Common integrals in quantum field theory are all variations and generalizations of
Gaussian integrals to the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
and to multiple dimensions. Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.
Variations on a simple Gaussian integral
Gaussian integral
The first integral, with broad application outside of quantum field theory, is the Gaussian integral.
In physics the factor of 1/2 in the argument of the exponential is common.
Note:
Thus we obtain
Slight generalization of the Gaussian integral
where we have scaled
Integrals of exponents and even powers of ''x''
and
In general
Note that the integrals of exponents and odd powers of x are 0, due to
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
symmetry.
Integrals with a linear term in the argument of the exponent
This integral can be performed by completing the square:
Therefore:
Integrals with an imaginary linear term in the argument of the exponent
The integral
is proportional to the
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 ...
of the Gaussian where is the
conjugate variable
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation— ...
of .
By again completing the square we see that the Fourier transform of a Gaussian is also a Gaussian, but in the conjugate variable. The larger is, the narrower the Gaussian in and the wider the Gaussian in . This is a demonstration of the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
.
This integral is also known as the
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 ...
used in field theory.
Integrals with a complex argument of the exponent
The integral of interest is (for an example of an application see
)
We now assume that and may be complex.
Completing the square
By analogy with the previous integrals
This result is valid as an integration in the complex plane as long as is non-zero and has a semi-positive imaginary part. See
Fresnel integral
250px,
Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below).
The Fresnel integrals and are two transcendental functions n ...
.
Gaussian integrals in higher dimensions
The one-dimensional integrals can be generalized to multiple dimensions.
Here is a real positive definite
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
.
This integral is performed by
diagonalization of with an
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we h ...
where is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
and is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
. This decouples the variables and allows the integration to be performed as one-dimensional integrations.
This is best illustrated with a two-dimensional example.
Example: Simple Gaussian integration in two dimensions
The Gaussian integral in two dimensions is
where is a two-dimensional symmetric matrix with components specified as
and we have used the
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
.
Diagonalize the matrix
The first step is to
diagonalize the matrix. Note that
where, since is a real
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
, we can choose to be
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
, and hence also a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
. can be obtained from the
eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of . We choose such that: is diagonal.
=Eigenvalues of ''A''
=
To find the eigenvectors of one first finds the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of given by
The eigenvalues are solutions of the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
which are found using the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
:
=Eigenvectors of ''A''
=
Substitution of the eigenvalues back into the eigenvector equation yields
From the characteristic equation we know
Also note
The eigenvectors can be written as:
for the two eigenvectors. Here is a normalizing factor given by,
It is easily verified that the two eigenvectors are orthogonal to each other.
=Construction of the orthogonal matrix
=
The orthogonal matrix is constructed by assigning the normalized eigenvectors as columns in the orthogonal matrix
Note that .
If we define
then the orthogonal matrix can be written
which is simply a rotation of the eigenvectors with the inverse:
=Diagonal matrix
=
The diagonal matrix becomes
with eigenvectors
=Numerical example
=
The eigenvalues are
The eigenvectors are
where
Then
The diagonal matrix becomes
with eigenvectors
Rescale the variables and integrate
With the diagonalization the integral can be written
where
Since the coordinate transformation is simply a rotation of coordinates the
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
* Jacobian matrix and determinant
* Jacobian elliptic functions
* Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähle ...
determinant of the transformation is one yielding
The integrations can now be performed.
which is the advertised solution.
Integrals with complex and linear terms in multiple dimensions
With the two-dimensional example it is now easy to see the generalization to the complex plane and to multiple dimensions.
Integrals with a linear term in the argument
Integrals with an imaginary linear term
Integrals with a complex quadratic term
Integrals with differential operators in the argument
As an example consider the integral
where
is a differential operator with
and functions of
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 ...
, and
indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is
where
and , called the
propagator
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. ...
, is the inverse of
, and
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 ...
.
Similar arguments yield
and
See
Path-integral formulation of virtual-particle exchange for an application of this integral.
Integrals that can be approximated by the method of steepest descent
In quantum field theory n-dimensional integrals of the form
appear often. Here
is the
reduced Planck's constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and f is a function with a positive minimum at
. These integrals can be approximated by the
method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in r ...
.
For small values of Planck's constant, f can be expanded about its minimum
Here
is the n by n matrix of second derivatives evaluated at the minimum of the function.
If we neglect higher order terms this integral can be integrated explicitly.
Integrals that can be approximated by the method of stationary phase
A common integral is a path integral of the form
where
is the classical
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 ...
and the integral is over all possible paths that a particle may take. In the limit of small
the integral can be evaluated in the
stationary phase approximation In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as k \to \infty .
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin.
It is closel ...
. In this approximation the integral is over the path in which the action is a minimum. Therefore, this approximation recovers the
classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
of
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
.
Fourier integrals
Dirac delta distribution
The
Dirac delta distribution
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 entire ...
in
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 ...
can be written as 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 ...
In general, for any dimension
Fourier integrals of forms of the Coulomb potential
Laplacian of 1/r
While not an integral, the identity in three-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 ...
where
is a consequence of
Gauss's theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
and can be used to derive integral identities. For an example see
Longitudinal and transverse vector fields
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
.
This identity implies that the
Fourier integral
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, ...
representation of 1/r is
Yukawa Potential: The Coulomb potential with mass
The
Yukawa potential
In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form:
:V_\text(r)= -g^2\frac,
where is a ...
in three dimensions can be represented as an integral over 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 ...
where
See
Static forces and virtual-particle exchange for an application of this integral.
In the small m limit the integral reduces to .
To derive this result note:
Modified Coulomb potential with mass
where the hat indicates a unit vector in three dimensional space. The derivation of this result is as follows:
Note that in the small limit the integral goes to the result for the Coulomb potential since the term in the brackets goes to .
Longitudinal potential with mass
where the hat indicates a unit vector in three dimensional space. The derivation for this result is as follows:
Note that in the small limit the integral reduces to
Transverse potential with mass