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Concept
Wick's theorem
Summary
Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. This allows for the use of Green's function methods, and consequently the use of Feynman diagrams in the field under study. A more general idea in probability theory is Isserlis' theorem.
In perturbative quantum field theory, Wick's theorem is used to quickly rewrite each time ordered summand in the Dyson series as a sum of normal ordered terms. In the limit of asymptotically free ingoing and outgoing states, these terms correspond to Feynman diagrams.
For two operators and we define their contraction to be
where denotes the normal order of an operator . Alternatively, contractions can be denoted by a line joining and , like .
We shall look in detail at four special cases where and are equal to creation and annihilation operators. For particles we'll denote the creation operators by and the annihilation operators by .
They satisfy the commutation relations for bosonic operators , or the anti-commutation relations for fermionic operators where denotes the Kronecker delta.
We then have
where .
These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
We can use contractions and normal ordering to express any product of creation and annihilation operators as a sum of normal ordered terms. This is the basis of Wick's theorem. Before stating the theorem fully we shall look at some examples.
Suppose and are bosonic operators satisfying the commutation relations:
where , denotes the commutator, and is the Kronecker delta.
We can use these relations, and the above definition of contraction, to express products of and in other ways.
Note that we have not changed but merely re-expressed it in another form as
In the last line we have used different numbers of symbols to denote different contractions.
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Wick's theorem
Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. This allows for the use of Green's function methods, and consequently the use of Feynman diagrams in the field under study. A more general idea in probability theory is Isserlis' theorem.
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Explores the Renormalization Group in field theory, discussing scaling functions, critical exponents, and Gaussian fixed points.
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Covers normal ordered product, Wick's theorem, creation and destruction fields, and efficient computation methodology.
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Covers cross section, lifetime, quantum fluid, asymptotic states, discrete symmetries, and normal ordering in quantum field theory.