Quantum Techniques for Reaction Networks

Fans of the network theory series might like to look at this paper:

• John Baez, Quantum techniques for reaction networks.

and I would certainly appreciate comments and corrections.

This paper tackles a basic question we never got around to discussing: how the probabilistic description of a system where bunches of things randomly interact and turn into other bunches of things can reduce to a deterministic description in the limit where there are lots of things!

Mathematically, such systems are given by ‘stochastic Petri nets’, or if you prefer, ‘stochastic reaction networks’. These are just two equivalent pictures of the same thing. For example, we could describe some chemical reactions using this Petri net:

but chemists would use this reaction network:

C + O2 → CO2
CO2 + NaOH → NaHCO3
NaHCO3 + HCl → H2O + NaCl + CO2

Making either of them ‘stochastic’ merely means that we specify a ‘rate constant’ for each reaction, saying how probable it is.

For any such system we get a ‘master equation’ describing how the probability of having any number of things of each kind changes with time. In the class I taught on this last quarter, the students and I figured out how to derive from this an equation saying how the expected number of things of each kind changes with time. Later I figured out a much slicker argument… but either way, we get this result:

Theorem. For any stochastic reaction network and any stochastic state \Psi(t) evolving in time according to the master equation, then

\displaystyle{ \frac{d}{dt} \langle N \Psi(t) \rangle } =  \displaystyle{\sum_{\tau \in T}} \, r(\tau) \,  (s(\tau) - t(\tau)) \;  \left\langle N^{\underline{s(\tau)}}\, \Psi(t) \right\rangle

assuming the derivative exists.

Of course this will make no sense yet if you haven’t been following the network theory series! But I explain all the notation in the paper, so don’t be scared. The main point is that \langle N \Psi(t) \rangle is a vector listing the expected number of things of each kind at time t. The equation above says how this changes with time… but it closely resembles the ‘rate equation’, which describes the evolution of chemical systems in a deterministic way.

And indeed, the next big theorem says that the master equation actually implies the rate equation when the probability of having various numbers of things of each kind is given by a product of independent Poisson distributions. In this case \Psi(t) is what people in quantum physics call a ‘coherent state’. So:

Theorem. Given any stochastic reaction network, let
\Psi(t) be a mixed state evolving in time according to the master equation. If \Psi(t) is a coherent state when t = t_0, then \langle N \Psi(t) \rangle obeys the rate equation when t = t_0.

In most cases, this only applies exactly at one moment of time: later \Psi(t) will cease to be a coherent state. Then we must resort to the previous theorem to see how the expected number of things of each kind changes with time.

But sometimes our state \Psi(t) will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago:

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

We wrote this first, but logically it comes after the one I just finished now!

All this material will get folded into the book I’m writing with Jacob Biamonte. There are just a few remaining loose ends that need to be tied up.

42 Responses to Quantum Techniques for Reaction Networks

  1. Phillip Harris says:

    Typo in paper: Hydronium is H3O+ not H2O+

  2. Blake Stacey says:

    Grammar nit:

    The big surprise one meets at the very beginning of this subject is that the canonical commutation relations between annihilation and creation operators, long viewed as a hallmark of quantum theory, is also perfectly suited to systems of identical classical particles interacting in a probabilistic way.

    If the subject is plural (which sounds a little better to me), the verb should be too.

  3. Blake Stacey says:

    For the HIV reaction network in Equation 1, if we also use the Greek letters names of the reactions as names for their rate constants, we get this rate equation:

    “Greek letters names” -> “Greek-letter names”

    • John Baez says:

      Thanks for catching these errors! I hope you like the math when you get to that part.

      So you don’t have to redo the work Brendan did, here is his list of corrections:

      — p1, figure and end of first paragraph: hydronium should be H_3O^+
      — p4, top line: double word “…we have have used multi-…”
      — p4, second paragraph: missing space “…rest of thispaper we…”
      — p6, last paragraph of proof: the ‘because’ in “This last step is valid even though… because…” might be ambiguous/confusing for a less mathematical audience — it’s not clear from the grammar whether you’re giving a reason why the last step is valid, or whether you’re giving a reason why the monomials don’t form a basis. It’s clear if you understand the content of course, but it’s a technical point in a less technical paper, so maybe it’d be nice to be clearer.
      — p7, before and after lemma 7: you use the vector number operator N in the line before the lemma, but only define it after.
      — p7, proof of theorem 8: not sure if you want to specify $n \le \ell$, or where that’s taken care of in the notation. Also, I think the falling power should be n in the displayed maths, not m.
      — p9, last sentence: maybe you should add “it” in “and follows that…”
      — p10, last displayed maths in proof of theorem 9: I don’t think you define the overdot notation for time derivative.
      — p10, end of first line of last paragraph: “…continues to be s0…”

    • John Baez says:

      I’ve made all of Brendan and Blake’s suggested corrections, and a new improved version can be found at the same place, namely here.

    • Blake Stacey says:

      I hope I’ll get a chance to go over the equations in a systematic way soon; this week is a mess of writing and lecturing obligations.

  4. Jon Rowlands says:

    Few more nits:
    p3: we make the follow/i/ng definition
    p3: that the rate at which a reaction/s/ occurs
    p4: the concentration of each input species i of tau/,/ raised to
    p4: Definition 4. The rate equation for a reaction network (S;K;/R/;s;t) with rate constants r : /R/ (R should be T?)
    p5: Our notation here follows that used /that used/ in quantum
    p7: in def of Nm, is Nm well defined because the Ni commute?

    • John Baez says:

      Thanks for the extra corrections! I’ve fixed all those things now. You and Blake are now in the acknowledgements.

      Yes, what I once called the set R of ‘reactions’ should now always be the set T of ‘transitions’… though in this paper I call them ‘reactions'; the name ‘transitions’ comes from Petri net theory. And yes, the number operators N_i all commute, since annihilation and creation operators with different indices commute. I should mention that.

  5. Blake Stacey says:

    There’s an interesting connection between the creation/annihilation operator formalism and the representation theory of SU(2), which goes back to Schwinger. I haven’t made use of it myself, but it might be useful in cases where the number of molecules per site is limited; or, perhaps, where the key quantity is the difference between the sizes of two populations.

    Schwinger’s idea was to relate the angular momentum Lie algebra to the algebra of two independent harmonic oscillators. If the oscillator algebras are given in the normal way by
    [a,a^\dag] = 1
    and
    [b,b^\dag] = 1,
    and if we define the operators
    J_+ = a^\dag b,
    J_- = b^\dag a,
    J_z = \frac{1}{2}\left(a^\dag a - b^\dag b\right),
    then it follows that
    [J_z,J_\pm] = \pm J_\pm,
    and
    [J_+,J_-] = 2J_z.
    This is just the algebra we use when manipulating angular momentum eigenstates.

    In other words, instead of labeling an angular momentum eigenket |j,m\rangle by the eigenvalues of J_z and J^2, we can label it by the eigenvalues of the number operators a^\dag a and b^\dag b. The correspondence is

    N_a = j + m,\ N_b = j - m.

    Sakurai and Napolitano’s book covers this in the quantum-mechanical context in section 3.9. The different normalization conventions we use in quantum and in stochastic mechanics turn out not to matter.

    Analogues of the Schwinger oscillator construction can apparently be done for other Lie algebras, too; e.g., using six independent oscillators for SU(3).

    • Blake Stacey says:

      Hmm. Maybe the “Schwinger boson” construction could be used to give a groupoidification of SU(2)?

      • John Baez says:

        Yes, I did something on that once and was going to have a grad student work on it, but they got distracted. I think you could use this to better understand Penrose’s spin networks, which are all about representations of SU(2) on the one hand, and the combinatorics of graph colorings (and thus groupoidification) on the other hand. Someone should do that! The ultimate dream was to groupoidify loop quantum gravity…

        When it comes to algebra, I think that by now most of the interesting categorified representation theory has been worked out using the Khovanov approach to categorification, which generalizes groupoidification. I don’t think the Jordan–Schwinger stuff has been categorified yet, but it could be and should be.

        The relation between groupoidification and Khovanov-style categorification is not as well-known as it should be. It was nicely worked out by Jeff Morton and Jamie Vicary. They showed how Khovanov’s ‘categorified Heisenberg algebra’ could be described by taking my groupoidification of the algebra of annihilation and creation operators and hitting it with a functor that takes groupoids to their categories of representations (which can also be seen as categories of representations of certain algebras) and spans of groupoids to certain functors between these categories (which can also be seen as coming from certain bimodules of algebras). So, groupoids and spans of groupoids give Khovanov’s algebras and bimodules.

        But crucially, Jeff and Jamie take groupoidification one step further by looking at spans of spans of groupoids, which correspond to morphisms of bimodules. And this lets them get a nice purely combinatorial explanation of the interesting new stuff in Khovanov’s categorified Heisenberg algebra—the relations that are the ‘next layer of structure’ after the canonical commutation relations:


        For an easy intro, see my n-Café article.

        Anyway, since there’s by now a huge mob of smart people working on categorified Lie algebras and quantum groups, I want to steer clear of that stuff. A bit more interesting would be to find applications of these in the stochastic setting, as you point out. We could look at a population moving around on a lattice where each lattice site can have just 0 or 1 individuals, say, and try to write down a stochastic Hamiltonian for this population using the Jordan–Schwinger formalism, and study that. If you’re interested, let me know!

        Over on the pure math side, someone should look at spans of spans of spans of groupoids, and see the next layer of relations after the canonical commutation relations and Khovanov’s relations. And on the physics side, it would be interesting to understand what the heck all these higher relations mean for physics. But I haven’t been able to get anyone interested in these questions.

      • John Baez says:

        Blake wrote:

        Hmm. Maybe the “Schwinger boson” construction could be used to give a groupoidification of SU(2)?

        By the way, for some fun, Google “categorified Jordan-Schwinger”.

      • Blake Stacey says:

        One of the things I’ve been juggling this past semester has been working out what happens in spatially-extended reactions where there is a limit to the number of reactant molecules at each location. For example, we can conceive of a square lattice of cells, where inside each cell the Brusselator dynamics are taking place, and in addition, molecules can pass between neighbouring cells. Limiting the carrying capacity per cell causes anomalous cross-diffusion terms to appear in the mean-field rate equations. One can plow through the van Kampen expansion and deduce that the terms are there, but I expect there’s a more direct way to see it.

        I think there’s a lot of fancy math which would become more meaningful to me if we could port it into the stochastic-mechanics context. For example, can the Hopf-algebraic approach to renormalization say anything useful about RG calculations for reaction-diffusion phenomena?

        (Look at the mess which starts in section 3 of Janssen and Taeuber (2004) if you need to understand my motivations here.)

        • John Baez says:

          Those renormalization group calculations for reaction-diffusion phenomena look really impressive and intimidating!

          As far as I know, the only people who use the Hopf-algebraic approach to renormalization to do renormalization, as opposed to prove things about renormalization, are the mad scientists who compute all Feynman diagrams up to 5 loops (say), and need to be very careful about subtleties like overlapping divergences. The Hopf-algebraic approach deals with these subtleties in a very systematic way and keeps you from going insane (though unfortunately too late, since you have to be insane to want to do these calculations in the first place).

          I could be wrong here; I’m not an expert on this stuff. It looks like Janssen and Taeuber are going up to 2 loops, which is complicated enough to be a bit of a nightmare, but probably not complicated enough to require one to automate the process of going through all Feynman diagrams and systematically keep track of overlapping divergences.

          I’d really like to find some subject we could work on together, but this hard-core field theory stuff is too removed from my current interests to want to work on it. I’m trying to focus on ‘networks’ and avoid issues that show up when you try to take some continuum limit. I used to think about field theory, but I’ve got a new schtick now.

          The idea of using a Jordan–Schwinger representation to describe systems where each cell has a maximum carrying capacity—that’s more something I can see myself doing.

        • Blake Stacey says:

          John Baez wrote:

          I’m trying to focus on ‘networks’ and avoid issues that show up when you try to take some continuum limit. I used to think about field theory, but I’ve got a new schtick now.

          I can appreciate that. Certainly, the network theory area is interesting—and I know a few people around my city who are into continuum limits, so on the days when I feel like going in that direction, I still have knowledgeable folks to talk with. :-)

          I’m trying to keep myself as focused as possible these days. It’s a skill I’ve never really learned: either I had immediate deadlines which locked me in to particular projects, or I wasn’t interacting with other people who were always coming up with their own crazy ideas, or I just plain didn’t know enough to be able to start anything.

          So, to make things as concrete as possible:

          First, as I said, I’m working on the spatial stochastic Brusselator, which has interesting finite-carrying-capacity behaviours. There may be something interesting in looking at this in a Jordan–Schwinger way, perhaps as a multi-cell discrete system.

          Second, I’m in the midst of revising arXiv:1110.3845 [nlin.CG]. This paper plots a lot of curves which we found by numerical simulations. I’d like to get analytical predictions for as many of those curves as possible. This is what’s been pushing me into stochastic field theory. In addition, approximation methods which people already use for the kinds of models we studied can be expressed in network-theory language. So, there are both field-theoretic and Petri-net-theoretic things to do for the model I’d really like to understand.

        • John Baez says:

          Here’s something that might be concrete enough to become a paper yet abstract enough to keep me happy.

          The Jordan–Schwinger trick might give a systematic way to reinterpret very general stochastic systems resembling stochastic Petri nets, but having limits on the number of particles of various species, as ordinary stochastic Petri nets. This might allow the instant application of theorems about stochastic Petri nets (or chemical reaction networks) to these other systems.

          Here’s a nice intuitive way to think about the Jordan–Schwinger trick in the stochastic situation. Say we have a kind of particle—I need a name for it, so I’ll call it a klingon—for which there can’t be more than n particles of this kind. We can’t force that constraint directly in the stochastic Petri net framework. But we can do it using this trick. We make up a stochastic Petri net two species of particles, 1 and 2, which we think of as ‘existent’ and ‘nonexistent’ klingons.

          We use the operator a_1^\dagger a_2 to simulate the creation of a klingon: in reality, it converts a ‘nonexistent’ klingon into an ‘existent one. Similarly, we use a_2^\dagger a_1 to simulate the annihilation of a klingon: in reality, it converts an ‘existent’ one into an ‘existent’ one. In reality, the total number of klingons, existent and nonexistent:

          N = N_1 + N_2 = a_1^\dagger a_1 + a_2^\dagger a_2

          is conserved by both these processes:

          [N, a_1^\dagger a_2] = [N, a_2^\dagger a_1] = 0

          So, if we start out with a state in the sector

          \{ \psi : \; N\psi = n \psi \}

          processes built using the operators a_1^\dagger a_2 and a_2^\dagger a_1 will keep us in this sector. So, the total number of existent klingons, measured by the operator N_1, can never exceed n.

          Of course this is just another way to describe the raising and lowering operators J_{\pm} for the n-dimensional representation of \mathrm{SU}(2) in terms of ordinary annihilation and creation operators. But I think it brings the idea down to earth a bit, and suggests how we can use stochastic Petri nets to study processes where different kinds of particles have different upper limits on their total particle number.

        • Blake Stacey says:

          Typo:

          in reality, it converts an ‘existent’ one into an ‘existent’ one.

          I like this idea. We might have to make it one level more involved in order to connect with the models people like to study. In the stochastic Brusselator model, one has

          x_i + y_i + e_i = N \forall i,

          that is, the sum total of type-X molecules, type-Y molecules and available empty slots within each cell is the same number N for all cells.

          In your terminology, this is like saying that the dynamics conserve the total number of smooth klingons, bumpy klingons and nonexistent klingons.

        • John Baez says:

          Blake wrote:

          In your terminology, this is like saying that the dynamics conserve the total number of smooth klingons, bumpy klingons and nonexistent klingons.

          That’s even more fun! And not just because I like Klingons. It’s because now we’re getting into the Jordan–Schwinger representation of \mathrm{SU}(3)!

          If you look at all the triples (n_1,n_2,n_3) of natural numbers obeying

          n_1+ n_2 + n_3 = N

          you’ll get a finite set of dots arranged in a triangular pattern. If you take N = 1, you get Gell–Mann’s famous weight diagram for up, down and strange quarks:

          If you take N = 3, you get his famous weight diagram for the ‘baryon decuplet’:

          You can see that up, down and strange quarks are the three species that take the place of smooth, bumpy and nonexistent klingons.

          In general you can do this for any N, and you’ll get the weight diagram for an irreducible representation of SU(3). This is the symmetrized Nth tensor power of its fundamental representation on \mathbb{C}^3. It’s symmetrized because we’re dealing with identical particles here. Its dimension is always a triangle number.

          We get the Jordan–Schwinger representation by looking at linear combinations of operators that annihilate one kind of particle and then create another:

          a_i^\dagger a_j , \qquad i , j = 1,2,3

          There are 9 of these. There’s a special operator that’s a linear combination of these:

          a_1^\dagger a_1 + a_2^\dagger a_2 + a_3^\dagger a_3

          and this is the total number operator: it takes value N on the space we’re talking about. It commutes with all the rest. So, we get an 8-dimensional space of other operators, and this is the Lie algebra \mathfrak{su}(3).

          (Actually we get an 8-dimensional complex space, which is \mathfrak{sl}(3), the complexification of \mathfrak{su}(3). Physicists are often a bit relaxed about this distinction, but we should be clear about it.)

          All this stuff works with more than 3 kinds of particles, too.

          Looking fun yet?

          We are really getting into the groupoidified Jordan–Schwinger idea here. We’re seeing a relation between N-colored sets and representations of \mathrm{SU}(N), which is typical of groupoidification: combinatorics on the one hand, linear algebra on the other.

        • Blake Stacey says:

          Another idea which might strike the right balance between concreteness and abstraction concerns the relations among different Petri-net descriptions of the same system. This is territory in between spatial and nonspatial modeling, where we incorporate information more detailed than overall averages or total population sizes, while still coarse-graining away many details which we hope won’t matter. For example, instead of considering only the total numbers of Susceptible, Infected and Recovered individuals, we can keep track of how many adjacent pairs of each possible combination of types there are. Both levels of description can have stochastic-Petri-net descriptions, but the species and transitions are different.

          I left a note for myself about this here:

          A Petri net specifies a symmetric monoidal category (Lerman et al. 2011). Each truncation of the moment-dynamics hierarchy for a system yields a Petri net, and so successive truncations of the moment-dynamics hierarchy yield mappings between categories. Going from a pair approximation to a mean-field approximation, for example, transforms a Petri net whose circles are labelled with pair states to one labelled by site states. Category theory might be able to say something interesting here. Anything which can tame the horrible spew of equations which arises in these problems would be great to have. Ought we be considering, say, the strict 2-category whose objects are moment-closure approximations to an ecosystem, and whose morphisms are symmetric monoidal functors between them?

          It would be interesting to have some general results about, say, when coarse-graining a stochastic Petri net makes an equilibrium unviable, or makes a new equilibrium appear.

        • John Baez says:

          Blake wrote:

          It would be interesting to have some general results about, say, when coarse-graining a stochastic Petri net makes an equilibrium unviable, or makes a new equilibrium appear.

          That sounds interesting but a bit tough. Since I’m pretty good at abstract nonsense, I’d be tempted to start out by building some general infrastructure: namely, working out the correction notion(s) of morphism between stochastic Petri nets, and figuring out how the rate equations and master equations are related, when we have two stochastic Petri nets related by a morphism.

          Since a Petri net is a free symmetric monoidal category, one obvious notion of morphism, which you mentioned, is a symmetric monoidal functor. (If we want to intimidate people, we can note that a category of ‘free’ gadgets and the obvious morphisms between them is a Kleisli category. Knowing this can even be useful for things other than intimidation.)

          A stochastic Petri net is a free symmetric monoidal category equipped with a symmetric monoidal functor to the multiplicative monoid (0,\infty), viewed as a symmetric monoidal category. This again suggests an obvious notion of morphism between stochastic Petri nets, at least to us category geeks.

          But the interesting part is whether these morphisms give rise to some sort of map sending solutions of the master equation of one stochastic Petri net to solutions of the other. This would probably be ‘too perfect’ for anyone interested in realistic coarse-grainings… but I suspect that if you follow the tao of mathematics, it will quickly lead you to consider these ‘perfect’ coarse-grainings.

        • Blake Stacey says:

          I think the N = 1 case has been looked at, after a fashion. See N. Stollenwerk and M. Aguiar’s “The SIRI stochastic model with creation and annihilation operators” [arXiv:0806.4565]. They cook up a lattice model where each site can be in one of three states (Susceptible, Infected or Recovered), and to write a time-evolution operator, they bring in the Gell-Mann matrices. I think this is pretty much a special case of what we were saying. That is, their model has for each site

          S_i + R_i + I_i = 1,

          as indeed they say at the beginning of section 3.1.

        • John Baez says:

          Cool! I think the problems start with N > 1 and the precise coefficients of operators like a^\dagger_i a_j. You see, if we have such an operator that annihilates a ‘nonexistent’ particle and creates an ‘existent’ one, its matrix entries will depend on the number of nonexistent particles, since there are different possibilities for which nonexistent particle is destroyed. This is okay if there are a number of ‘empty spots’ and we want to keep track of which one is filled by a new particle. But in other situations, we may not get the answer we want.

      • Blake Stacey says:

        By the way, for some fun, Google “categorified Jordan-Schwinger”.

        The second Google hit for me at the moment is some blog page saying, “I don’t think the Jordan–Schwinger stuff has been categorified yet, but it could be and should be.”

        • John Baez says:

          And the first one is my grant proposal where I say:

          For example, it is known that the usual ‘Jordan–Schwinger’ representation of the group SU(n) on the Hilbert space of the quantum harmonic oscillator with n degrees of freedom can be q-deformed [22, 23, 24]; we have already seen how to categorify a substantial portion of this setup, and we hope to do more. We are also categorifying representations of other quantum groups, as well as the theory of Hecke algebras.

          I got that grant, and my grad students groupoidified Hecke and Hall algebras (which are q-deformed universal enveloping algebras of Lie algebras), but they never got around to Jordan–Schwinger!

    • John Baez says:

      By the way, I used to really love this Jordan–Schwinger stuff, so thanks for reminding me about it.

      The reason the Jordan–Schwinger stuff works for the groups SU(n) is that these groups are contained in symplectic groups. The Jordan–Schwinger idea applies to any symplectic group, and that’s where the idea really lives (in my opinion).

      The symplectic group Sp(2n) is the group of symmetries of a 2n-dimensional symplectic vector space. Equivalently, it’s the group of symmetries of the algebra generated by n creation and n annihilation operators. These operators span a 2n-dimensional symplectic vector space, say V. The symplectic group Sp(2n) acts on this vector space. So, the Lie algebra \mathfrak{sp}(2n) acts on V as well.

      And here’s the fun part: for each element T \in \mathfrak{sp}(2n), there’s a quadratic expression \rho(T) in the annihilation and creation operators such that

      T v = [\rho(T), v ]

      where at right I’m using the commutator bracket in the algebra generated by annihilation and creation operators. In physics lingo, \rho(T) is a way of ‘quantizing’ the infinitesimal symmetry T.

      How do you define \rho(T)? I forget but I think it’s something dumb like this. Since T is a linear transformation of V it’s an element of V \otimes V^*. But a symplectic vector space is canonically isomorphic with its dual so we get an element of V \otimes V. But V is the space spanned by annihilation and creation operators, so V \otimes V is the space of all bilinear expressions in annihilation and creation operators.

    • Jon Rowlands says:

      In case this is relevant. See section 7.

      http://arxiv.org/abs/cond-mat/0612198

  6. Arjun Jain says:

    In definitions 1 and 2, shouldn’t a graph be specified as s , t : E \to V \times V and s , t : T \to K \times K ?

    • John Baez says:

      s, t : E \to V is a pretty standard shorthand for saying we have two functions s and t that both map E to V. It’s just faster than writing s : E \to V and t : E \to V.

      It’s just like we often say x, y \in \mathbb{R} when we have two real numbers x and y, instead of saying x , y \in \mathbb{R}^2 (which would confuse everyone) or (x , y) \in \mathbb{R}^2 (which is sometimes good, but only when we want to make the reader think about (x,y) as a point in the plane).

  7. Arjun Jain says:

    In the section on the Master equation:

    To describe the matrix H(\tau), this we need `falling powers’.

    – should be corrected.

    • John Baez says:

      Thanks! I’m really glad you’re reading this paper; it presents some of the same material as the ‘course’, but more quickly, and it also does some new things. Please get a copy of the latest version when you read this, because I keep updating it. Yesterday I added a more detailed explanation of the ‘Fock space’, and in a minute I’ll probably rename it the ‘stochastic Fock space’ to distinguish it from the usual quantum one.

  8. Jon Rowlands says:

    I think the proof of theorem 5 is funky — the last line includes a^s(\tau) , which I think should be a^{s(\tau)} , and otherwise the first and last lines are the same.

  9. Arjun Jain says:

    At the end of the proof of Theorem 5,

    1.typo: You write, …..n be expressed as an convergent in finite linear combination of these monomials….- an -> a

    2. Does the convergence of each coefficient mean that each coefficient is a finite number?

    3. What does this sentence mean?,- .. The annihilation and creation operators, and indeed all the operators discussed in this paper, are continuous in this topology..

    In the proof of Theorem 8,

    4.typo: \displaystyle{ \frac{d}{dt} \langle N \Psi(t) \rangle } = \displaystyle{\sum_{\tau \in T}} \, r(\tau) \, (s(\tau) - t(\tau)) \; \left\langle N^{\underline{s(\tau)}}\, \Psi(t) \right\rangle -> \displaystyle{ \frac{d}{dt} \langle N \Psi(t) \rangle } = \displaystyle{\sum_{\tau \in T}} \, r(\tau) \, (t(\tau) - s(\tau)) \; \left\langle N^{\underline{s(\tau)}}\, \Psi(t) \right\rangle.

    5. You write that the rate equation for x(t) would then follow from the master equation for \psi (t) if we had \langle N \Psi(t)\rangle^m=\langle N^{\underline{m}} \Psi (t)\rangle for every multi-index m. Isn’t this being true only for the source complexes, sufficient?

    6. Can you elaborate on this? : ..However, it should hold approximately in a suitable limit of large numbers. ..

    Lemma 12:

    7. typo: \langle \displaystyle{\sum_{l \in \mathbb{N}^k}} \psi_l z^{k+l} \rangle -> \langle \displaystyle{\sum_{l \in \mathbb{N}^k}} \psi_l z^{m+l} \rangle.

    • John Baez says:

      Thanks for catching those typos.

      2. Does the convergence of each coefficient mean that each coefficient is a finite number?

      Yes \mathbb{R}[[z_1, \dots, z_k]] is the space of formal power series in the variables z_1, \dots, z_k where the coefficients are real numbers, which of course are finite.

      We’re giving this space the topology where a sequence of formal power series \Psi^i converges to a formal power series \Psi iff all the coefficients converge:

      {\psi^i}_\ell \to \psi_\ell

      for ever multi-index \ell \in \mathbb{N}^k. (Here {\psi^i}_\ell is the coefficient of z^\ell in the formal power series \Psi^i; I need to use superscripts i = 1, 2, 3, \dots to indicate a sequence of formal power series since I’m using a subscript \ell to indicate a coefficient.)

      3. What does this sentence mean? “The annihilation and creation operators, and indeed all the operators discussed in this paper, are continuous in this topology…”

      All these operators are maps from \mathbb{R}[[z_1, \dots, z_k]] to itself. I’m asserting that they’re all continuous in the topology I just described.

      5. You write that the rate equation for x(t) would then follow from the master equation for \psi (t) if we had \langle N \Psi(t)\rangle^m=\langle N^{\underline{m}} \Psi (t)\rangle for every multi-index m. Isn’t this being true only for the source complexes, sufficient?

      Yes.

      6. Can you elaborate on this? : “However, it should hold approximately in a suitable limit of large numbers.”

      Making this precise would be easy for me, but it would take a few pages. That’s why I didn’t make that sentence precise, and that’s why I won’t do it now. I’ll leave it as a puzzle:

      Suppose we are given \epsilon > 0 and T > 0. Find conditions saying that \langle N \Psi(t)\rangle^m is close to \langle N^{\underline{m}} \Psi (t)\rangle, which guarantee that if \Psi(t) is a solution of the master equation obeying these conditions at some time t = t_0 then

      x(t) = \langle N \Psi(t)\rangle^m

      stays within a distance \epsilon of a solution of the rate equation for t_0 \le t \le t_0 + T. This could be a short paper. This is sort of standard analysis stuff.

  10. Dan says:

    I finally got around to reading paper. I think I found a couple more minor typos:

    In the proof of Lemma 12, I think z^{k+\ell} should be z^{m+\ell}.

    In the penultimate sentence, “…as show by…” should be “…as shown by…”

    Hope that helps.

    • Dan says:

      Oops. I see Arjun caught the typo in lemma 12. It must not have been fixed on the version I read.

      • John Baez says:

        Actually I somehow missed Arjun’s correction of that typo—he’s been posting lots of comments, and I’m having trouble keeping up with them all. (He’s working with me here at the Centre for Quantum Technologies now.) So, thanks a lot for posting your comment!

        The version on the arXiv is now a bit out of date. The latest version, with all known corrections made, is on my website. But I’ll eventually update the arXiv version.

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