In
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 ...
, a quartic interaction is a type of
self-interaction
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
in a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
. Other types of quartic interactions may be found under the topic of
four-fermion interactions. A classical free scalar field
satisfies the
Klein–Gordon equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...
. If a scalar field is denoted
, a quartic interaction is represented by adding a potential energy term
to the
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
. The
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
is
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
in 4-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
This article uses the
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
for
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
.
The Lagrangian for a real scalar field
The
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
scalar field with a quartic interaction is
:
This Lagrangian has a global Z
2 symmetry mapping
.
The Lagrangian for a complex scalar field
The Lagrangian for a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
scalar field can be motivated as follows. For ''two'' scalar fields
and
the Lagrangian has the form
:
which can be written more concisely introducing a complex scalar field
defined as
:
:
Expressed in terms of this complex scalar field, the above Lagrangian becomes
:
which is thus equivalent to the SO(2) model of real scalar fields
, as can be seen by expanding the complex field
in real and imaginary parts.
With
real scalar fields, we can have a
model with a
global
Global means of or referring to a globe and may also refer to:
Entertainment
* ''Global'' (Paul van Dyk album), 2003
* ''Global'' (Bunji Garlin album), 2007
* ''Global'' (Humanoid album), 1989
* ''Global'' (Todd Rundgren album), 2015
* Bruno ...
SO(N)
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
symmetry given by the Lagrangian
:
Expanding the complex field in real and imaginary parts shows that it is equivalent to the SO(2) model of real scalar fields.
In all of the models above, the
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
must be positive, since otherwise the potential would be unbounded below, and there would be no stable vacuum. Also, the
Feynman path integral
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...
discussed below would be ill-defined. In 4 dimensions,
theories have a
Landau pole
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the phy ...
. This means that without a cut-off on the high-energy scale,
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
would render the theory
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
.
The
model belongs to the Griffiths-Simon class, meaning that it can be represented also as the
weak limit
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
of an
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
on a certain type of graph. The triviality of both the
model and the Ising model in
can be shown via a graphical representation known as the random current expansion.
Feynman integral quantization
The
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
expansion may be obtained also from the Feynman
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
. The
time ordered
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter:
:\mathcal P \left\
\equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_).
H ...
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
s of polynomials in φ, known as the ''n''-particle Green's functions, are constructed by integrating over all possible fields, normalized by the
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
with no external fields,
:
All of these Green's functions may be obtained by expanding the exponential in ''J''(''x'')φ(''x'') in the generating function
:
A
Wick rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that s ...
may be applied to make time imaginary. Changing the signature to (++++) then gives a φ
4 statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
integral over a 4-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
,
:
Normally, this is applied to the scattering of particles with fixed momenta, in which case, a
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is useful, giving instead
:
where
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
.
The standard trick to evaluate this
functional integral is to write it as a product of exponential factors, schematically,
:
The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically. The integral with λ = 0 can be treated as a product of infinitely many elementary Gaussian integrals, and the result may be expressed as a sum of
Feynman diagrams
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
, calculated using the following Feynman rules:
* Each field
in the ''n''-point Euclidean Green's function is represented by an external line (half-edge) in the graph, and associated with momentum ''p''.
* Each vertex is represented by a factor ''-λ''.
* At a given order λ
''k'', all diagrams with ''n'' external lines and ''k'' vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a factor 1/(''q''
2 + ''m''
2), where ''q'' is the momentum flowing through that line.
* Any unconstrained momenta are integrated over all values.
* The result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity.
* Do not include graphs containing "vacuum bubbles", connected subgraphs with no external lines.
The last rule takes into account the effect of dividing by