# Can the Heisenberg interpretation or path integrals apply to open quantum systems?

It has been claimed by some people that Schrödinger's picture is more misleading compared to the Heisenberg principle or path integrals, and that we would be better off abandoning the Schrödinger picture in favor of either the Heisenberg picture or path integrals. However, when it comes to open quantum systems in a constant interaction with the environment, which isn't fully modeled, it is not at all clear how to apply the Heisenberg picture or path integrals. The Heisenberg picture requires the operators of the system to evolve in such a way in time what it becomes extremely mixed up with the environmental degrees of freedom, and the only way this can be done is to fully model the environment as a whole, or at least all of the part of the environment which ever has a chance of interacting with the system in question. Similarly, how do you even go about adapting path integrals to open systems without including the entire environment, or at least all of those parts which can ever interact with the system?

Might it be the case that the Heisenberg picture and path integrals can only be strictly applied to the universe as a whole, or at the very least, causal diamonds?

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I suspect (and I may be wrong) that the suggestion that we abandon the Schrodinger picture is usually only made in a pedagogical context, i.e., the idea is that we shouldn't use the Schrodinger picture when trying to understand or teach the fundamental principles of quantum mechanics. This doesn't necessarily mean that we shouldn't make use of it in contexts where it makes the math easier. (The interaction picture is already treated in much the same way.) –  Harry Johnston Dec 15 '11 at 21:06
Some people celebrating the Heisenberg picture and path integrals were right! motls.blogspot.com/2011/12/celebrating-heisenberg-picture.html –  Luboš Motl Dec 16 '11 at 17:07

Yes, it can.

An example is Brownian motion, in which you are interested in the dynamics of a particle in contact with some external reservoir without being interested in the dynamics of said reservoir. What you want is to incorporate the effect of the external reservoir on the dynamics of the particle (or subsystem), which can --in principle-- be done by integrating out all degrees of freedom associated with the external reservoir.

A specific example for which this is done is the Caldeira-Leggett model. The model starts with the full action in a path integral formulation. The degrees of freedom associated with the external reservoir are then integrated out, which can be a bit tricky. The action that remains is one completely defined in terms of the degrees of freedom of the subsystem / particle, but with effective potentials that account for the influence of the environment.

This is also known as the influence functional, and you can find a chapter on it in the book by Feynman and Hibbs.

This is one approach which has proven to be quite succesfull. Another approach is to write down the equations of motions for the reduced density operator, i.e. the density operator obtained by, again, integrating out the reservoir degrees of freedom. This leads to the Lindblad equation.

In general you cannot keep track of the exact influence of the external reservoir on the subsystem. Some statistical averaging / assumptions have to be made and the effect of the reservoir is often mimiced by a random potential, in which the potential is a random variable rather than explicitly known.

I mentioned Brownian motion as an example of an open quantum system. There are many more applications, such as quantum transport and quantum optics. This basically means that some type of randomness is introduced into the state evolution.

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The question is to formulate a path integral for Libdblatt eqn. directly. –  Ron Maimon Dec 15 '11 at 20:21
The Lindblad equation is only an approximated equation of motion for the reduced density operator. –  juanrga Nov 16 '12 at 19:46

Since that the the Schrödinger picture and the Heisenberg picture are equivalent, you can start from the Schrödinger equation for simple cases such as a particle in an external field and obtain the Heisenberg equations of motion or vice verse. Idem about the path integrals formalism, which can be derived from the Schrödinger equation or vice verse.

A complete treatment of an open quantum systems interacting with an environment cannot be done in any of those formalisms. You have to resort to the more general density matrix formalism for a complete treatment of open systems. The idea of fully model the environment as a whole does not change this if the system is in a mixed state, for the which no wavefunction exists. Again you have to resort to the density matrix formalism to fully model the environment.

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