Multi-Saddle Sums and Dilute Ensembles

Widely separated localized saddle events can be summed as a dilute ensemble when their core size is much smaller than their mean separation and their residual interactions give parametrically smaller cluster corrections. The collective-coordinate integrals then generate an exponential or a transfer matrix. The approximation is not a license to exponentiate arbitrary multi-saddle configurations: sector constraints, event ordering, and quasi-zero-mode interactions determine the sum.

Required background. Zero modes, collective coordinates, and moduli measures supplies the one-event center measure and the quotient by permutations.

Helpful background. Connected, disconnected, and vacuum diagrams supplies the combinatorial reason connected clusters exponentiate.

The dilute-event expansion

Let an isolated event have core size $\xi$, reduced action $\mathcal S_I$, and one-loop fugacity

$$\kappa =J_I \left(\frac{\det{}'M_I}{\det M_0}\right)^{-1/2} e^{-\mathcal S_I/g},$$

where $J_I$ is the collective-coordinate Jacobian per unit Euclidean time. The dimension of $\kappa$ is inverse time. If $n$ identical, noninteracting events have centers in an interval of length $T$, then

$$\int_{0<\tau_1<\cdots<\tau_n<T} \prod_{i=1}^{n}\mathrm d\tau_i =\frac{T^n}{n!}.$$

Consequently,

$$\sum_{n=0}^{\infty}\frac{(\kappa T)^n}{n!} =e^{\kappa T}.$$

This exponential is a combinatorial result, not an extra semiclassical assumption. The physical assumptions entered earlier:

$$\kappa\xi\ll1, \qquad \frac{|S_{\rm int}(R)|}{g}\ll1 \quad\text{at typical }R\sim\kappa^{-1},$$

and the nonzero fluctuation spectrum must remain gapped after the event centers are projected. Since $\kappa\xi$ is the expected fraction of the interval occupied by cores, it is the direct diluteness parameter.

For several event species $a$ with fugacities $\kappa_a$, an unconstrained gas gives

$$Z_{\rm gas} =\exp\!\left(T\sum_a\kappa_a\right).$$

Topological charge, endpoint vacua, fermion zero-mode saturation, or Gauss-law constraints can restrict the allowed sequences. One must impose these constraints before summing. Coleman 1985, ch. 7, §2.3, pp. 276–280 gives the classic ordered-center derivation; Mariño 2015, §1.9, pp. 42–53 develops the corrections due to correlated instanton–anti-instanton configurations.

Double-well instanton gas as a transfer matrix

In the symmetric double well, label localized perturbative vacua by $|L\rangle$ and $|R\rangle$. An instanton flips $L\to R$ and an anti-instanton flips $R\to L$. Events must therefore alternate. If their magnitudes are equal to $\kappa$, the leading Euclidean evolution in this two-state subspace is

$$U(T) \simeq e^{-E_{\rm pert}T} \sum_{n=0}^{\infty} \frac{(\kappa T)^n}{n!}\,\sigma_x^n =e^{-E_{\rm pert}T}e^{\kappa T\sigma_x}.$$

Since $\sigma_x^{2m}=1$ and $\sigma_x^{2m+1}=\sigma_x$,

$$\begin{aligned} \langle L|U(T)|L\rangle &\simeq e^{-E_{\rm pert}T}\cosh(\kappa T),\\ \langle R|U(T)|L\rangle &\simeq e^{-E_{\rm pert}T}\sinh(\kappa T). \end{aligned}$$

Diagonalizing in parity eigenstates $|\pm\rangle=(|L\rangle\pm|R\rangle)/\sqrt2$ gives

$$E_\pm=E_{\rm pert}\mp\kappa, \qquad \Delta E=E_--E_+=2\kappa.$$

This is the first physical payoff of the multi-saddle sum: a contribution invisible at every finite order in $g$ produces a real level splitting.

For the normalization used earlier in this chapter,

$$\mathcal S_I=\frac43, \qquad \frac{\det{}'M_I}{\det M_0}=\frac1{48}, \qquad J_I=\left(\frac{\mathcal S_I}{2\pi g}\right)^{1/2}.$$

Hence

$$\kappa_{\rm 1\,loop} =4\sqrt{\frac{2}{\pi g}}\, e^{-4/(3g)}$$

in dimensionless Euclidean time, and the leading splitting is $2\kappa_{\rm 1\,loop}$. This numerical prefactor changes under a rescaling of time or potential, whereas the factorization into action, Jacobian, and determinant does not.

Interactions and connected clusters

Real multi-event configurations are not exactly additive at finite separation. Write

$$\mathcal S_n(\tau_1,\ldots,\tau_n) =n\mathcal S_I +\sum_{i<j}V_{a_i a_j}(\tau_i-\tau_j) +\cdots.$$

For a one-component gas, define the Mayer function

$$f(R)=e^{-V(R)/g}-1.$$

The first connected correction to the logarithm of the partition function is proportional to

$$b_2 =\frac12\int_{-\infty}^{\infty}\mathrm dR\,f(R), \qquad \frac1T\log Z =\kappa+\kappa^2 b_2+\cdots.$$

Thus the condition for naive exponentiation is not merely $e^{-\mathcal S_I/g}\ll1$; it also requires $|\kappa b_2|\ll1$. A long-range tail or an attractive region can invalidate that condition.

Instanton–anti-instanton separation is often a quasi-zero mode rather than an exact modulus. At small separation the pair can approach the perturbative vacuum, so the separation integral may be ambiguous on the naive real contour. Analytic continuation of the interaction and a matched lateral prescription relate this ambiguity to the large-order ambiguity of perturbation theory. Bogomolny 1980, pp. 431–435 and Zinn-Justin 1981, pp. 125–140 give the original correlated-pair prescription.

This interaction problem marks a boundary between the elementary dilute gas and a resurgent transseries. Treating the pair as two independent exact saddles misses both the quasi-zero-mode integral and its contour dependence.

When the gas approximation is controlled

A dilute ensemble should pass all of the following tests:

• Core separation: the distribution of nearest-neighbor separations is concentrated at $R\gg\xi$.
• Cluster convergence: connected coefficients satisfy $|\kappa^{n-1}b_n|\ll1$ through the retained order.
• Mode separation: the only parametrically soft directions are the collective or quasi-collective coordinates treated explicitly.
• Sector constraints: endpoint, charge, and insertion selection rules are imposed before the sum.
• Volume order: the thermodynamic or long-time limit is taken only after the intensive logarithm has been formed.

The approximation fails near high event density, near saddle coalescence, when a massless field mediates a nonintegrable interaction, or when renormalization drives the coupling strong at the event size. In those regimes, a correlated molecule expansion, exact moduli-space integral, effective field theory of the ensemble, or numerical method may be needed.

Shared calculation. The saddle-contribution anatomy shows which one-event factors are exponentiated. Shared comparison. The canonical saddle comparison states the dilute-overlap boundary explicitly.

Common pitfalls

Exponentiating without endpoint constraints. In the double well, instantons and anti-instantons alternate. Independent Poisson sums would include impossible paths and give the wrong matrix element.

Calling a separated superposition an exact saddle. Its residual field equation is exponentially small, not zero. That residual generates the quasi-zero-mode interaction and must be retained at the accuracy where it matters.

Using the one-event action as the only control test. A large $\mathcal S_I/g$ suppresses fugacity but does not bound a long-range or attractive pair interaction. Check cluster coefficients or the separation distribution.

Exercises

1. Derive the $\cosh$ and $\sinh$ sums for the double-well kernels.

Solution

Returning to the same well requires an even number of flips:

$$\langle L|U|L\rangle \simeq e^{-E_{\rm pert}T} \sum_{m=0}^{\infty}\frac{(\kappa T)^{2m}}{(2m)!} =e^{-E_{\rm pert}T}\cosh(\kappa T).$$

Ending in the other well requires an odd number:

$$\langle R|U|L\rangle \simeq e^{-E_{\rm pert}T} \sum_{m=0}^{\infty}\frac{(\kappa T)^{2m+1}}{(2m+1)!} =e^{-E_{\rm pert}T}\sinh(\kappa T).$$

2. Show that the parity eigenvalues of $U(T)$ give a splitting $2\kappa$.

Solution

Since $\sigma_x|\pm\rangle=\pm|\pm\rangle$,

$$U(T)|\pm\rangle =e^{-(E_{\rm pert}\mp\kappa)T}|\pm\rangle.$$

Thus $E_+=E_{\rm pert}-\kappa$ and $E_-=E_{\rm pert}+\kappa$, so $E_--E_+=2\kappa$.

3. Let $V(R)=A e^{-m|R|}$ with $|A|/g\ll1$. Estimate $b_2$ to first order in $A/g$.

Solution

Expand $f(R)=e^{-V/g}-1=-V/g+O(A^2/g^2)$. Then

$$b_2 \simeq-\frac1{2g}\int_{-\infty}^{\infty} \mathrm dR\,A e^{-m|R|} =-\frac{A}{gm}.$$

The leading independent-event sum requires $|\kappa A/(gm)|\ll1$ in addition to $\kappa\xi\ll1$.

References

• Bogomolny, E. B. “Calculation of Instanton–Anti-Instanton Contributions in Quantum Mechanics.” Physics Letters B 91 (1980): 431–435. DOI.
• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, pp. 265–350. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, pp. 3–61. DOI.
• Zinn-Justin, Jean. “Multi-Instanton Contributions in Quantum Mechanics.” Nuclear Physics B 192 (1981): 125–140. DOI.