Wick's theorem (in quantum electrodynamics)

Wick's theorem (in quantum electrodynamics) is a statement that allows one to calculate the elements $S$ ${\ displaystyle S}$ $S$ - matrices in $n$ ${\ displaystyle n}$ $n$ perturbation theory order.

Wick's theorem was formulated and proved by D. Wick in 1950. ^[1] ^[2]

As you know, the transition matrix element has the form:

\langle f|S^{(n)}|i\rangle ={\frac {1}{n!}}(-ie)^{n}\int d^{4}x_{1}\ldots d^{4}x_{n}\times \langle 0|\ldots b_{2f}b_{1f}...a_{1f}\ldots c_{1f}T({\bar {\psi }}_{1}\gamma A_{1}\psi _{1})\times \ldots \times ({\bar {\psi }}_{n}\gamma A_{n}\psi _{n})c_{1i}^{+}\ldots a_{1i}^{+}\ldots b_{1i}^{+}\ldots |0\rangle .

{\ displaystyle \ langle f | S ^ {(n)} | i \ rangle = {\ frac {1} {n!}} (- ie) ^ {n} \ int d ^ {4} x_ {1} \ ldots d ^ {4} x_ {n} \ times \ langle 0 | \ ldots b_ {2f} b_ {1f} ... a_ {1f} \ ldots c_ {1f} T ({\ bar {\ psi}} _ {1} \ gamma A_ {1} \ psi _ {1}) \ times \ ldots \ times ({\ bar {\ psi}} _ {n} \ gamma A_ {n} \ psi _ {n}) c_ { 1i} ^ {+} \ ldots a_ {1i} ^ {+} \ ldots b_ {1i} ^ {+} \ ldots | 0 \ rangle.}

{\ displaystyle \ langle f | S ^ {(n)} | i \ rangle = {\ frac {1} {n!}} (- ie) ^ {n} \ int d ^ {4} x_ {1} \ ldots d ^ {4} x_ {n} \ times \ langle 0 | \ ldots b_ {2f} b_ {1f} ... a_ {1f} \ ldots c_ {1f} T ({\ bar {\ psi}} _ {1} \ gamma A_ {1} \ psi _ {1}) \ times \ ldots \ times ({\ bar {\ psi}} _ {n} \ gamma A_ {n} \ psi _ {n}) c_ { 1i} ^ {+} \ ldots a_ {1i} ^ {+} \ ldots b_ {1i} ^ {+} \ ldots | 0 \ rangle.}

Indices $1i,2i,\ldots$ ${\ displaystyle 1i, 2i, \ ldots}$ ${\ displaystyle 1i, 2i, \ ldots}$ the initial particles are numbered, and $1f,2f,\ldots$ ${\ displaystyle 1f, 2f, \ ldots}$ ${\ displaystyle 1f, 2f, \ ldots}$ - final. Indices $1,2,\ldots$ ${\ displaystyle 1,2, \ ldots}$ $1,2, \ ldots$ at operators $\psi$ ${\ displaystyle \ psi}$ $\ psi$ and $A$ ${\ displaystyle A}$ $A$ mean $\psi _{1}=\psi (x_{1})$ ${\ displaystyle \ psi _ {1} = \ psi (x_ {1})}$ ${\ displaystyle \ psi _ {1} = \ psi (x_ {1})}$ etc. $T$ ${\ displaystyle T}$ $T$ - a symbol of the chronological work of operators.

Content

Wording

Wick's theorem states that the vacuum average of any number of boson operators is equal to the sum of the products of all possible pairwise means of these operators. In this case, in each pair, the factors should be in the same sequence as in the original product. For fermionic operators, each member of the sum comes in with a plus or minus sign, depending on whether the number of permutations is even or odd, necessary to put all averaged operators together ^[3] .

Proof

Define as the normal product of several operators $N(AB...YZ)$ ${\ displaystyle N (AB ... YZ)}$ , in which all creation operators are to the left of the annihilation operators, and the plus or minus sign depends on whether the even or odd permutation of the Fermi operators leads to this kind of product. We define as the double product of two operators $A^{*}B^{*}=T(AB)-N(AB)$ ${\ displaystyle A ^ {*} B ^ {*} = T (AB) -N (AB)}$ . Wick's theorem states that the chronological product of any number of operators can be represented as the sum of normal products with all possible doublings

T(AB\ldots YZ)=N(AB\ldots YZ)+N(A^{*}B^{*}CD\ldots YZ)+N(A^{*}BC^{*}D\ldots YZ)+\ldots +N(A^{*}B^{**}C^{***}D\ldots X^{*}Y^{**}Z^{***}).

{\ displaystyle T (AB \ ldots YZ) = N (AB \ ldots YZ) + N (A ^ {*} B ^ {*} CD \ ldots YZ) + N (A ^ {*} BC ^ {*} D \ ldots YZ) + \ ldots + N (A ^ {*} B ^ {**} C ^ {***} D \ ldots X ^ {*} Y ^ {**} Z ^ {***}) .}

Thus, the chronological product of the operators is equal to the normal product, plus the sum of the normal products with one doubling, where the pair should be chosen in all possible ways, plus the sum of the normal products with two doubling, where the two doubling pairs should be chosen in all possible ways, etc. In order to convert a chronological product into a normal one, it is necessary to rearrange all birth operators with the annihilation operators in front of them. This gives the formula of the above form. It will include doubling only those operators whose order in the chronological product is different from the order in the normal product. Since doublings of operators for which both orders are equivalent are equal to zero, we can assume that the right-hand side of the formula includes normal products with all possible doublings. ^[four]

Notes

↑ Wick G. C. The Evaluation of the Collision Matrix // Phys. Rev. - 1950. - V. 80. - P. 268—272. - URL: http://dx.doi.org/10.1103/PhysRev.80.268
↑ Vick D. Calculation of the collision matrix // The latest development of quantum electrodynamics. Collection of articles, ed. D. D. Ivanenko. — M.: IL. — 1954.— S. 245—253.
↑ Bilenky, 1971 , p. 83.
↑ Sadovsky MV, Lectures on Quantum Field Theory, 2003, 480 pp., ISBN 5-93972-241-5

Literature

Berestetsky V.V. , Lifshits E.M. , Pitaevsky L.P. Quantum electrodynamics.- M .: Fizmatlit. - 2001.— 720 p., ISBN 5-9221-0058-0
Schweber S., Bethe G. , Hoffman F. Mesons and Fields, Volume 1.— 1957.
Bilenky S. M. Introduction to the Feynman diagram technique. - M .: Atomizdat, 1971. - 213 p.

[1] Wick G. C. The Evaluation of the Collision Matrix // Phys. Rev. - 1950. - V. 80. - P. 268—272. - URL: http://dx.doi.org/10.1103/PhysRev.80.268

[2] Vick D. Calculation of the collision matrix // The latest development of quantum electrodynamics. Collection of articles, ed. D. D. Ivanenko. — M.: IL. — 1954.— S. 245—253.

[_16b6c69ab3a46f42-3] Bilenky, 1971 , p. 83.

[4] Sadovsky MV, Lectures on Quantum Field Theory, 2003, 480 pp., ISBN 5-93972-241-5