- #1
Ali Lavasani
- 54
- 1
In https://arxiv.org/pdf/quant-ph/0203049.pdf, which is in the realm of Bohmian mechanics, Antony Valentini claims that by having a "non-equilibrium" particle with arbitrarily accurate "known" position, we can measure another particle's position with arbitrary precision, violating Heisenberg's uncertainty principle.
In summary, the "sub-quantum measurement" method works as follows (see part 4 of the paper): In a simple case, we have y(t) = y(0) + a.x(0).t, where x and y are the positions of the measured and test (pointer) particles, respectively. We have kind of entanglement between the test particle's momentum and the measured particle's position (with Hamiltonian H^ = a.x^.p^y, where a is a coupling constant, py is the momentum canonically conjugate to y, and x is the measured particle's position). Here we have y(0) and y(t) with arbitrary precision, since the test particle is in quantum non-equilibrium.
The point is that, In order to predict the trajectory of the measured particle, x(t), we need BOTH x(0) and Ψ(x,y,t) (the wavefunction of the ensemble). This wavefunction would be 6-Dimensional in Bohmian mechanics and inaccessible to us in the 3-D world, so actually the theory would in general be useless. It's only in the limit a.t -> 0 that Ψ(x,y,t) ≈ ψ0(x)g0(y), and the total wavefunction would be approximately 3-D.
Now my question comes. Ψ(x,y,t) ≈ ψ0(x)g0(y) is only an approximation, being completely precise only when a.t = 0, in which case y(t) = y(0) and we lose connection to the measured particle's position (x(0)). In other words, the bigger the a.x(0).t, the more the error we have in our approximation of the ensemble's wavefunction as 3D. We can have a large a.t, in which case the wavefunction would be inaccurate and we would not be able to accurately calculate x(t) from x(0), OR we can have a very small a.t and a precise wavefunction, in which case the parameter a.x(0).t, our tool to determine x(0), would also be very small and maybe comparable to the uncertainty in y(0), which makes our measurement of x(0) inaccurate. So there doesn't seem to be any way to track the trajectory x(t) precisely, we need to make either x(0) or Ψ(x) inaccurate.
So is my discussion valid? Is the uncertainty in x(t) kind of irreducible, because of the uncertainty in any case we have, either in x(0) or in Ψ(x), which are in a kind of trade-off?
In summary, the "sub-quantum measurement" method works as follows (see part 4 of the paper): In a simple case, we have y(t) = y(0) + a.x(0).t, where x and y are the positions of the measured and test (pointer) particles, respectively. We have kind of entanglement between the test particle's momentum and the measured particle's position (with Hamiltonian H^ = a.x^.p^y, where a is a coupling constant, py is the momentum canonically conjugate to y, and x is the measured particle's position). Here we have y(0) and y(t) with arbitrary precision, since the test particle is in quantum non-equilibrium.
The point is that, In order to predict the trajectory of the measured particle, x(t), we need BOTH x(0) and Ψ(x,y,t) (the wavefunction of the ensemble). This wavefunction would be 6-Dimensional in Bohmian mechanics and inaccessible to us in the 3-D world, so actually the theory would in general be useless. It's only in the limit a.t -> 0 that Ψ(x,y,t) ≈ ψ0(x)g0(y), and the total wavefunction would be approximately 3-D.
Now my question comes. Ψ(x,y,t) ≈ ψ0(x)g0(y) is only an approximation, being completely precise only when a.t = 0, in which case y(t) = y(0) and we lose connection to the measured particle's position (x(0)). In other words, the bigger the a.x(0).t, the more the error we have in our approximation of the ensemble's wavefunction as 3D. We can have a large a.t, in which case the wavefunction would be inaccurate and we would not be able to accurately calculate x(t) from x(0), OR we can have a very small a.t and a precise wavefunction, in which case the parameter a.x(0).t, our tool to determine x(0), would also be very small and maybe comparable to the uncertainty in y(0), which makes our measurement of x(0) inaccurate. So there doesn't seem to be any way to track the trajectory x(t) precisely, we need to make either x(0) or Ψ(x) inaccurate.
So is my discussion valid? Is the uncertainty in x(t) kind of irreducible, because of the uncertainty in any case we have, either in x(0) or in Ψ(x), which are in a kind of trade-off?
