1. Einstein's Contrasting Standard Nonrelativistic Quantum Mechanics with Field Theory in 1948

Einstein's dissatisfaction with quantum mechanics (QM) and his attempts to show that QM is an incomplete theory is one of the best known and most analyzed facts in history and philosophy of QM. Specifically, it is well known that Einstein gave his argument against the completeness of QM (at least) four times after the publication of the EPR paper (Einstein, Podolsky, and Rosen Reference Einstein, Podolsky and Rosen1935). The first formulation is contained in his 1935 letter to Schrödinger, which he wrote just a few weeks after the EPR paper had appeared (see Howard [Reference Howard1985] and Jammer [Reference Jammer, Lahti and Mittelstaedt1985] for the details of the Einstein-Schrödinger correspondence). This was followed by three formulations (Einstein Reference Einstein1936, Reference Einstein1948, Reference Einstein and Schilpp1949).

In his 1948 Dialectica paper Einstein writes:

In the above passage, Einstein formulates (informally) three requirements for a physical theory to be compatible with a field-theorectical paradigm:

The above three requirements are not independent: Independence presupposes Spatiotemporality conceptually: only if physical things are assumed to be arranged in a space-time continuum can one ask whether the things so arranged have the feature Independence, and it also is more or less clear that the Local operations requirement is an independence condition—independence of system of (measurement) operations carried out on system .

It is true that standard (nonrelativistic) Hilbert space QM is not field theoretical in the sense that observables in nonrelativistic quantum theory are not “conceived of as being arranged in a spacetime continuum”: the observable quantities in quantum theory do not carry labels that would indicate their spatiotemporal location in a four-dimensional space-time continuum, and, accordingly, quantum measurements and operations are also not conceived of in QM as possessing spatiotemporal tags explicitly. Nor is Hilbert space QM covariant with respect to a relativistic symmetry group. Thus, QM does not meet requirements of relativistic locality interpreted in the sense of a field-theorectical paradigm.

But a local and relativistically covariant QM (quantum field theory) does exist: a particularly well-developed and mathematically exact version of it is local algebraic (relativistic) quantum field theory (AQFT). The core ideas of AQFT emerged years after Einstein's 1948 paper, during the early sixties (Haag and Schroer Reference Haag and Schroer1962; Haag and Kastler Reference Haag and Kastler1964), and in the past 40 years AQFT has developed into a rich theory (see Horuzhy Reference Horuzhy1990; Haag Reference Haag1992, for monographic presentations of AQFT).

I argue here that local AQFT can be viewed as a theory that satisfies the three major criteria of Spatiotemporality, Independence, and Local operations formulated by Einstein. In section 3 the standard axioms of AQFT are recalled, and it will be seen that Spatiotemporality is explicitly incorporated in AQFT in that it is a basic assumption that observables are explicitly regarded as localized in regions of the space-time continuum. Section 4 recalls results showing that systems localized (in the sense of AQFT) in causally independent (i.e., spacelike separated) space-time regions of the Minkowski space-time are independent—there is a rich hierarchy of notions of subsystems independence that are typically satisfied by local algebras of observables in AQFT. Consequently, AQFT does satisfy the Independence requirement as well. Section 5 interprets the Local operation requirement in terms of operations (understood as generalizations of the measurement) and recalls recent results showing that operations in AQFT can be and are typically local in the sense Einstein requires, although there are some open questions concerning the locality of operations in AQFT. First, section 2 recalls some notions of operator algebra theory, including the notion of operation.

2. Some Notions of Algebraic Quantum Mechanics

Throughout the article, denotes a unital algebra, are assumed to be subalgebras of (with common unit), denotes a von Neumann algebra, algebras are assumed to be von Neumann subalgebras of (with common unit), () denotes the (von Neumann) algebra generated by and ( and ), and is the (von Neumann) algebra formed by the set of all bounded operators on Hilbert space . For von Neumann algebra, , stands for the commutant of in , and is the state space of algebra . The self-adjoint elements in a algebra are interpreted as representatives of physical observables; the elements of the state space represent physical states. For the operator algebraic notions, see Kadison and Ringrose (Reference Kadison and Ringrose1986) or Blackadar (Reference Blackadar2005).

In what follows, T will denote a completely positive (CP) map on a algebra ; such a T will also be assumed to preserve the identity , where I is the unit of . A (unit-preserving) CP map is called a (nonselective) operation. An operation T on a von Neumann algebra is called a normal operation if it is weakly continuous.

The dual of an operation defined by

maps the state space into itself. If T is a normal operation on the von Neumann algebra , then takes normal states into normal states.

Operations are the mathematical representatives of physical operations: physical processes that take place as a result of physical interactions with the system. For a detailed description and physical interpretation of the notion of operation, see Kraus (Reference Kraus1983).

A special case of such operations are measurements. If one measures a (possibly unbounded) observable Q defined on the Hilbert space with purely discrete spectrum and corresponding spectral projections , then the “projection postulate” is described by the operation defined by

where is a normal operation from into the commutative von Neumann algebra generated by the spectral projections :

3. Algebraic Quantum Field Theory

The basic idea of AQFT is precisely what Einstein requires in Spatiotemporality: “physical things … are conceived of as being arranged in a spacetime continuum”: in AQFT, observables, interpreted as self-adjoint parts of algebras, are assumed to be localized in regions V of the Minkowski space-time M. The basic object in the mathematical model of a quantum field is thus the association of a algebra to (open, bounded) regions V of M, and all the physical content of the theory is assumed to be contained in the assignment . In particular, relativistic covariance of the theory also is expressed in terms of the net of algebras , the net of algebras of local observables.

The net is specified by imposing on it physically motivated postulates. Below, we list these postulates.

1. i) Isotony: is a subalgebra (with common unit) of if .

2. ii) Local commutativity (also called Einstein causality or microcausality): commutes with if and are spacelike separated. Let

   
   then is a normed “” algebra, completion of which (in norm) is a algebra, called the quasilocal algebra determined by the net .

3. iii) Relativistic covariance: there exists a continuous representation of the identity-connected component of the Poincaré group by automorphisms on such that

   
   for every and for every V.

4. iv) Vacuum: It also is postulated that there exists at least one physical representation of the algebra ; that is to say, it is required that there exist a Poincaré invariant state (vacuum) such that the spectrum condition (v below) is fulfilled in the corresponding cyclic (Gelfand-Neumark-Segal) representation . In this representation of the quasilocal algebra, the representation is given as a unitary representation, and there exist the generators () of the translation subgroup of the Poincaré group . The spectrum condition is formulated in terms of these generators as the spectrum condition.

5. v) Spectrum condition:

   
   Given a state , one can consider the net in the representation determined by . If the particular state is not important, then the local von Neumann algebras will be denoted by .

It is a remarkable feature of the above axioms that they are very rich in consequences: they entail a number of nontrivial features of the net. Independence of local algebras associated with spacelike separated space-time regions is such a feature, as we will see in the next section.

4. Independence

The Independence requirement demands that if and are spacelike separated regions, then the systems represented by algebras and must be “independent.” The Local commutativity postulate is such an independence requirement; however, as it turns out, there are many other, nonequivalent (but also nonindependent) concepts of independence that one can formulate for two and algebras. Below we list a few from the rich hierarchy, but the list is far from being complete (for an extensive review of the independence concepts, see Summers [Reference Summers1990b]).

Definition 1 A pair of subalgebras of algebra is called independent if for any state on and for any state on there exists a state on such that

The independence expresses that any two partial states (states on and ) are copossible as states of the larger system . The next independence condition is a strengthening of independence:

Definition 2 A pair of subalgebras of a algebra is called independent in the product sense, if the map defined by

extends to a isomorphism of with , where denotes the tensor product of and with the minimal norm.

If the algebras happen to be von Neumann algebras, then states can have stronger continuity properties, and this makes it possible to formulate independence notions that fit the von Neumann algebra category more naturally and are analogous to the above independence concepts:

Definition 3 Two von Neumann subalgebras of the von Neumann algebra are called independent, if for any normal state on and for any normal state on there exists a normal state on such that

The analogue of definition 2 in the von Neumann algebra setting is:

Definition 4 Two von Neumann subalgebras of the von Neumann algebra are called independent in the product sense, if for any normal state on and for any normal state on there exists a normal product state on , that is, a normal state on such that

The following notion is an even stronger independence property:

Definition 5 Two von Neumann subalgebras of the von Neumann algebra are called independent in the spatial product sense, if there exist faithful normal representations of and of such that the map

extends to a spatial isomorphism of with .

If are commuting von Neumann subalgebras of von Neumann algebra , then they are independent in the spatial product sense if and only if they satisfy the following so-called split property:

Definition 6 Subalgebras have the split property if there exists a type I factor such that

The next four definitions formulate an idea of operational independence; these definitions were proposed recently in Rédei and Summers (Reference Rédei and Summers2010), and they are closely related to the Local operation requirement, as we will see in the final section:

Definition 7 A pair of subalgebras of algebra is operationally independent in , if any two operations on and , respectively, have a joint extension to an operation on ; that is, if for any two CP unit-preserving maps,

there exists a CP unit-preserving map

such that

What follows is a natural strengthening of operational independence:

Definition 8 A pair of subalgebras of algebra is operationally independent in in the product sense, if any two operations on and , respectively, have a joint extension to an operation on that factorizes across the algebras in the sense that

The corresponding von Neumann algebra versions of the above two operational independence notions are:

Operational independence expresses that any operation (procedure, state preparation, etc.) on system is copossible with any such operation on system —if these systems are represented by algebras (similarly for algebras). For a more detailed motivation of operational independence, see Rédei and Summers (Reference Rédei and Summers2010).

The characterization of the interdependence relations of the above independence notions is a highly nontrivial task, and there are still a number of open problems related to this issue. We refer to Summers (Reference Summers1990b) for a review of the main results known by 1990 on this problem. What is important from the perspective of this article is that local algebras in AQFT pertaining to spacelike separated space-time regions typically do satisfy the above independence notions. We recall, in the form of two propositions, the status of the above independence properties in AQFT:

“Typically” in proposition 1 means “in all physically nonpathological models” and possibly with a spacelike distance of the double cones that is above a certain threshold. In those physically nonpathological models, the algebras associated with strictly spacelike separated double cones are split (definition 6), which entails all the other independence properties. (See Summers [Reference Summers, Accardi and Waldenfels2009] for a detailed, nontechnical discussion of the split property and for further discussion of what “(non)pathological” means in this context.)

5. Local Operations

Let and be spacelike separated space-time regions, and let and be local observable algebras in AQFT pertaining to space-time regions and and regarded as observables of a larger system localized in space-time region V, with . Let T be an operation on that can be regarded also as representing an operation carried out on system viewed as a subsystem of . Let be a state on . Call

a local system. The idea of the Local operations requirement is that local systems be such that the operation-conditioned state should coincide with on if T represents an operation carried out on . This idea is captured by the notion of operational separatedness introduced in Rédei and Valente (Reference Rédei and Valente2010) in the form of the following two definitions:

Definition 11 Let be a local system. The operation

is said to be based in if it is the extension of an operation on , that is, if is invariant under T:

And it is said to be based in if is invariant under T:

Note that if () is an operation on , then there may not exist an operation T on that extends (see remark 1).

Given a local system with the operation defined by the projection postulate (with or ; cf. eq. [1]), the local commutativity requirement of AQFT entails that the operation is the identity map on (or on ), so we recover the well-known fact:

Not all operations are of the form of the projection postulate, however; in fact, one can show what follows (Rédei and Valente Reference Rédei and Valente2010):

Thus, it would seem that the Local operations requirement is violated in AQFT—contrary to the claim in the introductory section. But this conclusion would be too quick. One can argue that the mere existence of operationally not separated local systems should not be interpreted as a genuine incompatibility of AQFT with the Local operations requirement because one cannot expect a theory such as AQFT to exclude causally non-well-behaving local systems necessarily. But it is reasonable to demand that AQFT allow a locally equivalent and causally acceptable description of an operationally not separated local system. In other words, one can say that it may happen that the possible causal bad behavior of the local system is due to the nonrelativistically conformed choice of the operation T on representing an operation based in , say, and there may exist another operation on that is based in and that has the same effect on as that of T (i.e., for all ) and such that the system is causally well behaving. This idea of reducibility of operational nonseparatedness is formulated explicitly in the form of the following two definitions, each weakening definition 12 in the and categories, respectively (see Rédei and Valente Reference Rédei and Valente2010):

Einstein's requirement of Local operations can be interpreted as the requirement that local systems in AQFT should be operationally and separable in the sense of definitions 13 and 14. To answer the question of whether local systems in AQFT are operationally separable, one can relate operational separability to operational independence:

Since operational independence in does hold for local algebras and associated with strictly spacelike separated double-cone regions , and double-cone (proposition 1), one concludes that AQFT typically satisfies the Local operations requirement—at least in the sense. As per remark 1, it is not clear at this point whether operational independence of the pair in also holds, so one cannot yet conclude in full generality that the Local operations condition also holds in AQFT typically, but it should be clear from these results that AQFT is a theory that displays exactly the features that Einstein thought were necessary for a physical theory to be compatible with a field-theorectical paradigm. So it seems fair to say that in his critique of standard nonrelativistic QM, Einstein formulated informally a research program that started off later, in the early sixties, in the form of what is known today as local AQFT.

6. Closing Comments

The interpretation given in this article of Einstein's critique of standard nonrelativistic QM is not the only one possible. If one regards Einstein's critique as an argument pointing out the incompatibility of the field-theorectical paradigm with entanglement, as Howard (Reference Howard1985) and Clifton and Halvorson (Reference Clifton and Halvorson2001) do, then one must conclude (as Clifton and Halvorson [Reference Clifton and Halvorson2001] do) that AQFT fares even worse than nonrelativistic QM because entanglement is even more endemic in AQFT than in nonrelativistic QM (Bell's inequality is violated even more dramatically in AQFT than in nonrelativistic QM; Summers and Werner Reference Summers1985; Summers Reference Summers, Doplicher, Longo, Roberts and Zsido1987a, Reference Summers and Werner1987b, Reference Summers and Werner1988, Reference Summers1990a, Reference Summers1997; Clifton and Halvorson Reference Clifton and Halvorson2000). But one can argue, as von Neumann did in his reply to Schrödinger's entanglement papers, that the presence of correlations between spacelike separated local systems does not in and by itself make AQFT unacceptable from a field-theorectical point of view—as long as it is possible, in principle at least, to give a causal explanation (in terms of local common causes) of the spacelike correlations (see von Neumann's letter to Schrödinger dated April 11, 1936, in Rédei [Reference Rédei2005, 211–13] and Rédei [Reference Rédei, Demopoulos and Pitowsky2006] for an analysis of von Neumann's position). Whether such local common cause explanations of the spacelike correlations in AQFT are possible is still an open question, and only unsatisfactory results are known on the problem (Rédei and Summers Reference Rédei2002; Rédei Reference Rédei and Summers2007).

One also can raise the question of whether the Local operations requirement is satisfiable with selective operations, that is, operations that do not preserve the identity . A typical selective measurement operation is the selective projection postulate:

where is the spectral projection belonging to the ith eigenvalue of an observable (cf. eq. [1]). The dual of a selective operation T on does not take states into states; the selective operation-conditioned state is given by

If , , and are local observable algebras and T is a selective operation on representing a (selective) operation on , then there is an such that , and since the set of states on a algebra is separating, there is a state on such that . The state can be extended from to a state on , and it is clear then that the system

is not operationally separated. One can modify the definition of operational separability (definitions 13 and 14) by allowing selective operations and ask when operational separability holds and whether operational separability holds in AQFT for selective operations in particular. Unfortunately, in this case, one cannot invoke proposition 5 to answer this question because operational independence cannot hold for two subalgebras and of a algebra for selective operations: if and are two selective operations on the respective algebras, they can assign different elements to the identity, and this prevents a common extension T on to exist. Thus, it is unclear under what conditions operational separability with respect to selective operations can hold. But operational separability with respect to selective operations seems to have limited interest only: one can argue that “a selective operation involves performing a physical operation on an ensemble followed by a purely conceptual operation in which one makes a selection of a subensemble on the basis of the outcome of a physical operation (assigning ‘state’ 0 to the reminder). Non-selective operations, by contrast, always elicit a ‘yes’ response from any state, hence the final state is not obtained by selection but purely as a result of the physical interaction between object system and the device that effects the operation” (Clifton and Halvorson Reference Clifton and Halvorson2001, 10).

So a possible violation of the Local operations requirement by selective operations should not be regarded as troublesome physically; hence, it should not entail a physical incompatibility of the field-theorectical paradigm with QM. One also can take the position that proposition 4 does entail that the Local operations requirement is violated in AQFT; hence, AQFT is not really compatible with the field-theorectical paradigm; that is to say, one can regard definitions 13 and 14 as too weak. It would be interesting to explore how these definitions can be strengthened and to see whether physically well-motivated stronger definitions of operational separability are satisfied in AQFT.