An explicit formula for perturbation theory at any order with infinitely many perturbations

This paper introduces a systematic, single matrix equation based on integer partitions that explicitly generates high-order perturbation theory corrections for both eigenvalues and eigenvectors, accommodating infinitely many perturbations while simplifying traditionally tedious derivations.

The Gist

Imagine you are trying to predict the weather. You have a perfect model for a calm, sunny day (the "unperturbed" state). But in reality, the wind is blowing, clouds are forming, and a storm is brewing. These are the "perturbations"—the little things that mess up your perfect model.

For decades, physicists have used a method called Perturbation Theory to fix their models. They start with the perfect sunny day and add corrections: "Okay, add a little wind," then "add a little more wind," then "add a storm."

The problem? As you try to predict the weather further into the future (higher orders), the math becomes a nightmare. The equations get so long and tangled that even supercomputers struggle, and scientists usually stop after just a few steps because it's too hard to write down the next step.

This paper is like discovering a master key that unlocks the door to any level of complexity, instantly.

Here is the breakdown of what the authors, Joseph Jones and M. W. Long, have achieved, using simple analogies:

1. The Infinite Buffet of Problems

Usually, physicists only deal with one problem at a time (e.g., just the wind). But in the real world, you often have an infinite buffet of problems happening at once (wind, rain, temperature shifts, humidity, etc.).

• The Old Way: Trying to calculate the weather with an infinite buffet meant you had to write a new, massive, unique recipe for every single dish. It was impossible to keep track of.
• The New Way: The authors found a way to handle infinite perturbations all at once. They didn't just solve for one storm; they solved for every possible storm simultaneously.

2. The "Integer Partition" Recipe Book

The core of their discovery is a mathematical concept called Integer Partitions.

Imagine you have a number, say 4. You want to break it down into smaller numbers that add up to 4.

• 4
• 3 + 1
• 1 + 3
• 2 + 2
• 2 + 1 + 1
• 1 + 2 + 1
• 1 + 1 + 2
• 1 + 1 + 1 + 1

In the old days, figuring out how these combinations interacted in physics was like trying to solve a jigsaw puzzle where the pieces kept changing shape. The authors realized that every single possible way to break down a number (a partition) corresponds to a specific step in the physics calculation.

They created a systematic formula (a recipe) that says: "To get the answer for step 4, just look at all the ways you can break the number 4 apart, and plug those into this specific matrix equation."

3. The "One-Stop Shop" Equation

Traditionally, calculating the energy of a system and the shape of the wave (the eigenvector) required two separate, tedious, and different sets of calculations.

The authors found a single matrix equation that does both jobs at once.

• Analogy: Imagine you are building a house. Usually, you need one blueprint for the foundation (energy) and a completely different one for the walls (shape).
• The Innovation: This paper gives you one single blueprint that automatically tells you both the foundation and the walls. If you know the foundation, the walls are already there in the same equation.

4. Why This Matters (The "Magic" of the Formula)

The most exciting part is that this formula is explicit.

• Before: To get the 10th step of the calculation, you had to manually derive it from the 9th, then the 8th, and so on. It was like climbing a ladder where you had to build each rung as you went up.
• Now: You can jump straight to the 10th, 50th, or 100th step. You just plug the number into their formula, and it spits out the answer. It turns a months-long algebraic struggle into a simple computer script.

5. The Real-World Connection: The "Baker-Campbell-Hausdorff" Cake

The paper mentions a specific, complex mathematical cake called the Baker-Campbell-Hausdorff (BCH) formula. This formula is used in quantum physics to mix two things that don't play nicely together (non-commuting variables).

Usually, this cake is so complex it's cut into tiny, messy slices. The authors realized that this cake is actually made of an infinite series of layers. Their new formula is the perfect knife to slice through all those infinite layers at once, making it much easier to study things like phase transitions in materials or complex quantum systems.

Summary

Think of this paper as the GPS for the mathematical wilderness of quantum physics.

• Before: You were walking blind, trying to map the terrain step-by-step, getting lost in the algebraic jungle after a few miles.
• Now: You have a satellite map. You can see the entire terrain (infinite perturbations) at once. You can instantly calculate the path to any destination (any order of correction) without getting lost in the weeds.

It takes a process that was traditionally "tedious and messy" and turns it into a "clean, algorithmic, and ready-to-use" tool for scientists to solve the hardest problems in physics.

Technical Summary

1. Problem Statement

Perturbation theory is a fundamental tool in theoretical physics for finding approximate solutions to problems where exact analytical results are inaccessible. However, traditional applications are often limited to low orders (typically second order) due to the algebraic complexity and tediousness of deriving higher-order corrections.

• The Challenge: Existing explicit formulas for arbitrary orders are rare. While methods like Rayleigh-Schrödinger (RSPT) and Brillouin-Wigner (BWPT) exist, they become increasingly cumbersome at high orders.
• The Gap: Previous formulations generally assume a single perturbing Hamiltonian. There is a lack of systematic, closed-form expressions that can handle an infinite set of perturbations (e.g., arising from power series expansions like the Baker-Campbell-Hausdorff formula) while simultaneously providing corrections for both eigenvalues and eigenvectors.

2. Methodology

The authors develop a systematic approach based on ordered integer partitions (also known as compositions) to generate perturbation theory corrections at any arbitrary order $N$.

Key Definitions and Framework

• Generating Functions: The problem is framed using generating functions for the Hamiltonian $H(z)$, energy $E(z)$, and eigenvector $\psi(z)$, expanded in powers of a parameter $z$.
• The Central Quantity ($\Delta H(n)$): The authors define a modified perturbation operator:
  $$\Delta H(n) \equiv H(n) - E(n)Q$$
  where $H(n)$ is the $n$-th order perturbation, $E(n)$ is the energy correction, and $Q = 1 - |0\rangle\langle 0|$ is the projector onto the subspace orthogonal to the unperturbed ground state $|0\rangle$. This definition unifies the energy and eigenvector corrections into a single matrix quantity.
• The Denominator Matrix ($\Gamma$): Defined as $\Gamma \equiv Q(\epsilon_0 - H_0)^{-1}$, representing the resolvent operator projected onto the orthogonal subspace.
• Recursive Relation: The authors derive a recurrence relation for a matrix quantity $M(n)$:
  $$M(n) = \Delta H(n) + \sum_{n'=1}^{n-1} \Delta H(n') \Gamma M(n-n')$$
  In generating function form: $M(z) = \Delta H(z) + \Delta H(z)\Gamma M(z)$.

The Solution via Integer Partitions

By solving the recurrence relation, the authors express $M(N)$ explicitly as a sum over all ordered partitions of the integer $N$. If $N$ is partitioned into $p$ integers $(n_1, n_2, \dots, n_p)$ such that $\sum n_i = N$, the term corresponds to a product of perturbations separated by $\Gamma$ operators.

3. Key Contributions

1. Generalization to Infinite Perturbations:
   The formulation naturally accommodates an infinite number of perturbing Hamiltonians ($H(1), H(2), \dots$). This is crucial for physical scenarios involving power series expansions (e.g., the Baker-Campbell-Hausdorff formula) where the perturbation is inherently an infinite series.

2. Unified Matrix Equation:
   The method provides a single matrix equation ($M(N)$) from which both the eigenvalue correction $E(N)$ and the eigenvector correction $|\psi(N)\rangle$ are derived:

   • Eigenvalue: $E(N) = \langle 0 | M(N) | 0 \rangle$
   • Eigenvector: $|\psi(N)\rangle = \Gamma M(N) |0\rangle$
     This eliminates the need for separate, disjointed derivations for energy and state corrections.

3. Explicit Closed-Form Formula:
   The paper provides an explicit formula for $M(N)$ as a sum over ordered partitions:
   $$M(N) = \sum_{p=1}^{N} \sum_{n_1+\dots+n_p=N} \Delta H(n_1) \Gamma \Delta H(n_2) \dots \Gamma \Delta H(n_p)$$
   This replaces the recursive, step-by-step derivation with a direct algorithmic generation of terms.

4. Algorithmic Tractability:
   The authors provide Mathematica code to generate these terms automatically. They demonstrate the expansion up to the 8th order for infinite perturbations and the 10th order for a single perturbation, showing the method's scalability.

4. Results

• High-Order Formulas: The paper explicitly lists the terms for $M(N)$ up to $N=4$ in the main text and up to $N=8$ (infinite perturbations) and $N=10$ (single perturbation) in the Appendix.
  • Example (4th Order): $M(4)$ consists of 8 terms corresponding to the 8 ordered partitions of 4: $(4), (1,3), (2,2), (3,1), (1,1,2), (1,2,1), (2,1,1), (1,1,1,1)$.
• Recovery of Standard Theory: In the limit of a single perturbation (where $H(n)=0$ for $n>1$), the formula reduces exactly to standard Rayleigh-Schrödinger perturbation theory. The authors show how their compact notation maps to the familiar, lengthy expressions involving sums over intermediate states.
• Non-Self-Consistent Nature: Unlike Brillouin-Wigner theory, which requires solving a self-consistent equation, this RSPT-based approach is strictly iterative. The correction at order $N$ depends only on previously calculated lower-order terms, avoiding numerical convergence issues associated with self-consistency.

5. Significance

• Streamlining Derivations: The primary significance is the drastic simplification of high-order perturbation theory. What traditionally requires tedious, error-prone algebraic manipulation is reduced to a combinatorial problem of partitioning integers.
• Physical Applicability: The ability to handle infinite perturbations makes this formalism directly applicable to advanced problems in condensed matter physics and quantum chemistry, particularly those involving the Baker-Campbell-Hausdorff expansion or transfer matrix methods in statistical mechanics.
• Computational Efficiency: By providing a direct "sum-over-partitions" formula, the method facilitates the development of computer algorithms for high-order calculations, which are increasingly relevant in modern many-body physics where high precision is required.
• Conceptual Clarity: The use of the $\Delta H(n)$ quantity effectively separates "connected" contributions (which survive projection) from "disconnected" ones, offering a clearer physical interpretation of the perturbative hierarchy.

In summary, Jones and Long provide a powerful, generalizable framework that transforms perturbation theory from a case-by-case algebraic exercise into a systematic, algorithmic procedure capable of handling complex, multi-order, and infinite-perturbation scenarios.