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Approximation methods in physics: foundational, philosophical and historical questions

James D. Fraser, Pablo Ruiz de Olano, Alexander S. Blum

Published Feb 17, 2026• 0 citations
Publisher

Abstract

When attempting to describe the world using mathematical models, we almost inevitably encounter a problem of tractability: the equations that most faithfully describe the real-world system of interest are too complex to be solved exactly. There are different ways to respond to this situation. One approach is to move to a simpler set of equations; if all goes well, we may hit upon a more tractable representation of the target system which is nevertheless predictive, explanatory or otherwise useful. Another approach is to retain the more complex equations but broaden the range of methods used to draw inferences from them. Applied mathematicians have developed a dizzying array of approximation methods, which allow quantitative results to be obtained in the absence of exact solutions. In the context of quantum physics alone, the list of approximation methods is long and ever-expanding: perturbative expansions, WKB semi-classical expansions, Hartree-Fock variational approximations, Born-Oppenheimer approximations, numerical integration methods, Monte Carlo simulations, and the list goes on.

DOI
10.1007/s11229-026-05447-7

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When attempting to describe the world using mathematical models, we almost inevitably encounter a problem of tractability: the equations that most faithfully describe the real-world system of interest are too complex to be solved exactly. There are different ways to respond to this situation. One approach is to move to a simpler set of equations; if all goes well, we may hit upon a more tractable representation of the target system which is nevertheless predictive, explanatory or otherwise useful. Another approach is to retain the more complex equations but broaden the range of methods used to draw inferences from them. Applied mathematicians have developed a dizzying array of approximation methods, which allow quantitative results to be obtained in the absence of exact solutions. In the context of quantum physics alone, the list of approximation methods is long and ever-expanding: perturbative expansions, WKB semi-classical expansions, Hartree-Fock variational approximations, Born-Oppenheimer approximations, numerical integration methods, Monte Carlo simulations, and the list goes on.
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The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
In this thesis, an approximate method introduced to solve time-dependent many-body problems known as time-dependent many-body perturbation theory is studied. Many-body perturbation theory for interacting electrons and phonons is reviewed. In particular, the electron propagator G and an unconventional two-component phonon propagator, which satisfy coupled integral Dyson equations, are introduced. In practice, the associated integral kernels known as the electron Σ and phonon self-energies need to be approximated. The conserving approximations known as the Hartree (-Fock) and the first and second Born approximations, which respect the continuity equation between the electron density and current, are considered in this work. Time-dependent systems of interest are studied in this thesis by using the integro-differential forms of the Dyson equations referred to as the Kadanoff-Baym Equations (KBE). The Kadanoff-Baym equations are introduced for the electron and phonon propagators unconventionally as coupled firstorder integro-differential equations. It is reviewed how these equations are solved numerically by describing an integration rule, a class of practical methods and a parallel implementation of the numerical method. In addition, documentation of how the Kadanoff-Baym equations allow to solve the Bethe-Salpeter Equation (BSE) with the kernel δΣ/δG for the density response function, is provided. In two of the enclosed publications, we benchmarked observables obtained by using the Hartree and partially and fully self-consistent Born approximations against numerically exact results for the two- site, two-electron Holstein model. In this model, the two electrons which are constrained to move between two lattice sites interact indirectly via the electron-phonon coupling. It was observed that only the fully self-consistent Born approximation could cope qualitatively correctly with the competition between the delocalizing and localizing effects of the kinetic and interaction energies. For the other two approximations, spurious bifurcative symmetry breaking with an associated unbounded density response was observed. Nevertheless, also the self-consistent Born approximation was concluded to fail in describing the bipolaronic behavior of the true system. In the third publication, we benchmarked the Hartree-Fock and second Born approximations against an exact method for the few-site Hubbard and Pariser-Parr-Pople models in which the underlying lattice is inert and the electrons interact amongst themselves directly. It was found that the second Born approximation is capable of describing the so-called correlation induced doubly-excited states. This is not possible for time-local approximations such as Hartree-Fock. In addition to the qualitative results, which highlight successes of the applied simple self-energy approximations, the approximate and exact results were also compared on a more quantitative level. It is the quantitative and qualitative results combined which are used in this thesis to assess the quality of the many-body approximations used, with the aim to better understand when these methods are predictive.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.
In quantum mechanics, there exist a lot of problems from the domain of atomic, molecular, solid-state physics, physics of magnetic phenomena, nuclear physics, etc. The number of problems which may be solved exactly are few such as some one-dimensional problems, problems in the rectangular well (barrier), the oscillator, problems of motion in a central field such as motion in a Coulomb field. These methods are well formulated and are used in evaluations to a greater approximation. However, in some cases, these may create problems that are very difficult, if not impossible, to be solved exactly. In such problems, the Hamiltonian cannot be exactly diagonalized in a simple manner. For such problems, we resort to approximate methods for finding the eigenvalues and eigenfunctions of the Hamiltonian. Besides, these approximate methods yield results closer to experimental ones. Amongst the approximate methods, we examine the variational method.