Exploring the Quantum Theory of the Past

[0xDEADBEEF]

Does anyone know of a quantum theory of the past. If I measure an observable, this gives me certain information of the possible states it has been in before the wave function collapse.
I can exclude those states without any overlap with the observed eigenstate.

But it is more complicated than this. Often we have prepared a state so we know exactly what state it was in before. Whatever a single measurement yields. We seem to be pretty sure about the past, and that it is classical. Decoherence doesn't work well backwards in time, and wave function collapse doesn't look too compatible with SRT.

Is there any theory about this. Can someone point out some papers or books?

 

[Fra]

Does anyone know of a quantum theory of the past. If I measure an observable, this gives me certain information of the possible states it has been in before the wave function collapse.
I can exclude those states without any overlap with the observed eigenstate.

But it is more complicated than this. Often we have prepared a state so we know exactly what state it was in before. Whatever a single measurement yields. We seem to be pretty sure about the past, and that it is classical. Decoherence doesn't work well backwards in time, and wave function collapse doesn't look too compatible with SRT.

Is there any theory about this. Can someone point out some papers or books?

I may not have understood your idea exactly but to add how I think about "the past" from my somewhat personal information theoretic interpretation, the past in the sense of "objective history records" is not consistent with the idea that a given observer only holds and encodes finite information.

IMO, history is not only observer dependent, it is also uncertain in the sense that only certain information about the past is retained. I guess one can also say that eventually the past is "ereased" simply because no physical observers can retain an actual "time history".

So at least as far as I'm concerened, I'm not sure what you mean with this: "We seem to be pretty sure about the past, and that it is classical"

I am not certain about the past, and I wouldn't say it's classical.

But there is probably no consensus of answer to the good points you raise. I just add my voice here to object to past beeing classical or certain. My view is rather that each "observer dependent" incomplete, compressed memory record of the past by impliciation (here I think along the lines of statistical implications of "counting" evidence for "expected" possisble futures) yields expectations of the future. The observer dependens here, is made sense by introducing interactions to make up for the differences. So instead of thinking that interactions + observer dependence, recovers observer invariance, one can turn it around and think that the deviations from observer invariance, defines or maps out interactions interactively.

/Fredrik

 

[Truecrimson]

The family of approaches like decoherence histories and consistent histories by Gell-mann, Omnes and Robert Griffiths may say something about this, but I don't know them myself.

Edit: Form a quick look at Schlosshauer's review of decoherence, the consistent histories approach have their own problems as well.

I'd like to hear more about retrodiction in quantum theory as well.

Regarding the classical past, I think the OP means that, in laboratories, we prepare quantum states by classical information i.e. configurations of experimental apparatus.

 

[MaxwellsDemon]

Before the measurement, the particle exists in a linear combination of all the basis eigenstates. I don't understand what you mean when you say that there is no overlap with the observed eigenstate...because the past state of the particle included a probability of being in the eigenstate you found it in. Remember that there is no classical cause and effect in quantum physics. The observed eigenstate isn't predictable using knowledge of the measurement process and the initial state. According to QM there was nothing that caused the particle to be thrown into that particular eigenstate. Suppose you've made your measurement and the body in question is in some eigenstate. Inserting -t into the time dependent Schrodinger equation...the best you can do is talk about the probability that it was in a particular eigenstate prior to measurement. It doesn't have a well defined classical history. The best you can do is talk about the probability that it was in some eigenstate prior to measurement.

 

[0xDEADBEEF]

Thanks for the interesting replies so far.
About the classical past I mean, that in a normal experiment like a Stern-Gerlach type, we might have prepared a polarized state, and when we measure we know the possible outcomes.
If we have measured a spin at 45 degrees (1) and we proceed by measuring along 0 degrees (2), whatever the outcome of the second measurement is, we know what it was before that measurement, because we know the result of (1). So we cannot naively back-propagate from the second measurement, because we know the intermediate state. And knowing this is classical, because there are no probabilities involved at all.
Saying that the observer can only hold finite information is interesting, but I am still not sure if it gives one enough of a cool person to break open this problem.

 

[Truecrimson]

I think I see what you mean there. I agree that knowing is classical. But if we don't know anything about the outcome of the first measurement then quantum theory can't retrodict the value of spin before the second measurement.

However, there's an attempt to formulate time-symmetric quantum theory by Aharonov et al. (which actually generates the example below) but I don't know much about it other than the keywords: time-symmetric quantum theory and pre- and post-selection paradox, which you can search. There's also a reply from Asher Peres (who adheres to standard quantum theory, but not necessarily Copenhagen,collapse etc.) http://arxiv.org/abs/quant-ph/9501005.

There's one homework problem in my intro QM class that may be interesting to you.
We first prepare a particle in the state [tex]|\psi \rangle=\frac{|A\rangle +|B \rangle +|C\rangle}{\sqrt{3}}[/tex] where the three states correspond to the particle being in one of three boxes A,B, or C.

Then another person performs one of the two experiment: open the box A, or open the box B, but we don't know which one.

Finally, we perform a measurement on the system and find the outcome to be a state [tex]|\phi \rangle=\frac{|A\rangle +|B \rangle -|C\rangle}{\sqrt{3}}[/tex]. What can we conclude?

1. The overlap [tex]\langle A|\phi \rangle =\frac{1}{\sqrt{3}}[/tex] while [tex](\langle B|+\langle C|)|\phi \rangle =0[/tex]. Thus if the person opened the box A, he certainly found a particle there.

2. The overlap [tex]\langle B|\phi \rangle =\frac{1}{\sqrt{3}}[/tex] while [tex](\langle A|+\langle C|)|\phi \rangle =0[/tex]. Thus if the person opened the box B, he certainly found a particle there.

We encounter a paradox because of counterfactual reasoning.

 

[Fredrik]

The ABL rule was discussed here. It tells you the probability of getting a specific result in a measurement, given that we know that the system was determined to be in state [itex]|\alpha\rangle[/itex] before the measurement and [itex]|\beta\rangle[/itex] after the measurement. It can be derived from the Born rule and Bayes theorem. The derivation is in the thread I linked to.

I don't remember what I said there about how to interpret this stuff, but I think I was pretty confused and probably said several dumb things, so please ignore that. The derivation should be OK though.

I think the consistent histories approach takes the ABL rule as the starting point instead of the Born rule. It might have something to say about QM predicting the past, but I think it has a lot of problems, so I'm not taking it very seriously at this point. Here's something that Lee Smolin said in https://www.amazon.com/dp/0465078362/?tag=pfamazon01-20 (page 44): "if the consistent-histories interpretation is correct, we have no right to deduce from the existence of fossils now that dinosaurs roamed the planet a hundred million years ago."