Common integrals in quantum field theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/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.

$${\displaystyle G\equiv {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{x}^2}dx}$$

In physics the factor of 1/2 in the argument of the exponential is common.

Note:

$${\displaystyle G^2=\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{x}^2}dx\right)\cdot \left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{y}^2}dy\right)=2\pi {\int }_0^{\mathrm{\infty }}r{e}^{-\frac{1}{2}{r}^2}dr=2\pi {\int }_0^{\mathrm{\infty }}{e}^{-w}dw=2\pi .}$$

Thus we obtain

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}{x}^2}dx=\sqrt{2\pi }.}$$

Slight generalization of the Gaussian integral

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}a{x}^2}dx=\sqrt{\frac{2\pi }{a}}}$$

where we have scaled

$${\displaystyle x\to \frac{x}{\sqrt{a}}.}$$

Integrals of exponents and even powers of x

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^2{e}^{-\frac{1}{2}a{x}^2}dx=-2\frac{d}{da}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}a{x}^2}dx=-2\frac{d}{da}{\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\frac{1}{a}}$$

and

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^4{e}^{-\frac{1}{2}a{x}^2}dx=\left(-2\frac{d}{da}\right)\left(-2\frac{d}{da}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{1}{2}a{x}^2}dx=\left(-2\frac{d}{da}\right)\left(-2\frac{d}{da}\right){\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\frac{3}{a^2}}$$

In general

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^{2n}{e}^{-\frac{1}{2}a{x}^2}dx={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\frac{1}{a^n}\left(2n-1\right)\left(2n-3\right)\cdots 5\cdot 3\cdot 1={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\frac{1}{a^n}\left(2n-1\right)!!}$$

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{2}a{x}^2+Jx\right)dx}$$

This integral can be performed by completing the square:

$${\displaystyle \left(-\frac{1}{2}a{x}^2+Jx\right)=-\frac{1}{2}a\left(x^2-\frac{2Jx}{a}+\frac{J^2}{a^2}-\frac{J^2}{a^2}\right)=-\frac{1}{2}a{\left(x-\frac{J}{a}\right)}^2+\frac{J^2}{2a}}$$

Therefore:

$${\displaystyle \begin{array}{rl}& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{2}a{x}^2+Jx\right)dx\\ & =\exp \left(\frac{J^2}{2a}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left[-\frac{1}{2}a{\left(x-\frac{J}{a}\right)}^2\right]dx\\ & =\exp \left(\frac{J^2}{2a}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{2}a{w}^2\right)dw\\ & ={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\exp \left(\frac{J^2}{2a}\right)\end{array}}$$

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

The integral

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{2}a{x}^2+iJx\right)dx={\left(\frac{2\pi }{a}\right)}^{\frac{1}{2}}\exp \left(-\frac{J^2}{2a}\right)}$$

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)

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\frac{1}{2}ia{x}^2+iJx\right)dx.}$$

We now assume that a and J may be complex.

Completing the square

$${\displaystyle \left(\frac{1}{2}ia{x}^2+iJx\right)=\frac{1}{2}ia\left(x^2+\frac{2Jx}{a}+{\left(\frac{J}{a}\right)}^2-{\left(\frac{J}{a}\right)}^2\right)=-\frac{1}{2}\frac{a}{i}{\left(x+\frac{J}{a}\right)}^2-\frac{i{J}^2}{2a}.}$$

By analogy with the previous integrals

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\frac{1}{2}ia{x}^2+iJx\right)dx={\left(\frac{2\pi i}{a}\right)}^{\frac{1}{2}}\exp \left(\frac{-i{J}^2}{2a}\right).}$$

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.

$${\displaystyle \int \exp \left(-\frac{1}{2}x\cdot A\cdot x+J\cdot x\right)d^{n}x=\sqrt{\frac{(2\pi {)}^n}{detA}}\exp \left(\frac{1}{2}J\cdot A^{-1}\cdot J\right)}$$

Here A is a real positive definite symmetric matrix.

This integral is performed by diagonalization of A with an orthogonal transformation

$${\displaystyle D=O^{-1}AO=O^{T}AO}$$

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

$${\displaystyle \int \exp \left(-\frac{1}{2}{A}_{ij}{x}^i{x}^j\right)d^{2}x=\sqrt{\frac{(2\pi {)}^2}{detA}}}$$

where A is a two-dimensional symmetric matrix with components specified as

$${\displaystyle A=\left[\begin{array}{cc}a& c\\ c& b\end{array}\right]}$$

and we have used the Einstein summation convention.

Diagonalize the matrix

The first step is to diagonalize the matrix. Note that

$${\displaystyle A_{ij}{x}^i{x}^j\equiv x^{T}Ax=x^T\left(O{O}^T\right)A\left(O{O}^T\right)x=\left(x^{T}O\right)\left(O^{T}AO\right)\left(O^{T}x\right)}$$

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 ≡ O^TAO is diagonal.

Eigenvalues of A

To find the eigenvectors of A one first finds the eigenvalues λ of A given by

$${\displaystyle \left[\begin{array}{cc}a& c\\ c& b\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\lambda \left[\begin{array}{c}u\\ v\end{array}\right].}$$

The eigenvalues are solutions of the characteristic polynomial

$${\displaystyle (a-\lambda )(b-\lambda )-c^2=0}$$

$${\displaystyle {\lambda }^2-\lambda (a+b)+ab-c^2=0}$$

which are found using the quadratic equation:

$${\displaystyle \begin{array}{rl}{\lambda }_{\pm }& =\frac{1}{2}(a+b)\pm \frac{1}{2}\sqrt{(a+b{)}^2-4(ab-c^2)}.\\ & =\frac{1}{2}(a+b)\pm \frac{1}{2}\sqrt{a^2+2ab+b^2-4ab+4c^2}.\\ & =\frac{1}{2}(a+b)\pm \frac{1}{2}\sqrt{(a-b{)}^2+4c^2}.\end{array}}$$

Eigenvectors of A

Substitution of the eigenvalues back into the eigenvector equation yields

$${\displaystyle v=-\frac{\left(a-{\lambda }_{\pm }\right)u}{c},v=-\frac{cu}{\left(b-{\lambda }_{\pm }\right)}.}$$

From the characteristic equation we know

$${\displaystyle \frac{a-{\lambda }_{\pm }}{c}=\frac{c}{b-{\lambda }_{\pm }}.}$$

Also note

$${\displaystyle \frac{a-{\lambda }_{\pm }}{c}=-\frac{b-{\lambda }_{\mp }}{c}.}$$

The eigenvectors can be written as:

$${\displaystyle \left[\begin{array}{c}\frac{1}{\eta }\\ -\frac{a-{\lambda }_{-}}{c\eta }\end{array}\right],\left[\begin{array}{c}-\frac{b-{\lambda }_{+}}{c\eta }\\ \frac{1}{\eta }\end{array}\right]}$$

for the two eigenvectors. Here η is a normalizing factor given by,

$${\displaystyle \eta =\sqrt{1+{\left(\frac{a-{\lambda }_{-}}{c}\right)}^2}=\sqrt{1+{\left(\frac{b-{\lambda }_{+}}{c}\right)}^2}.}$$

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

$${\displaystyle O=\left[\begin{array}{cc}\frac{1}{\eta }& -\frac{b-{\lambda }_{+}}{c\eta }\\ -\frac{a-{\lambda }_{-}}{c\eta }& \frac{1}{\eta }\end{array}\right].}$$

Note that det(O) = 1.

If we define

$${\displaystyle \sin (\theta )=-\frac{a-{\lambda }_{-}}{c\eta }}$$

then the orthogonal matrix can be written

$${\displaystyle O=\left[\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right]}$$

which is simply a rotation of the eigenvectors with the inverse:

$${\displaystyle O^{-1}=O^T=\left[\begin{array}{cc}\cos (\theta )& \sin (\theta )\\ -\sin (\theta )& \cos (\theta )\end{array}\right].}$$

Diagonal matrix

The diagonal matrix becomes

$${\displaystyle D=O^{T}AO=\left[\begin{array}{cc}{\lambda }_{-}& 0\\ 0& {\lambda }_{+}\end{array}\right]}$$

with eigenvectors

$${\displaystyle \left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right]}$$

Numerical example

$${\displaystyle A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]}$$

The eigenvalues are

$${\displaystyle {\lambda }_{\pm }=\frac{3}{2}\pm \frac{\sqrt{5}}{2}.}$$

The eigenvectors are

$${\displaystyle \frac{1}{\eta }\left[\begin{array}{c}1\\ -\frac{1}{2}-\frac{\sqrt{5}}{2}\end{array}\right],\frac{1}{\eta }\left[\begin{array}{c}\frac{1}{2}+\frac{\sqrt{5}}{2}\\ 1\end{array}\right]}$$

where

$${\displaystyle \eta =\sqrt{\frac{5}{2}+\frac{\sqrt{5}}{2}}.}$$

Then

$${\displaystyle \begin{array}{rl}O& =\left[\begin{array}{cc}\frac{1}{\eta }& \frac{1}{\eta }\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)\\ \frac{1}{\eta }\left(-\frac{1}{2}-\frac{\sqrt{5}}{2}\right)& \frac{1}{\eta }\end{array}\right]\\ O^{-1}& =\left[\begin{array}{cc}\frac{1}{\eta }& \frac{1}{\eta }\left(-\frac{1}{2}-\frac{\sqrt{5}}{2}\right)\\ \frac{1}{\eta }\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)& \frac{1}{\eta }\end{array}\right]\end{array}}$$

The diagonal matrix becomes

$${\displaystyle D=O^{T}AO=\left[\begin{array}{cc}{\lambda }_{-}& 0\\ 0& {\lambda }_{+}\end{array}\right]=\left[\begin{array}{cc}\frac{3}{2}-\frac{\sqrt{5}}{2}& 0\\ 0& \frac{3}{2}+\frac{\sqrt{5}}{2}\end{array}\right]}$$

with eigenvectors

$${\displaystyle \left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right]}$$

Rescale the variables and integrate

With the diagonalization the integral can be written

$${\displaystyle \int \exp \left(-\frac{1}{2}{x}^{T}Ax\right)d^{2}x=\int \exp \left(-\frac{1}{2}\sum _{j=1}^2{\lambda }_j{y}_j^2\right)d^{2}y}$$

where

$${\displaystyle y=O^{T}x.}$$

Since the coordinate transformation is simply a rotation of coordinates the Jacobian determinant of the transformation is one yielding

$${\displaystyle d^{2}y=d^{2}x}$$

The integrations can now be performed.

$${\displaystyle \begin{array}{rl}\int \exp \left(-\frac{1}{2}{x}^{\mathsf{T}}Ax\right)d^{2}x=& \int \exp \left(-\frac{1}{2}\sum _{j=1}^2{\lambda }_j{y}_j^2\right)d^{2}y\\ =& \prod _{j=1}^2{\left(\frac{2\pi }{{\lambda }_j}\right)}^{\frac{1}{2}}\\ =& {\left(\frac{(2\pi {)}^2}{\prod _{j=1}^2{\lambda }_j}\right)}^{\frac{1}{2}}\\ =& {\left(\frac{(2\pi {)}^2}{det\left(O^{-1}AO\right)}\right)}^{\frac{1}{2}}\\ =& {\left(\frac{(2\pi {)}^2}{det\left(A\right)}\right)}^{\frac{1}{2}}\end{array}}$$

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

$${\displaystyle \int \exp \left(-\frac{1}{2}x\cdot A\cdot x+J\cdot x\right)d^{n}x=\sqrt{\frac{(2\pi {)}^n}{detA}}\exp \left(\frac{1}{2}J\cdot A^{-1}\cdot J\right)}$$

Integrals with an imaginary linear term

$${\displaystyle \int \exp \left(-\frac{1}{2}x\cdot A\cdot x+iJ\cdot x\right)d^{n}x=\sqrt{\frac{(2\pi {)}^n}{detA}}\exp \left(-\frac{1}{2}J\cdot A^{-1}\cdot J\right)}$$

Integrals with a complex quadratic term

$${\displaystyle \int \exp \left(\frac{i}{2}x\cdot A\cdot x+iJ\cdot x\right)d^{n}x=\sqrt{\frac{(2\pi i{)}^n}{detA}}\exp \left(-\frac{i}{2}J\cdot A^{-1}\cdot J\right)}$$

Integrals with differential operators in the argument

As an example consider the integral

$${\displaystyle \int \exp \left[\int d^{4}x\left(-\frac{1}{2}\phi \hat{A}\phi +J\phi \right)\right]D\phi }$$

where $\hat{A}$ is a differential operator with $\phi$ and J functions of spacetime, and $D\phi$ indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is

$${\displaystyle \int \exp \left[\int d^{4}x\left(-\frac{1}{2}\phi \hat{A}\phi +J\phi \right)\right]D\phi \propto \exp \left(\frac{1}{2}\int d^{4}x{d}^{4}yJ(x)D(x-y)J(y)\right)}$$

where

$${\displaystyle \hat{A}D(x-y)={\delta }^4(x-y)}$$

and D(x − y), called the propagator, is the inverse of $\hat{A}$, and ${\delta }^4(x-y)$ is the Dirac delta function.

Similar arguments yield

$${\displaystyle \int \exp \left[\int d^{4}x\left(-\frac{1}{2}\phi \hat{A}\phi +iJ\phi \right)\right]D\phi \propto \exp \left(-\frac{1}{2}\int d^{4}x{d}^{4}yJ(x)D(x-y)J(y)\right),}$$

and

$${\displaystyle \int \exp \left[i\int d^{4}x\left(\frac{1}{2}\phi \hat{A}\phi +J\phi \right)\right]D\phi \propto \exp \left(-\frac{i}{2}\int d^{4}x{d}^{4}yJ(x)D(x-y)J(y)\right).}$$

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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{\hslash }f(q)\right)d^{n}q}$$

appear often. Here $\hslash$ is the reduced Planck's constant and f is a function with a positive minimum at $q=q_0$. 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

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left[-\frac{1}{\hslash }\left(f\left(q_0\right)+\frac{1}{2}{\left(q-q_0\right)}^2{f}^{\mathrm{\prime }\mathrm{\prime }}\left(q-q_0\right)+\cdots \right)\right]d^{n}q.}$$

Here $f^{\mathrm{\prime }\mathrm{\prime }}$ 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.

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left[-\frac{1}{\hslash }(f(q))\right]d^{n}q\approx \exp \left[-\frac{1}{\hslash }\left(f\left(q_0\right)\right)\right]\sqrt{\frac{(2\pi \hslash {)}^n}{det{f}^{\mathrm{\prime }\mathrm{\prime }}}}.}$$

Integrals that can be approximated by the method of stationary phase

A common integral is a path integral of the form

$${\displaystyle \int \exp \left(\frac{i}{\hslash }S\left(q,\dot{q}\right)\right)Dq}$$

where $S\left(q,\dot{q}\right)$ is the classical action and the integral is over all possible paths that a particle may take. In the limit of small $\hslash$ 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

$${\displaystyle \int \frac{d^{4}k}{(2\pi {)}^4}\exp (ik(x-y))={\delta }^4(x-y).}$$

In general, for any dimension $N$

$${\displaystyle \int \frac{d^{N}k}{(2\pi {)}^N}\exp (ik(x-y))={\delta }^N(x-y).}$$

Fourier integrals of forms of the Coulomb potential

Laplacian of 1/r

While not an integral, the identity in three-dimensional Euclidean space

$${\displaystyle -\frac{1}{4\pi }{\mathrm{\nabla }}^2\left(\frac{1}{r}\right)=\delta \left(\mathbf{r}\right)}$$

where

$${\displaystyle r^2=\mathbf{r}\cdot \mathbf{r}}$$

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

$${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2}=\frac{1}{4\pi r}.}$$

Yukawa Potential: The Coulomb potential with mass

$${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}=\frac{e^{-mr}}{4\pi r}}$$

where

$${\displaystyle r^2=\mathbf{r}\cdot \mathbf{r},k^2=\mathbf{k}\cdot \mathbf{k}.}$$

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:

$${\displaystyle \begin{array}{rl}\int \frac{d^{3}k}{(2\pi {)}^3}\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}=& {\int }_0^{\mathrm{\infty }}\frac{k^{2}dk}{(2\pi {)}^2}{\int }_{-1}^{1}du\frac{e^{ikru}}{k^2+m^2}\\ =& \frac{2}{r}{\int }_0^{\mathrm{\infty }}\frac{kdk}{(2\pi {)}^2}\frac{\sin (kr)}{k^2+m^2}\\ =& \frac{1}{ir}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{kdk}{(2\pi {)}^2}\frac{e^{ikr}}{k^2+m^2}\\ =& \frac{1}{ir}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{kdk}{(2\pi {)}^2}\frac{e^{ikr}}{(k+im)(k-im)}\\ =& \frac{1}{ir}\frac{2\pi i}{(2\pi {)}^2}\frac{im}{2im}{e}^{-mr}\\ =& \frac{1}{4\pi r}{e}^{-mr}\end{array}}$$

Modified Coulomb potential with mass

$${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}{\left(\hat{\mathbf{k}}\cdot \hat{\mathbf{r}}\right)}^2\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}=\frac{e^{-mr}}{4\pi r}\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}}$$

where the hat indicates a unit vector in three dimensional space. The derivation of this result is as follows:

$${\displaystyle \begin{array}{rl}& \int \frac{d^{3}k}{(2\pi {)}^3}{\left(\hat{\mathbf{k}}\cdot \hat{\mathbf{r}}\right)}^2\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}\\ =& {\int }_0^{\mathrm{\infty }}\frac{k^{2}dk}{(2\pi {)}^2}{\int }_{-1}^{1}du{u}^2\frac{e^{ikru}}{k^2+m^2}\\ =& 2{\int }_0^{\mathrm{\infty }}\frac{k^{2}dk}{(2\pi {)}^2}\frac{1}{k^2+m^2}\left\{\frac{1}{kr}\sin (kr)+\frac{2}{(kr{)}^2}\cos (kr)-\frac{2}{(kr{)}^3}\sin (kr)\right\}\\ =& \frac{e^{-mr}}{4\pi r}\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}\end{array}}$$

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

$${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}\hat{\mathbf{k}}\hat{\mathbf{k}}\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}=\frac{1}{2}\frac{e^{-mr}}{4\pi r}\left(\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right]+\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\right)}$$

where the hat indicates a unit vector in three dimensional space. The derivation for this result is as follows:

$${\displaystyle \begin{array}{rl}& \int \frac{d^{3}k}{(2\pi {)}^3}\hat{\mathbf{k}}\hat{\mathbf{k}}\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}\\ =& \int \frac{d^{3}k}{(2\pi {)}^3}\left[{\left(\hat{\mathbf{k}}\cdot \hat{\mathbf{r}}\right)}^2\hat{\mathbf{r}}\hat{\mathbf{r}}+{\left(\hat{\mathbf{k}}\cdot \hat{\theta }\right)}^2\hat{\theta }\hat{\theta }+{\left(\hat{\mathbf{k}}\cdot \hat{\varphi }\right)}^2\hat{\varphi }\hat{\varphi }\right]\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}\\ =& \frac{e^{-mr}}{4\pi r}\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}\left\{\mathbf{1}-\frac{1}{2}\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\right\}+{\int }_0^{\mathrm{\infty }}\frac{k^{2}dk}{(2\pi {)}^2}{\int }_{-1}^{1}du\frac{e^{ikru}}{k^2+m^2}\frac{1}{2}\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\\ =& \frac{1}{2}\frac{e^{-mr}}{4\pi r}\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right]+\frac{e^{-mr}}{4\pi r}\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}\left\{\frac{1}{2}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\right\}\\ =& \frac{1}{2}\frac{e^{-mr}}{4\pi r}\left(\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right]+\left\{1+\frac{2}{mr}-\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)\right\}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\right)\end{array}}$$

Note that in the small m limit the integral reduces to

$${\displaystyle \frac{1}{2}\frac{1}{4\pi r}\left[\mathbf{1}-\hat{\mathbf{r}}\hat{\mathbf{r}}\right].}$$

Transverse potential with mass

$${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}\left[\mathbf{1}-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\frac{\exp \left(i\mathbf{k}\cdot \mathbf{r}\right)}{k^2+m^2}=\frac{1}{2}\frac{e^{-mr}}{4\pi r}\left\{\frac{2}{(mr{)}^2}\left(e^{mr}-1\right)-\frac{2}{mr}\right\}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]}$$

In the small mr limit the integral goes to

$${\displaystyle \frac{1}{2}\frac{1}{4\pi r}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right].}$$

For large distance, the integral falls off as the inverse cube of r

$${\displaystyle \frac{1}{4\pi m^2{r}^3}\left[\mathbf{1}+\hat{\mathbf{r}}\hat{\mathbf{r}}\right].}$$

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

$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos (\phi )\right)=J_0(p)}$$

and

$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\cos (\phi )\exp \left(ip\cos (\phi )\right)=i{J}_1(p).}$$

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

$${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{J}_0\left(kr\right)=K_0(mr).}$$

See Abramowitz and Stegun.

For ${\displaystyle mr\ll 1}$, we have

$${\displaystyle K_0(mr)\to -\ln \left(\frac{mr}{2}\right)+\mathrm{0.5772.}}$$

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

$${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{J}_1^2(kr)=I_1(mr)K_1(mr).}$$

For small mr the integral becomes

$${\displaystyle {\int }_o^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{J}_1^2(kr)\to \frac{1}{2}\left[1-\frac{1}{8}(mr{)}^2\right].}$$

For large mr the integral becomes

$${\displaystyle {\int }_o^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{J}_1^2(kr)\to \frac{1}{2}\left(\frac{1}{mr}\right).}$$

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

In general

$${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{J}_{\nu }^2(kr)=I_{\nu }(mr)K_{\nu }(mr)\mathrm{\Re }(\nu )>-1.}$$

Integration over a magnetic wave function

The two-dimensional integral over a magnetic wave function is

$${\displaystyle \frac{2a^{2n+2}}{n!}{\int }_0^{\mathrm{\infty }}dr{r}^{2n+1}\exp \left(-a^2{r}^2\right)J_0(kr)=M\left(n+1,1,-\frac{k^2}{4a^2}\right).}$$

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

Action (physics)

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Action_(physics)

In physics, action is a scalar quantity describing how a physical system has changed over time (its dynamics). Action is significant because the equations of motion of the system can be derived through the principle of stationary action.

In the simple case of a single particle moving with a constant velocity (uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is twice the particle's kinetic energy times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined.

More formally, action is a mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant h).

Introduction

Main article: Hamilton's principle

Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.

It applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.

Solution of differential equation

Empirical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion.

Minimization of action integral

Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is stationary. In other words, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.

This simple principle provides deep insights into physics, and is an important concept in modern theoretical physics.

History

Action was defined in several now obsolete ways during the development of the concept.

• Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defined the action for light as the integral of its speed or inverse speed along its path length.
• Leonhard Euler (and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space.
• Pierre Louis Maupertuis introduced several ad hoc and contradictory definitions of action within a single article, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.

Mathematical definition

Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action.

Several different definitions of "the action" are in common use in physics. The action is usually an integral over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.

The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:

$${\displaystyle \mathcal{S}={\int }_{t_1}^{t_2}Ldt,}$$

where the integrand L is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.

Action has the dimensions of [energy] × [time], and its SI unit is joule-second, which is identical to the unit of angular momentum.

Action in classical physics

In classical physics, the term "action" has a number of meanings.

Action (functional)

Most commonly, the term is used for a functional $\mathcal{S}$ which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution q(t) of the system between two times t₁ and t₂, where q represents the generalized coordinates. The action $\mathcal{S}[\mathbf{q}(t)]$ is defined as the integral of the Lagrangian L for an input evolution between the two times:

$${\displaystyle \mathcal{S}[\mathbf{q}(t)]={\int }_{t_1}^{t_2}L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)dt,}$$

where the endpoints of the evolution are fixed and defined as ${\mathbf{q}}_1=\mathbf{q}(t_1)$ and ${\mathbf{q}}_2=\mathbf{q}(t_2)$. According to Hamilton's principle, the true evolution q_true(t) is an evolution for which the action $\mathcal{S}[\mathbf{q}(t)]$ is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.

Abbreviated action (functional)

The abbreviated action is also a functional. It is usually denoted as ${\mathcal{S}}_0$. Here the input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated action ${\mathcal{S}}_0$ is defined as the integral of the generalized momenta along a path in the generalized coordinates:

$${\displaystyle {\mathcal{S}}_0=\int \mathbf{p}\cdot d\mathbf{q}=\int p_{i}d{q}_i.}$$

Spelled out concretely, this is

$${\displaystyle {\mathcal{S}}_0={\int }_{t_1}^{t_2}\mathbf{p}(t)\cdot \dot{\mathbf{q}}(t)dt={\int }_{t_1}^{t_2}{p}_i(t)\frac{d{q}_i(t)}{dt}dt.}$$

According to Maupertuis' principle, the true path is a path for which the abbreviated action ${\mathcal{S}}_0$ is stationary.

Hamilton's principal function

Main article: Hamilton–Jacobi equation

Hamilton's principal function ${\displaystyle S=S(q,t;q_0,t_0)}$ is obtained from the action functional $\mathcal{S}$ by fixing the initial time $t_0$ and the initial endpoint ${\displaystyle q_0,}$ while allowing the upper time limit $t$ and the second endpoint $q$ to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.

Hamilton's characteristic function

When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:

$${\displaystyle S(q_1,\dots ,q_N,t)=W(q_1,\dots ,q_N)-E\cdot t,}$$

where the time-independent function W(q₁, q₂, ..., q_N) is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative

$${\displaystyle \frac{dW}{dt}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{q}_i}{\dot{q}}_i=p_i{\dot{q}}_i.}$$

This can be integrated to give

$${\displaystyle W(q_1,\dots ,q_N)=\int p_i{\dot{q}}_{i}dt=\int p_{i}d{q}_i,}$$

which is just the abbreviated action.

Other solutions of Hamilton–Jacobi equations

The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., S_k(q_k), are also called an "action".

Action of a generalized coordinate

This is a single variable J_k in the action-angle coordinates, defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:

$${\displaystyle J_k=\oint p_{k}d{q}_k}$$

The variable J_k is called the "action" of the generalized coordinate q_k; the corresponding canonical variable conjugate to J_k is its "angle" w_k, for reasons described more fully under action-angle coordinates. The integration is only over a single variable q_k and, therefore, unlike the integrated dot product in the abbreviated action integral above. The J_k variable equals the change in S_k(q_k) as q_k is varied around the closed path. For several physical systems of interest, J_k is either a constant or varies very slowly; hence, the variable J_k is often used in perturbation calculations and in determining adiabatic invariants.

Action for a Hamiltonian flow

See tautological one-form.

Euler–Lagrange equations

Main article: Euler–Lagrange equations

In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.

The action principle

Main article: principle of stationary action

Classical fields

See also: Einstein–Hilbert action

The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.

The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.

The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic.

Conservation laws

Main article: Conservation laws

Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely). This deep connection requires that the action principle be assumed.

Quantum mechanics and quantum field theory

Main articles: Quantum mechanics and quantum field theory

In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes.

Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.

Maxwell's equations can also be derived as conditions of stationary action.

Single relativistic particle

Main articles: Relativistic Lagrangian mechanics and Theory of relativity

When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time $\tau$ is

$${\displaystyle S=-m{c}^2{\int }_{C}d\tau .}$$

If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t₁ to t₂, then the action becomes

$${\displaystyle S={\int }_{t1}^{t2}Ldt,}$$

where the Lagrangian is

$${\displaystyle L=-m{c}^2\sqrt{1-\frac{v^2}{c^2}}.}$$

Modern extensions

The action principle can be generalized still further. For example, the action need not be an integral, because nonlocal actions are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.