# Path integrals, Ensembles, and Information theory in QFT and statistical mechanics (POV of a mathematician)

I apologize in advance for any stupid/wrong/enraging remarks in the question. I am a mathematician and not a physicist. Consequently this question was written entirely from the perspective of a mathematician.

The theme of this question is the attempt to provide a certain philosophical ground for path integrals based on information theoretic considerations.

Lets consider a classical dynamical system with a finite number of degrees of freedom (like a an ideal gas). Then the famous derivation of the ideal gas law from the principle of maximal entropy actually derives a measure on the space of states which describes the canonical ensemble. It looks something like $$e^{-\beta H}$$. All we needed to assume was

1. The average energy of the system is constant
2. The entropy is as large as it can be subject to $$(1)$$

Moving one step further we can consider finite dimensional dynamical systems (by which I mean closed symplectic manifolds equipped with a hamiltonian function). Statistical states are measures (or probability distributions) on the manifold and observables are functions on the manifold. Integrating an observable against a measure gives the average/expectation value of the observable in the given state.

In other words to come up with some prediction we need to specify 2 things, the observable we wish to measure and the (statistical) state in which we want to measure it in. We would like to do the same as above to get some "canonical ensemble" but now our measures are continuous and it looks a bit tricky. Here comes the first question:

Question 1: Is there a derivation of something like a continuous Gibbs measure (for the "continuous canonical ensemble") from the two assumptions I mentioned? (fixed average energy and maximum entropy).

Lets consider now statistical field theory. By analogy with the finite dimensional statistical mechanics we can say that the Gibbs measure for the grand canonical ensemble is still this $$e^{-\beta H}$$ (times a non-existent Lebesgue measure on the infinite dimensional space of fields) and we can try to justify this by a variational argument. Here comes the second question:

Question 2: Is there some characterization (at physics level of rigor) of this $$e^{-\beta H}$$ term in path integrals in statistical field theory coming from considerations of (Fixed average Energy)+(Maximum Entropy)?

Lets consider now (perturbative) relativistic quantum field theory. We are not in statistical mechanics anymore but the path integral formulation of the theory makes it look suspiciously familiar. The "Feynman measure" term $$e^{\frac{i}{\hbar} S}$$ resembles the Gibbs measure and seems to carry the role of "Macrostate of the universe". The $$h$$ is like temperature while $$S$$ is like energy (I have no intuition for why the $$i$$ is stuck there apart from "quantum mechanics is $$\mathbb{C}$$omplex").

The similarities with the previous setups leads me to wonder whether one could arrive at the "Feynman measure" from purely information theoretic principles similar to the previous cases. In other words:

Question 3 (Main): Is the "Feynman measure" $$e^{\frac{i}{\hbar} S}D\phi$$ uniquely determined by:

1. Principle of maximal entropy

2. Principle of least action

At this point I have to admit (as a mathematician) I have a bit of a problem with the principle of least action (as a conceptual ground) as it feels a bit like a tautology. Axiom (1) can be put on a very solid philosophical basis as an information theoretic assumption of maximum ignorance. The only justification I can come up with for $$(2)$$ is that it comes from classical Lagrangian mechanics which is a theory that works very well.

• I am unsure what you are asking in Q1: Are you looking for a derivation of the canonical probability $p = e^{-\beta H}/Z$? – Themis Sep 1 '19 at 20:55
• @Themis Yes that's basically it. – Saal Hardali Sep 2 '19 at 3:29

## 1 Answer

Canonical probability

We take the microstate of a system to be a random variable $$i=1,2\cdots$$ and write the mean energy as $$\tag{1} \bar E = \sum_i E_i p_i,$$ where $$p_i$$ is the unknown probability of microstate. We seek the probability distribution $$\{p_i\}$$ that maximizes the entropy functional $$\tag{2} S[p_i] = -\sum_i p_i \log p_i$$ given Eq. (1) and the normalization requireemnt $$\tag{2} \sum_i p_i = 1 .$$ Using the method of Lagrange multiplies we maximize $$-\sum_i p_i \log p_i - \lambda_0\left(\sum_i p_i - 1\right) - \lambda_1\left(\sum_i E_i p_i - \bar E\right)$$ with respect to $$p_i$$ at fixed $$\bar E$$. The solution is $$p_i^* = e^{-1-\lambda_0 - \lambda_1 E_i}$$ With $$\lambda_1=\beta$$, $$e^{1+\lambda_0}=Z$$ the result is written in the usual form $$\boxed{ p_i = \frac{e^{-\beta E_i}}{Z} }$$

Continuous case

The are is no fundamental difficulty in generalizing this result to continuous distributions. The problem is to maximize the continuous entropy functional $$S[p] = -\int p(x) \log p(x) dx$$ under the constraints $$\int p(x) dx = 1,\quad \int E(x)\, p(x)\, dx = \bar E.$$ where $$E(x)$$ is some function of $$x$$. By the same procedure using Lagrange multipliers the solution is $$\boxed{ p^*(x) = \frac{e^{-\beta E(x)}}{Z} }$$

Variable $$x$$ ay be a multidimensional state vector. In the case of a classical molecular system $$\Gamma = (\mathbf r_1\cdots; \mathbf q_1\cdots)$$, where $$\mathbf r_i$$ is the position and $$\mathbf q_i$$ is the momentum of particle $$i$$.

A useful reference is Jaynes, Information Theory and Statistical Mechanics.