Common integrals in quantum field theory
Common integrals in quantum field theory 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. Variations on a simple Gaussian integral. Gaussian integral. The first integral, with broad application outside of quantum field theory, is the Gaussian integral. formula_1 In physics the factor of 1/2 in the argument of the exponential is common. Note: formula_2 Thus we obtain formula_3 Slight generalization of the Gaussian integral. formula_4 where we have scaled formula_5  Integrals of exponents and even powers of "x". formula_6 and formula_7 In general formula_8 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. formula_9 This integral can be performed by completing the square: formula_10 Therefore: formula_11 Integrals with an imaginary linear term in the argument of the exponent. The integral formula_12 is proportional to the Fourier transform of the Gaussian where is the conjugate variable 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. 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) formula_13 We now assume that and may be complex. Completing the square formula_14 By analogy with the previous integrals formula_15 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. Gaussian integrals in higher dimensions. The one-dimensional integrals can be generalized to multiple dimensions. formula_16 Here is a real positive definite symmetric matrix. This integral is performed by diagonalization of with an orthogonal transformation formula_17 where is a diagonal matrix and is an orthogonal matrix. 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 formula_18 where is a two-dimensional symmetric matrix with components specified as formula_19 and we have used the Einstein summation convention. Diagonalize the matrix. The first step is to diagonalize the matrix. Note that formula_20 where, since is a real symmetric matrix, we can choose to be orthogonal, and hence also a unitary matrix. can be obtained from the eigenvectors of . We choose such that: is diagonal. Eigenvalues of "A". To find the eigenvectors of one first finds the eigenvalues of given by formula_21 The eigenvalues are solutions of the characteristic polynomial  formula_22 formula_23 which are found using the quadratic equation: formula_24 Eigenvectors of "A". Substitution of the eigenvalues back into the eigenvector equation yields formula_25 From the characteristic equation we know formula_26 Also note formula_27 The eigenvectors can be written as: formula_28 for the two eigenvectors. Here is a normalizing factor given by formula_29 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 formula_30 Note that . If we define formula_31 then the orthogonal matrix can be  written formula_32 which is simply a rotation of the eigenvectors with the inverse: formula_33 Diagonal matrix. The diagonal matrix becomes formula_34 with eigenvectors formula_35 Numerical example. formula_36 The eigenvalues are formula_37 The eigenvectors are formula_38 where formula_39 Then formula_40 The diagonal matrix becomes formula_41 with eigenvectors formula_35 Rescale the variables and integrate. With the diagonalization the integral can be written formula_43 where formula_44 Since the coordinate transformation is simply a rotation of coordinates the Jacobian determinant of the transformation is one yielding formula_45 The integrations can now be performed. formula_46 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. formula_47 Integrals with an imaginary linear term. formula_48 Integrals with a complex quadratic term. formula_49 Integrals with differential operators in the argument. As an example consider the integral formula_50 where formula_51 is a differential operator with formula_52 and functions of spacetime, and formula_53 indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is formula_54 where formula_55 and , called the  propagator, is the inverse of formula_56, and formula_57 is the Dirac delta function. Similar arguments yield formula_58 and formula_59 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 formula_60 appear often. Here formula_61 is the reduced Planck's constant and f is a function with a positive minimum at formula_62. 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 formula_63Here formula_64 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. formula_65 Integrals that can be approximated by the method of stationary phase. A common integral is a path integral of the form formula_66 where formula_67 is the classical action and the integral is over all possible paths that a particle may take. In the limit of small formula_68 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 formula_69 In general, for any dimension formula_70 formula_71 Fourier integrals of forms of the Coulomb potential. Laplacian of 1/r. While not an integral, the identity in three-dimensional Euclidean space formula_72 where formula_73 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 formula_74 Yukawa Potential: The Coulomb potential with mass. The Yukawa potential in three dimensions can be represented as an integral over a Fourier transform formula_75 where formula_76 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: formula_77 Modified Coulomb potential with mass. formula_78 where the hat indicates a unit vector in three dimensional space. The derivation of this result is as follows: formula_79 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. formula_80 where the hat indicates a unit vector in three dimensional space. The derivation for this result is as follows: formula_81 Note that in the small limit the integral reduces to formula_82 Transverse potential with mass. formula_83 In the small mr limit the integral goes to formula_84 For large distance, the integral falls off as the inverse cube of r formula_85 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 formula_86 and formula_87 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. formula_88 See Abramowitz and Stegun. For formula_89, we have formula_90 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 formula_91 For small mr the integral becomes  formula_92 For large mr the integral becomes formula_93 For applications of this integral see Magnetic interaction between current loops in a simple plasma or electron gas. In general formula_94 Integration over a magnetic wave function. The two-dimensional integral over a magnetic wave function is formula_95 Here, M is a confluent hypergeometric function. For an application of this integral see Charge density spread over a wave function.