Appendix A

Wick’s Theorem

The quantum kinetic equations in Chapter 2 for the mean field Eq. (2.70) and fluctuationsEq. (2.71) and Eq. (2.72) were derived from the Born-Markov form of the generalizedkinetic equation Eq. (2.60) for a general relevant observable q . We do not provide thedetails in getting from Eq. (2.60) to Eq. (2.70) through Eq. (2.72), however it is importantto understand the main steps, which involve evaluating the averages of multiple operatorproducts. Due to the Gaussian structure of our reference distribution �(0)(t) in Eq. (2.9),we can utilize Wick’s theorem to simplify the calculation [42].

We are interested in calculating the average of a product of s operators

hA1A2 � � � Asit(0) = Trf�(0)(t) A1A2 � � � Asg; (A.1)

where we are denoting Ai to represent either a raising ayi or a lowering operator ai. Theseaverages arise in the kinetic equations. For example, in the kinetic equation for the normalfluctuations, the second order collisional terms involve a product of ten operators due toterms of the form� hV 2 ~fiy

(0), since the interaction V contains a product of four operators

V � ayi ay

j akal and the normal fluctuations have the form ~f � ayj ai.Wick’s theorem is valid for averages taken over a Gaussian distribution that can involve

both normal ayj ai and anomalous pairs aiaj in the exponent, as our reference distributiondoes in Eq. (2.9). We do not derive Wick’s theorem here, but simply state the results. Theinterested reader can find derivations of Wick’s theorem in Louisell [119] (p. 182) for thecase of bosons, and in Zubarev [39] (p. 172) for fermions. Both of these derivations assumea Gaussian form with only normal pairs ayj ai in the exponent; one can find a more generalderivation including anomalous pairs (for a squeezed vacuum) in Vaglica [43]. To includethe anomalous pairs in the proof, one can simply make a canonical transformation of theoperators ai and ayi so that only normal pairs appear in the exponent for the new set ofoperators; doing this allows one to use the standard proof, which treats only normal pairs.

Wick’s theorem states that the average value of a product of creation and annihilationoperators is equal to the sum of all complete systems of pairings, which can be stated moreformally as

hA1A2 � � � Asit(0) =XPd

hA1A2it(0)hA3A4it(0) � � � hAs�1Asit(0); (A.2)

where the sum runs over all Pd distinct permutations of the s indices. An alternative way

Appendix A Wick’s Theorem

to write Eq. (A.2) is

hA1A2 � � � Asit(0) = hA1A2it(0)hA3A4 � � � Asit(0) + hA1A3it(0)hA2A4 � � � Asit(0)+ hA1Asit(0)hA2A3 � � � As�1it(0); (A.3)

and then applying this relation recursively to all of the multiple operator averages until onlypairs of operators remain. If the number of operators s is odd, Wick’s theorem reduces tothe simple result

hA1A2 � � � Asit(0) = 0 if s is odd: (A.4)

Wick’s theorem is augmented by the further rules for averages of pairs of operators, whichhold for bosons

ha1ay2it(0) = Æ12 + hay1a2it(0);ha1a2it(0) = ha2a1it(0);hay1ay2it(0) = hay2ay1it(0): (A.5)

As an example, we consider the product of four operators hA1A2A3A4it(0), which can bereduced according to Wick’s theorem to

hA1A2A3A4it(0) = hA1A2it(0)hA3A4it(0) + hA1A3it(0)hA2A4it(0) + hA1A4it(0)hA2A3it(0)(A.6)

The application of Wick’s theorem to the kinetic equation Eq. (2.60) for our set ofrelevant operators is complicated by the fact that the operators ai are shifted by the c-number mean-field i. The evaluation of the average of a product of ten shifted operatorsinvolves 210 terms, each of which must be reduced according to Wick’s theorem. Even ademonstration on the example in Eq. (A.6) for the case four operators would involve onthe order of � 3 � 24 = 48 terms. A symbolic algebra package using Mathematica wasdeveloped by Reinhold Walser in our group to carry out all of these steps.

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