FTL Tunneling? No not really!
Posted Jul6-09 at 02:11 PM by jambaugh
Tags ftl, quantum, relativity, tunneling
I keep running across claims of FTL travel based on quantum tunneling. Here I try to explain why this sort of thing is a gross misinterpretation of the physical process.
Specifically I am addressing the Nimtz experiment http://en.wikipedia.org/wiki/G%C3%BCnter_Nimtz
The critical error in Nimtz's experiment and in all other FTL tunnelling claims is in confusing the trajectory of the peak of a the probability amplitude distribution with the trajectory of the physical particle.
In considering a classical traveling wave we consider two quantities, phase velocity and group velocity. Given the speed of (causal) propagation c of a wave then either the group and phase velocities are both also c or if one is less the other is necessarily greater. This is true for sound, light, or any other waves. There is no contradiction here with causal physics of the system propagation.
In quantum mechanics we also have propagating waves (wave functions) but their proper interpretation is not as physical waves but rather they represent square-roots of relative probabilities for observing a quantum at a given position.
(By "proper interpretation" I mean the operational interpretation as in how a laboratory scientist uses the wave function to predict outcomes of experiments.)
We needn't delve deeply into the quantum mechanics or the tunneling phenomenon, we need only consider how one can err by misreading a probability distribution as a physical object.
Below is an analogy to the Nimtz's oblique tunneling experiment though it does not get into the details of the quantum tunneling itself it is sufficient to show the mistakes made in interpretation.
Imagine a runner leaves a strip of road laying in a north-east direction and the runner is traveling southeast. You are not quite sure from what part of the road he left but you have a probability distribution in the shape of a bell curve with center at point A. The runner if he keeps going straight will come to another highway running north and south. Now imagine that along this barrier road cars are traveling so the runner either gets hit by a car or misses all the cars and keeps on going south east.
Now you also observe a peculiar statistical phenomenon in collecting data on many runners. Whereas the line the runner most likely takes before crossing the road passes through the point A the line the runner most likely takes given he makes it across lies a bit east of point A. This is to say there is a correlation between how far east of A the runner started and how likely it is he crossed the road. (If you want to be more humane you can assume the runner always bounces off the car and is dazed but unhurt so that he starts running SW instead of SE)
Now assume that given the runner crosses the barrier highway he will reach another strip of road lying also in a north-east direction. Noting that the probability distribution for crossing runners will be shifted but still a bell shaped curve. Lets say the center of this bell curve at the second NE road is at point B.
Assume in all cases that the runner takes a straight line path and travels at a fixed speed we will call c. Say the two oblique strips of road are 1000 meters apart and the runner runs at 1 meter per second so his travel time is always 1000 seconds.
But you mistakenly try to calculate this speed by measuring the distance from point A to the point (A') on the left side of the barrier highway directly SE of pt A, then you measure the distance from point B to the point (B') on the right side of the barrier directly NW of point B. You then add the distance between these two points A' and B' noting that the slope is a bit shallower than the 45degrees of the lines A to A' and the line B to B'.
You incorrectly assume the runner traveled along line A to A' then A' to B' then B' to B. This is a distance greater than 1000m and so his average travel time is greater than 1m/s. You further assume that while running from A to A' and running B' to B he maintains his measured speed of c. You then incorrectly calculate the speed he traveled while crossing the highway. You in fact may find this speed to be very much larger than 1m/s or even infinite or negative (he traveled backwards through time).
The error is in taking the center of the probability as an actual position. Given the runner did start from point A he will not have taken the path B' to B but rather will be traveling parallel but west of this line. (He also is slightly less likely to have even made it across the road compared to the runner who starts east of A and ends up on the line B' to B.).
Now this exposition does not get into the details of quantum tunneling and why the center of probability shifts during a tunneling event. For that you'll have to get more deeply into the quantum mechanics. But it does show how misreading probability distributions as the actual object they describe leads to fantastic mistakes.
If you want you can modify the analogy by taking the north direction as time and instead of a Gaussian probability about where the runner left you take a Gaussian probability about when the runner left. The result will be the same, a mistaken greater than c calculation as to how fast he crosses the road. This covers the more traditional 1dim tunneling example.
Specifically I am addressing the Nimtz experiment http://en.wikipedia.org/wiki/G%C3%BCnter_Nimtz
The critical error in Nimtz's experiment and in all other FTL tunnelling claims is in confusing the trajectory of the peak of a the probability amplitude distribution with the trajectory of the physical particle.
In considering a classical traveling wave we consider two quantities, phase velocity and group velocity. Given the speed of (causal) propagation c of a wave then either the group and phase velocities are both also c or if one is less the other is necessarily greater. This is true for sound, light, or any other waves. There is no contradiction here with causal physics of the system propagation.
In quantum mechanics we also have propagating waves (wave functions) but their proper interpretation is not as physical waves but rather they represent square-roots of relative probabilities for observing a quantum at a given position.
(By "proper interpretation" I mean the operational interpretation as in how a laboratory scientist uses the wave function to predict outcomes of experiments.)
We needn't delve deeply into the quantum mechanics or the tunneling phenomenon, we need only consider how one can err by misreading a probability distribution as a physical object.
Below is an analogy to the Nimtz's oblique tunneling experiment though it does not get into the details of the quantum tunneling itself it is sufficient to show the mistakes made in interpretation.
Imagine a runner leaves a strip of road laying in a north-east direction and the runner is traveling southeast. You are not quite sure from what part of the road he left but you have a probability distribution in the shape of a bell curve with center at point A. The runner if he keeps going straight will come to another highway running north and south. Now imagine that along this barrier road cars are traveling so the runner either gets hit by a car or misses all the cars and keeps on going south east.
Now you also observe a peculiar statistical phenomenon in collecting data on many runners. Whereas the line the runner most likely takes before crossing the road passes through the point A the line the runner most likely takes given he makes it across lies a bit east of point A. This is to say there is a correlation between how far east of A the runner started and how likely it is he crossed the road. (If you want to be more humane you can assume the runner always bounces off the car and is dazed but unhurt so that he starts running SW instead of SE)
Now assume that given the runner crosses the barrier highway he will reach another strip of road lying also in a north-east direction. Noting that the probability distribution for crossing runners will be shifted but still a bell shaped curve. Lets say the center of this bell curve at the second NE road is at point B.
Assume in all cases that the runner takes a straight line path and travels at a fixed speed we will call c. Say the two oblique strips of road are 1000 meters apart and the runner runs at 1 meter per second so his travel time is always 1000 seconds.
But you mistakenly try to calculate this speed by measuring the distance from point A to the point (A') on the left side of the barrier highway directly SE of pt A, then you measure the distance from point B to the point (B') on the right side of the barrier directly NW of point B. You then add the distance between these two points A' and B' noting that the slope is a bit shallower than the 45degrees of the lines A to A' and the line B to B'.
You incorrectly assume the runner traveled along line A to A' then A' to B' then B' to B. This is a distance greater than 1000m and so his average travel time is greater than 1m/s. You further assume that while running from A to A' and running B' to B he maintains his measured speed of c. You then incorrectly calculate the speed he traveled while crossing the highway. You in fact may find this speed to be very much larger than 1m/s or even infinite or negative (he traveled backwards through time).
The error is in taking the center of the probability as an actual position. Given the runner did start from point A he will not have taken the path B' to B but rather will be traveling parallel but west of this line. (He also is slightly less likely to have even made it across the road compared to the runner who starts east of A and ends up on the line B' to B.).
Now this exposition does not get into the details of quantum tunneling and why the center of probability shifts during a tunneling event. For that you'll have to get more deeply into the quantum mechanics. But it does show how misreading probability distributions as the actual object they describe leads to fantastic mistakes.
If you want you can modify the analogy by taking the north direction as time and instead of a Gaussian probability about where the runner left you take a Gaussian probability about when the runner left. The result will be the same, a mistaken greater than c calculation as to how fast he crosses the road. This covers the more traditional 1dim tunneling example.
Total Comments 0
