Historical research guide · 1156 · 1991
Weinberg's Nonlinear Quantum Mechanics and the Einstein-Podolsky-Rosen Paradox
Joseph Polchinski
In one minute
In ordinary quantum mechanics two entangled systems are correlated, yet nothing done to one can send a message to the other. Joseph Polchinski asked whether that survives Weinberg's proposal to let quantum mechanics be slightly nonlinear. A nonlinear theory has more observables than the linear one, which suggests the wave function carries more usable information, and so the question is sharp: could an Einstein–Podolsky–Rosen apparatus become a telephone? Polchinski works out the exact condition under which a distant system receives no signal. Every observable in that system must depend only on its own density matrix. Weinberg's original treatment of separated systems does not satisfy this, and so allows EPR communication. Polchinski then shows that the only way to forbid it opens a different, equally unusual channel: communication between different branches of the wave function, demonstrated with a four-step Stern–Gerlach thought experiment. Nonlinearity changes what quantum correlations can carry; the question is which way.
Connection
Chapter 17 teaches why self-consistent time loops are a real subject rather than a puzzle for fiction. The door to signalling through entanglement is precisely the linearity of quantum mechanics, and this paper states exactly what a small nonlinearity would do to that door. It turns an argument about paradoxes into a statement about a measurable property of nature.
From the source
The no-signal condition. Assuming the observables include all the usual linear Hermitian observables, the necessary and sufficient condition for an isolated system not to receive information through EPR correlations is that every observable in that system depends only on the density matrix for that system.The first main result, following equations 6 to 9
Weinberg's separated systems. The nonlinear observables Weinberg originally proposed for separated systems are not functions of the density matrix alone, except when linear, and therefore allow communication through EPR correlations.Following equation 10
The other channel. Nonlinear observables that do satisfy the no-signal condition lead to communication of another sort, between different branches of the wave function, shown with a four-step process in which a spin-one-half ion passes a Stern–Gerlach device, a macroscopic observer acts on the outcome, the ion meets a field coupled to a nonlinear observable, and the observer measures again. Forbidding EPR communication in nonlinear quantum mechanics therefore necessarily produces this second kind.Introduction; the four-step process and the partial density matrices that follow it
The way in
https://doi.org/10.1103/PhysRevLett.66.397How to cite it
Joseph Polchinski (1991) Weinberg's Nonlinear Quantum Mechanics and the Einstein-Podolsky-Rosen Paradox doi:10.1103/PhysRevLett.66.397