Path Integral over Markov Histories

Let us represent the product $ \Pi_N $ of the transitional probabilities of the particular history of the Markov processes as follows (with $ n_\pm \equiv n_\pm(l), \Delta n_\pm = -1 $):

$$\Pi_N = \exp\Bigl( N \Lambda_N\Bigr);$$

$$\Lambda_N = \frac{1}{N}\sum_l G_l;$$

$$G_l = \Delta n_+ \log\!\Bigl(\frac{n_+}{n_+ + n_-}\Bigr) + \Delta n_- \log\!\Bigl(\frac{n_-}{n_+ + n_-}\Bigr);$$

$$n_+ = \sum_{k\le l} \nu_k;$$

$$n_- = \sum_{k\le l} (1-\nu_k);$$

These $ n_\pm $ are net numbers of $ \eta=\pm1 $ in terms of Ising spins or occupation numbers $ \nu_k=(1,0) $ in the Fermi representation. There is an extra constraint on the Markov process:

$$n_+ + n_- = l \quad;\; \forall l$$

which follows from the above definition in terms of the occupation numbers. We can redefine $ n_\pm $ as $ N $ times piecewise constant functions.

$$n_\pm = N f_\pm(\xi);$$

$$f_\pm(\xi) = \sum_{k=1}^{\lfloor N \xi\rfloor}\frac{\nu_k}{N};$$

$$f'_\pm(\xi) = \sum_{l=1}^N\delta\!\Bigl(\xi-\frac{l}{N}\Bigr) \sum_{k=1}^{l}\frac{\nu_k}{N};$$

$$0<\xi<1;$$

The sums can be rewritten as Lebesgue integrals:

$$\Lambda_N = \int_1^0\Bigl( d f_+(\xi)\log\!\Bigl(\frac{f_+(\xi)}{\xi}\Bigr) + d f_-(\xi)\log\!\Bigl(\frac{f_-(\xi)}{\xi}\Bigr) \Bigr)$$

The sum over histories of the Markov process will become a path integral over the difference $ \phi = f_+(\xi)-f_-(\xi) $:

$$\sum_{\eta_.=\pm1}\exp\Bigl(N\Bigl(\Lambda_N+\imath\Lambda_N^{(1)}\Bigr)\Bigr) \to \int D\phi\,\exp\Bigl(N\Bigl(\Lambda_N+\imath\Lambda_N^{(1)}\Bigr)\Bigr)$$

This path integral will be dominated by the “classical history,” maximizing the product of transitional probabilities if such a classical trajectory exists.

The first term (without the circulation term) brings the variational problem:

$$\max_{\phi}\Lambda_N[\phi];$$

$$\Lambda_N[\phi] = \int_1^0 d\xi\,\Biggl( \frac{d f_+}{d\xi}\log\!\Bigl(\frac{f_+}{\xi}\Bigr) + \frac{d f_-}{d\xi}\log\!\Bigl(\frac{f_-}{\xi}\Bigr) \Biggr);$$

$$f_\pm(\xi)=\frac{1}{2}\Bigl(\xi\pm\phi(\xi)\Bigr);$$

$$f_\pm(\xi)\ge0;$$

This problem is, however, a degenerate one, as the functional reduces to the integral of a total derivative:

$$\frac{\delta\Lambda_N[\phi]}{\delta\phi(\xi)}=0;$$

$$ \begin{aligned} \Lambda_N[\phi] &= \int_1^0 d\Bigl(f_+\log f_+ + f_-\log f_-\Bigr) + 1 + \int_0^1 d\xi\,\log\xi\\[1ex] &= -\frac{1}{2}(1-s)\log(1-s) -\frac{1}{2}(1+s)\log(1+s)+\log(2); \end{aligned} $$

It depends on the boundary condition $ \phi(0)=0,\; \phi(1)=s $ but not on the shape of $ \phi(\xi) $. This expression matches the Stirling formula for the logarithm of the binomial coefficient in the combinatorial solution [1] for the Euler ensemble:

$$ \lim_{N\to\infty}\frac{\log\binom{N}{N(1+s)/2}}{N} = \log(2)-\frac{1}{2}(1-s)\log(1-s)-\frac{1}{2}(1+s)\log(1+s); $$

This $ \Lambda(s)=\Lambda_\infty(s) $ is a smooth even function of $ s $ taking positive values from $ \Lambda(\pm1)=0 $ to the maximal value $ \Lambda(0)=\log(2) $.

Now, let us add the circulation term to the exponential of the partition function (see the trace formula). This term can be directly expressed in terms of the difference between our two densities $ N\phi(\xi)=Nf_+(\xi)-Nf_-(\xi) $:

$$\imath N\Lambda^{(1)}_N[\phi,C_\Omega] = \frac{\imath}{\sqrt{2\nu t}} \int_0^1 d\vec{C}_\Omega(\xi)\cdot\vec{F}(\xi);$$

$$\vec{F}(\xi)=\frac{\Bigl\{\sin(\beta N\phi(\xi)),\, \cos(\beta N\phi(\xi)),\,0\Bigr\}}{2\sin(\beta/2)};$$

$$\vec{C}_\Omega(\theta)=\hat{\Omega}\cdot\vec{C}(\theta);$$

We remember that the last component of the vector $ \vec{F}(\xi) $ does not contribute to the circulation integral in (Λ1) with the closed loop $ \vec{C}_\Omega(\xi) $. This is why we replaced it with zero. The key assumption is the existence of the smooth limit of the charge density $ \phi(\xi) $ of these Fermions when they densely cover this loop.

We are working with $ \alpha(\xi)=\beta N\phi(\xi) $ in the following.

The measure for paths $ [D\alpha] $ is undetermined. The derivatives of these alphas were quantized in the original Fermi theory: each step $ \alpha'(\xi)\approx N\Delta\alpha=N\beta\sigma=\pm N\beta $.

As we demonstrate below, in continuum theory, this discrete distribution can be replaced by a Gaussian distribution with the same mean square:

$$\sum_{\alpha'=\pm N\beta}\leftrightarrow\int d\alpha' \exp\Bigl(-\frac{(\alpha')^2}{2N^2\beta^2}\Bigr);$$

To demonstrate that, we consider in the critical region $ \beta^2\sim N^{-1}\to0 $ the most general term that arises in the moments of the circulation in (Λ1):

$$2^{-N}\sum_{\sigma_i=\pm1}\exp\Bigl(\imath\beta\sum_i k_i\sigma_i\Bigr) = \prod_i\cos(\beta k_i) \to \exp\Bigl(-\frac{\beta^2}{2}\sum_i k_i^2\Bigr);$$

where $ k_i $ are some integers. With a large number $ N $ of these integers, the sum in the exponential becomes an integral, equivalent to a Gaussian integral:

$$ \begin{aligned} \exp\Bigl(-\frac{\beta^2}{2}\sum_i k_i^2\Bigr) &= \prod_i\int_{-\infty}^\infty\frac{d\sigma_i}{\sqrt{2\pi}} \exp\Bigl(-\frac{\sigma_i^2}{2}\Bigr) \exp\Bigl(\imath\beta k_i\sigma_i\Bigr)\\[1ex] &\to \exp\Bigl(-\frac{N\beta^2}{2}\int_0^1 d\xi\, k(\xi)^2\Bigr); \end{aligned} $$

We arrive at the standard path integral measure:

$$\int [D\alpha]=\int D\alpha(\xi) \exp\Bigl(-\int_0^1d\xi\,\frac{(\alpha')^2}{2N\beta^2}\Bigr);$$

$$ \frac{\int [D\alpha]\exp\Bigl(\imath N\int_0^1d\xi\,\alpha(\xi)K(\xi)\Bigr)} {\int [D\alpha]} = \exp\Bigl(-\frac{N^2}{2}\iint d\xi_1 d\xi_2\, K(\xi_1)K(\xi_2)G_{1,2}\Bigr); $$

$$G_{1,2} = \VEV{\alpha(\xi_1)\alpha(\xi_2)};$$

The next Appendix will compute this Green’s function $ G_{1,2}=G(\xi_1,\xi_2) $.

Thus, we arrive at the following path integral in the continuum limit:

(a) Ψ[C] =

∑_{p<q; (p,q)} ∫_{Ω ∈ O(3)} dΩ ∫ [Dα] ∑_{p<q; (p,q)} |O(3)| ∫ [Dα]

× exp {

i ∫₀¹ dξ ℑ(C′_Ω(ξ) exp{iα(ξ)}) 2 sin(πp/q) √(2ν(t + t₀))

}

$$\textbf{(b)}\quad \mathcal C_\Omega(\theta)=\vec{C}(\theta)\cdot\hat{\Omega}\cdot\{\imath,1,0\};$$

We get the $ U(1) $ statistical model with the boundary condition $ \alpha(1)=\alpha(0)+\beta N s $. The period $ \beta N s=2\pi p r $ is a multiple of $ 2\pi $, which is irrelevant at $ N\to\infty $. The effective potential for this theory is a linear function of the loop slope $ \vec{C}'(\xi) $.

This model is yet another representation of the Euler ensemble, suitable for the continuum limit.

References

Alexander Migdal, “To the Theory of Decaying Turbulence”, Fractal and Fractional, vol. 7, no. 10, pp. 754, Oct 2023. DOI: 10.3390/fractalfract7100754 .