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.

(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

(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 ∑

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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