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\author{Tilman Sauer\\
{\small Einstein Papers Project}\\%[-0.5cm]
{\small California Institute of Technology 20-7}\\%[-0.5cm]
{\small Pasadena, CA 91125, USA}\\%[-0.5cm]
{\small \makeatletter tilman@einstein.caltech.edu \makeatother}}
\title{An Einstein manuscript on the EPR paradox for spin observables}
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\begin{abstract}
A formulation by Einstein of the Einstein-Podolsky-Rosen incompleteness argument found in his scientific
manuscripts is presented and briefly commented on. It is the only known version in which Einstein
discussed the argument for spin observables.
The manuscript dates, in all probability, from late 1954 or early 1955 and hence also
represents Einstein's latest version of the incompleteness argument and one of his last
statements on quantum theory in general. A puzzling formulation raises the question
of Einstein's interpretation of space quantization and the non-classical spin degree of freedom.
\end{abstract}
\section*{Introduction}
It is well-known that the EPR paper
\cite[Einstein, Podolsky, and Rosen, 1935]{EPR35} was actually written by
Boris Podolsky ``after many discussions,'' and that Einstein, as he confided in
correspondence with Erwin Schr\"odinger, was discontent
with the way it came out since he thought that ``the main point was, so to speak, buried by
erudition.''\footnote{Einstein to Schr\"odinger, 19 June 1935, Albert Einstein Archives (AEA)
call number 22-047, cited
in \cite[Howard, 1985, p.~175]{How85},
see also the discussion by Fine \cite[1996, p.~35]{Fin96} who was the
first to point out the significance of this letter.
\label{note:Podolskywrote}}
Einstein gave his own version of the EPR incompleteness argument in
the same letter to Schr\"odinger and shortly thereafter in print in his essay
on ``Physics and Reality,'' \cite[1936]{Ein36} where it is explicitly referred to as a
``paradox''\cite[p.~376]{Ein36}.\footnote{Note that, as pointed out by Fine
\cite[1996, p.~47, n.~11]{Fin96}, it is explicitly but erroneously denied by
Jammer \cite[1974, p.~186]{Jam74} that the authors of the EPR
paper ever considered their argument as ``paradoxical.''}
There are three more published versions of the EPR incompleteness argument
by Einstein. One was included in his ``Autobiographical Notes,'' drafted in 1947 and
published in Schilpp's {\it Albert Einstein Philosopher--Scientist} in 1949, another in his
``Reply to Criticisms'' in the same volume, the third in his paper
on ``Quantum-Mechanics and Reality'' which appeared in 1948 in a special issue of the
journal {\it Dialectica} %
\cite[Einstein, 1949a, pp.~83--87; 1949b, pp.~681--682; 1948]{Ein49a,Ein49b,Ein48}.
Other versions of the argument are found in his
correspondence.\footnote{See, e.g., Einstein to Schr\"odinger, 19 June 1935
(cp.~note~\ref{note:Podolskywrote}) and Einstein to J.L.B.~Cooper, 31 October 1949 and
18 December 1949 (AEA~8-412, 8-414, cited in part by Stachel \cite[2002, p.~391, 410f]{Sta02}).
Discussions of Einstein's incompleteness argument
with extensive references to his correspondence are given by Fine \cite[1996, ch.~4]{Fin96},
Howard \cite[1985]{How85},
\cite[1990]{How90}, and also Stachel \cite[2002, ch.~VI]{Sta02}.
\label{note:correspondence}}
All of Einstein's own formulations of the EPR argument as known
discuss the problem either directly
for position and momentum observables, or else in terms of some set of non-specified
canonically conjugate observables.
In these known formulations Einstein illustrated the
incompleteness argument by referring
to a hypothetical experiment in which the values of the position and momentum variables of two
spatially separated but quantum-mechanically entangled particles are being measured resp.\ inferred.
Modern discussions of the EPR
argument, however, routinely reformulate the EPR experiment for particle observables
of a finite, usually $2$-dimensional, Hilbert space and illustrate the experiment in terms of
spin observables or for the polarization degrees of freedoms of photons. It is also for
correlated photons
that the EPR thought experiment was first realized in actual experiments.%
\footnote{Shimony \cite[2006]{Shi06} provides a recent review of both the theory and the
experiments performed.}
The first published reformulation of the EPR argument for spin observables seems to
have been given in 1951 by David Bohm in his textbook on quantum
mechanics \cite[1951, pp.~614--622]{Boh51}.\footnote{See \cite[Jammer, 1974, p.~235]{Jam74}.
It is clear from the extant Einstein-Bohm correspondence that
Einstein knew {\it of} Bohm's book but it is unclear to what extent if at all he had actually
read it, see the discussion below.}
Bohm considered his reformulation ``conceptually equivalent'' to the EPR version
but argued that it was ``considerably easier to treat
mathematically'' \cite[p.~614]{Boh51}. Bohm's reformulation also was
presented just shortly
after a mathematical difficulty of the original EPR argument had been pointed
out by the mathematician J.L.B.~Cooper (1950).\footnote{See
\cite[Jammer, 1974, pp.~236--238]{Jam74}. Jammer also discusses other early contentions of the EPR
argument that amount to the charge of a wrong or inconsistent application of the
mathematical formalism, such as Paul Epstein's objection that the argument does not take into
account the time-dependence of the wave functions explicitly.}
Cooper had argued that spatial
separation of two particles would imply an infinitely high potential barrier between the two
spatially separated particles, leading to a vanishing of the joint wave-function at some
place, say $z=0$. Such vanishing would render the momentum operator no longer
self-adjoint (but still Hermitian), since its domain would have to be restricted to
the positive (resp.\ negative) half line $[0,\infty]$ (resp. $[-\infty,0]$).
Therefore, Cooper argued, EPR were not allowed to invoke the
respresentation theorem for the restricted momentum operator, which would have
no (non-trivial) eigenfunctions at all. Einstein's response to this argument, given in
correspondence after having read a manuscript version of Cooper's paper,
was to repeat the incompleteness argument,
emphasizing that the spatial separation of the two particles only demanded the vanishing
of the wave function at $z=0$ in some limiting sense, and that hence the question of an
infinite potential
barrier was ``of no interest'' for the question.\footnote{See note \ref{note:correspondence}.}
We now present and briefly comment on a manuscript in which Einstein
gives another formulation of the EPR incompleteness argument. This appears to be the only version
in which he discusses the EPR argument explicitly for spin observables.
\section*{The manuscript}
Einstein's version of the EPR incompleteness argument for spin observables
is found on the lower half of the verso of a sheet of
calculations (AEA~62-575r) that is part of a considerable batch of some 1800 pages of
manuscript calculations. The manuscript pages turned up when Einstein's papers
were being packed up to be shipped from Princeton to Jerusalem in 1982.\footnote{The
pages were then added to the
Albert Einstein Archives at the Hebrew University of Jerusalem and given archival numbers
62-001 through 63-416 \cite[Sauer, 2004]{Sau04}.} Most of the manuscript pages
show calculations in the context of
Einstein's search for a unified field theory or else on problems of conventional general
relativity. A few pages deal with other problems, but I found no other instance in the set
that appears to be related to the quantum incompleteness argument.
The whole batch contains manuscripts dating from the late twenties until the very
end of Einstein's life. The dating of the particular sheet to be discussed here is unclear.
The handwriting suggests a date late in Einstein's life
as does the context of the unified field theory calculations.\footnote{The surrounding
calculations on the recto and verso of the same sheet are dealing with the problems of
finding a diagonal representation of the metric tensor and the positivity of the respective
metric components, and with variational calculations involving the curvature tensor.
Without further context, one might perhaps argue that these calculations may well have been done in
the framework of conventional general relativity. However, notational continuities with the
surrounding pages suggest that they were rather part of considerations in the context
of Einstein's unified field theory approach based on an asymmetric metric.} The most concrete hint
is the fact that the particular sheet is found right after a sheet that
can be dated explicitly as after 30 November 1954. This sheet (AEA~62-574) is a
letter of Serge Moguillanes to Einstein, of that date, with calculations by Einstein on the
back.\footnote{The letter (AEA~62-574-1) was typewritten in French and sent from Paris.
With this letter, Moguillanes sent Einstein a copy of a book he had written, entitled
``N\'eod\'emocratie,'' and asked for comments. I have not seen a copy of this book. The
library catalogue of the Biblioth\'eque Nationale lists one book by S.~Moguillanes
that carries this title, but the
bibliographic reference in the library catalogue lists the item as being self-published
without a date. I have no other information about the author or his book.}
It is unclear whether the two sheets are related in any other way that would entail a temporal
proximity but a preliminary global
assessment of the full batch of manuscript calculations has shown that, in general,
proximity in the physical sequence of the sheets also reflects temporal proximity
\cite[Sauer, 2004, pp.~161--163]{Sau04}. If we accept the date suggested by the
proximity to the dated letter, late 1954 or early 1955, the manuscript would,
most likely, represent Einstein's latest version of the incompleteness argument and one of his last
statements on quantum mechanics in general. In any case, it is
likely that the manuscript was written after Einstein's last published discussion of
quantum mechanics \cite[1953]{Ein53} and a fortiori later than Bohm's
reformulation of the EPR argument in 1951.
The following is an English translation of the relevant passage of the
manuscript:\footnote{For ease of understanding, the following English translation
incorporates words and characters that I have added to the original text in order to
render the words unambiguous and the sentences grammatically complete. The full
German text is given in the appendix where all added characters and words are
indicated by square brackets.}
\begin{quote}
Composite system of total spin $0$.
1) The description is assumed to be complete.\\
2) A coupling of distant things is excluded.
If the spin of the subsystem I is measured along the $x$-axis, it is
found to be either $1$ or $-1$ in that direction. It then follows that the spin
of the subsystem II equals 0 along the $y$-direction. But if instead the spin
of subsystem I is measured along the $y$-direction, it follows that the spin of the
subsystem II is equal to $1$ or $-1$.
If there is no coupling, then the result of a measurement of the spin of subsystem II
may in no way depend on whether a measurement was taken of subsystem I (or on what kind
of measurement).
The two assumptions therefore cannot be combined.
If the description is {\it not} assumed to be complete for the individual system, then that
what is being described is not a single system but an ensemble of systems. Then a measurement
of subsystem I amounts to the selection of a subensemble of the ensemble of the total
system. Then the prediction for a measurement of subsystem II can depend on the choice of
the measurement of subsystem I.
The conclusion is valid under the assumption that the assertion of quantum
theory is correct, which we can hardly put into doubt.
\end{quote}
The following lines were written at the right margin of the page:
\begin{quote}
a) the description by the quantum theory is an incomplete one with respect to
the individual system, or\\
b) there is an immediate coupling of states of spatially separated things.
\end{quote}
\section*{Discussion}
Einstein's known versions of the EPR incompleteness argument have been discussed
at length in the literature.\footnote{See note \ref{note:correspondence} and further literature
cited in these references.}
To the extent that the argument is, in fact, quite similar
to other versions, it is not the purpose of this note to add
to the existing historical commentary.
A puzzling feature of the argument is Einstein's assertion that the $y$-component
of the spin of particle II would be ``equal to 0'' (``=0'') if the $x$-component of the spin
of particle I is being measured. The assertion is puzzling since it is made for
an individual system rather than for an ensemble of systems. If the two subsystems
are in an entangled state with total spin 0 and the $x$-component $s^{\rm I}_x$ of the spin
is measured on subsystem I, subsystem II will be in an eigenstate of $s^{\rm II}_x$
and not in an eigenstate of $s^{\rm II}_y$.
Hence, a measurement of the $y$-component of the spin of subsystem II cannot
give $s^{\rm II}_y=0$,
which can emerge only as an average over an ensemble.
Even though Einstein explictly speaks about the ``result of a measurement of the spin of subsystem II''
in the following sentence, he presumably
has in mind not what measurements would yield, but what is the ``real'' state of affairs.
If the total spin of system I is in the $x$-direction, the total spin of system II is also
in the $x$-direction; which implies that it ``really'' cannot have a component in the $y$-direction.
The passage thus raises the question
of Einstein's understanding of space quantization and the non-classical spin degree of freedom.
Somewhat suprisingly, we have very little textual evidence that would shed light on
this question. Without pretending to provide an answer, I will briefly comment on relevant
evidence that I am aware of.
The classical experiment demonstrating space quantization
is the Stern-Gerlach experiment \cite[Gerlach and Stern, 1922]{GS22}.%
\footnote{The sequence of authors on the relevant papers
is alphabetical. It is unclear to me why the experiment is usually referred to as ``Stern-Gerlach,''
as is done, e.g., already in the title of \cite[Einstein and Ehrenfest, 1922]{EE22}.
One possible reason is that the experiment realized
an idea originally published by Stern alone. To avoid confusion, I will continue to refer to the
experiment in the usual parlance.} In this experiment silver atoms are deflected in two directions
upon traversing an inhomogeneous magnetic field. The results showed that the spatial orientation of
the magnetic moments of silver atoms in a magnetic field was quantized. Numerically, the quantization
was later explained in terms of an electron spin that gives rise to the magnetic moment of the silver atoms.
Immediately after Gerlach and Stern published their results, Einstein and Paul Ehrenfest
\cite[1922]{EE22} critically
discussed some implications of the experiment and some possible ways of accounting for the result
without, however,
providing themselves a positive explanation. Einstein and Ehrenfest understood that the magnetic
moments of {\it all} silver atoms were aligned along the axis of the magnetic field and that
they would do so without any mutual interaction \cite[p.~31]{EE22}.
Several years later, in a letter to Ehrenfest dated 21 January 1928, (AEA 10~173)
i.e.\ after the development of
mature quantum mechanics and the recognition of the existence of the electron's spin,
Einstein comments on the Stern-Gerlach experiment again. In the letter, he reported that
he had suggested to Stern to do two new experiments. In the first one, which he calls
``curious'' (``lustig''), two Stern-Gerlach magnets would be put next to each other in such a way
that the magnetized molecules would run first through one inhomogenous magnetic field, then through
a stretch of empty space, and then through a second Stern-Gerlach device. The point of the
experiment is that Einstein suggested that one would see no effect if the direction of the
second magnetic field
were reversed with respect to the first.\footnote{The experiment is strongly reminiscent of
Rabi's molecular beam apparatus, except for the lack of focussing diaphragms and, more importantly,
of an oscillatory magnetic field that would induce the spin flips.} The reasoning here is
as follows. ``In a magnetic field, let the molecular axis be aligned and follow
the magnetic field if it changes slowly.''\footnote{``Im Magnetfeld soll die Molek\"ul-Axe orientiert
sein und dem Magnetfeld bei dessen langsamer Ver\"anderung nachfolgen.''} Apparently Einstein
assumed that the molecule would follow the reversal of the two magnetic fields between the two
Stern-Gerlach devices adiabatically: ``If a molecule traverses two inhomogeneous fields
that point in opposite directions, it will reverse itself on travelling through the interval
between the two fields.''\footnote{``Durchl\"auft ein Molek\"ul zwei entgegengesetzte inhomogene
Felder, so kehrt es sich beim Durchlaufen des Intervalls zwischen beiden Feldern um.''}
In a second experiment, Einstein suggested to Stern that he should try to separate the effects
of a magnetic field {\it gradient} and of the magnetic field itself. ``The paradoxical result
to be expected is that, for a given field {\it gradient}, an {\it arbitrarily small field}
should determine Stern's
deflections (the plane of splitting).''\footnote{``Das paradoxe zu erwartende Resultat ist, dass
bei gegebenem Feld{\it gradienten} ein {\it beliebig schwaches Feld} f\"ur die Stern'schen Ablenkungen
massgebend sein soll (Aufspaltungsebene).'' (Einstein's emphasis)}
In late 1933 and early 1934, Einstein published several papers together with Walther Mayer on the
use of the so-called semi-vectors for an alternate equivalent representation of the Dirac
equation \cite[van Dongen, 2004]{Don04}.
These semi-vectors are a technique for representing spinors. The episode therefore at least indicates that
Einstein was aware of contemporary mathematical techniques to represent spinors. Other than that the
episode seems to shed little light on our specific question.
Let me finally add a few comments concerning David Bohm's work and Einstein's
reaction to it.
The extant correspondence between Einstein and Bohm dates from ca.\ 1951 to November 1954. I
have found no direct indication in this correspondence that Bohm's reformulation of the incompleteness
argument for spin observables was explicitly discussed. Einstein certainly knew of Bohm's ``excellent
book about quantum theory''%
\footnote{Einstein to Nathan Rosen, March 11, 1954, AEA 8-042.}
\cite[Bohm, 1951]{Boh51}, and Bohm's ``causal''
interpretation of quantum mechanics that he developed soon after publishing this book
is discussed to some detail.\footnote{An appreciable fraction of the Einstein-Bohm correspondence
(AEA 8-001 to 8-058) deals with Einstein's support for Bohm when the latter had left the United States,
after refusing to ``answer official questions concerning colleagues'' (AEA 8-003). A number of items of
correspondence in the cited archival call number range are letters of recommendation for Bohm.}
There is, however, an indirect hint to the problem which also corroborates our dating of
Einstein's manuscript. On June 8, 1954, Bohm sent Einstein the manuscript of a paper,
co-authored with Jean-Pierre Vigier, which he had just submitted for publication.%
\footnote{\cite[Bohm and Vigier, 1954]{BV54} was received by {\it Physical Review}
on June 14, 1954. The typescript
version sent to Einstein carries the archival number AEA 8-046. I did not see any
marginalia in (a copy of) the manuscript, and the two footnotes to be discussed below (as well as the
text in general) is the same in both the manuscript and the published paper.}
In the accompanying letter, Bohm also announced ``some interesting new results'' which he promised
to communicate to Einstein as soon as they would be ready. These new results, he added in brackets,
concerned ``the theory of the spinning electron'' (Bohm to Einstein, June 8, 1954, AEA 8-045).
There are no indications in the further correspondence about these new results but the paper itself
has two footnotes that are interesting in our context. In the paper, the
authors discuss and extend a hydrodynamic model
in which the Schr\"odinger equation is interpreted in terms of a continuous fluid and ``the
quantum potential may be thought of as arising in the effects of an internal stress in the fluid''
\cite[Bohm and Vigier, 1954, p.~209]{BV54}.
A little later in the paper,
the authors discuss the possibility that the velocity field may not be derivable
from a potential and then add a footnote, in which they observe that ``vortex
components of the velocity may also explain the appearance of `spin' ,''
which, however, would be neglected at the present level of precision.
In the concluding paragraph of this paper, Bohm and Vigier discuss the difference
between the ``usual'' and the ``causal'' interpretation of quantum mechanics with respect to
``the irregular statistical fluctuations in the observed results [...] when we make very precise
measurements on {\it individual} atomic systems.'' According to the usual interpretation, these
fluctuations are assumed to be ``fundamental elements of reality'' and ``cannot be traced to anything
else.'' At this point, we find the following footnote:
\begin{quote}
For example, they cannot in general be ascribed to the uncontrollable actions of the measuring
apparatus, as demonstrated by Einstein, Rosen, and Podolsky, Phys. Rev. 47,
774 (1933) [sic!] and
also D.~Bohm, {\it Quantum Theory} (Prentice Hall Publications, New York, 1951), p.~614.
As Bohr has made clear [Phys. Rev. 48, 696 (1935)] the measuring apparatus plus observed
object must be regarded as a single indivisible system which yields a statistical
aggregate of irregularly fluctuating observable phenomena. It would be incorrect, however, to suppose
that these fluctuations originate in anything at all. They must simply be accepted as fundamental
and not further analyzable elements of reality, which do not come from anything else but just
exist in themselves. For a complete discussion of this problem, see, {\it Albert Einstein,
Philosopher-Scientist}, Paul Arthur Schilpp, Editor (Library of Living Philosophers, Evanston, 1949).
\cite[p.~215]{BV54}
\end{quote}
Einstein who was asked by Bohm for comments on this paper must surely have pondered over this
footnote which puts together the relevant references, including a reference of the original
EPR paper right next to the relevant page in Bohm's book.
The existing later correspondence between Einstein and Bohm---two letters by
Bohm, dated October 18 and November 14, 1954, and two letters by
Einstein, dated October 28 and November 24, 1954, respectively---indeed
discusses differences in opinion regarding the foundations of quantum mechanics.
But these differences are expressed on a very general level, notably with
respect to the question of whether one should give up the notion of a
continuum, and the letters do not make any explicit reference to the EPR incompleteness
argument or the problem of the spin degree of freedom.
A closer comparison of Einstein version of the incompleteness argument
presented here and the discussion in \cite[Bohm, 1951]{Boh51} should be embedded in a
more detailed and comprehensive discussion of both Einstein's and Bohm's
perspectives on quantum mechanics. The purpose of this note was to present an important
document that adds to our understanding of these issues.
It remains a deplorable fact, that Einstein himself would never explain his understanding of the
quantum incompleteness argument in more explicit mathematical terms. But the
succinctness of Einstein's incompleteness argument in the manuscript discussed here may also
carry a positive aspect. It provides another concise statement of what, in Einstein's understanding,
the EPR argument is about.
\bigskip
\section*{Appendix: Partial transcription of AEA~62-575r}
Zus[ammengesetztes] System von Ges[amt] Spin $0$.
1) [Die] Beschr[eibung ist als] vollst[\"andig] vorausges[etzt].
2) [Eine] Koppelung distanter Din[g]e [wird] ausgeschl[ossen].
Wenn an [dem] Teilsystem I [eine] Mess[ung] des Spin[s] in [der] $X$ Axe
vorgenommen [wird], dann ist [der] Spin dieses Teilsystems entweder $1$
oder $-1$ in dieser Richtung[.] Dann folgt, das[s der] Spin des II-Teilsystem[s] $=0$
in der $Y$-Richtung [ist]. Wenn aber statt dessen an [dem Teilsystem] I [der]
Spin in der $Y$-Richtung gemessen wird, so folgt, dass hierauf [der] Spin des
II-Teilsystems [in der $Y$-Richtung] $-1$ oder $+1$ ist.
Wenn [es] keine Kopplung [gibt], so darf das Ergebnis\footnote{The words ``das Ergebnis''
are interlineated.} einer Messung des Spins an
[dem Teilsystem] II \"uberhaupt nicht davon abh\"angen[,] ob \"uberhaupt eine Messung
an [dem Teilsystem] I vorgenommen war (bezw.\ was f\"ur eine Messung).
Beide Annahmen lassen sich also nicht vereinigen.
Wenn [die] Beschr[eibung] {\it nicht} als vollst[\"andig] f\"ur das individuelle
System\footnote{The words ``als'' and ``f\"ur das individuelle System'' are interlineated.}
vorausgesetzt [wird], dann ist das Beschriebene nicht ein System,
sondern eine System-Gesamtheit. Dann bedeutet [eine] Messung an [dem Teilsystem]
I die Aussonderung einer Teilgesamtheit f\"ur das Ensemble das Gesamtsystem.\footnote{The words
``das Ensemble'' are interlineated, and ``das Gesamtsystem'' should probably read ``des
Gesamtsystems''.}
Dann kann die Voraussage f\"ur [eine] Messung an [dem Teilsystem] II von der Wahl
der Messung an [dem Teilsystem] I abh\"angen.
[Der] Schluss [gilt] unter der Voraussetzung, dass die Aussage der Quantentheorie
richtig ist, woran wir kaum zweifeln k\"onnen[.]
[The following text was written at the right margin of the document:]
a) die Beschreibung durch die Quantentheorie ist eine unvollst\"andige
inbezug auf das individuelle System
oder
b) es gibt eine unmittelbare Kopplung von Zust\"anden r\"aumlich getrennter
Dinge.
\bigskip
\bigskip
\noindent
{\it Acknowledgment}. I wish to thank the anonymous referees for their
comments and Diana Buchwald for a critical reading of an earlier version of
this paper. The manuscript as well as unpublished correspondence by Einstein
are quoted with kind permission from the Albert Einstein Archives, The
Hebrew University of Jerusalem.
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