Wednesday, January 29, 2014

Locality and Nonlinear Quantum Mechanics

New paper!
Locality and Nonlinear Quantum Mechanics

Chiu Man Ho, Stephen D.H. Hsu

Nonlinear modifications of quantum mechanics generically lead to nonlocal effects which violate relativistic causality. We study these effects using the functional Schrodinger equation for quantum fields and identify a type of nonlocality which causes nearly instantaneous entanglement of spacelike separated systems.
Some excerpts:
The linear structure of quantum mechanics has deep and important consequences, such as the behavior of superpositions. One is naturally led to ask whether this linearity is fundamental, or merely an approximation: Are there nonlinear terms in the Schrodinger equation?

Nonlinear quantum mechanics has been explored in [1–6]. It has been observed that the fictitious violation of locality in the Einstein-Podolsky-Rosen (EPR) experiment in conventional linear quantum mechanics might become a true violation due to nonlinear effects [7, 8] (in [8] signaling between Everett branches is also discussed). This might allow superluminal communication and violate relativistic causality. These issues have subsequently been widely discussed [9].

Properties such as locality or causality are difficult to define in non-relativistic quantum mechanics (which often includes, for example, “instantaneous” potentials such as the Coulomb potential). Therefore, it is natural to adopt the framework of quantum field theory: Lorentz invariant quantum field theories are known to describe local physics with relativistic causality (influences propagate only within the light cone), making violations of these properties easier to identify. ...

... Our results suggest that nonlinearity in quantum mechanics is associated with violation of relativistic causality. We gave a formulation in terms of factorized (unentangled) wavefunctions describing spacelike separated systems. Nonlinearity seems to create almost instantaneous entanglement of the two systems, no matter how far apart. Perhaps our results are related to what Weinberg [11] meant when he wrote “... I could not find any way to extend the nonlinear version of quantum mechanics to theories based on Einstein’s special theory of relativity ... At least for the present I have given up on the problem: I simply do not know how to change quantum mechanics by a small amount without wrecking it altogether.”


Michael Bacon said...

"I simply do not know how to change quantum mechanics by a small amount without wrecking it altogether"

Hear hear!!

Waldemar Puszkarz said...

That's an interesting paper. As someone who did some work in the area of nonlinear modifications of the Schroedinger equation, I am glad to see it.

The papers you mention in the references are hardly the only ones that deal with the non-linear modifications of the Schroedinger equation. In fact, these are rather invasive (or crude) modifications in that they are bound to alter the energy levels of quantum systems.

That's true also about the scheme proposed by Weinberg that is pretty general, but I don't think it's exhaustive.

I designed a modification that leaves the energy levels of linear QM untouched and yet has a non-trivial dynamics and even has a free soliton solution. It modifies the Schroedinger equation via its phase rather than the amplitude of the wave function as is the case with many other modifications of this equation.

I discuss this modification in this paper:

Its solitonic solution is presented in another paper:

Finally, you can use this modification to study nonlocality as well. I have done it in this paper:

steve hsu said...

Thanks for the references!

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