Wick's theorem

In plain terms

Quantum field theory computes almost everything from one bookkeeping rule. A correlation function is the average of a product of several field operators, quantities such as the value of the field at chosen places and times, and Wick's theorem says any such average reduces to a sum over the ways of pairing the operators up, with each pair contributing one basic building block, the propagator, which is the correlation between just two of them. The rule is usually proved with an infinite ladder of energy levels available for every mode of the field.

A finite arena must cut that ladder off at some number of quanta, the units of excitation, per mode, so the question becomes quantitative, whether the pairing formula survives and where exactly it breaks. The corpus's answer is that it survives word for word below an explicit threshold, and that at the threshold it fails by a computed amount. The reason is combinatorial, as a correlation of a given order is a sum over walks up and down the ladder that return to where they started, and a walk with a fixed number of steps can climb only so far, so it never reaches the missing rungs until the order is large enough to get there.

An approximation with an error bar is replaced by exactness with a boundary.

In CSD

This is the field track, the part of the programme that treats CSD's arena as a finite quantum field, at its most textbook. Four-point functions at equal and at distinct times close into the pairing sum, the two-time kernel carries the time dependence, and the full moment ladder, the double-factorial law for arbitrary even powers of a field mode (a double factorial is the product of every second whole number), is proved by a recursion in which the defect caused by the cutoff is eliminated by the walk argument rather than ignored.

The boundary is stated plainly. The one-shot sum-over-matchings formula for arbitrary interleaved products of operators isn't claimed, and the double-factorial law carries that content on the mode diagonal, the case of repeated powers of a single mode.

Mathematically

The theorem timeFourPoint_wick states that the distinct-times four-point function equals the pairing sum of two-time kernels over every mode pattern, for cutoffs above two.

The moment ladder states that the vacuum expectation of Q to the 2n is (2n-1) double factorial times one-half to the n, exactly, for n below the cutoff, via the recursion against the truncated commutation relation, and odd moments vanish by walk parity while grouped mode patterns factorise by clustering.

The name

Gian Carlo Wick (1909-1992) was born in Turin, worked alongside Fermi in Rome in the 1930s, and after the war held positions at Berkeley, which he left rather than sign the loyalty oath, then at Carnegie Tech, Brookhaven and Columbia. The 1950 theorem on evaluating products of field operators, published in the Physical Review, systematised the calculations of the newly renormalised quantum electrodynamics and has been the daily tool of the subject ever since.

Related entries

• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• KMS conditionThe algebraic signature of thermal equilibrium holds exactly on CSD's finite field arena, and cooling the thermal state all the way down recovers the vacuum.
• No exact finite-dimensional CCRIn a world with only finitely many distinguishable states, the founding equation of quantum mechanics can't hold exactly, and one line of arithmetic proves it while CSD says precisely where the equation gives way.
• Lieb-Robinson boundEven without relativity, influence in a quantum system with local interactions has a speed limit, and on CSD's finite field arena that limit is a theorem with the velocity written out.
• Gibbs stateThe state a system settles into at a given temperature is the unique solution to a trade-off between energy and disorder.