A problem with David Deutsch’s model of time travel
September 2, 2015 5 Comments
In 1991, David Deutsch published a paper on the quantum mechanics of time travel. This model appeared to solve many of the ‘paradoxes’ of time travel and Deutsch used the same model to discuss time travel in The Fabric of Reality. But there are problems with this model.
A summary of Deutsch’s model
(1) To travel back in time, you go along a path which takes you from a particular region in space and time, back to that same region: this is called a closed timelike curve (CTC). In everyday life, you can go along a path that takes you back to the same region in space, but not in time. For example, you leave home to go to work, and when you have finished your work, you come back home. If you went along a CTC, you would leave your home for work at 9am, work a full day and then arrive at home at 9am. Such paths don’t exist in everyday life, but the current theory of space and time, general relativity, predicts that they are possible.
(2) The paradoxes are popularly alleged to be inconsistencies that could be produced by stuff you could do. For example, you could go back to when you left for work at 9am, and then persuade the earlier version of yourself to go play computer games instead. As a result, you would never have left for work and so would not have come back home along the CTC at 9am. As Deutsch points out in the paper, this sort of thing is just inconsistent, and so it couldn’t happen. There is no paradox in the sense that no inconsistent set of events would happen. Rather, you simply could not go back in time and persuade your earlier version to stay home. So the CTC just constrains what it is possible for you to do in a way that you would not be constrained in the CTC’s absence.
(3) The discussion of the paradoxes usually assumes that reality is described by classical physics. Reality is described by quantum physics, which might have different implications for the constraints on the actions you could take near a CTC. Quantum physics describes physical reality in terms of the multiverse. Every object exists in multiple versions that can interfere with one another under some circumstances. These versions are sorted into layers. Each layer behaves like the universe as described by classical physics to some approximation, i.e. – in some approximation it is a collection of parallel universes. Since reality as described by quantum physics is different from classical physics, the paradoxes might not arise. Deutsch invented a quantum mechanical model in which you could go back in time and persuade an earlier yourself to play video games without any inconsistency. When you travelled back in time, you would give rise to a new universe, in which you and your earlier version stay at home playing video games.
(4) There was another problem with time travel: the knowledge paradox. You could take a mathematical paper back in time and give it to the paper’s author before he wrote it. The author could then copy the paper and send it to the journal that originally published it. But this would mean that a mathematical result came into existence without anyone doing the work needed to create it. This ‘paradox’ did not produce a logical inconsistency, but it is incompatible with the evolutionary principle. All knowledge has to come into existence by processes that involve producing variations on previous knowledge and then selecting among those variations. There is no such thing as free knowledge that you can get without going through such a process. But the above time travel scenario does give you free knowledge. Nobody invented the result in the paper. If quantum mechanics solved this problem it might do it by making a new universe if you brought such information back in time and showed it to the author. The knowledge would then have been created in one universe and transferred to another. However, the physics didn’t give that result. Deutsch added additional assumptions to try to solve the knowledge problem but it was not clear whether he succeeded.
So what’s the problem with Deutsch’s model? In quantum mechanics, a system carries information about what versions of other systems it will interact with. I am interacting with a version of my computer that is in a particular location. I am not interacting with another version one millimetre to the right of the version I interact with. What prevents me from interacting with other versions of the computer is that I contain information about what version of my computer I can interact with. That information doesn’t allow this version of me to interact with other versions of my computer. Another version of me is interacting with the version of my computer one millimetre to the right. The information that specifies which version of a system can interact with which version of some other system is called entanglement information. Deutsch’s model implicitly assumes that when I travel back in time all of my entanglement information is erased .
If you use a model in which the entanglement information is not erased, the version of the system that goes back in time can’t interact with the past versions of itself or anything else it had interacted with in the past. The reason is as follows. A measurable quantity can be set up so that it is the same across the section of the multiverse in which an experiment is taking place: such a quantity is said to be sharp. If that quantity is not the same across that section of the multiverse, it is unsharp. In general, if one quantity associated with a system is sharp, then others must be unsharp. For example, if the position of a system is sharp, its momentum must be unsharp and vice versa. As a result, if you wanted to transfer information in a systems’s momentum into its position, you could only do that by erasing that information in the momentum. In general my future self’s measurable quantities that depend on those of my past self, so that copying information from one to the other is not possible. The same will be true of all of the other objects in the past that I interacted with. So the future version of me won’t be able to share information with anything in the past that I had interacted with . As a result, the paradoxes don’t arise because the relevant interactions can’t take place. This includes the knowledge paradox, which was not solved by Deutsch’s model. If you can’t transfer information to earlier versions of systems you interacted with, then you can’t give out free mathematical results.
 Technical note. The system that comes out of the CTC is assumed to have a Schrodinger picture state that is the partial trace over its state when it went into the CTC. The partial trace leaves out all of the entanglement information.
 Technical note. This issue can be discussed in the Heisenberg picture of quantum computation. Suppose you are entering the region in which the CTC is present at time . If you go in to the CTC, you come out at . You can either go in or not, and whether you will go in can be represented by a qubit . This qubit can be represented by a triple of observables satisfying the Pauli algebra. can be represented by , where and $latex \sigma_a$ is the ath Pauli matrix. In the Heisenberg picture state , starts out with the obervable having expectation value 1, where 1 means you will enter the CTC and 0 means you won’t.
Now, whether a version of you comes out of the CTC or not can be represented by a qubit . This qubit is represented by some triple that we have to work out. But what we would like to have happen is that if its value is 1, i.e. – if somebody came out of the CTC, that should set ‘s value to 0 if it was 1. This corresponds the situation in which the version of you that leaves the CTC persuades you not to go in. A controlled not gate with as the target fits the bill. Suppose had interacted with a qubit represented by in a controlled not gate, with as the target. This gives the result . Since this is the qubit that goes into the CTC, we have , which is not in the same form as the qubit used to construct the controlled not. The “controlled not” would be written as . The controlled not between two qubits and would usually be of the form . As a function of and , the gate is not of this form and so it is not a controlled NOT between them. Also, at has value 0, which means it represents a situation in which you didn’t go in to the CTC, so the qubits don’t represent the experiment we wanted to do. Finally, the gate is its own inverse, so when goes through the gate it ends up with the value had at $t=0$. So it doesn’t seem to be the case that any exchange of information has taken place between and . Adding more qubits and more interactions would raise the same problems, and also would not allow any exchange of information between past and future versions of the same system.