2

Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all annihilation operators. This is the definition given in many physics textbooks and also on the normal order Wikipedia page. The reason for introducing this ordering is to have a finite expectation value for the vacuum (e.g. the Casimir effect).

In mathematically rigorous treatments of quantum field theory Wick ordering is given as a probabilistic tool and is defined recursively. For example on p.9 in Simon's book on Euclidean quantum field theory he says:

Let $f$ be a random variable with finite moments. Then $:f^n:$, $n = 0, 1, \ldots$ is defined recursively by: $$:f^0: ~=~ 1 \tag{I.14a} $$ $$ \frac{\partial}{\partial f} :f^n: ~=~ n : f^{n-1}: \qquad n = 1, 2, \ldots \tag{I.14b} $$ $$ \langle :f^n: \rangle ~=~ 0 \qquad n = 1, 2, \ldots \tag{I.14c}$$

I have also seen this arXiv paper by Wurum and Berg titles "Wick Calculus" and this MathOverflow question. They give some brief motivation and give an analogy between Wick ordering and Hermite polynomials, but I am struggling to see how the probabilistic definition of Wick ordering given above has anything to do with the physicists idea of putting creating operators before annihilation operators. How does this version of Wick ordering help make things finite?

Qmechanic
  • 220,844
CBBAM
  • 4,632

1 Answers1

2

  1. For what it's worth, given a family $(\hat{A}_i)_{i\in I}$ of operators $\hat{A}_i\in{\cal A}$ in a (super) operator algebra ${\cal A}$ there seems to be an implicit/tacit assumption that the $n$th operator product $$ \hat{A}_{i_1}\ldots\hat{A}_{i_n}~=~:P_n(\hat{A}_i): \tag{A}$$ is a normal-ordered $n$-order polynomial in the $\hat{A}_i$s. (Or some equivalent assumption.)

  2. Here the $0$th-order term on the RHS of eq. (A) is proportional to the identity operator $\hat{\bf 1}$, due to eq. (I.14a).

  3. The coefficient of the $0$th-order term is the $n$-point functions $\langle \hat{A}_{i_1}\ldots\hat{A}_{i_n} \rangle$, due to eq. (I.14c).

  4. Eq. (I.14b) is used when we apply the (super) commutator $[\frac{\partial}{\partial \hat{A}_k}, \cdot]$ on both sides of eq. (A).

  5. Example 1. For $n=1$, we have a 1st-order polynomial $$ \hat{A}_i~=~ \sum_ja_i^j :\hat{A}_j: ~+~ \langle \hat{A}_i\rangle\hat{\bf 1} \tag{B}$$ From eq. (I.14b) it follows that the coefficients $a_i^j=\delta_i^j$, so that $$ \hat{A}_i~=~:\hat{A}_i: ~+~ \langle \hat{A}_i\rangle\hat{\bf 1} \tag{C}$$

  6. Example 2. For $n=2$, we have a 2nd-order polynomial $$\begin{align} \hat{A}_i\hat{A}_j~=~& \sum_{k\ell}a_{ij}^{k\ell} :\hat{A}_k\hat{A}_{\ell}: \cr ~+~&\sum_kb_{ij}^k :\hat{A}_k: ~+~ \langle \hat{A}_i\hat{A}_j\rangle\hat{\bf 1} \end{align}\tag{D}$$ When we apply the (super) commutator $[\frac{\partial}{\partial \hat{A}_k}, \cdot]$ on both sides of eq. (D), we get $$\begin{align} \delta^k_i\underbrace{\hat{A}_j}_{:\hat{A}_j: ~+~ \langle \hat{A}_j\rangle\hat{\bf 1}}~+~&(-1)^{|i||j|}(i \leftrightarrow j)\cr ~=~& \sum_{\ell}\underbrace{\left( a_{ij}^{k\ell}~+~(-1)^{|k||\ell|}a_{ij}^{\ell k}\right)}_{=2a_{ij}^{k\ell}}:\hat{A}_{\ell}:\cr ~+~& b_{ij}^k \hat{\bf 1}\end{align} \tag{E}$$ It follows that the coefficients are $$\begin{align}b_{ij}^k~=~&\delta^k_i \langle \hat{A}_j\rangle~+~(-1)^{|i||j|}(i \leftrightarrow j),\cr a_{ij}^{k\ell}~=~&\frac{1}{2}\delta^k_i\delta^{\ell}_j~+~(-1)^{|k||\ell|}(k \leftrightarrow \ell),\end{align}\tag{F}$$ so that $$\begin{align} \hat{A}_i\hat{A}_j~=~&:\hat{A}_i\hat{A}_j: ~+~\langle \hat{A}_i\rangle :\hat{A}_j: \cr ~+~& :\hat{A}_i: \langle \hat{A}_j\rangle ~+~ \langle \hat{A}_i\hat{A}_j\rangle\hat{\bf 1}\end{align}\tag{G}$$

  7. Normally one also assumes that the $1$-point functions $\langle \hat{A}_i\rangle$ vanish: $$\forall i\in I~: \qquad \langle \hat{A}_i\rangle~=~0, \tag{H}$$ and that the commutators $[\hat{A}_i,\hat{A}_j]$ belong to the center $Z({\cal A})$ of the operator algebra $$ \forall i,j\in I~: \qquad [\hat{A}_i,\hat{A}_j] ~\in~Z({\cal A}).\tag{I}$$

  8. Eqs. (C) and (G) are then similar to the usual Wick's theorems, cf. e.g. this & this Phys.SE posts.

Qmechanic
  • 220,844