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The connections between standard theoretical tools used to study open quantumsystems can sometimes seem opaque. Whether it is a Lindblad master equation,the equation of motion for the Wigner function or a dissipative Keldysh action,features evident in one formalism are often masked in another. Here, wereformulate the technique of third quantization in a way that explicitlyconnects all three methods. We first show that our formulation reveals afundamental dissipative symmetry present in all quadratic bosonic or fermionicLindbladians. This symmetry can then be used to easily diagonalize thesemodels, and provides a intuitive way to demonstrate the separation ofdissipation and fluctations in linear systems. For bosons, we then show thatthe Wigner function and the characteristic function can be thought of as''wavefunctions'' of the density matrix in the eigenbasis of thethird-quantized superoperators we introduce. The field-theory representation ofthe time-evolution operator in this basis is then the Keldysh path integral. Tohighlight the utility of our approach, we apply our version of thirdquantization to a dissipative non-linear oscillator, and use it to obtain newexact results.

Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations