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When considering diffusion of chemicals, the reaction part is business of chemical kinetics, where the relevant characteristics of different substances come from collision theory together with some classical statistics. If one want to go deep down, one can try to compute the collision rates with quantum theoretical methods.

For some time now I wonder if there is a reason that it might not be possible to just compute the whole chemical reaction process using a path integral. Can't one encode chemical substances (atoms to molecules) in a Hilbert space'ish manner and come up with some sort of Lagrangian, mirroring the change of species concentration from one equilibirum (with seperated chemicals) to another?

$$``\ |\text{CH}_4,2 \text O_2 \rangle\ \ \overset{\text {burn}}\longrightarrow \ \ |\text{CO}_2, 2 \text H_2\text O\rangle\ "$$

Instead of iterating non-linear differential equations, which arise from classical statistical consideration, and which involve empircal or quantum chemically computed rate constants, can't one transfer all of this to computing Feynman diagrams? (Not implying that that would be easier.)

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3 Answers 3

In principle, yes, you can compute all chemical reactions by Feynman diagrams, since the underlying theory is QED.

In practice, to a reasonable accuracy, the mechanism of a chemical reaction can be describe by the transition state theory, ref. Atkins, "Physical Chemistry". The theory uses many assumptions of the transition state related to the equilibrium state statistical mechanics and usually it works.

Along this line, we may use density operator/matrix, than a single state vector to describe a reaction. It is physically plausible since there are $10^{23}$ molecules in a realistic scale.

Still, it is possible to use path integral with statistical mechanics, maybe wiki and references therein http://en.wikipedia.org/wiki/Path_integral_molecular_dynamics

There is an underlying issue in both transition state theory and path integral molecular dynamics, namely constructing the potential energy surface (defined by the Born-Oppenheimer approximation). There are some techniques with Feynmann diagrams as well, ref. Rev. Mod. Phys. 79, 291–352 (2007) .

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If you take the reaction $$``\ |\text{CH}_4,2 \text O_2 \rangle\ \ \overset{\text {burn}}\longrightarrow \ \ |\text{CO}_2, 2 \text H_2\text O\rangle\ "$$ There are 7 nuclei and 42 electrons. In nonrelativistic quantum mechanics, a state of this system is a function on a ~150-dimensional space. It's essentially impossible to do any calculations on such a function. Even if you just try to write down the function in terms of adjustable parameters, it's too many parameters to store on a computer. Forget about doing integrals etc. You would have to make severe approximations like, oh let's say, using classical physics when possible with semi-heuristic quantum mechanics inserted here and there. So that's exactly what people do.

In relativistic quantum mechanics, you replace a function on a complex 150-dimensional space with something even more complicated, I guess a bunch of complex linear operators at every point? So the calculation becomes even more impossible in practice. (In principle, of course, you would get the correct answer.)

The fundamental issue is that the transition is a complicated process. First the carbon atom wiggles a little this way, then the valence electrons get distorted in a certain way, then the hydrogen wiggles a little that way, and on and on. Transition path sampling is a powerful technique to figure out this chain of events, and it works within the framework of ordinary classical or semi-classical statistical mechanics. If you tried to use Feynman diagrams, the calculation would be completely intractable.

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The mathematician John Baez recently wrote a long series of blog posts about using quantum techniques for non-quantum stochastic systems, in which chemical reaction networks played a central role as an important special case. This culminated in a paper entitled Quantum Techniques for Reaction Networks, which might be something close to what you're looking for. He does end up drawing Feynman diagrams and that sort of thing.

(I apologise for posting an answer that's really just links rather than a full explanation - I've dipped into this stuff but I haven't yet found the time to really get to grips with it, so I don't feel confident trying to summarise it.)

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Thanks for the answer. (I know the papers.) (sidenote: If you're interested in networky problems with an applications to physics, on the math board I have an open question http://math.stackexchange.com/questions/420467/what-is-the-function-space-gener‌​ated-by-addition-and-a-b-mapsto-ab-1-cd which turns out to be tough. My current formulation of a sub-problem is if two sequences of s's and p's can lead to the same resistor.) –  NikolajK Jul 5 '13 at 15:08
    
Just out of curiosity, where do those papers fall short in answering your question? (I'm not sure I understand the question fully.) –  Nathaniel Jul 5 '13 at 15:11
    
Essentially, writing down the suitable space. Or giving a description to go from a detailed physical model to one that looks like computational chemistry in action. –  NikolajK Jul 5 '13 at 15:14
    
What's a "suitable space", and in what way does Baez fail to write it down? I'm not saying he doesn't, just trying to understand what you're asking for. –  Nathaniel Jul 5 '13 at 15:17
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In other words, what would you consider to be a suitable space in which to perform such a calculation? Currently your question reads as "I have some very vague ideas about using quantum field theory to solve problems in chemistry", and if you're not willing to be a bit more specific about where your ideas differ from existing approaches I doubt anyone will be able to give you an answer. –  Nathaniel Jul 6 '13 at 5:19

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