# Doubling of degrees of freedom in the Schwinger-Keldysh formalism

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I am studying the Schwinger-Keldysh formalism. Basically, we double the number of degrees of freedom for the upper and lower branches.

Let´s consider the case where we have a certain field, coupled to an external current, given by the Lagrangian:

$$L \equiv (\partial_\mu \varphi)(\partial^\mu\varphi) - U(\varphi) + j\varphi$$

Now, in the Schwinger-Keldysh technique, we are going to have the fields and currents:

$$\varphi_+;\varphi_-;j_+;j_-$$

After we perform whatsoever the calculations we want to, typically something like:

$$\langle 0_{in}|P \space φ_{−}(x_1) · · · φ_{−}(x_{n}) \space φ_{+}(y_1) · · · φ_{+}(y_{p})|0_{in}\rangle$$ ($P$ stands for the ordering in the Schwinger-Keldysh contour, that is to say normal ordering in the uppper branch and anti-time ordering in the lower branch) we take the external fields in both branches to be identical.

Does it have any physical meaning to keep the external fields  different in our final result?

Yes, they are identified only at the very end - in the final equations for the correlation functions. otherwise one doesn't get the right equations.

Thank you for your comment. I know this is the way computations are made. I just wondered whether in some special circumstances, keeping the external fields different made any sense, or simply it´s a completely unphysical thing. Namely, if you put the external fields different in the final result, the theory is non unitary, so I didn´t know if that could be related somehow to the non-unitary evolution for the reduced density matrix.

The fields must be different initially so that one can take partial derivatives with respect to the two parts separately. Otherwise one misses the dissipative part of the reduced dynamics.

Thank you very much again. I don´t know anything about the dynamics of reduced systems (Lynbladian, Feynman-Vernon influence functionals), so I would very much appreciate if you could provide some papers where that´s addressed, preferably in the Schwinger-Keldysh formalism.

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Yes, they are identified only at the very end - in the final equations for the correlation functions. otherwise one doesn't get the right equations. The fields must be different initially so that one can take partial derivatives with respect to the two parts separately. Otherwise one misses the dissipative part of the reduced dynamics.

A good treatment is in

E. Calzetta and B.L. Hu,
Nonequilibrium quantum field theory,
Cambridge Univ. Press, New York 2008.

An earlier paper is

E. Calzetta and B.L. Hu,
Nonequilibrium quantum fields: Closed-time-path effective action,
Wigner function, and Boltzmann equation,
Phys. Dev. D 37 (1988), 2878--2900.

answered Jan 15, 2020 by (15,488 points)

Thank you very much, I´ll read them.

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