Common integrals in quantum field theory
There are common integrals in quantum field theory that appear repeatedly. 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.
- 1 Variations on a simple gaussian integral
- 1.1 Gaussian integral
- 1.2 Slight generalization of the gaussian integral
- 1.3 Integrals of exponents and even powers of x
- 1.4 Integrals with a linear term in the argument of the exponent
- 1.5 Integrals with an imaginary linear term in the argument of the exponent
- 1.6 Integrals with a complex argument of the exponent
- 2 Gaussian integrals in higher dimensions
- 2.1 Example: Simple gaussian integration in two dimensions
- 2.2 Integrals with complex and linear terms in multiple dimensions
- 2.3 Integrals with differential operators in the argument
- 3 Integrals that can be approximated by the method of steepest descent
- 4 Integrals that can be approximated by the method of stationary phase
- 5 Fourier integrals
- 5.1 Dirac delta function
- 5.2 Fourier integrals of forms of the Coulomb potential
- 6 Bessel functions
- 7 See also
- 8 References
Variations on a simple 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.
Thus we obtain
Slight generalization of the gaussian integral
where we have scaled
Integrals of exponents and even powers of x
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:
Integrals with an imaginary linear term in the argument of the exponent
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 has a positive imaginary part.
Gaussian integrals in higher dimensions
The one-dimensional integrals can be generalized to multiple dimensions.
Here A is a real symmetric matrix.
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
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
Eigenvectors of A
Substitution of the eigenvalues back into the eigenvector equation yields
From the characteristic equation we know
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:
The diagonal matrix becomes
The eigenvalues are
The eigenvectors are
The diagonal matrix becomes
Rescale the variables and integrate
With the diagonalization the integral can be written
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
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
Similar arguments yield
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
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 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.
Dirac delta function
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
This identity implies that the Fourier integral representation of 1/r is
Yukawa Potential: The Coulomb potential with mass
See Static forces and virtual-particle exchange for an application of this integral.
In the small m limit the integral reduces to 1/. 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
Angular integration in cylindrical coordinates
For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas.
Integration of the cylindrical propagator with mass
First power of a Bessel function
See Abramowitz and Stegun.
For , we have
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
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.
Integration over a magnetic wave function
The two-dimensional integral over a magnetic wave function is
- A. Zee (2003). Quantum Field Theory in a Nutshell. Princeton University. ISBN 0-691-01019-6. pp. 13-15
- 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.
- Zee, pp. 21-22.
- Zee, p. 23.
- Zee, p. 26, 29.
- I. S. Gradshteyn and I. M. Ryzhik (1965). Tables of Integrals, Series, and Products. Academic Press. LCCN 65029097. p. 402
- Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. p. 113
- M. Abramowitz and I. Stegun (1965). Handbook of Mathematical Functions. Dover. ISBN 0486-61272-4. Section 11.4.44
- Jackson, p. 116
- I. S. Gradshteyn and I. M. Ryzhik, p. 679
- Abramowitz and Stegun, Section 11.4.28