Common integrals in quantum field theory

Common integrals in quantum field theory are set of formulas that are useful for computation of various types in quantum field theory such as partition function, integrals of loop diagrams, etc. == Gaussian integrals == The following Gaussian integrals are useful in calculating path integrals appearing in path integral formulation of quantum field theory: ∫ − ∞ ∞ e − 1 2 a x 2 + J x d x = ( 2 π a ) 1 / 2 exp ⁡ ( J 2 2 a ) , a , J ∈ C , Re ⁡ ( a ) > 0 ∫ − ∞ ∞ exp ⁡ ( i ( 1 2 ( a + i ε ) x 2 + J x ) ) d x = ( 2 π i a + i ε ) 1 / 2 exp ⁡ ( − i 2 J 2 a + i ε ) , a , J , ε ∈ R , ε → 0 + ∫ exp ⁡ ( ∑ i , j = 1 n − 1 2 x i A i j x j + J i x i ) d n x = ( 2 π ) n det A exp ⁡ ( 1 2 ∑ i , j = 1 n J i A i j − 1 J j ) , A , J ∈ R , A i j = A j i positive definite ∫ exp ⁡ ( i ( ∑ i , j = 1 n 1 2 x i ( A + i ε I ) i j x j + J i x i ) ) d n x = ( 2 π ) n det ( A + i ε I ) exp ⁡ ( − i 2 ∑ i , j = 1 n J i ( A + i ε I ) i j − 1 J j ) , A , J , ε ∈ R , A i j = A j i , ε → 0 + {\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }e^{-{1 \over 2}ax^{2}+Jx}\,dx&=\left({2\pi \over a}\right)^{1/2}\exp \left({J^{2} \over 2a}\right),&a,J\in \mathbb {C} ,\,\operatorname {Re} (a)>0\\\int _{-\infty }^{\infty }\exp \left(i\left({\frac {1}{2}}(a+i\varepsilon )x^{2}+Jx\right)\right)dx&=\left({2\pi i \over a+i\varepsilon }\right)^{1/2}\exp \left(-{\frac {i}{2}}{J^{2} \over a+i\varepsilon }\right),&a,J,\varepsilon \in \mathbb {R} ,\,\varepsilon \rightarrow 0^{+}\\\int \exp \left(\sum _{i,j=1}^{n}-{\frac {1}{2}}x_{i}A_{ij}x_{j}+J_{i}x_{i}\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det A}}}\exp \left({1 \over 2}\sum _{i,j=1}^{n}J_{i}A_{ij}^{-1}J_{j}\right),&A,J\in \mathbb {R} ,\,A_{ij}=A_{ji}{\text{ positive definite}}\\\int \exp \left(i\left(\sum _{i,j=1}^{n}{\frac {1}{2}}x_{i}(A+i\varepsilon I)_{ij}x_{j}+J_{i}x_{i}\right)\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det {(A+i\varepsilon I)}}}}\exp \left(-{\frac {i}{2}}\sum _{i,j=1}^{n}J_{i}(A+i\varepsilon I)_{ij}^{-1}J_{j}\right),&A,J,\varepsilon \in \mathbb {R} ,\,A_{ij}=A_{ji},\,\varepsilon \rightarrow 0^{+}\\\end{aligned}}} === Integrals with differential operators in the argument === As an example consider the integral ∫ exp ⁡ [ ∫ d 4 x ( − 1 2 φ A ^ φ + J φ ) ] D φ {\displaystyle \int \exp \left[\int d^{4}x\left(-{\frac {1}{2}}\varphi {\hat {A}}\varphi +J\varphi \right)\right]D\varphi } where A ^ {\displaystyle {\hat {A}}} is a Hermitian differential operator with positive spectra for convergence, φ {\displaystyle \varphi } and J functions of spacetime, and D φ {\displaystyle D\varphi } indicates integration over all possible paths.

Source: Wikipedia — Common integrals in quantum field theory (CC BY-SA 4.0)

Common integrals in quantum field theory

Common integrals in quantum field theory are set of formulas that are useful for computation of various types in quantum field theory such as partition function, integrals of loop diagrams, etc. == Gaussian integrals == The following Gaussian integrals are useful in calculating path integrals appearing in path integral formulation of quantum field theory: ∫ − ∞ ∞ e − 1 2 a x 2 + J x d x = ( 2 π a ) 1 / 2 exp ⁡ ( J 2 2 a ) , a , J ∈ C , Re ⁡ ( a ) > 0 ∫ − ∞ ∞ exp ⁡ ( i ( 1 2 ( a + i ε ) x 2 + J x ) ) d x = ( 2 π i a + i ε ) 1 / 2 exp ⁡ ( − i 2 J 2 a + i ε ) , a , J , ε ∈ R , ε → 0 + ∫ exp ⁡ ( ∑ i , j = 1 n − 1 2 x i A i j x j + J i x i ) d n x = ( 2 π ) n det A exp ⁡ ( 1 2 ∑ i , j = 1 n J i A i j − 1 J j ) , A , J ∈ R , A i j = A j i positive definite ∫ exp ⁡ ( i ( ∑ i , j = 1 n 1 2 x i ( A + i ε I ) i j x j + J i x i ) ) d n x = ( 2 π ) n det ( A + i ε I ) exp ⁡ ( − i 2 ∑ i , j = 1 n J i ( A + i ε I ) i j − 1 J j ) , A , J , ε ∈ R , A i j = A j i , ε → 0 + {\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }e^{-{1 \over 2}ax^{2}+Jx}\,dx&=\left({2\pi \over a}\right)^{1/2}\exp \left({J^{2} \over 2a}\right),&a,J\in \mathbb {C} ,\,\operatorname {Re} (a)>0\\\int _{-\infty }^{\infty }\exp \left(i\left({\frac {1}{2}}(a+i\varepsilon )x^{2}+Jx\right)\right)dx&=\left({2\pi i \over a+i\varepsilon }\right)^{1/2}\exp \left(-{\frac {i}{2}}{J^{2} \over a+i\varepsilon }\right),&a,J,\varepsilon \in \mathbb {R} ,\,\varepsilon \rightarrow 0^{+}\\\int \exp \left(\sum _{i,j=1}^{n}-{\frac {1}{2}}x_{i}A_{ij}x_{j}+J_{i}x_{i}\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det A}}}\exp \left({1 \over 2}\sum _{i,j=1}^{n}J_{i}A_{ij}^{-1}J_{j}\right),&A,J\in \mathbb {R} ,\,A_{ij}=A_{ji}{\text{ positive definite}}\\\int \exp \left(i\left(\sum _{i,j=1}^{n}{\frac {1}{2}}x_{i}(A+i\varepsilon I)_{ij}x_{j}+J_{i}x_{i}\right)\right)d^{n}x&={\sqrt {\frac {(2\pi )^{n}}{\det {(A+i\varepsilon I)}}}}\exp \left(-{\frac {i}{2}}\sum _{i,j=1}^{n}J_{i}(A+i\varepsilon I)_{ij}^{-1}J_{j}\right),&A,J,\varepsilon \in \mathbb {R} ,\,A_{ij}=A_{ji},\,\varepsilon \rightarrow 0^{+}\\\end{aligned}}} === Integrals with differential operators in the argument === As an example consider the integral ∫ exp ⁡ [ ∫ d 4 x ( − 1 2 φ A ^ φ + J φ ) ] D φ {\displaystyle \int \exp \left[\int d^{4}x\left(-{\frac {1}{2}}\varphi {\hat {A}}\varphi +J\varphi \right)\right]D\varphi } where A ^ {\displaystyle {\hat {A}}} is a Hermitian differential operator with positive spectra for convergence, φ {\displaystyle \varphi } and J functions of spacetime, and D φ {\displaystyle D\varphi } indicates integration over all possible paths.

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