"This universe is constantly splitting into a stupendous number of branches, all resulting from the measurementlike interactions between its myriads of components. Moreover, every quantum transition taking place on every star, in every galaxy, in every corner of the universe is splitting our local world into myriads of copies of itself." (Bryce De Witt, 1970).
The many-worlds interpretation of quantum mechanics famously holds that if a physical system is prepared into a state which is a superposition with respect to the possible values of a quantity A, then when the value of A is measured the universe splits into multiple branches, each of which realises one of the different possible definite values of quantity A. The superposed state Ψ is a sum
Ψ = c1 ψ1 + ⋅ ⋅ ⋅ + cn ψn ,
where each ψi is a quantum state in which quantity A possesses a definite value. The many-worlds interpretation proposes that when the universe branches, it branches into all the different states ψi from the superposition.
Extending this interpretation to the macroscopic world of human experience, it is postulated that the universe branches every time a choice is made, and that all the different possible lives we may have led if only our choices had been different, do in fact exist as different branches of the quantum universe.
The naive manner of picturing this process is to represent the different histories at each branching point as if they were the leaves of a book, radiating outwards from the spine. The spine in this analogy corresponds to a spacelike three-dimensional hypersurface, and the leaves radiating from it correspond to different four-dimensional space-time histories. John Earman illustrates this concept in the first diagram reproduced here, for the simple case where only two histories radiate from each branching point.
However, the notion that the entire universe branches in this style every time there is a measurement-like interaction, renders such branching a highly non-local process, and tacitly supposes that there is a unique global time coordinate for the universe. Treating a measurement-like interaction as a point event in space-time, there will be many spacelike hypersurfaces which pass through that point; the selection of only one of these as the branching hypersurface requires one to accept that there is a preferential time coordinate for the universe.
To avoid these difficulties, one can suggest that the universe only branches locally as the result of a measurement-like interaction. To be specific, one can suggest that the future light-cone of the interaction event has multiple branches, one for each possible outcome of the interaction. If one imagines such a universe as a two-dimensional sheet, then the image is one in which there are numerous pockets in the sheet, formed by the multiple branches of the future light cones. Roger Penrose drew just such an image of a branching universe in 1979, reproduced as the second diagram here.
Returning to the many-worlds interpretation, it is important to note that the different branches ψi of the overall wave-function Ψ, do not themselves correspond to different classical universes. Whilst the states ψi do indeed bestow definite values upon the quantity A, they are still quantum states in their own right, and as such, they fail to assign definite values to all the quantities possessed by the physical system under consideration.
This is crucial, because advocates of the many-worlds interpretation can often be found claiming that a quantum universe is a universe whose basic ontological fabric consists of interfering classical universes. One could conceivably subscribe to the many-worlds interpretation of quantum measurement without endorsing this stronger ontological claim. The many-worlds interpretation of quantum measurement requires one to accept that the universe is continually branching into components of the quantum wave-function, whilst the stronger ontological claim requires one to accept that the entire quantum wave-function and its branches, consist of bifurcating, interfering and merging classical histories.
The stronger claim seems to be fuelled by the 'sum-over-histories', or path-integral formulation of quantum mechanics, in which each branch of the quantum wave-function corresponds to a different set of interfering classical histories, and in which the different branches of the wave-function interfere when there is interference between these different sets of classical histories.
For example, in the famous double-slit experiment, there are two possible sets of classical histories. If we label the slits as Slit A and Slit B, then one set consists of all the possible trajectories through Slit A, and the other consists of all the possible trajectories through Slit B. When both slits are open, the paths through the different slits duly interfere with each other to produce the overall wave-function on the distant screen.
(The interference between the different branches of the wave-function are purportedly removed by decoherence on macroscopic scales, thereby explaining why we never observe quantum superpositions on such length scales).
In this sense, the classical histories form the warp and weft of the quantum fabric of the universe. Whilst this is a stronger ontological claim than the basic many-worlds interpretation, in many ways this is a more coherent picture than one in which an overall quantum state branches by fiat into its component quantum states.