# Time Reversal Invariance in Quantum Mechanics

I thought of a thought experiment that had me questioning how time reversal works in quantum mechanics and the implications. The idea is this ... you are going forward in time when you decide to measure a particle. The particle then collapses to the observed state. Now if physics were to be the same in reverses time, then if we stop and reverse time then measure that very same particle again ... then I would imagine that since the wave function has collapsed we ought to measure the same thing. What this says to me is that given some time evolution in the + direction, if we measure a particle and it collapses the wave function, then if you reverse the arrow of time to go in the - direction we ought to get the same answer as before. The future/present effects the past. This means if we theoretically had a time machine and went back in time, we would have traveled into a different past.

Another implication of this thought experiment is that the future would be indistinguishable from the past and would hence forth be the same. I would imagine that this is consistent with the 2nd law of thermodynamics since physics dictates that entropy only increases ... going in the reverse direction of time to decrease entropy would violate the laws of physics. Has anyone else out there thought about this?

From my studies in quantum mechanics, I don't remember any postulates stating anything like this, but this all makes sense to me. Are there any theories out there that go along these lines?

• This reminds me a lot of scattering. The scattering operators $\Omega^{\pm}=\lim_{t\rightarrow \pm \infty}e^{-iHt}e^{iH_{0}t}$ are time reversals of each other. But this just says we either trace the after-interaction trajectories back or take the incoming trajectories forward, at least that's where I'm at now (I'm still learning). I'm confident someone will give you a good answer, and if not, I'll read a little more and give you mine. – kηives May 23 '12 at 1:24
• Thanks. I'm thinking about getting back into research, I haven't heard of anyone tackling a question such as this. This might be something that I would like to take on since I'm unemployed with ample time on my hand. Its interesting that you mentioned this to be an observed characteristic in scattering. – Dr.Knowitall May 23 '12 at 1:41
• I suppose that my criticism would be that: you seemingly treat the problem quantum mechanically going forward in time, but switch back to classical treatment when you go backwards. As an analogy, you had a quantum egg and let it rupture quantum mechanically, but when you went backwards, you rebuild up the egg as if it was simple newton trajectories. I think the trick would be to treat both directions quantum mechanically, and let the "un-measurement" process be as quantum mechanical as the normal measurement process. Just a thought. – kηives May 25 '12 at 1:48
• @knives : Ya, I did treat the problem quantum mechanically going back in time. My question was spurred because I think most people think of going back into time as a classical newtonian problem as you said. My thoughts were, what have you if going back to your original wave-function you had to go through quantum mechanics as well. I was thinking of it the way you just put it. – Dr.Knowitall May 30 '12 at 19:43

Just a few pointers for you to explore more on this. Check out Aharonov's paper the time symmetric formulation of quantum mechanics: http://arxiv.org/abs/quant-ph/9501011

From my studies in quantum mechanics, I don't remember any postulates stating anything like this, but this all makes sense to me. Are there any theories out there that go along these lines?

Indeed there is such a theory. It's called decoherence.

You mention the comparison with thermodynamics, and this is basically the same way decoherence works. The physics behind thermodynamics is perfectly reversible, but we don't see broken eggs reassembling themselves because it's extremely improbable for this to happen.

To quote the article on decoherence, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior. Quantum decoherence gives the appearance of wave function collapse. In effect information about the quantum system is dissipated into the rest of the universe. The information isn't lost, and in principle it could recombine to form the original quantum superposition, but this is even less likely than a broken egg spontaneously reassembling itself.

• I see your point and I never really thought of it from the stand point of a broken eggshell fixing itself. My previous argument I suppose is that once broken, its not possible to go back to the previous states. So if we were traveling back in time wouldn't we have to perform operations that would undue the effects from measuring (breaking the eggshell)? I can't think of a way measurement operations can be interverted and reapplied to the particle to undue the measurement ... not unless we prepare the particle the way we previously had. – Dr.Knowitall May 23 '12 at 19:32

|Dear Mr Student, the time-reversal symmetry in conventional QM at best (e.g. no magnetic fields) only applies to the unitary evolution of a quantum system; the measurement process is not time-reversal symmetric. Also, the second law of Thermodynamics says (very roughly speaking) the Entropy should decrease if you go back in time; again there is no time-reversal symmetry. The fact that your current understanding of the subject makes sense to you, means that you have been smart enough to find seemingly convincing arguments to make it make sense to you; but it's not correct I believe.

• I can see what your saying if you apply a time-dependent potential into the equation. Actually, between your point, Spot's article, and Rennie's response my question seems a bit more clear. Given a time-independent equation, the wavefunction should be constant for all time which agrees with my statement ... however the measurement itself is not a time independent problem as you said. Spots article however shows that quantum info is never lost, its just dispersed throughout the universe which is on par to Rennie's response. It makes sense to me now, thanks. – Dr.Knowitall May 23 '12 at 20:08

No!

Time invariance holds in quantum mechanics ONLY when wave function does not collapse. This means once you did any measurements, the time invariance is destroyed. There is no time invariance in the presence of observer.

If I'm reading your question correctly, it is at least in part whether the nominal event of "wave collapse" (please note that different schools of thought describe that event differently!) is reversible in time. I won't try to address the schools, but rather whether what you ask has any experimental meaning.

This is not a complete answer, but the concept of quantum erasure does seems to support the idea that "wave collapse," if approached carefully, can be reversed.

The general prescription for reversing such an event is this: You must return all of the information about the event, from the entire universe, back to the point of origin. By all I really do mean all, including for example any photons that high-tailed outwards at the speed of light, and any phonons that left as vibrations.

If you think about that a little, it tells you very quickly that reversal of any "wave collapse" must necessarily be extraordinarily unlikely if the event has been touched by any of the equipment of a classical observer. It's only likely to happen within very small systems where information is extremely limited, and at the lower limit such ideas of erasure fold smoothly into the QED (Feynman) concept of quantum waves as sums of the amplitudes of all possible histories by which the event could have unfolded.

I should point out also that the recipe I just described of returning all information cannot really return that information to the original location in space time, since classical equipment can only work in the present. So, what you really end up doing is making do with a very similar location embedded within present time. Nonetheless, if you follow the return-all-data prescription, you should be able to create a new wave function from which some entirely new future could emerge, as you also speculated.

The catch is that since your new wave function is just a replica of the past, you don't really change the past if your new "identical but moved in time" wave function decides to unfold in some entirely new way. A bit of a Catch 22, that.

Also, one follow-up question could be "what about time entanglement? can you change the past that way?" Nope. Even there, causality has this remarkable ways of keeping temporal paradoxes at bay. While I would strongly agree that time entanglement is possible and even go further to assert that it is a normal component of spatial entanglement, I would also say that all forms of entanglement depend on there being no contradictory information available anywhere in the current universe about how the various components of the entangled event unfolded, at least not until you check them out in the present or future.

• I see what your saying about returning quantum information from the universe to create the original wave function, but I still can't get past the catch 22. If we ran 3 consecutive experiments, one after another ... you know that you would get random results with each measurement. Given that we reversed time and redid the experiment, quantum mechanics would fail if the results were the same. I can however see that there is always the off chance in going forward in the same future with the same results .. its just probability. Though doing unmeasurements, I agree would spit the original wavefxn – Dr.Knowitall May 30 '12 at 19:55

The second law of thermodynamics is only valid within a precondition of linear time. If time itself is multi-dimensional (Many Worlds) the increase or decrease of entropy is relative.

Two recent experiments demonstrate that you can't think classically about quantum events: in both of these experiments entanglement was in operation until all the observables were measured - the order of measurements made no difference. See "Entanglement Swapping between Photons that have Never Coexisted" , Megidish, et al.,PRL 110, 210403 (2013), DOI: 10.1103/PhysRevLett.110.210403 and "Quantum erasure with causally disconnected choice", X.S.Ma, et al., PNAS , January 22, 2013 , vol. 110 , no. 4 , 1221–1226, DOI: 10.1073/pnas.1213201110

so from the viewpoint of a linear time line you have a symetrical system wether you run the experiment forward or backwards - unentangled photons become entangled do some quantum mechanical operations with other photons that changes the nature of the entanglement and then return to an unentangled state - before and after you have quantum soup - in between either forward or backward you have unobserved states. I don't believe you can consider the in-between state(s) to violate the second law of thermodynamics looking forward or backward in time because the entangled photons exist in a isolated quantum phase space until they become unentangled.