Open QFT and Driven Dynamics

An open quantum field theory describes selected field degrees of freedom after other degrees of freedom have been ignored, monitored, driven, or traced out. The resulting evolution is not fixed by a complex dispersion relation alone: it depends on the system–environment split, the initial state, bath correlations, coarse-graining, and the algebraic constraints imposed on the reduced map. This chapter develops that chain from a microscopic influence functional to local or nonlocal evolution, then tests trace preservation, positivity, causality, and renormalization before assigning a physical interpretation.

Helpful background. Open-system effective theory introduces reduced density matrices and the influence-action viewpoint. The influence-functional derivation is also a useful entry point if the microscopic split is already known.

Enter this chapter

The central object is the reduced state

$$\rho_S(t)=\operatorname{Tr}_E\!\left[U(t,t_0)\rho_{SE}(t_0)U^\dagger(t,t_0)\right].$$

This formula defines a completely positive trace-preserving map when the initial total state is factorized with a fixed environment state. It does not say that the map forms a semigroup, has a time-local generator, or can be represented by local field jump operators. Initial system–environment correlations can even prevent a state-independent map from being defined on the full system state space.

A reliable reduction therefore follows a sequence of distinct questions:

1. Which variables constitute the system, and is the split compatible with constraints and gauge symmetry?
2. What environment state and initial correlations are assumed?
3. What retarded and noise kernels follow from tracing out the environment?
4. Are weak coupling, short memory, and secular frequency separation controlled?
5. If a local generator is proposed, does it preserve trace, Hermiticity, and complete positivity on its declared domain?
6. Does its Schwinger–Keldysh action have the required normalization, reality, causal, and noise structure?
7. Are the operator basis and consistency constraints closed under renormalization?

Each step can fail while the preceding ones remain valid. In particular, an exact influence functional may be nonlocal in time; a Redfield equation may approximate expectation values yet fail complete positivity; and a non-Hermitian Hamiltonian may correctly describe a postselected no-jump trajectory while being incomplete as an unconditional reduced evolution.

Choose the description by scale

Let $\tau_E$ be a bath correlation time, $\tau_S$ a system evolution time, $\Delta\omega$ a relevant Bohr-frequency separation, and $\gamma$ a relaxation rate. The following routes are useful only with their hypotheses attached.

Question	Natural description	Necessary control	What it does not guarantee
What does a specified microscopic split imply?	Influence functional	initial state, coupling, contour normalization, regulator	time locality or a unique split
Can the bath be eliminated perturbatively?	Microscopic master equation	weak coupling, $\tau_E\ll\tau_S$, spectral input	complete positivity of Redfield form
Is there a Markovian CPTP semigroup?	Lindblad field dynamics	positive rate matrix, operator domain, regulator	microscopic derivation or relativistic locality
How are response and noise organized for fields?	Open Schwinger–Keldysh action	$S[\phi_r,\phi_a=0]=0$, reality and causal structure	complete positivity from normalization alone
Does the environment obey thermal detailed balance?	Noise and dissipation	KMS state and operator conventions	a constant effective temperature out of equilibrium
Does finite bath memory matter?	Non-Markovian dynamics	kernel tails, initial slip, witness dependence	one universal measure of memory
What steady state and scaling follow from drive and loss?	Driven criticality	Liouvillian gap, finite size, symmetry and noise	equilibrium universality without an emergent symmetry
Is a proposed evolution physically admissible?	Consistency tests	independent algebraic and dynamical checks	ultraviolet completion
Does the effective theory survive changing scale?	Open-system renormalization	closed doubled operator basis and matching	a continuum Lindblad theory from a finite truncation

The Markov and secular limits are separate. Short bath memory allows one to replace a delayed state by a present state over a controlled interval. Secularization additionally averages terms oscillating with frequency differences, requiring roughly $\lvert\omega-\omega'\rvert\gg\gamma$ for the discarded interference. Near degeneracies can invalidate the latter even when the former is accurate.

The non-Hermitian boundary

For a Lindblad generator with jumps $L_\alpha$,

$$\dot\rho=-i(H_{\mathrm{eff}}\rho-\rho H_{\mathrm{eff}}^\dagger) +\sum_\alpha L_\alpha\rho L_\alpha^\dagger, \qquad H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\alpha L_\alpha^\dagger L_\alpha.$$

The first term alone reduces the trace at the total jump rate. It is the unnormalized evolution conditioned on observing no jump. The recycling term restores probability in the unconditional density matrix. Consequently, a complex mass or absorbing potential may be useful for resonances, response poles, or postselection, but it is not by itself a CPTP theory of an unobserved environment.

Standards of evidence

Open-system conclusions must name the object being tested. Agreement of one expectation value does not certify the map; positivity on a small family of Gaussian states does not establish complete positivity; and a finite-size steady state with a small Liouvillian gap does not by itself establish a thermodynamic phase transition. The chapter’s canonical classification of approximations, guarantees, and evidence is the open-dynamics consistency and evidence matrix.

For evolving research claims, record at least the regulator, volume, operator truncation, environment spectrum, initial state, drive protocol, time window, and error estimate. A numerical positivity scan is evidence on the scanned state set, not a proof over an infinite-dimensional field algebra. Conversely, a proof of GKSL form at fixed cutoff does not establish a controlled cutoff removal.

Review the chapter

Before using an open-QFT result, be able to answer all of the following.

• Is the reduced map exact, perturbative, Markovian, secular, or phenomenological?
• Which bath correlator fixes damping, and which fixes noise?
• Is complete positivity proved, assumed through GKSL form, or only tested on selected states?
• If a non-Hermitian Hamiltonian appears, what measurement record or recycling term completes the interpretation?
• Which constraint, symmetry, or conservation law must the jump operators respect?
• Does the regulator preserve the identities used to establish causality and trace normalization?
• Which counterterms and dissipative operators are generated at the claimed order?

The next chapter turns to hot non-Abelian plasmas, where integrating out scales also produces dissipation and noise but gauge constraints and the hierarchy $T$, $gT$, and $g^2T$ select more specialized effective descriptions.

References

• Breuer, Heinz-Peter, and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford: Oxford University Press, 2002. doi:10.1093/acprof:oso/9780199213900.001.0001.
• Feynman, Richard P., and Frank L. Vernon Jr. “The Theory of a General Quantum System Interacting with a Linear Dissipative System.” Annals of Physics 24 (1963): 118–173. doi:10.1016/0003-4916(63)90068-X.
• Gorini, Vittorio, Andrzej Kossakowski, and E. C. George Sudarshan. “Completely Positive Dynamical Semigroups of N-Level Systems.” Journal of Mathematical Physics 17 (1976): 821–825. doi:10.1063/1.522979.
• Lindblad, Göran. “On the Generators of Quantum Dynamical Semigroups.” Communications in Mathematical Physics 48 (1976): 119–130. doi:10.1007/BF01608499.
• Sieberer, Lukas M., Michael Buchhold, and Sebastian Diehl. “Keldysh Field Theory for Driven Open Quantum Systems.” Reports on Progress in Physics 79 (2016): 096001. doi:10.1088/0034-4885/79/9/096001. Open preprint.