Web essay: www.zeh-hd.de  -- (2006 - last revised: May 2011)

## Quantum nonlocality vs. Einstein locality

H. D. Zeh

Quantum theory is kinematically nonlocal, while the theory of relativity (including relativistic quantum field theory) requires dynamical locality ("Einstein locality"). How can these two elements of the theory (well based on experimental results) be simultaneously meaningful and compatible? How can dynamical locality even be defined in terms of kinematically nonlocal concepts?

Dynamical locality in conventional terms means that there is no action at a distance: states "here" cannot directly influence states "there". Relativistically this has the consequence that dynamical effects can only arise within the forward light cones of their causes. However, generic quantum states are "neither here nor there", nor are they simply composed of "states here and states there" (with a logical "and" that would in the quantum formalism be represented as a direct product). Quantum systems at different places are usually entangled, and thus do not possess any states of their own. Therefore, quantum dynamics must in general describe the dynamics of global states. It may thus appear to be necessarily nonlocal.

This discrepancy is often muddled by insisting that reality is made up of local events or phenomena only. However, quantum entanglement does not merely represent statistical correlations that would represent incomplete information about a local reality. Individually observable quantities, such as the total angular momentum of composed systems, or the binding energy of the He atom, can not be defined in terms of local quantities. This nonlocality has been directly confirmed by the violation of Bell's inequalities or the existence of Greenberger-Horne-Zeilinger relations. If there were kinematically local concepts completely describing reality, they would indeed require some superluminal "spooky action at a distance" (in Einstein's words). Otherwise, however, such a picture is questionable or meaningless. In particular, nothing has to be teleported in so-called quantum teleportation experiments. In terms of nonlocal quantum states, one has to carefully prepare an appropriate entangled state that contains, among its components, all states to be possibly teleported (or their dynamical predecessors) already at their final destination – similar to the hedgehog's wife in the Grimm brothers' story of Der Hase und der Igel (see Quantum teleportation and other quantum misnomers).

These kinematical properties characterize quantum nonlocality. But what about Einstein locality in this description? Why does the change of a global quantum state not allow superluminal signals, for example? The concept of locality in quantum theory requires more than a formal Hilbert space structure (relativistically as well as non-relativistically). It presumes a local Hilbert space basis (for example consisting of spatial fields and/or particles). Dynamical locality then means that the Hamiltonian is a sum over local terms, or an integral over a local Hamiltonian density in space, while all dynamical propagators for these local elements must relativistically obey the light cone structure.

This framework is most successfully represented by quantum field theory. It may be characterized by the following program:
(1) Define an underlying set of local "classical" fields (including a spatial metric) on a three-dimensional (or more general) manifold.
(2) Define quantum states as wave functionals of these fields (that is, nonlocal superpositions of different spatial fields).
(3) Assume that the Hamiltonian operator H (acting on wave functionals) is defined as an integral over a Hamiltonian density, written in terms of these fields at each space point.
(4) Using this Hamiltonian, write down a time-dependent Schrödinger equation for the wave functionals, or, in order to allow the inclusion of quantum gravity, a Wheeler-DeWitt equation: H \Psi = 0.
The dynamics is then local (in the classical sense) for all local basis elements, which, according to this construction, must span the space of all states. This concept defines the quantum version of Einstein locality. (I have here not discussed complications resulting from nonlocal gauge degrees of freedom.)

The local (additive) form of the Hamiltonian has an important dynamical consequence for nonlocal states. If two distant systems \phi and \psi are entangled, assuming the form ∑n√pn\phin\psin in their Schmidt decomposition, all matrix elements of H between components with different n must vanish, since the individual, local terms of H can only act on \phi or \psi. Such "dislocalized superpositions" arise unavoidably by means of decoherence, while their relocalization ("recoherence") would require an improbable accident in a causal universe (see The Physical Basis of the Direction of Time). The factorizing Schmidt components thus describe dynamically autonomous "worlds", which must contain separate observers, and which permanently branch by means of measurement-like processes.  This dynamical argument, based on nothing else but the Schrödinger equation with its local Hamiltonian, justifies Everett's collapse-free interpretation of quantum theory – see How Decoherence can solve the measurement problem. (Note that the linearity of dynamics by itself would not be sufficient for this purpose, since it would not be able to describe quantum measurements and other processes that lead to nonlocal entanglement.)

If, in the case of a Wheeler-DeWitt equation, a WKB approximation (based on a Born-Oppenheimer expansion in terms of the Planck mass) applies, orbit-like "wave tubes" in the "superspace" of spatial geometries (the configuration space of general relativity) may define quasi-classical spacetimes (such as solutions of the Einstein equations). The corresponding matter states obey a derived time-dependent Schrödinger equation with respect to a "WKB time" parameter along these quasi-classical orbits of spatial geometries (see C. Kiefer: Quantum Gravity, Cambridge UP, 2007). Wave tubes on the configuration space of geometry are then decohered from one another by the matter states (which thereby act as an environment to quantum geometry) according to the Wheeler-DeWitt equation. This decoherence along quasi-trajectories in superspace may lead to further quasi-classical fields, and possibly other quasi-local variables, which are robust in the sense that their different values define dynamically autonomous components ("branches"). Einstein locality then holds up to remaining quantum uncertainties of the spacetime metric (resulting from the non-vanishing widths of the wave packets in superspace).

In "effective" (phenomenological) quantum field theories, dynamical locality is often formulated by means of a condition of microcausality. It requires that commutators between field operators at spatially different spacetime points vanish. This condition is partially kinematical (as it presumes a local reference basis of quantum states), partially dynamical (as it uses the Heisenberg picture for field operators), and partially a matter of definition (as it requires a decomposition of the field operators in terms of "particles and antiparticles", which may depend on the effective vacuum, for example). The dynamical consistency of this microcausality condition is nontrivial. In principle, the properties of (anti-)commutators of (effective) field operators at different times should represent a deterministic consequence from those on an arbitrary simultaneity, t = t', caused by the given relativistic dynamics (Hamiltonian). They cannot be independently postulated for all times.

In his foundation of quantum field theory, Steven Weinberg derived microcausality and the locality of the Hamiltonian from his cluster decomposition principle. This is a phenomenological constraint to the S-matrix, which requires that "distant experiments give uncorrelated results". However, such a principle cannot form a fundamental element of the quantum theory, since (a) observable correlations may exist or controllably be prepared either as statistical correlations or as entanglement between distant systems, and (b) the concept of an S-matrix is (approximately) applicable only to sufficiently isolated (microscopic) systems. Macroscopic systems never cease to interact uncontrollably with their environment; this fact is known as the source of decoherence, and hence of the classical phenomena and the appearance of "quantum events" (see How decoherence may solve the measurement problem). Only such apparent events justify the probability interpretation of the S-matrix – even for microscopic objects. So I feel that instead of going beyond the empirically founded effective theories when searching for mathematical consistency of hypothetical theories (in the hope for finding the final universal theory), physicsts should first analyze the physical consistency and meaning of effective field theories (see also Chap. 6 of The Physical Basis of the Direction of Time).

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