Local Physical Magnitudes
Non-locality, as the term suggests, is best approached via the notion of locality. As it will be seen, the notion of locality as it appears in physics has several components, but the foundational component is that of a local state in space-time. If one conceives of space-time as a four-dimensional (or eleven- or twenty-six-dimensional) manifold, one can think of covering the manifold with overlapping open neighborhoods, such that every point is contained within at least one neighborhood. One can also imagine indefinitely shrinking the size of the neighborhoods and indefinitely increasing their number. For any particular neighborhood, the intuitive notion of a neighborhood-local state is a physical state that depends only on what is inside the neighborhood. To get a more formal handle on this, a necessary condition for a neighborhood-local state is that the values of quantities for such a state put no constraints on the values of neighborhood-local states for any nonoverlapping neighborhoods. By this criterion, familiar physical properties, like the locations and velocities of particles or the values of electric fields, are neighborhood-local, while global physical properties, like the total charge of the universe, are not. (It is tempting to try to take this notion to the limit, where the neighborhoods become punctate, but this leads to many technical problems that are unrelated to the basic notion.)
Classical physics has many neighborhood-local quantities: for example, mass and charge densities, field strengths, velocities and accelerations, and the relativistic space-time metric. Anything represented by a tensor in a classical theory will be, by this account, neighborhood-local. Indeed, in classical physics it appears that all nonneighborhood-local quantities, such as the total charge of the universe, are functions of the neighborhood-local ones in the following sense: Cover the space-time manifold with open neighborhoods in any way one likes and specify the neighborhood-local quantities in each neighborhood and the neighborhood-local quantities in all intersections of neighborhoods, and one will thereby fix the value of the global quantities. Given the charge in every little patch, and in the intersections of all the little patches, the total charge of the universe follows.
Physics textbooks do not typically present the notion of neighborhood-locality in this way: They rather get at it via an account of coordinatizing the manifold. Rather than demanding a single, global coordinate system that completely covers a manifold (which in many cases will not exist), one is rather required only to break up the manifold into overlapping neighborhoods (each of which is topologically simple) and to coordinatize each neighborhood. The coordinatization of each neighborhood is called a chart, and a collection of charts for neighborhoods that cover the manifold is called an atlas. In addition, one is required to specify how the coordinates assigned to a point in one chart are related to the coordinates assigned by any other chart in which the point occurs. That is, one is required to specify how the different coordinate systems relate to one another where they overlap. The assumption that all the physics is ultimately neighborhood-local is then essentially the supposition that physical states can be assigned to each charted neighborhood such that the total physical state of the universe is determined by the information in the atlas. One can say that in such a case the total physics is neighborhood-local.
The neighborhood-locality of physics accepts the physical reality of many global properties, such as the total charge. It also accepts the physical reality of more subtle global properties. Consider, for example, a cylinder and a Möbius strip. In a certain sense, a cylinder and a Möbius strip can be made to match locally: each can be divided into overlapping neighborhoods such that every neighborhood of the Möbius strip is exactly like the corresponding neighborhood of the cylinder. In this sense, the twist in the Möbius strip is not located anywhere in particular: it is a global rather than local feature of the space. Nonetheless, one could tell from an atlas whether one was dealing with a cylinder or a Möbius strip. Begin, for example, by drawing an F on one chart. The chart contains enough information to determine how the F could move rigidly in the neighborhood covered by the chart. So one could move it into a region that overlaps another chart, and the functions relating the chart would show how the F shows up in the new region. Continuing in this way, one could determine from the information in the atlas what the result of any rigid motion of the F would be. On a Möbius strip, some such motion will bring the F back to the original neighborhood mirror-reflected, while on a cylinder this can never happen.
The notion of neighborhood-locality is therefore quite broad: all of classical physics and relativity theory (both special and general) count as neighborhood-local in this way. One's commonsense picture of the world is also neighborhood-local. Albert Einstein powerfully expressed the notion of neighborhood-locality this way:
It is … characteristic of … physical objects that they are thought of as arranged in a space-time continuum. An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects "are situated in different parts of space." Unless one makes this kind of assumption about the existence (the "being-thus") of objects which are far apart from one another in space—which stems in the first place from everyday thinking—physical thinking in the familiar sense would not be possible. … This principle has been carried to extremes in the field theory by localizing the elementary objects on which it is based and which exist independently of each other, as well as the elementary laws which have been postulated for it, in the infinitely small (four-dimensional) elements of space.
(in Born 1971, p. 170)
If one reads "situated in different parts of space" as "situated in nonoverlapping neighborhoods," and understands the "existence (the being-thus)" as the demand that the physical state defined on one neighborhood puts no constraint on the physical state in a nonoverlapping neighborhood, one sees that Einstein is expressing the same idea.
Suppose that physics is neighborhood-local in the sense that the physical information provided in any atlas is complete (determines all the physical properties of the universe). This appears to be a mild constraint, seeing as it takes in all of classical physics and relativity. It is hard to see, in fact, how the postulate of neighborhood locality puts any real empirical constraint on a theory: Could not any set of phenomena be accounted for by a neighborhood-local physics? As it will be seen, this is correct: To get an empirical constraint one will have to add on to neighborhood-locality in this sense. However, the postulate of neighborhood-locality does do something: It implies that, for any region in space-time, there is something that counts as the physical state of that region. Recall the twin requirements: The physical state in any neighborhood should not put any constraints on the physical state in a nonoverlapping neighborhood and the totality of physical states in an atlas (including appropriate information about overlapping charts) should determine the total physical state of the universe. Meeting these requirements demands that many well-defined quantities cannot count as local. For example, although the center of mass of the solar system is, one may suppose, always located at some particular point in space, it does not count as a part of the local physical state of that space. For taking a small neighborhood that contains that point, one cannot specify that the center of mass of the solar system occupies that point without thereby constraining the physical state of the nonoverlapping neighborhoods that contain the sun and planets.
If the requirement of neighborhood-locality is so mild, why was Einstein concerned with it? Because, taken at face value, the quantum theory rejects neighborhood-locality. Consider a pair of particles in the singlet state
1/√2|x −up>R|x −down>L − 1/√2|x −down>R|x −up>L
where the particle on the right and the particle on the left are far apart, in different regions of space. Then one's atlas could contain a neighborhood that includes particle R but not particle L, and a nonoverlapping neighborhood that contains L but not R. And the requirement of neighborhood-locality would then demand the existence of some physical state that can be assigned to (the neighborhood containing) R that puts no constraint on the state of L, and a state that can be assigned to L that puts no constraint on the state of R, such that from these two local states the singlet state for the pair can be recovered.
The singlet state itself cannot be used for this purpose: It makes reference to both particles and requires for its existence the existence of both particles. There is a well-defined state that quantum mechanics associates with particle R alone: It is called the reduced state for R from the singlet state. The reduced state supplies enough information to make quantum-mechanical (probabilistic) predictions for the result of any experiment carried out on R alone. There is a similar reduced state for L. These states are, mathematically speaking, mixed quantum mechanical states.
Why, then, can one not take the reduced state for R to be its neighborhood-local state, and the reduced state for L to be its neighborhood-local state, and do the physics using these? The reason is because different joint quantum mechanical states for the pair of particles give rise to exactly the same pair of reduced states for R and L, and these different joint states make different predictions for some measurements that involve both particles. For example, the singlet state is mathematically distinct from the m = 0 triplet state
1/√2|x −up>R|x −up>L + 1/√2|x −down>R|x −down>L.
Furthermore, the m = 0 triplet state makes different predictions for the pair: If one measures the spin of both particles in the x -direction, the singlet state predict that the outcomes on the two sides will be different, while the m = 0 triplet state predicts they will come out the same. Even so, the reduced states for R and L that can be derived from these are identical (they both predict a 50 percent chance for the measurement of x -spin to be up). So one cannot use the joint state as a neighborhood-local state, and one cannot use the reduced states as neighborhood-local states (and recover the full physical state of the pair from the atlas), and quantum mechanics provides no other states one can use.
What Einstein saw was that quantum mechanics is not neighborhood-local on account of the entanglement of states for spatially separated systems. And since Einstein thought that physics must be neighborhood-local, he thought quantum mechanics must not be giving one a complete account of the physical states of things.
Non-Locality and Experiment
So far, all one has is a remark about the formalism of quantum mechanics, not about the empirical predictions of quantum mechanics. However, Einstein saw that the peculiar entanglement of quantum-mechanical states forced another kind of non-locality on the standard quantum mechanical accounts of experiments.
Consider a pair of separated particles in the singlet state. Given only that state, the quantum formalism permits no definite predictions about the outcome of an x -spin measurement on either side: For each individual particle, quantum mechanics assigns a 50 percent probability for each possible outcome. If the quantum description is complete and leaves no physical facts about the particle out of account, then these probabilities must reflect objective indeterminacy in nature: Nothing in the universe determines which outcome will occur. Nonetheless, as Einstein saw, quantum theory does make a perfectly definite prediction: Whatever the outcome of the experiments on the two particles, the results for the pair will be opposite—one will yield x -spin up and the other x -spin down (in the m = 0 triplet state, the results are instead guaranteed to be the same). So the question is: If nothing in the whole universe determines what the result of measuring the particle on the right will be, and if the particle on the left can be arbitrarily far away, what could possibly ensure that the outcome on the left will be the opposite of that on the right?
In the standard quantum formalism, this correlation between the outcomes is secured by the collapse of the wave function: when the particle on the right displays, for example, x -spin up, then the overall quantum state for the pair suddenly changes from 1/√2|x −up>R|x −down>L − 1/√2|x −down>R|x −up>L to |x −up>R|x −down>L. Because of the non-locality of the wave function, this change is a change not only in the physical state of particle R but a change in the state of particle L as well. When particle R displays x -spin up, particle L changes from a state of indefinite x -spin to a state of definite x -spin down. It is by this "spooky action-at-a-distance" that the standard quantum interpretation manages to secure the correlation in spins between distant particles, neither of which is initially in a definite spin state.
There are several ways in which the wave collapse is "spooky." One is that is it unmediated: The measurement on the right influences the state on the left without the aid of any particles or waves traveling between the two sides. However, more important, the collapse is instantaneous: Even if there were mediating particles or waves, they would have to travel faster than light. This last property seems to contradict the theory of relativity. Einstein rejected the quantum theory because of this feature. He saw that, in this particular case, the spooky action-at-a-distance is not required by the empirical phenomena: The perfect correlations can be easily explained in a neighborhood-local physics without resorting to any direct causal connection between the two sides. One need only suppose (as the quantum theory does not) that the results of the spin measurements are predetermined by the local state of each electron and that the electrons are created in states in which they are disposed to give the opposite outcomes to all spin measurements.
Putting Einstein's two requirements together, one can now specify what it is for a theory to be simply local: First, all the fundamental physical properties of the theory should be neighborhood-local, and second, no physical influences in the theory should be allowed to propagate faster than light. (One could also add that causal connections between events should be mediated by continuous processes, but that is not needed for the sequel.) Einstein's argument against quantum theory as complete is that taking it to be complete requires that one treat the physics as non-local, even though the phenomena do not force non-locality on the theory. Einstein thought it perverse to insist that the theory is complete instead of trying to supersede it by a local theory that recovered all the same empirical predictions.
A local theory can be either deterministic or indeterministic. In a deterministic theory, every event is determined by the physical state that precedes it, and in a local deterministic theory, those determining factors cannot be so far away that it would require a superluminal influence for them to have their effect. Putting these together, it follows that in a local deterministic theory, every event is determined by the neighborhood-local state on its past light cone.
In an indeterministic local theory, an event need not be determined by the physical state of its past light cone, but the probability for the event will be. Furthermore, nothing outside the past light cone can have any influence on the event. That is, conditionalizing an event on the state of its past light cone should yield a probability that is screened off from any further information about events at space-like separation. (The probability will not be screened off from events in the future light cone, which can be effects of the event in question.) So positing that a theory is local is not the same as positing that it is deterministic, but it puts definite mathematical constraints on the nature of any local theory, whether deterministic or indeterministic. What Einstein had argued, in the 1935 Einstein, Boris Podolsky, and Nathan Rosen (EPR) paper, was that quantum mechanics is an indeterministic, non-local theory, but the sorts of correlations he discussed admit a deterministic, local explanation. And despite Einstein's oft-cited remarks about God's gambling habits, it was the spooky action-at-a-distance, the non-locality, that was the focus of his criticism in the EPR paper. As it turns out, if one is to recover the perfect EPR correlations with a local theory, it must also be a local deterministic theory (otherwise the correlations will not be perfect), but recovering determinism is not the main issue.
Bell's Theorem and Locality
What Einstein did not realize is that although the perfect correlations he discussed can be recovered by a local theory, the full range of quantum mechanical predictions cannot be recovered by any local theory. This was proven in 1964 by John Bell. Bell demonstrated that the predictions of any local theory, deterministic or indeterministic, must satisfy a certain statistical constraint called Bell's inequality. Furthermore, the predictions of the quantum theory violate that inequality, and the violations have been experimentally confirmed in the laboratory. So the non-locality of quantum theory is not just an artifact of the quantum formalism: It is a physical aspect of nature.
Although in principle a neighborhood-local theory could predict violations of Bell's inequality (by use of neighborhood-local items that travel faster than light), the only presently existing accounts of physical non-locality employ the quantum wave function, which is not a neighborhood-local object. The role of the wave function differs from interpretation to interpretation, but in every case it is the wave function that secures the violation of locality and the superluminal physical connection between the distant particles.
It is a first-order technical problem to reconcile the non-locality of quantum theory with the space-time structure postulated by the theory of relativity. The simplest way to construct a non-local theory is to add a preferred foliation of space-time to the relativistic picture, thereby violating the spirit of relativistic physics. Such a foliation also allows for a straightforward causal account of the phenomena: Intraction with one of the particles is the cause of a change of behavior in the other. It is, however, feasible (although quite tricky) to construct theories that achieve non-locality but employ only the relativistic space-time structure. In these cases, it appears that standard causal locutions cannot the recovered: there is a real physical connection between space-like separated events, but one cannot identify one of the events particularly as a cause and the other as an effect.
Born, M. The Born-Einstein Letters. Translated by I. Born. New York: Walker, 1971.
Einstein, Albert, Boris Podolsky, and Nathan Rosen. "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Physical Review 47 (1935): 777–780.
Maudlin, Tim Quantum Non-locality and Relativity: Metaphysical Intimations of Modern Physics. Oxford, U.K.: Basil Blackwell, 1994.
Tim Maudlin (2005)