The question narrowly posed is:

What is the accepted name of the bracket operation that is obtained by replacing the (antisymmetric) symplectic structure of the Poisson bracket with a (symmetric) Riemann metric?

In the (likely?) event that this bracket operation is known by various names in various disciplines, preferred answer(s) will relate to algebras that are associated to Hamiltonian flows on (low-dimension) Kählerian varieties that are naturally immersed in a (large-dimension) Hilbert space, whose dynamical potentials are the operator symbol functions pulled back from that Hilbert space.

Specifically, in practical applications, the Kählerian varieties generically are (low-dimension) rank-$r$ secant varieties of $n$-factor (equivalently, $n$-particle) Segre varieties.

As explained below, this question is stimulated by Michael Nielsen's recent weblog post Survey Notes on Fermi algebras and the Jordan-Wigner Transform, which points to Michaels's GitHub release titled The-Fermionic-canonical-commutation-relations-and-the-Jordan-Wigner-transform

Michael's write-up poses this broader question:

For what algebraic/geometric reason(s) are fermionic quantum dynamical flows (seemingly) harder to simulate on low-dimension varieties than bosonic quantum dynamical flows?


The key ideas of Charles Slichter's classic 1963 textbook Principles of Magnetic Resonance (still in print, and presently in its 3rd edition) are developed in Chapter 3: Magnetic Dipolar Broadening In Rigid Lattices.

Mathematically speaking, a lot has happened since 1963, and it proves to be very instructive to supplement Slichter's Chapter 3 with material describing quantum spin dynamics in the language of Hamiltonian flow, and quantum simulation in the language of pullback (draft of "Slichter redux" here)

The key to reconciling the old and new ways of thinking about spin dynamics is summarized in the following theorem, which physically speaking, asserts that pullback onto low-dimension varieties preserves much of the algebraic and thermodynamic "quantum goodness" of Hilbert space:

quantum pullback theorem

(the above graphic is hosted on GitHub).

This theorem describes how to pullback operator commutator algebras onto low-dimension simulation varieties, and so it is natural to ask (inspired by Michael's GitHub notes):

Onto what algebraic varieties do canonical (fermionic) anticommutators pullback naturally?

A partial answer is supplied by the following lemma, namely, the above theorem goes through if in the Poisson bracket $\langle ds_{\mathcal{H}},dh_{\mathcal{H}}\rangle_{\phi^{-1}_\omega}$ the simple replacement $\omega\to g$ is made, that is, if we simply replace the canonical Kählerian symplectic structure with the canonical Kählerian metric structure. Algebraically speaking, this means that anticommutation relations pullback just as naturally as commutation relations.

So in essence, we would like to extend our quantum pullback theorem, and its umbral discussion of practical applications, to encompass the fermionic dynamics of Michael Nielsen's notes, as well the spin dynamics of Charlie Slichter's textbook.

The practical problem is, it's not so easy (for me) to construct low-dimension algebras that realize (even approximately) the canonical anticommutation relations … whereas low-dimension constructions are easy for (say) angular momentum commutators … this is where expert mathematical advice (even starting name(s) for the symmetric bracket operation) would be welcome.


The above may seem pretty dry, but these algebraic/geometric considerations are of central importance in the practical pursuit of a goal set in the 1940s and 1950s by von Neumann, Wiener and Feynman (among many mathematicians and scientists of that generation), namely (in von Neumann's words) "to look at an $H$ atom." Very broadly speaking, the relevance to the question asked is that bosonic (commutator-respecting) pullbacks characterize the quantum communication channels by which we see the atoms, while fermionic (anticommutator-respecting) pullbacks characterize their chemical dynamics.

To appreciate the attraction this challenge had for von Neumann, Wiener, and Feynman, it is instructive to start with Feynman's celebrated question: What good would it be to see individual atoms distinctly? and restrict it to: Which is numerically the greater challenge, to catalogue every individual star in the universe, or to catalogue every individual atom in the human body?

Then it is easy to compute, that if a 1.7 meter human body were scaled to the size of the observable universe (at present $\sim45.7\times10^9$ light-years), then the individual atoms would be separated by 4 light-years ... and so the answer is, the two great challenges are numerically comparable.

As Stephane Guisard's and Jose Salgado's awe-inspiring VLT timelapse video shows, the astronomers have made a very good start at their challenge … and as our practical mathematical understanding of spin-and-atom dynamics approaches the astronomer's practical understanding of photon dynamics (hopefully with help from MathOverflow) … well, it will be mighty interesting to participate in both of these great challenges.

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    $\begingroup$ Canonical anticommutation relations with $n$ generators has a standard description using complex Clifford algebras with $2n$ generators, i.e. $a_n = (e_{2n}+ i e_{2n+1})/2$, $a^*_n = (e_{2n}- i e_{2n+1})/2$. Could it help? $\endgroup$ Jun 1 '11 at 19:12
  • $\begingroup$ The practical difficulty associated to canonical anticommutation relations is that (as Nielsen's manuscript discusses) these operators have exponentially large dimension ... this is bad for code efficiency! Canonical commutators are even worse ... they formally require infinite-dimension Hilbert spaces. For the case of commutators, a well-known technique is to modify the commutation relation by putting $I_z$ on the right, thus converting it to an SU(N) algebra (where N can be small). I know of no similar code-speeding trick for anti-commutators... that's one practical point of the question. $\endgroup$ Jun 1 '11 at 21:43
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    $\begingroup$ Yes, the Clifford algebra are isomorphic with algebra of $2^n \times 2^n$ complex matixes, but it is not always necessary to write down this matrices. Expressions with products of few generators correspond to some subspaces of the exponential space, e.g. products of two generators -- is simply Lie algebra so(2n). $\endgroup$ Jun 1 '11 at 22:42
  • $\begingroup$ Similar operator product expansions, with implicit rather than explicit evaluation, are discussed in (e.g.) Hogben, Hore, and Kuprov "Strategies for state space restriction in densely coupled spin systems with applications to spin chemistry" (2010). But these restriction methods don't work so well in integrating long-time trajectories of very large, grossly inhomogeneous dynamical systems, in essence because these methods restrict the density matrix rather than the trajectories themselves. The motivation of the question is to leave Hilbert space (and thus density matrices) wholly behind... $\endgroup$ Jun 2 '11 at 14:06

I will answer only your first question --- I hope that you get a more thorough answer from someone with more expertise than I have. Your other questions, I think, are deeper, and I haven't had a chance yet to process them.

The short answer to your first question, I think, is that although there are many texts with words like "antibracket", whenever you come across a seemingly "symmetric" "Poisson bracket", it is probably still truly antisymmetric, for the right notion of "antisymmetric". The reason is that it's hard to imagine a condition on a metric tensor $g^{ij}$ so that the bracket (biderivation)

$$ g^{ij}(x) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j} $$

(always Einstein's summation convention) satisfies any sort of "Jacobi identity" --- at least, doing so is hard for me. (Maybe this is some "flatness" condition? An expert might be able to answer quickly.)

Much more common, I think, is the situation when $x^i$ are Fermionic variables, and the geometric structure is a symplectic one. Note that from the "external" point of view of doing computations with homogeneous coordinates, a symplectic form on a Fermionic manifold looks "symmetric". But the correct perspective is an internal one, where "symplectic" means, among other things, antisymmetric in the way that something acting on fermions should be. If from an internal perspective the geometric structure is symplectic, then you should not invent extra language for it. For details on the "internal" (aka "functor of points") approach to sign rules and Fermions and all that jazz, see the article "Notes on Supersymmetry" by Deligne and Morgan in QFT and Strings.

As an aside, let me also mention one more place where "symmetric" "antibrackets" show up, and try to explain why again there really should not be a new word for them. Namely, in the Batalin-Vilkovisky approach to perturbative integrals, one thing that happens is that you have a ($\mathbb Z$-graded) supermanifold (choose local coordinates $x^i$) with a biderivation $P = P^{ij}(x) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}$. This biderivation is required to satisfy that $P$ is symmetric in the internal sense, and that it satisfy a Jacobi-like requirement, and most importantly that $P$ itself is an odd operator for the Fermionic grading. (In $\mathbb Z$-graded world you can do more interesting things with choice of which odd number.) A typical example, where I denote by $\Pi$ the "parity reversal operator", are "odd cotangent bundles" $\Pi T^\ast X$ for $X$ any supermanifold (i.e. this is the supermanifold you get as the total space of a vector bundle over $X$, where the fiber over $x$ is the parity-reversal of the cotangent vector space $T^\ast_x X$). Then $P$ is the "canonical Poisson bracket".

I claim that in an appropriate sense $P$ is still "antisymmetric". Let $A$ denote the algebra of functions on your supermanifold --- it is a commutative algebra in the category of supervector spaces ("commutative" in the "internal" sense). Note that the biderivation $P$ is not a morphism in the category of supervector spaces $A \otimes A \to A$, because the morphisms are only the "even" ones, and $P$ is odd. It does define a morphism $\Pi A \otimes \Pi A \to \Pi A$; the conditions on $P$ above are that this morphism is a Lie bracket (in the internal sense).

In any case, the most basic statements about the geometry of an operator $P$ as above are very similar to the classical Poisson geometry. So it's reasonable to say that "odd Poisson bracket" means something symmetric in the internal sense.

  • $\begingroup$ Thank you Theo, for this graceful answer ... which will take me a considerable time to digest. The chemistry literature had been pointing me in a direction that was more geometric and less algebraic, namely, that the natural varieties on which to pullback fermionic trajectories are the varieties that chemists use, which they call Slater determininants (and their rank-$r$ secant joins). Now, it seems (to me) that this geometric approach should be wholly equivalent to your algebraic approach... but it will take awhile to appreciate how this works. Meanwhile, sincere thanks are extended. $\endgroup$ Jun 1 '11 at 18:24
  • $\begingroup$ @John: You're welcome. I am definitely not familiar with the chemistry literature, and for me "geometry" and "algebra" are very close. I expect that the approaches are essentially equivalent if phrased correctly; coming at the question as a category theorist has led me to see the similarities between bosonic and fermionic constructions, but of course there are also qualitative differences. $\endgroup$ Jun 1 '11 at 22:41

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