In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, the numerical sign problem is the problem of numerically evaluating the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of a highly
oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of a large number of variables.
Numerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high
precision
Precision, precise or precisely may refer to:
Science, and technology, and mathematics Mathematics and computing (general)
* Accuracy and precision, measurement deviation from true value and its scatter
* Significant figures, the number of digit ...
in order for their difference to be obtained with useful
accuracy
Accuracy and precision are two measures of ''observational error''.
''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
.
The sign problem is one of the major unsolved problems in the physics of
many-particle system
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
s. It often arises in calculations of the properties of a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
system with large number of strongly interacting
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, or in field theories involving a non-zero density of strongly interacting fermions.
Overview
In physics the sign problem is typically (but not exclusively) encountered in calculations of the properties of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions. Because the particles are strongly interacting,
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
is inapplicable, and one is forced to use brute-force numerical methods. Because the particles are fermions, their
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
changes sign when any two fermions are interchanged (due to the anti-symmetry of the wave function, see
Pauli principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
). So unless there are cancellations arising from some symmetry of the system, the quantum-mechanical sum over all multi-particle states involves an integral over a function that is highly oscillatory, hence hard to evaluate numerically, particularly in high dimension. Since the dimension of the integral is given by the number of particles, the sign problem becomes severe in the
thermodynamic limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
. The field-theoretic manifestation of the sign problem is discussed below.
The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas:
*
Condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
— It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the
Hubbard model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each el ...
.
*
Nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
— It prevents the ''
ab initio
''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning").
Etymology
Circa 1600, from Latin, literally "from the beginning", from ab ...
'' calculation of properties of
nuclear matter
Nuclear matter is an idealized system of interacting nucleons ( protons and neutrons) that exists in several phases of exotic matter that, as of yet, are not fully established.
It is ''not'' matter in an atomic nucleus, but a hypothetical s ...
and hence limits our understanding of
nuclei and
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
s.
*
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
— It prevents the use of
lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
to predict the phases and properties of
quark matter
Quark matter or QCD matter (quantum chromodynamics, quantum chromodynamic) refers to any of a number of hypothetical phase (matter), phases of matter whose degrees of freedom (physics and chemistry), degrees of freedom include quarks and gluons, of ...
.
(In
lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly sol ...
, the problem is also known as the complex action problem.)
The sign problem in field theory
In a field-theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
. One evaluates the
partition function by summing over all classical field configurations, weighted by
, where
is the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic fields
(which may have been originally part of the theory, or have been produced by a
Hubbard–Stratonovich transformation The Hubbard–Stratonovich (HS) transformation is an exact mathematical 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 respect ...
to make the fermion action quadratic)
:
where
represents the measure for the sum over all configurations
of the bosonic fields, weighted by
:
where
is now the action of the bosonic fields, and
is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable