Common integrals in quantum field theory

There are common integrals in quantum field theory that appear repeatedly.[1] These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussian integral. Fourier integrals are also considered.

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Variations on a simple gaussian integralEdit

Gaussian integralEdit

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 integralEdit

 

where we have scaled

 .

Integrals of exponents and even powers of xEdit

 

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 exponentEdit

 

This integral can be performed by completing the square:

 

Therefore:

 

Integrals with an imaginary linear term in the argument of the exponentEdit

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 exponentEdit

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 has a positive imaginary part.

Gaussian integrals in higher dimensionsEdit

The one-dimensional integrals can be generalized to multiple dimensions.[2]

 

Here A is a real 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 dimensionsEdit

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 matrixEdit

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: DOTAO is diagonal.

Eigenvalues of AEdit

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 AEdit

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 matrixEdit

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 matrixEdit

The diagonal matrix becomes

 

with eigenvectors

 
Numerical exampleEdit
 

The eigenvalues are

 

The eigenvectors are

 

where

 

Then

 

The diagonal matrix becomes

 

with eigenvectors

 

Rescale the variables and integrateEdit

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 dimensionsEdit

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 argumentEdit

 

Integrals with an imaginary linear termEdit

 

Integrals with a complex quadratic termEdit

 

Integrals with differential operators in the argumentEdit

As an example consider the integral[4]

 

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(xy), 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 descentEdit

In quantum field theory n-dimensional integrals of the form

 

appear often. Here   is the reduced Planck's 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 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 phaseEdit

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 integralsEdit

Dirac delta distributionEdit

The Dirac delta distribution in spacetime can be written as a Fourier transform[5]

 

In general, for any dimension  

 

Fourier integrals of forms of the Coulomb potentialEdit

Laplacian of 1/rEdit

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 massEdit

The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform[6]

 

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 massEdit

 

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 massEdit

 

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 massEdit

 

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 coordinatesEdit

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[7][8]

 

and

 

For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas.

Bessel functionsEdit

Integration of the cylindrical propagator with massEdit

First power of a Bessel functionEdit

 

See Abramowitz and Stegun.[9]

For  , we have[10]

 

For an application of this integral see Two line charges embedded in a plasma or electron gas.

Squares of Bessel functionsEdit

The integration of the propagator in cylindrical coordinates is[7]

 

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 functionEdit

The two-dimensional integral over a magnetic wave function is[11]

 

Here, M is a confluent hypergeometric function. For an application of this integral see Charge density spread over a wave function.

See alsoEdit

ReferencesEdit

  1. ^ A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6. pp. 13-15
  2. ^ Frederick W. Byron and Robert W. Fuller (1969). Mathematics of Classical and Quantum Physics. Addison-Wesley. ISBN 0-201-00746-0.
  3. ^ Herbert S. Wilf (1978). Mathematics for the Physical Sciences. Dover. ISBN 0-486-63635-6.
  4. ^ Zee, pp. 21-22.
  5. ^ Zee, p. 23.
  6. ^ Zee, p. 26, 29.
  7. ^ 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 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.
  8. ^ Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. p. 113
  9. ^ M. Abramowitz and I. Stegun (1965). Handbook of Mathematical Functions. Dover. ISBN 0486-61272-4. Section 11.4.44
  10. ^ Jackson, p. 116
  11. ^ Abramowitz and Stegun, Section 11.4.28