observed by $X_H,X_A,X_B,\ldots$.

Of course, for purely aesthetic reasons the community of “quantum postulators” is interested in precisely the same goals.

Then a marvelously harmonizing aspect of the Kalai/Harrow debate — an aspect that is still-mysterious and even near-miraculous (as it seems to me) — is that the set of operators $H,A,B,\ldots$ that Nature provides, and varietal manifolds $\mathcal{M}$ that Platonic Heaven provides, are so sufficiently well-suited to realize (Eqs. 4-5) as identities, that we engineers can hope to sustain transformatively rapid progress regarding (for example) medical capabilities that are of the greatest humanitarian, economic, and political consequence.

Our 2013 seminar notes give additional details (the PDF download is fully hyperlinked, we hope), and needless to say, here is much that we understand incompletely (if at all) regarding the interplay of geometry, dynamics, infomration, and combinatorics upon these varietal state-spaces, that (seeming) realize your postulates so marvelously. Hopefully by the end of 2014 we will all understand much more than we do at present, regarding these open questions.

Meanwhile, thank you again Gil, for your sustained commitment to enriching the width, depth, and context of quantum discourse.