We address conservation laws associated with current, momentum, and energy and show how they can be satisfied within many body theories which focus on pair correlations. Of interest are two well known $T$-matrix theories which represent many body theories which incorporate pairing in the normal state. The first of these is associated with the Nozieres Schmitt-Rink theory, while the second involves the $T$ matrix of a BCS-Leggett-like state as identified by Kadanoff and Martin. $T$-matrix theories begin with an ansatz for the single particle self energy and are to be distinguished from $\Phi$-derivable theories which introduce an ansatz for a particular contribution to the thermodynamical potential. Conservation laws are equivalent to Ward identities which we address in some detail here. Although $\Phi$-derivable theories are often referred to as “conserving theories,” a consequence of this work is the demonstration that these two $T$-matrix approaches similarly can be made to obey all conservation laws. When simplifying approximations are made in $\Phi$-derivable or other theories, one has to take care that the end results are not incompatible with conservation.

• Received 30 August 2013
• Revised 14 November 2013

DOI:https://doi.org/10.1103/PhysRevB.89.035106