Normal order

Since quantum mechanical Hamiltonians consist of operators, they depend on the order of these. When quantizing a classical Hamiltonian one therefore has some freedom when choosing the operator order, and these choices lead to differences in ground state energy.

Normal order only applies to free field theories. The normal order of the operators is the choice that leads to zero ground state energy. It puts all annihilation operators to the right, and all creation operators to the left, leading to a ground state expectation value of 0:

<math>\langle 0 | H | 0 \rangle = 0<math>

if H is in normal order.

The operator for putting an expression in normal order is N. It has the effect of moving (by commutation relations) all creation operators to the left and all annihilation operators to the right with all commutators (for bosons) and anticommutators (for fermions) temporarily set to 0.

No matter what order some expression K is in,

<math>0 = \langle 0 | N(K) | 0 \rangle <math>.

Another notation is :K:.

With two free fields φ and χ,

<math>:\phi(x)\chi(y):=\phi(x)\chi(y)-<\Omega|\phi(x)\chi(y)|\Omega><math>

where |Ω> is the vacuum state. Each of the two terms on the right hand side typically blows up in the limit as y approaches x but the difference between them has a well-defined limit. This allows us to define :φ(x)χ(x):.

Wick's theorem states that

<math>\phi_{i_1}(x_1)\cdots \phi_{i_N}(x_N)=\sum_{all possible pairs of contractions}:\phi_{i_1}(x_1)\cdots \phi_{i_N}(x_N):<math>

(with contractions).