Multiverse interpretation of quantum mechanics

We argue that the many worlds of quantum mechanics and the many worlds of the multiverse are the same thing, and that the multiverse is necessary to give exact operational meaning to probabilistic predictions from quantum mechanics. Decoherence—the modern version of wave-function collapse—is subjective in that it depends on the choice of a set of unmonitored degrees of freedom, the environment. In fact decoherence is absent in the complete description of any region larger than the future light cone of a measurement event. However, if one restricts to the causal diamond—the largest region that can be causally probed—then the boundary of the diamond acts as a one-way membrane and thus provides a preferred choice of environment. We argue that the global multiverse is a representation of the many worlds (all possible decoherent causal diamond histories) in a single geometry. We propose that it must be possible in principle to verify quantum-mechanical predictions exactly. This requires not only the existence of exact observables but two additional postulates: a single observer within the Universe can access infinitely many identical experiments; and the outcome of each experiment must be completely definite. In causal diamonds with a finite surface area, holographic entropy bounds imply that no exact observables exist, and both postulates fail: experiments cannot be repeated infinitely many times; and decoherence is not completely irreversible, so outcomes are not definite. We argue that our postulates can be satisfied in hats (supersymmetric multiverse regions with vanishing cosmological constant). We propose a complementarity principle that relates the approximate observables associated with finite causal diamonds to exact observables in the hat.

Figure 1

Decoherence and causality. At the event $M$, a macroscopic apparatus $A$ becomes correlated with a quantum system $S$. Thereafter, environmental degrees of freedom $E$ interact with the apparatus. In practice, an observer viewing the apparatus is ignorant of the exact state of the environment and so must trace over this Hilbert space factor. This results in a mixed state which is diagonal in a particular “pointer” basis picked out by the interaction between $E$ and $A$. The state of the full system $SAE$, however, remains pure. In particular, decoherence does not take place, and no preferred bases arises, in a complete description of any region larger than the future light cone of $M$.

Figure 2

The future domain of dependence, $D({\Sigma }_0)$, (light or dark shaded) is the spacetime region that can be predicted from data on the time slice ${\Sigma }_0$. If the future conformal boundary contains spacelike portions, as in eternal inflation or inside a black hole, then the future light cones of events in the dark shaded region remain entirely within $D({\Sigma }_0)$. Pure quantum states do not decohere in this region, in a complete description of $D({\Sigma }_0)$. This is true even for states that involve macroscopic superpositions, such as the locations of pocket universes in eternal inflation (dashed lines), calling into question the self-consistency of the global picture of eternal inflation.

Figure 3

Environmental degrees of freedom entangled with an observer at $O$ remain within the causal future of the causal past of $O$, $J^{+}[J^{-}(O)]$ (cyan/shaded). They are not entangled with distant regions of the multiverse. Tracing over them will not lead to decoherence of a bubble nucleated at $P$, for example, and hence will fail to reproduce the standard global picture of eternal inflation.

Figure 4

The causal diamond (pink/shaded) spanned by two events $p$ and $q$ is the set of points that lie on causal curves from $p$ to $q$. $p$ is called the origin and $q$ the tip of the causal diamond. In the example shown, $p$ lies on the initial surface and $q$ on the future conformal boundary of the spacetime. The causal diamond is the largest spacetime region that can be causally probed by an observer travelling from $p$ to $q$.

Figure 5

Causal diamond spanned by the worldline (green line) of an observer. Environmental degrees of freedom (purple dashed line) that leave the observer’s past light cone (blue line) at some finite time can be recovered using mirrors.

Figure 6

The two-dimensional surface $\beta$ divides the future boundary of the causal diamond into two portions $B_{\pm }$. Degrees of freedom that passed through $B_{-}$ are forever inaccessible from within the diamond. Tracing over them defines a density matrix at the time $\gamma$. The pure states that diagonalize this matrix can be represented as branches. As more degrees of freedom leave the causal diamond, a branching tree is generated that represents all possible decoherent histories within the diamond.

Figure 7

In this diagram, the standard global picture of eternal inflation is taken as a starting point. Geodesics (thin vertical lines) emanating from an initial surface ${\Sigma }_0$ define an ensemble of causal patches (the leftmost is shaded grey/light) with a particular mix of initial conditions. In the limit of a dense family of geodesics, the global spacetime is thus deconstructed into overlapping causal patches. The number of patches overlapping at $Q$ is determined by the number of geodesics entering the future light-cone of $Q$. In the continuum limit, this becomes the volume $ϵ(Q)$ from which the relevant geodesics originate, which in turn defines the light-cone time at $Q$. This relation implies an exact duality between the causal patch measure and the light-cone time cutoff for the purpose of regulating divergences and computing probabilities in eternal inflation.

Figure 8

Construction of a global spacetime from decoherent causal diamond histories. Green squares indicate the origins of causal diamonds that tile the global de Sitter space. Red dots indicate discrete time steps of order $t_{\Lambda }$ at which decoherence occurs within individual causal diamonds. For example, a nonzero amplitude for bubble nucleation in the region bounded by $ACDEFG$ entangles the null segment $A$ with $C$: either both will contain a bubble or neither will. Similarly, $D$ is entangled with $E$. The density matrix for $CD$ is diagonal in a basis consisting of pure states in which a bubble either forms or does not form. This eliminates superpositions of the false (white) and true (yellow) vacuum. Initial conditions for new causal diamonds at green squares are controlled by entanglements such as that of $A$ with $C$. Further details are described in the text.

Figure 9

Three optical interference experiments: $m$’s denote mirrors, $S$ is a screen, and $D$ means detector. In the first setup, the coherence of the two paths is maintained. In the second setup, a detector is placed at one mirror. If the detector is treated as an environment and its Hilbert space is traced over, then the coherence of the superposition of photon paths is lost and the interference pattern disappears. In the third setup, the detector reverts to its original state when the photon passes it a second time. During the (arbitrarily long) time when the photon travels between $m_{1}D$ and $m_3$, tracing over the detector leads to decoherence. But after the second pass through $m_{1}D$, coherence is restored, so the screen will show the same pattern as in the first setup.

Figure 10

The shaded region is the late time portion of a causal diamond that ends in a supersymmetric vacuum with vanishing cosmological constant (left). We will think of this region as the asymptotic causal past of an “observer” in this vacuum, called the Census Taker. A portion of the boundary of such causal diamonds has infinite cross-sectional area. This portion, called a hat, coincides with a lightlike portion of the future conformal boundary in the standard global picture of the multiverse. $\Lambda =0$ bubbles that form late look small on the conformal diagram (right). But they always have infinite area. Hence, the hat regions can admit exact observables and can allow things to happen, with enough room for arbitrarily good statistics.

Figure 11

A black hole formed by gravitational collapse. The horizontal shaded region is the causal diamond of an observer (red) who remains forever outside the black hole. The vertical shaded region is the causal diamond of an observer (blue) who falls into the black hole. Note that there are infinitely many inequivalent in-falling observers, whose diamonds have different endpoints on the conformal boundary (the singularity inside the black hole); on the other hand, all outside observers have the same causal diamond.

Figure 12

An observable $A$ inside the black hole can be approximately defined and evolved using local field theory. In this way, it can be propagated back out of the black hole to an operator $A_{\mathrm{in}}$ defined on the asymptotic past boundary. From there, it is related by an exact $S$ matrix to an operator $A_{\mathrm{out}}$ that can be measured in the Hawking radiation (wiggly dashed line) by a late time observer.

Figure 13

Hat complementarity states that operators in finite causal diamonds (right) have an exact counterpart in the hat. Over time, the census taker receives an unbounded amount of information. The past light cones of the Census Taker at different times along his worldline define a natural cutoff on his information. Because there are only finitely many different diamonds, there must be infinitely many copies of every piece of information in the hat. They are related by hat complementarity to the infinite number of causal diamonds that make up the global spacetime; see also Fig. 8. At late times, increasing the Census Taker’s cutoff is related to increasing the light-cone time cutoff, which adds a new (redundant) layer of diamonds.