How does the interpretation of consistent histories fit with the path integral? The interpretation of consistent histories states that different histories of a quantum particle have different probabilities, but in fact only one of them happens. If we take the trajectory of the particle as a history, it turns out that only one trajectory is real, that is, the particle passes only one, quite definite trajectory, but we do not know which one. This makes quantum mechanics similar to classical ones. But how does this agree with the path integral, in which it is necessary to take into account the interference effects between different trajectories?
 A: This answer is essentially paraphrasing parts of Sections 3 and 4 of Gell-Mann and Hartle's classic paper on the sum-over-histories point of view on the consistent histories approach given in:

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*Proceedings of the Santa Fe Institute Workshop on Complexity, Entropy, and the Physics of Information, May 1989

*Proceedings of the 3rd International Symposium on The Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, Japan, August 1989,

*Also available on the web as a pdf here.

Decoherence plays a key role in any interpretation of quantum mechanics. For example, in the double slit experiment, you will destroy the interference pattern if you measure the slit through which the particle went through. The word "measurement" can be misleading because there doesn't need to be a person involved in this process; a macroscopic object placed near one of the slits which interacts with the electron, will cause the same effect to occur.
The key idea is that without a measurement (or decoherence), the quantum state is a coherent superposition $\psi_1 + \psi_2$ (where $\psi_1$ is the probability amplitude for the particle to go through slit 1, and similarly for $\psi_2$). Then the probability to observer the particle on the screen, $P = |\psi_1|^2 + |\psi_2|^2 + 2{\rm Re}(\psi_1^2 \psi_2)$ contains the interference term, and is not simply the sum of the probabilities for the particle to go through either slit, $|\psi_1|^2 + |\psi_2|^2$. On the other hand, if a measurement observes the particle to have gone through slit 1, then the probability is $|\psi_1|^2$, with no interference term. The outcome of the double slit experiment does not depend on your interpretation of quantum mechanics. Perhaps you will assign different words to this calculation, but in any approach to quantum mechanics (including the consistent history approach), when no measurement is made with the object going through the slits, there is no way to say which slit the particle went through. (I suppose in a pilot wave theory there are hidden variables that could tell you which slit, but in that case I'm assuming we don't have access to the hidden variables).
What the consistent histories approach does is to build up a set of possible histories that led to the current state of the Universe. In the case of the double slit experiment, we can say there are two histories:

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*History 1 The particle went through slit 1.

*History 2 The particle went through slit 2.

The set of histories $\{{\rm History\ 1}, {\rm History\ 2}\}$ is a set of consistent histories, if it is possible to assign probabilities to each history in the set in a consistent way. The exact conditions for when this is possible, for a given set of histories, is described in Gell-Mann and Hartle's paper.
Crucially, a set of histories that is not consistent, is to give the value of the position at every point in time (or the value of the momentum, or to take some set of commuting observables and give their values at every point in time). There needs to be some level of coarse graining to obtain a set of consistent histories.
In the path integral approach, you

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*First define a set of histories. These are the values of some set of commuting observables at some discrete set of times. Y

*Further, you coarse grain the histories, by allowing for there to be some range of possible values in each observable at each time

*You assign a probability to a given coarse-grained history, by summing the amplitudes for each trajectory in a given history consistent with the constraints laid out by the values (including coarse graining) specified in steps 1 and 2, using the decoherence functional defined in their paper.

The set of probabilities you obtain in this way need to satisfy the usual laws of probability. In particular, there cannot be cross terms in which different histories interfere with each other.This puts a minimum bound on how much coarse graining is needed in a given set of histories. In the double slit example, this means that if you see an interference pattern, your coarse graining must be large enough to include the trajectories where the particle passed through both slits.
In the end, you do not end up with a unique trajectory for a quantum particle, but rather a set of possible (coarse-grained) histories for that particles, with a probability you can assign to each history. Note that even picking out one history from the set, you do not know exactly what trajectory the particle took, (a) because of the coarse graining, and (b) because in between the discrete points at which observables are defined, the particle can have any property. Essentially, you have projected the set of all possible histories a particle could have taken, onto a smaller (but still infinite) set of consistent, coarse-grained histories.
