Many-sorted logic

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Definition of contraction

For two operators ${\hat {A}}$ and ${\hat {B}}$ we define their contraction to be

${\hat {A}}^{\bullet }\,{\hat {B}}^{\bullet }\equiv {\hat {A}}\,{\hat {B}}\,-{\mathopen {:}}{\hat {A}}\,{\hat {B}}{\mathclose {:}}$

where ${\mathopen {:}}{\hat {O}}{\mathclose {:}}$ denotes the normal order of an operator ${\hat {O}}$.

Alternatively, contractions can be denoted by a line joining ${\hat {A}}$ and ${\hat {B}}$.

We shall look in detail at four special cases where ${\hat {A}}$ and ${\hat {B}}$ are equal to creation and annihilation operators. For $N$ particles we'll denote the creation operators by ${\hat {a}}_{i}^{\dagger }$ and the annihilation operators by ${\hat {a}}_{i}$ ($i=1,2,3\ldots ,N$). They satisfy the usual commutation relations $[{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }]=\delta _{ij}$, where $\delta _{ij}$ denotes the Kronecker delta.

We then have

${\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\bullet }={\hat {a}}_{i}\,{\hat {a}}_{j}\,-{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}\,{\mathclose {:}}\,=0$

${\hat {a}}_{i}^{\dagger \bullet }\,{\hat {a}}_{j}^{\dagger \bullet }={\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}^{\dagger }\,-\,{\mathopen {:}}{\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}\,=0$

${\hat {a}}_{i}^{\dagger \bullet }\,{\hat {a}}_{j}^{\bullet }={\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}\,-{\mathopen {:}}\,{\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}\,{\mathclose {:}}\,=0$

${\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }={\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,-{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}\,=\delta _{ij}$

where $i,j=1,\ldots ,N$.

These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.

Examples

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 ${\hat {a}}_{i}$ and ${\hat {a}}_{i}^{\dagger }$ are bosonic operators satisfying the commutation relations:

$\left[{\hat {a}}_{i}^{\dagger },{\hat {a}}_{j}^{\dagger }\right]=0$

$\left[{\hat {a}}_{i},{\hat {a}}_{j}\right]=0$

$\left[{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }\right]=\delta _{ij}$

where $i,j=1,\ldots ,N$, $\left[{\hat {A}},{\hat {B}}\right]\equiv {\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}$ denotes the commutator, and $\delta _{ij}$ is the Kronecker delta.

We can use these relations, and the above definition of contraction, to express products of ${\hat {a}}_{i}$ and ${\hat {a}}_{i}^{\dagger }$ in other ways.

Example 1

${\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }=\,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }$

Note that we have not changed ${\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }$ but merely re-expressed it in another form as $\,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }$

Example 2

${\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}=({\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}){\hat {a}}_{k}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{k}+\delta _{ij}{\hat {a}}_{k}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{k}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }{\hat {a}}_{k}=\,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }{\hat {a}}_{k}\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}{\mathclose {:}}$

Example 3

${\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }=({\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij})({\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl})$

$={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}$

$={\hat {a}}_{j}^{\dagger }({\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{i}+\delta _{il}){\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}$

$={\hat {a}}_{j}^{\dagger }{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{i}{\hat {a}}_{k}+\delta _{il}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}$

$=\,{\mathopen {:}}{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger \bullet }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}^{\bullet }\,{\hat {a}}_{l}^{\dagger \bullet }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }\,{\mathclose {:}}+\,{\mathopen {:}}{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}^{\bullet \bullet }\,{\hat {a}}_{l}^{\dagger \bullet \bullet }{\mathclose {:}}$

In the last line we have used different numbers of $^{\bullet }$ symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express ${\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }$ in the form of a sum of normally ordered products. It is an even lengthier calculation for more complicated products.

Luckily Wick's theorem provides a shortcut.

Statement of the theorem

A product of creation and annihilation operators ${\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots$ can be expressed as

${\begin{aligned}{\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots &={\mathopen {:}}{\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\sum _{\text{singles}}{\mathopen {:}}{\hat {A}}^{\bullet }{\hat {B}}^{\bullet }{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\sum _{\text{doubles}}{\mathopen {:}}{\hat {A}}^{\bullet }{\hat {B}}^{\bullet \bullet }{\hat {C}}^{\bullet \bullet }{\hat {D}}^{\bullet }{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\ldots \end{aligned}}$

In other words, a string of creation and annihilation operators can be rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.

Applying the theorem to the above examples provides a much quicker method to arrive at the final expressions.

A warning: In terms on the right hand side containing multiple contractions care must be taken when the operators are fermionic. In this case an appropriate minus sign must be introduced according to the following rule: rearrange the operators (introducing minus signs whenever the order of two fermionic operators is swapped) to ensure the contracted terms are adjacent in the string. The contraction can then be applied (See Rule C″ in Wick's paper).

Example:

If we have two fermions ($N=2$) with creation and annihilation operators ${\hat {f}}_{i}^{\dagger }$ and ${\hat {f}}_{i}$ ($i=1,2$) then

${\begin{array}{ll}{\hat {f}}_{1}\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger }\,&=\,{\mathopen {:}}{\hat {f}}_{1}\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger }\,{\mathclose {:}}\\&-\,{\mathopen {:}}{\hat {f}}_{1}^{\bullet }\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger \bullet }\,{\hat {f}}_{2}^{\dagger }\,{\mathclose {:}}+\,{\mathopen {:}}{\hat {f}}_{1}^{\bullet }\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger \bullet }\,{\mathclose {:}}+\,{\mathopen {:}}{\hat {f}}_{1}\,{\hat {f}}_{2}^{\bullet }\,{\hat {f}}_{1}^{\dagger \bullet }\,{\hat {f}}_{2}^{\dagger }\,{\mathclose {:}}-{\mathopen {:}}{\hat {f}}_{1}\,{\hat {f}}_{2}^{\bullet }\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger \bullet }\,{\mathclose {:}}\\&-{\mathopen {:}}{\hat {f}}_{1}^{\bullet \bullet }\,{\hat {f}}_{2}^{\bullet }\,{\hat {f}}_{1}^{\dagger \bullet \bullet }\,{\hat {f}}_{2}^{\dagger \bullet }\,{\mathclose {:}}+{\mathopen {:}}{\hat {f}}_{1}^{\bullet \bullet }\,{\hat {f}}_{2}^{\bullet }\,{\hat {f}}_{1}^{\dagger \bullet }\,{\hat {f}}_{2}^{\dagger \bullet \bullet }{\mathclose {:}}\end{array}}$

Note that the term with contractions of the two creation operators and of the two annihilation operators is not included because their contractions vanish.

Wick's theorem applied to fields

${\mathcal {C}}(x_{1},x_{2})=\left\langle 0|{\mathcal {T}}\phi _{i}(x_{1})\phi _{i}(x_{2})|0\right\rangle ={\overline {\phi _{i}(x_{1})\phi _{i}(x_{2})}}=i\Delta _{F}(x_{1}-x_{2})=i\int {{\frac {d^{4}k}{(2\pi )^{4}}}{\frac {e^{-ik(x_{1}-x_{2})}}{(k^{2}-m^{2})+i\epsilon }}}.$

Which means that ${\overline {AB}}={\mathcal {T}}AB-{\mathopen {:}}AB{\mathclose {:}}$

In the end, we arrive at Wick's theorem:

The T-product of a time-ordered free fields string can be expressed in the following manner:

${\mathcal {T}}\Pi _{k=1}^{m}\phi (x_{k})={\mathopen {:}}\Pi \phi _{i}(x_{k}){\mathclose {:}}+\sum _{\alpha ,\beta }{\overline {\phi (x_{\alpha })\phi (x_{\beta })}}{\mathopen {:}}\Pi _{k\not =\alpha ,\beta }\phi _{i}(x_{k}){\mathclose {:}}+$

${\mathcal {+}}\sum _{(\alpha ,\beta ),(\gamma ,\delta )}{\overline {\phi (x_{\alpha })\phi (x_{\beta })}}\;{\overline {\phi (x_{\gamma })\phi (x_{\delta })}}{\mathopen {:}}\Pi _{k\not =\alpha ,\beta ,\gamma ,\delta }\phi _{i}(x_{k}){\mathclose {:}}+\cdots .$

Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on vacuum state give a null contribution to the sum. We conclude that m is even and only completely contracted terms remain.

$F_{m}^{i}(x)=\left\langle 0|{\mathcal {T}}\phi _{i}(x_{1})\phi _{i}(x_{2})|0\right\rangle =\sum _{\mathrm {pairs} }{\overline {\phi (x_{1})\phi (x_{2})}}\cdots {\overline {\phi (x_{m-1})\phi (x_{m}}})$

$G_{p}^{(n)}=\left\langle 0|{\mathcal {T}}{\mathopen {:}}v_{i}(y_{1}){\mathclose {:}}\dots {\mathopen {:}}v_{i}(y_{n}){\mathclose {:}}\phi _{i}(x_{1})\cdots \phi _{i}(x_{p})|0\right\rangle$

where p is the number of interaction fields (or, equivalently, the number of interacting particles) and n is the development order (or the number of vertices of interaction). For example, if $v=gy^{4}\Rightarrow {\mathopen {:}}v_{i}(y_{1}){\mathclose {:}}={\mathopen {:}}\phi _{i}(y_{1})\phi _{i}(y_{1})\phi _{i}(y_{1})\phi _{i}(y_{1}){\mathclose {:}}$

This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.

Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the vacuum expectation values (VEV's) of fields. (Wick's theorem provides as a way of expressing VEV's of n fields in terms of VEV's of two fields.) There are any other possible definitions of normal ordering, and Wick's theorem is valid irrespective. However Wick's theorem only simplifies computations if the definition of normal ordering used is changed to match the type of expectation value wanted. That is we always want the expectation value of the normal ordered product to be zero. For instance in thermal field theory a different type of expectation value, a thermal trace over the density matrix, requires a different definition of normal ordering.

References

• G.C. Wick, The Evaluation of the Collision Matrix, Phys. Rev. 80, 268 - 272 (1950)
• Silvan S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York (1962). (Chapter 13, Sec c)
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books (1995). (§4.3)
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• T.S. Evans, D.A. Steer, Wick's theorem at finite temperature, Nucl. Phys B 474, 481-496 (1996) arXiv:hep-ph/9601268