Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions.[1]: 13–15 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 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 of the Gaussian where J is the conjugate variable of x.
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 a is, the narrower the Gaussian in x and the wider the Gaussian in J. This is a demonstration of the uncertainty principle.
This integral is also known as the Hubbard–Stratonovich transformation used in field theory.
Integrals with a complex argument of the exponent
The integral of interest is (for an example of an application see Relation between Schrödinger's equation and the path integral formulation of quantum mechanics)
We now assume that a and J 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 a is non-zero and has a semi-positive imaginary part. See Fresnel integral.
Gaussian integrals in higher dimensions
The one-dimensional integrals can be generalized to multiple dimensions.[2]
Here A is a real positive definite symmetric matrix.
This integral is performed by diagonalization of A with an orthogonal transformation
where D is a diagonal matrix and O is an orthogonal matrix. This decouples the variables and allows the integration to be performed as n 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 A is a two-dimensional symmetric matrix with components specified as
and we have used the Einstein summation convention.
Diagonalize the matrix
The first step is to diagonalize the matrix.[3] Note that
where, since A is a real symmetric matrix, we can choose O to be orthogonal, and hence also a unitary matrix. O can be obtained from the eigenvectors of A. We choose O such that: D ≡ OTAO is diagonal.
Eigenvalues of A
To find the eigenvectors of A one first finds the eigenvalues λ of A given by
The eigenvalues are solutions of the characteristic polynomial
which are found using the quadratic equation:
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 η 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 det(O) = 1.
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 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[1]: 21‒22
where
is a differential operator with
and J functions of spacetime, and
indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is
where
and D(x − y), called the propagator, is the inverse of
, and
is the Dirac delta function.
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 constant and f is a function with a positive minimum at
. These integrals can be approximated by the method of steepest descent.
For small values of the Planck 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 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 this approximation the integral is over the path in which the action is a minimum. Therefore, this approximation recovers the classical limit of mechanics.
Fourier integrals
Dirac delta distribution
The Dirac delta distribution in spacetime can be written as a Fourier transform[1]: 23
In general, for any dimension
Laplacian of 1/r
While not an integral, the identity in three-dimensional Euclidean space
where
is a consequence of Gauss's theorem and can be used to derive integral identities. For an example see Longitudinal and transverse vector fields.
This identity implies that the Fourier integral representation of 1/r is
Yukawa potential: the Coulomb potential with mass
The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform[1]: 26, 29
where
See Static forces and virtual-particle exchange for an application of this integral.
In the small m limit the integral reduces to 1/4πr.
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 m limit the integral goes to the result for the Coulomb potential since the term in the brackets goes to 1.
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 m limit the integral reduces to
Transverse potential with mass
In the small mr limit the integral goes to
For large distance, the integral falls off as the inverse cube of r
For applications of this integral see Darwin Lagrangian and Darwin interaction in a vacuum.
Angular integration in cylindrical coordinates
There are two important integrals. The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind[4][5]: 113
and
For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas.
Bessel functions
Integration of the cylindrical propagator with mass
First power of a Bessel function
See Abramowitz and Stegun.[6]: §11.4.44
For
, we have[5]: 116
For an application of this integral see Two line charges embedded in a plasma or electron gas.
Squares of Bessel functions
The integration of the propagator in cylindrical coordinates is[4]
For small mr the integral becomes
For large mr the integral becomes
For applications of this integral see Magnetic interaction between current loops in a simple plasma or electron gas.
In general,
Integration over a magnetic wave function
The two-dimensional integral over a magnetic wave function is[6]: §11.4.28
Here, M is a confluent hypergeometric function. For an application of this integral see Charge density spread over a wave function.
See also
References
- ^ a b c d A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6.
- ^ Frederick W. Byron and Robert W. Fuller (1969). Mathematics of Classical and Quantum Physics. Addison-Wesley. ISBN 0-201-00746-0.
- ^ Herbert S. Wilf (1978). Mathematics for the Physical Sciences. Dover. ISBN 0-486-63635-6.
- ^ a b Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
- ^ a b Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
- ^ a b M. Abramowitz; I. Stegun (1965). Handbook of Mathematical Functions. Dover. ISBN 0486-61272-4.