## Time dispersion in time-of-arrival measurements

I will be presenting a paper “Time dispersion in time-of-arrival measurements” at the International Assocation for Relativistic Dynamics this coming Wednesday (6/3/2010). The conference was originally scheduled to be held in Prague but has been moved online because of COVID-19. It may still be held as a physical conference as well, we will see.

My own paper is a follow up to my “Time dispersion in quantum mechanics“, published last year as part of the Institute of Physics Conference Series. That took the hypothesis: *the quantum wave function should extend in time as it does in space* & worked out the implications. The new paper is about experimental tests of the hypothesis: how would we determine if this hypothesis is true. Since it is real science however I turned the question around & made it “how do we prove that the wave function does not extend in time”.

In the new paper I shift focus to the Heisenberg uncertainty principle (HUP), specifically to the Heisenberg uncertainty principle in time and energy. Einstein & Bohr both held it was true, in fact essential if quantum mechanics was to be consistent with relativity. Bohr’s demonstration that it was was the end of Einstein’s direct attempts to falsify quantum mechanics.

Note that the formulation “the Heisenberg uncertainty principle applies to time/energy as it does to space/momentum” is loosely equivalent to “the wave function extends in time as it does in space”. If the wave function extends in time, then we would get the HUP in time/energy as a side-effect. And the most direct tests of the wave function extending in time are really tests of the HUP in time/energy.

The test I primarily focus on is that if the wave function extends in time all measurements in the time dimension would be just a bit fuzzier. In particular, if you are measuring when a particle is detected, if you are measuring the time-of-arrival, then if the wave function is extended in time you expect to see it both sooner & later than otherwise expected.

The advantage of this as a test is that the additional fuzziness if present at all must be present everywhen. Any time-varying experimental setup can potentially serve as a test.

The main problem — somewhat to my surprise — was that we really don’t know how to predict the time-of-arrival in standard quantum mechanics, let alone quantum mechanics with time in play as well! I’m trying to make a pincer attack on time: left jaw — standard quantum mechanics (SQM), right jaw — quantum mechanics with time (TQM). I was focused on the right jaw, but found that actually it was the left jaw that was weak. So I had to backtrack & deal with this problem. Interesting. And this turned out to be the single trickiest bit in the paper.

After getting the left jaw in better shape, good enough to take a punch anyway, I did a recap of TQM. This was probably the 2nd trickiest bit of the paper: how do you describe a hypothesis that took over a hundred pages and nearly five hundred equations to work out in a just a few pages? I found the core ideas coming a bit clearer in my own head at least. That’s gotta be worth something.

Then the payoff bit, the actual tests, is only the last quarter of the paper. And after working out how the additional fuzziness in time plays out, I got to my favorite test: the single slit in time. This is the single cleanest test of the idea. Not an easy experiment however.

Really the best part of tests of TQM is that if it is proved true, great. But if it proved false it will be taking down one or two of its neighbors with it. TQM is built by applying the quantum rules to relativity (or applying relativity to the quantum rules). If it is false, one (or both) of those two has a problem. And that in turn means there are really no null experiments.

And if I know my experimentalists, there is nothing they like more than proving a bunch of theorists wrong. If I have setup the arguments correctly — we’ll see — then they are sure to break something. As the well-known quantum experimentalist Nicholas Gisin said to me a long time ago (I paraphrase, it was quite a long time ago) “Look, I don’t care what your theory of time is. Just give me something I can prove wrong!”