The Hubbard–Stratonovich (HS) transformation is an exact

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respective field theory by linearizing the

density operator In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while thos ...

in the many-body interaction term of the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

and introducing an auxiliary scalar field. It is defined via the integral identity :

\exp \left( - \frac x^2 \right) =
\sqrt \; \int_^\infty
\exp \left( - \frac - i x y \right) \, dy,

where the real constant

a > 0

. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in

polymer physics Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation of polymers and polymerisation of monomers.P. Flory, ''Principles of Polymer Che ...

, classical particle physics, spin glass theory, and electronic structure theory.

Calculation of resulting field theories

The resulting field theories are well-suited for the application of effective approximation techniques, like the mean field approximation. A major difficulty arising in the simulation with such field theories is their highly oscillatory nature in case of strong interactions, which leads to the well-known

numerical sign problem In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and n ...

. The problem originates from the repulsive part of the interaction potential, which implicates the introduction of the complex factor via the HS transformation.

References

Functions and mappings Transforms {{CMP-stub