A TRANSLATION OF SCHR=D6DINGER'S "CAT PARADOX PAPER"

This translation was originally published in Proceedings of the = American=20 Philosophical Society, 124, 323-38. [And then appeared as Section = I.11 of=20 Part I of Quantum Theory and Measurement (J.A. Wheeler and = W.H.=20 Zurek, eds., Princeton university Press, New Jersey 1983).]=20

- Introducto= ry=20 Note=20
- 1. The=20 Physics of Models=20
- 2.=20 Statistics of Model Variables in Quantum Mechanics=20
- 3.=20 Examples of Probability Predictions=20
- 4. Can = One=20 Base the Theory on Ideal Ensembles?=20
- 5. Are = the=20 Variables Really Blurred?=20
- 6. The=20 Deliberate About-face of the Epistemological Viewpoint=20
- 7. The=20 Psi-function as Expectation-catalog=20
- 8. Theory = of Measurement, Part One=20
- 9. The=20 Psi-function as Description of State=20
- 10.=20 Theory of Measurement, Part Two=20
- 11.=20 Resolution of the "Entanglement" Result Dependent on the = Experimenter's=20 Intention=20
- 12. An=20 example=20
- 13.=20 Continuation of the Example: All Possible Measurements are Entangled=20 Unequivocally=20
- 14.=20 Time-dependence of the Entanglement. Consideration of the Special Role = of=20 Time=20
- 15.=20 Natural Law or Calculating Device?=20
- Notes =

TRANSLATION=20

Of natural objects, whose observed behavior one might treat, one set =
sup a=20
representation - based on the experimental data in one's possession but =
without=20
handcuffing the intuitive imagination - that is worked out in all =
details=20
exactly, *much* more exactly than any experience, considering its =
limited=20
extent, can ever authenticate. The representation in its absolute =
determinacy=20
resembles a mathematical concept or a geometric figure which can be =
completely=20
calculated from a number of *determining parts*; as, e.g., a =
triangle's=20
one side and two adjoining angles, as determining parts, also determine =
the=20
third angle, the other two sides, the three altitudes, the radius of the =
inscribed circle, etc. Yet the representation differs intrinsically from =
a=20
geometric figure in this important respect, that also in *time* =
as fourth=20
dimension it is just as sharply determined as the figure is in the three =
space=20
dimensions. Thus it is a question (as is self-evident) always of a =
concept that=20
changes with time, that can assume different *states*; and if a =
state=20
becomes known in the necessary number of determining parts, then not =
only are=20
all other parts also given for this moment (as illustrated for the =
triangle=20
above), but likewise all parts, the complete state, for any given later =
time;=20
just as the character of a triangle on its base determines its character =
at the=20
apex. It is part of the inner law of the concept that it should change =
in a=20
given manner, that is, if left to itself in a given initial state, that =
it=20
should continuously run through a given sequence of states, each one of =
which it=20
reaches at a fully determined time. That is its nature, that is the =
hypothesis,=20
which, as I said above, one builds on a foundation of intuitive =
imagination.=20

Of course one must not think so literally, that in this way one =
learns how=20
things go in the real world. To show that one does not think this, one =
calls the=20
precise thinking aid that one has created, an *image* or a=20
*model*. With its hindsight-free clarity, which cannot be =
attained=20
without arbitrariness, one has merely insured that a fully determined =
hypothesis=20
can be tested for its consequences, without admitting further =
arbitrariness=20
during the tedious calculations required for deriving those =
consequences. Here=20
one has explicit marching orders and actually works out only what a =
clever=20
fellow could have told directly from the data! At least one then knows =
where the=20
arbitrariness lies amd where improvement must be made in case of =
disagreement=20
with experience: in the initial hypothesis or model. For this one must =
always be=20
prepared. If in many various experiments the natural object behaves like =
the=20
model, one is happy and thinks that the image fits the reality in =
essential=20
features. If it fails to agree, under novel experiments or with refined=20
measuring techniques, it is not said that one should *not* be =
happy. For=20
basically this is the means of gradually bringing our picture, i.e., our =
thinking, closer to the realities.=20

The classical method of the precise model has as principal goal =
keeping the=20
unavoidable arbitrariness neatly isolated in the assumptions, more or =
less as=20
body cells isolate the nucleoplasm, for the historical process of =
adaptation to=20
continuing experience. Perhaps the method is based on the belief that=20
*somehow* the initial state *really* determines uniquely =
the=20
subsequent events, or that a *complete* model, agreeing with =
reality in=20
*complete exactness* would permit predictive calculation of =
outcomes of=20
all experiments with complete exactness. Perhaps on the other hand this =
belief=20
is based on the method. But it is quite probable that the adaptation of =
thought=20
to experience is an infinite process and that "complete model" is a=20
contradiction in terms, somewhat like "largest integer".=20

A clear presentation of what is meant by *classical model*, =
its=20
*determining parts*, its *state*, is the foundation for =
all that=20
follows. Above all, a *determinate model* and a *determinate =
state of=20
the same* must not be confused. Best consider an example. The =
Rutherford=20
model of the hydrogen atom consists of two point masses. As determining =
parts=20
one could for example use the two times three rectangular coordinates of =
the two=20
points and the two times three components of their velocities along the=20
coordinate axes - thus twelve in all. Instead of these one could also =
choose:=20
the coordinates and velocity components of the *center of mass*, =
plus the=20
*separation* of the two points, *two angles* that =
establish the=20
direction in space of the line joining them, and the speeds (=3D time =
derivatives)=20
with which the separation and the two angles are changing at the =
particular=20
moment; this again adds up of course to twelve. It is *not* part =
of the=20
concept "R-model of the H-atom" that the determining parts should have=20
particular numerical values. Such being assigned to them, one arrives at =
a=20
*determinate state* of the model. The clear view over the =
totality of=20
possible states - yet without relationship among them - constitutes "the =
model"=20
or "the model in *any* state *whatsoever*". But the =
concept of the=20
model then amounts to more than merely: the two points in certain =
positions,=20
endowed with certain velocities. It embodies also knowledge for =
*every*=20
state how it will change with time in absence of outside interference.=20
(Information on how one half of the determining parts will change with =
time is=20
indeed given by the other half, but how this other half will change must =
be=20
independently determined.) *This* knowledge is implicit in the=20
assumptions: the points have the masses m, M and the charges -e, +e and=20
therefore attract each other with force e^2/r^2, if their separation is =
r.=20

These results, with definite numerical values for m, M, and e (but of =
course=20
*not* for r), belong to the description *of the model* =
(not first=20
and only to that of a definite state). m, M, and e are *not* =
determining=20
parts. By contrast, separation r is one. It appears as the seventh in =
the second=20
"set" of the example introduced above. And if one uses the first, r is =
not an=20
independent thirteenth but can be calculated from the six rectangular=20
coordinates:=20

r =3D | (x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2 | ^ (1/2) .=20

The number of determining parts (which are often called =
*variables* in=20
contrast to *constants of the model* such as m, M, e) is =
unlimited.=20
Twelve conveniently chosen ones determine all others, or the =
*state*. No=20
twelve have the privilege of being *the* determining parts. =
examples of=20
other especially important determining parts are: the energy, the three=20
components of angular momentum relative to center of mass, the kinetic =
energy of=20
center of mass motion. These just named have, however, a special =
character. They=20
are indeed *variable*, i.e., they have different values in =
different=20
states. But in every *sequence* of states, that is actually =
passed=20
through in the course of time, they retain the same value. So they are =
also=20
called *constants of the motio*n - differing from constants of =
the model.=20

One might think that for anyone believing this, the classical models =
have=20
played out their roles. But this is not the case. Rather one uses =
precisely=20
*them*, not only to express the negative of th new doctrine, but =
also to=20
describe the diminished mutual determinacy remaining afterwards as =
though=20
obtaining among the same variables of the same models as were used =
earlier, as=20
follows:=20

*A*. The classical concept of *state* becomes lost, in =
that at=20
most a well-chosen *half* of a complete set of variables can be =
assigned=20
definite numerical values; in the Rutherford example for instance the =
six=20
rectangular coordinates *or* the velocity components (still other =
groupings are possible). the other half then remains completely =
indeterminate,=20
while supernumerary parts can show highly varying degrees of =
indeterminacy. In=20
general, of a complete set (for the R-model twelve parts) *all* =
will be=20
known only uncertainly. One can best keep track of the degree of =
uncertainty by=20
following classical mechanics and choosing variables arranged in =
*pairs*=20
of so-called canonically-conjugate ones. The simplest example is a space =
coordinate x of a point mass and the component p_x along the same =
direction, its=20
linear momentum (i.e., mass times velocity). Two such constrain each =
other in=20
the precision with which they may be simultaneously known, in that the =
product=20
of their tolerance- or variation-widths (customarily designated by =
putting a=20
Delta ahead of the quantity) cannot fall *below* the magnitude of =
a=20
certain universal constant,[4] =
thus:=20

Delta-x . Delta-p_x >=3D hbar.=20

(Heisenberg uncertainty relation.)=20

*B*. If even at any given moment not all variables are =
determined by=20
some of them, then of course neither are they all determined for a later =
moment=20
by data obtainable either. This may be called a break with causality, =
but in=20
view of *A.*, it is nothing essentially new. If a classical state =
does=20
not exist at any moment, it can hardly change causally. What do change =
are the=20
*statistics* or *probabilities*, *these* moreover =
causally.=20
Individual variables meanwhile may become more, or less, uncertain. =
Overall it=20
may be said that the total precision of the description does not change =
with=20
time, because the principle of limitations decsribed under *A.* =
remains=20
the same at every moment.=20

Now what is the meaning of the terms "uncertain", "statistics",=20
"probability"? Here Q.M. gives the following account. It takes over=20
unquestioningly from the classical model the entire infinite roll call =
of=20
imaginable variables or determining parts and proclaims each part to be=20
*directly measurable*, indeed measuravle to arbitrary precision, =
so far=20
as it alone is concerned. If through a well-chosen, constrained set of=20
measurements one has gained that maximal knowledge of an object which is =
just=20
possible according to *A.*, then the mathematical apparatus of =
the new=20
theory provides means of assigning, for the same or for any later =
instant of=20
time, a fully determined *statistical distribution* to =
*every*=20
variable, that is, an indication of the fraction of cases it will be =
found at=20
this or that value, or within this or that small interval (which is also =
called=20
probability.) The doctrine is that this is in fact the probability of=20
encountering the relevant variable, if one measures it at the relevant =
time, at=20
this or that value. By a single trial the correctness of this =
*probability=20
prediction* can be given at most an approximate test, namely in the =
case=20
that it is comparatively sharp, i.e., declares possible only a small =
range of=20
values. To test it thoroughly one must repeat the entire trial *ab =
ovo*=20
(i.e., including the orientational or preparatory measurements) =
*very*=20
often and may use only those cases in which the *preparatory*=20
measurements gave exactly the same results. For these cases, then, the=20
statistics of a particular variable, reckoned forward from the =
preparatory=20
measurements, is to be confirmed by measurement - this is the doctrine.=20

One must guard against criticizing this doctrine because it is so =
difficult=20
to express; this is a matter of language. But a different criticism =
surfaces.=20
Scarcely a single physicist of the classical era would have dared to =
believe, in=20
thinking about a model, that its determining parts are measurable on the =
natural=20
object. Only much remoter consequences of the picture were actually open =
to=20
experimental test. And all experience pointed towards one conclusion: =
long=20
before the advancing experimental arts had bridged the broad chasm, the =
model=20
would have substantially changed through adaptation to new facts. --Now =
while=20
the new theory calls the classical model incapable of specifying all =
details of=20
the *mutual interrelationship of the determining parts* (for =
which its=20
creators intended it), it nevertheless considers the model suitable for =
guiding=20
us as to just which measurements can in principle be made on the =
relevant=20
natural object. This would have seemed to those who thought up the =
picture a=20
scandalous extension of their thought-pattern and an unscrupulous =
proscription=20
against future development. Would it not be pre-established harmony of a =
peculiar sort if the classical-epoch researchers, those who, as we hear =
today,=20
had no idea of what *measuring* truly is, had unwittingly gone on =
to give=20
us as legacy a guidance scheme revealing just what is fundamentally =
measurable=20
for instance about a hydrogen atom!?=20

I hope later to make clear that the reigning doctrine is born of = distress.=20 Meanwhile I continue to expound it.=20

If one measures the energy of a Planck oscillator, the probability of =
finding=20
for it a value between E and E' cannot possibly be other than zero unles =
between=20
E and E' there lies at least one value from the series 3.pi.hbar.nu,=20
5.pi.hbar.nu, 7.pi.hbar.nu, 9.pi.hbar.nu,... For any interval containing =
none of=20
these values the probability is zero. In plain English: other =
measurement=20
results are excluded. The values are odd multiples of the *constant =
of the=20
model* pi.hbar.nu=20

(Planck constant) / 2.pi, nu =3D frequency of the oscillator.=20

Two points stand out. First, no account is taken of preceding = measurements -=20 these are quite unnecessary. Second, the statement certainly doesn't = suffer an=20 excessive lack of precision - quite to the contrary it is sharper than = any=20 actual measurement could ever be.=20

Figure 1. /|\ | |M . . O........F . . .

Angular momentum. M is a material point, O a geometric reference =
point. The=20
vector arrow represents the momentum (=3D mass times velocity) of M. =
Then the=20
*angular momentum* is the product of the length of the arrow by =
the=20
length OF.=20

Another typical example is magnitude of angular momentum. In Fig. 1 =
let M be=20
a moving point mass, with the vector representing, in magnitude and =
direction,=20
its momentum (mass times velocity). O is any arbitrary fixed point in =
space, say=20
the origin of coordinates; thus not a physically significant point, but =
rather a=20
geometric reference point. As magnitude of the angular momentum of M =
about O=20
classical mechanics designates the product of the length of the momentum =
vector=20
by the length of the *normal OF*. In Q.M. the magnitude of =
angular=20
momentum is governed much as the energy of the oscillator. Again the =
probability=20
is zero for any interval not containing some value(s) from the following =
series=20

hbar(2)^(1/2), hbar(2x3)^(1/2), hbar(3x4)^(1/2), hbar(4x5)^(1/2),...; =

that is, only one of these values is allowed. Again this is true =
without=20
reference to preceding measurements. And one readily conceives how =
important is=20
this precise statement, *much* more important than knowing which =
of these=20
values, or what probability for each of them, would actually pertain to =
a given=20
case. Moreover it is also noteworthy here that there is no mention of =
the=20
reference point: however it is chosen one will get a value from the =
series. This=20
assertion seems unreasonable for the model, because the normal OF =
changes=20
*continuously* as the point O is displaced, if the momentum =
vector=20
remains unchanged. In this example we see how Q.M. does indeed use the =
model to=20
read of those quantities which one can measure and for which it makes =
sense to=20
predict results, but finds the classical model inadequate for =
explicating=20
relationships among these quantities. Now in both examples does one not =
get the=20
feeling that the essential content of what is being said can only with =
some=20
difficulty be forced into the Spanish boot of a prediction of =
probability of=20
finding this or that measurement result for a variable of the classical =
model?=20
Does one not get the impression that here one deals with fundamental =
properties=20
of *new* classes of characteristics, that keep only the name in =
common=20
with classical ones? And by no means do we speak here of exceptional =
cases,=20
rather it is precisely the truly valuable statements of the new theory =
that have=20
this character. There are indeed problems more nearly of the type for =
which the=20
mode of expression is suitable. But they are by no means equally =
important.=20
Moreover of no importance whatever are those that are naively set up as =
class=20
exercises. "Given the position of the elctron in the hydrogen atom at =
time t=3D0,=20
find the statistics of its position at a later time." No one cares about =
that.=20

The big idea seems to be that all statements pertain to the intuitive = model.=20 But the useful statements are scarcely intuitive in it, and its = intuitive=20 aspects are of little worth.=20

The second interpretation is especially appealing to those acquainted =
with=20
the *statistical viewpoint* that came up in the second half of =
the=20
preceding century; the more so, considering that on the eve of the new =
century=20
quantum theory was born *from it*, from a central problem in the=20
statistical theory of heat (Max Planck's Theory of Heat =
Radiation,=20
December, 1899). The essence of this line of thought is precisely this, =
that one=20
practically never knows all the determining parts of the system, but =
rather=20
*much* fewer. To describe an actual body at a given moment one =
relies=20
therefore not on *one* state of the model but on a so-called =
*Gibbs=20
ensemble*. By this is meant an ideal, that is, merely imagined =
ensemble of=20
states, that accurately reflects our limited knowledge of the actual =
body. The=20
body is then considered to behave as though in a single state =
*arbitrarily=20
chosen from this ensemble*. This interpretation had the most =
extensive=20
results. Its highest triumphs were in those cases for which *not* =
all=20
states appearing in the ensemble led to *the same* observable =
behavior.=20
Thus the body's conduct is now this way, now that, just as foreseen=20
(thermodynamics fluctuations). At first thought one might well attempt =
likewise=20
to refer back the always uncertain statements of Q.M. to an ideal =
ensemble of=20
states, of which a quite specific one applies in any concrete instance - =
but one=20
does not know which one.=20

That this won't work is shown by the *one* example of angular=20
momentum, as one of many. Imagine in Fig. 1 the point M to be situated =
at=20
various positions relative to O and fitted with various momentum =
vectors, and=20
all these possibilities to be combined into an ideal ensemble. Then one =
can=20
indeed so choose these positions and vectors that in every case the =
product of=20
vector length by length of normal OF yields one or the other of the =
acceptable=20
values - relative to the particular point O. But for an arbitrarily =
different=20
point O', of course, unacceptable values occur. Thus appeal to the =
ensemble is=20
no help at all. --Another example is the oscillator energy. Take the =
case that=20
it has a sharply determined value, e.g., the lowest, 3.pi.hbar.nu. The=20
separation of the two point masses (that constitute the oscillator) then =
appears=20
very *unsharp*. To be able to refer this statement to a =
statistical=20
collective of states would require the distribution of separations to be =
sharply=20
limited, at least toward large values, by that separation for which the=20
*potential energy* alone would equal or exceed the value =
3.pi.hbar.nu.=20
But that's not the way it is - arbitrarily large separations occur, even =
though=20
with markedly reduced probability. And this is no mere secondary =
calculation=20
result, that might in some fashion be circumvented, without striking at =
the=20
heart of the theory: along with many others, the quantum mechanical =
treatment of=20
radioactivity (Gamow) rests on this state of affairs. --One could go on=20
indefinitely with more examples. One shoudl note that there was no =
question of=20
any time-dependent changes. It would be of no help to permit the model =
to vary=20
quite "unclassically", perhaps to "jump". Already for the single instant =
things=20
go wrong. At no moment does there exist an ensemble of classical states =
of the=20
model that squares with the totality of quantum mechanical statements of =
this=20
moment. The same can also be said as follows: if I wish to ascribe to =
the model=20
at each moment a definite (merely not exactly known to me) state, or =
(which is=20
the same) to *all* determining parts definite (merely not eactly =
known to=20
me) numerical values, then there is no supposition as to these numerical =
values=20
*to be imagined* that would not conflict with some portion of =
quantum=20
theoretical assertions.=20

That is not quite what one expects, on hearing that the = pronouncements of the=20 new theory are always uncertain compared to the classical ones.=20

That it is in fact not impossible to express the degree and kind of =
blurring=20
of *all* variables in *one* perfectly *clear* =
concept=20
follows at once from the fact that Q.M. as a matter of fact has and uses =
such an=20
instrument, the so-called wave function or psi-function, also called =
system=20
vector. Much more is to be said about it further on. That it is an =
abstract,=20
unintuitive mathematical construct is a scruple that almost always =
surfaces=20
against new aids to thought and that carries no great message. At all =
events it=20
is an imagined entity that images the blurring of all variables at every =
moment=20
just as clearly and faithfully as does the classical model its sharp =
numerical=20
values. Its equation of motion too, the law of its time variation, so =
long as=20
the system is left undisturbed, lags not one iota, in clarity and =
determinacy,=20
behind the equations of motion of the classical model. So the latter =
could be=20
straight-forwardly replaced by the psi-function, so long as the blurring =
is=20
confined to atomic scale, not open to direct control. In fact the =
function has=20
provided quite intuitive and convenient ideas, for instance the "cloud =
of=20
negative electricity" around the nucleus, etc. But serious misgivings =
arise if=20
one notices that the uncertainty affects macroscopically tangible and =
visible=20
things, for which the term "blurring" seems simply wrong. The state of a =
radioactive nucleus is presumably blurred in such a degree and fashion =
that=20
neither the instant of decay nor the direction, in which the emitted=20
alpha-particle leaves the nucleus, is well-established. Inside the =
nucleus,=20
blurring doesn't bother us. The emerging particle is described, if one =
wants to=20
expain intuitively, as a spherical wave that continuously emanates in =
all=20
directions and that impinges continuously on a surrounding luminescent =
screen=20
over its full expanse. The screen however does not show a more or less =
constant=20
uniform glow, but rather lights up at *one* instant at =
*one* spot=20
- or, to honor the truth, it lights up now here, now there, for it is =
impossible=20
to do the experiment with only a single radioactive atom. If in place of =
the=20
luminescent screen one uses a spatially extended detector, perhaps a gas =
that is=20
ionised by the alpha-particles, one finds the ion pairs arranged along=20
rectilinear columns,[5] =
that=20
project backwards on to the bit of radioactive matter from which the=20
alpha-radiation comes (C.T.R. Wilson's cloud chamber tracks, made =
visible by=20
drops of moisture condensed on the ions).=20

One can even set up quite ridiculous cases. A cat is penned up in a =
steel=20
chamber, along with the following device (which must be secured against =
direct=20
interference by the cat): in a Geiger counter there is a tiny bit of =
radioactive=20
substance, *so* small, that *perhaps* in the course of the =
hour=20
one of the atoms decays, but also, with equal probability, perhaps none; =
if it=20
happens, the counter tube discharges and through a relay releases a =
hammer which=20
shatters a small flask of hydrocyanic acid. If one has left this entire =
system=20
to itself for an hour, one would say that the cat still lives =
*if*=20
meanwhile no atom has decayed. The psi-function of the entire system =
would=20
express this by having in it the living and dead cat (pardon the =
expression)=20
mixed or smeared out in equal parts.=20

It is typical of these cases that an indeterminacy originally =
restricted to=20
the atomic domain becomes transformed into macroscopic indeterminacy, =
which can=20
then be *resolved* by direct observation. That prevents us from =
so=20
naively accepting as valid a "blurred model" for representing reality. =
In itself=20
it would not embody anything unclear or contradictory. There is a =
difference=20
between a shaky or out-of-focus photograph and a snapshot of clouds and =
fog=20
banks.=20

Now this sheds some light on the origin of the proposition that I = mentioned=20 at the end of Sect. = 2 as=20 something very far-reaching: that all model quantities are measurable in = principle. One can hardly get along without this article of belief if = one sees=20 himself constrained, in the intersts of physical methodology, to call in = as=20 dictatorial help the above-mentioned philosophical principle, which no = sensible=20 person can fail to esteem as the supreme protector of all empiricism.=20

Reality resists imitation through a model. So one lets go of niave =
realism=20
and leans directly on the indubitable proposition that *actually* =
(for=20
the physicist) after all is said and done there is only observation,=20
measurement. Then all our physical thinking thenceforth has as sole =
basis and as=20
sole object the results of measurements which can in principle be =
carried out,=20
for we must now explicitly *not* relate our thinking any longer =
to any=20
other kind of reality or to a model. All numbers arising in our physical =
calculations must be interpreted as measurement results. But since we =
didn't=20
just now come into the world and start to build up our science from =
scratch, but=20
rather have in use a quite definite shceme of calculation, from which in =
view of=20
the great progress in Q.M. we would less than ever want to be parted, we =
see=20
ourselves forced to dictate from the writing-table which measurements =
are in=20
principle possible, that is, must be possible in order to support =
adequately our=20
reckoning system. This allows a sharp value for each single variable of =
the=20
model (indeed for a whole "half set") and so each single variable must =
be=20
measurable to arbitrary exactness. We cannot be satisfied with less, for =
we have=20
lost our naively realistic innocence. We have nothing but our reckoning =
scheme,=20
i.e., what is a *best possible* knowledge of the object. And if =
we=20
couldn't do that, then indeed would our measurement reality become =
highly=20
dependent on the diligence or laziness of the experimenter, how much =
trouble he=20
takes to inform himself. We must go on to tell him how far he could go =
if only=20
he were clever enough. Otherwise it would be seriously feared that just =
there,=20
where we forbid further questions, there might well still be something =
worth=20
knowing that we might ask about.=20

*The systematically arranged interaction of two systems (measured =
object=20
and measuring instrument) is called a measurement on the first system, =
if a=20
directly-sensible variable feature of the second (pointer position) is =
always=20
reproduced within certain error limits when the process is immediately =
repeated=20
(on the same object, which in the meantime must not be exposed to any =
additional=20
influences)*.=20

This statement will require considerable added comment: it is by no = means a=20 faultless definition. Empirics is more complicated than mathematics and = is not=20 so easily captured in polished sentences.=20

*Before* the first measurement there might have been an =
arbitrary=20
quantum-theory prediction for it. *After* it the prediction=20
*always* runs: within error limits again the same result. The=20
expectation-catalog (=3D psi-function) is therefore changed by the =
measurement in=20
respect to the variable being measured. If the measurement procedure is =
known=20
from beforehand to be *reliable*, then the first measurement at =
once=20
reduces the theoretical expectation within error limits on to the value =
found,=20
regardless of whatever the prior expectation may have been. This is the =
typical=20
abrupt change of the psi-function discussed above. But the =
expectation-catalog=20
changes in unforeseen manner not only for the measured variable itself, =
but also=20
for others, in particular for its "canonical conjugate". If for instance =
one has=20
a rather sharp prediction for the *momentum* of a particle and =
proceeds=20
to measure its *position* more exactly than is compatible with =
Theorem A=20
of Sec. 2, then the *momentum* prediction must change. The =
quantum=20
mechanical reckoning moreover takes care of this automatically; there is =
no=20
psi-function whatsoever that would contradict Theorem A when one deduces =
from it=20
the combined expectations.=20

Since the expectation-catalog changes radically during measurement, =
the=20
object is then no longer suited for testing, in their full extent, the=20
statistical predictions made earlier; at the very least for the measured =
variable itself, since for it now the (nearly) same value would occur =
over and=20
over again. *That* is the reason for the prescription already =
given in Sect. =
2: one=20
can indeed test the probability predictions completely, but for this one =
must=20
repeat the entire experiment *ab ovo*. One's prior treatment of =
the=20
measured object (or one identical to it) must be exactly the same as =
that given=20
the first time, in order that the same expectation-catalog (=3D =
psi-function)=20
should be valid as before the first measurement. Then one "repeats" it. =
(This=20
repeating now means of course something quite other than earlier!) All =
this one=20
must do not twice but very often. Then the predicted statistics are =
established=20
- that is the doctrine.=20

One should note the difference between the error limits and the error =
distribution of the *measurement*, on the one hand, and the =
theoretically=20
predicted statistics, on the other hand. They have nothing to do with =
each=20
other. They are established by the two quite different types of=20
*repetition* just discussed.=20

Here there is opportunity to deepen somewhat the above-attempted =
delimitation=20
of *measuring*. There are measuring instruments that remain fixed =
on the=20
reading given by the measurement just made. Or the pointer could remain =
stuck=20
because of a defect. One would then repeatedly make exactly the same =
reading,=20
and according to our instruction that would be a spectacularly accurate=20
measurement. Moreover that would be true not merely for the object but =
also for=20
the instrment itself! As a matter of fact there is still missing from =
our=20
exposition an important point, but one which could not readily be stated =
earlier, namely what it is that truly makes the difference between=20
*object* and *instrument* (tat it is the latter on which =
the=20
reading is made, is more or less superficial). We have just seen that =
the=20
instrument under certain circumstances, as required, must be set back to =
its=20
neutral initial condition before any control measurement is made. This =
is well=20
known to the experimentalist. Theoretically the matter may best be =
expressed by=20
prescribing that on principle the instrument should be subjected to the=20
identical prior treatment before each measurement, so that *for =
it* each=20
time the same expectation-catalog (=3D psi-function) applies, as it is =
brought up=20
to the object. For the object it is just the other way around, any =
interference=20
being forbidden when a control measurement is to be made, a "repetition =
of the=20
first kind" (that leads to *error* statistics). That is the=20
characteristic difference between objectand instrument. It disappears =
for a=20
"repetition of the second kind" (that serves for checking the quantum=20
predictions). Here the difference between the two is actually rather=20
insignificant.=20

>From this we gather further that for a second measurement one may =
use a=20
similarly built and similarly prepared instrument - it need not =
necessarily be=20
*the same one*; this is in fact sometimes done, as a check on the =
first=20
one. it may indeed happen that two qute differently built instruments =
are so=20
related to each other that if one measures with them one after the other =
(repetition of the first kind!) their two indications are in one-to-one=20
correlation with each other. They then measure on the object =
esssentially the=20
same variable - i.e., the same for suitable calibration of the scales.=20

Thence it follows that two different catalogs, that apply to the same = system=20 under different circumstances or at different times, may well partially = overlap,=20 but never so that the one is entirely contained within the other. For = otherwise=20 it would be susceptible to completion through additional correct = statements,=20 namely through those by which the other one exceeds it. --The = mathematical=20 structure of the theory automatically satisfies this condition. There is = no=20 psi-function that furnishes exactly the same statements as another and = in=20 addition several more.=20

Therefore if a system changes, whether by itself or because of =
measurements,=20
there must always be statements missing from the new function that were=20
contained in the earlier one. In the catalog not just new entries, but =
also=20
deletions, must be made. Now knowledge can well be *gained*, but =
not=20
*lost*. So the deletions mean that the previously correct =
statements have=20
now become incorrect. A correct statement can become incorrect only if =
the=20
*object* to which it applies changes. I consider it acceptable to =
express=20
this reasoning sequence as follows:=20

Theorem 1: If different psi-functions are under discussion the system = is in=20 different states.=20

If one speaks only of systems for which a psi-function is in general=20 available, then the inverse of this theorem runs:=20

Theorem 2: For the same psi-function the system is in the same state. =

The inverse does not follow from Theorem 1 but independently of it, =
directly=20
from *completeness* or *maximality*. Whoever for the same=20
expectation-catalog would yet claim a difference is possible, would be =
admitting=20
that it (the catalog) does not give information on all justifiable =
questions.=20
--The language usage of almost all authors implies the validity of the =
above two=20
theorems. Of course, they set up a kind of new reality - in entirely =
legitimate=20
fashion, I believe. Moreover they are not trivially tautological, not =
mere=20
verbal interpretations of "state". Without presupposed maximality of the =
expectation-catalog, change of the psi-function could be brought about =
by mere=20
collecting of new information.=20

We must face up to yet another objection to the derivation of Theorem =
1. One=20
can argue that each individual statement or item of knowledge, under =
examination=20
there, is after all a probability statement, to which the category of=20
*correct*, or *incorrect* does not apply in any relation =
to an=20
individual case, but rather in relation to a collective that comes into =
being=20
from one's preparing the system a thousand times in identical fashion =
(in order=20
then to allow the same measurement to follow; cf. Sect. =
8.). That=20
makes sense, but we must specify all members of this collective to be=20
identically prepared, since to each the same psi-function, the same=20
statement-catalog applies and we dare not specify differences that are =
not=20
specified in the catalog (cf. the foundation of Theorem 2). Thus the =
collective=20
is made up of identical individual cases. If a statement is wrong for=20
*it*, then the individual case must have changed, or else the =
collective=20
too would again be the same.=20

So, using catchwords for emphasis, I try again to contrast: 1.) The=20
discontinuity of the expectation-catalog due to measurement is=20
*unavoidable*, for if measurement is to retain any meaning at all =
then=20
the *measured value*, from a good measurement, *must* =
obtain. 2.)=20
The discontinuous change is certainly *not* governed by the =
otherwise=20
valid causal law, since it depends on the measured value, which is not=20
predetermined. 3.) The change also definitely includes (because of =
"maximality")=20
some *loss* of knowledge, but knowledge cannot be lost, and so =
the object=20
*must* change - *both* along with the discontinuous =
changes and=20
*also*, during these changes, in an unforeseen, =
*different* way.=20

How does this add up? Things are not at all simple. It is the most = difficult=20 and most interesting point of the theory. Obviously we must try to = comprehend=20 objectively the interaction between measured object and measuring = instrument. To=20 that end we must lay out a few very abstract considerations.=20

This is the point. Whenever one has a complete expectation-catalog - = a=20 maximum total knowledge - a psi-function - for two completely separated = bodies,=20 or, in better terms, for each of them singly, then one obviously has it = also for=20 the two bodies together, i.e., if one imagines that neither of them = singly but=20 rather the two of them together make up the object of interest, of our = questions=20 about the future.[6]=20

But the converse is not true. *Maximal knowledge of a total system =
does=20
not necessarily include total knowledge of all its parts, not even when =
these=20
are fully separated from each other and at the moment are not =
influencing each=20
other at all.* Thus it may be that some part of what one knows may =
pertain=20
to relations or stipulations between the two subsystems (we shall limit=20
ourselves to two), as follows: if a particular measurement on the first =
system=20
yields *this* result, then for a particular measurement on the =
second the=20
valid expectation statistics are such and such; but if the measurement =
in=20
question on the first system should have *that* result, then some =
other=20
expectation holds for that on the second; should a third result occur =
for the=20
first, then still another expectation applies to the second; and so on, =
in the=20
manner of a complete disjunction of all possible measurement results =
which the=20
one specifically contemplated measurement on the first system can yield. =
In this=20
way, any measurement process at all or, what amounts to the same, any =
variable=20
at all of the second system can be tied to the not-yet-known value of =
any=20
variable of the first, and of course *vice versa* also. If that =
is the=20
case, if such conditional statements occur in the combined catalog, =
*then it=20
can not possibly be maximal in regard to the individual systems*. =
For the=20
content of two maximal individual catalogs would by itself suffice for a =
maximal=20
combined catalog; the conditional statements could not be added on.=20

These conditional predictions, moreover, are not something that has =
suddenly=20
fallen in here from the blue. They are in every expectation-catalog. If =
one=20
knows the psi-function and makes a particular measurement and this has a =
particular result, then one again knows the psi-function, *voila =
tout*.=20
It's jst that for the case under discussion, because the combined system =
is=20
supposed to consist of two fully separated parts, the matter stands out =
as a bit=20
strange. For thus it becomes meaningful to distinguish between =
measurements on=20
the one and measurements on the other subsystem. This provides to each =
full=20
title to a private maximal catalog; on the other hand it remains =
possible that a=20
portion of the attainable combined knowledge is, so to say, squandered =
on=20
conditional statements, that operate between the subsystems, so that the =
private=20
expectancies are left unfulfilled - even though the combined catalog is =
maximal,=20
that is even though the psi-function of the combined system is known.=20

Let us pause for a moment. This result in its abstractness actually = says it=20 all: Best possible knowledge of a whole does not necessarily include the = same=20 for its parts. let us translate this into terms of Sect. = 9: The=20 whole is in a definite state, the parts taken individually are not.=20

"How so? Surely a system must be in some sort of state." "No. State = is=20 psi-function, is maximal sum of knowledge. I didn't necessarily provide = myself=20 with this, I may have been lazy. Then the system is in no state."=20

"Fine, but then too the agnostic prohibition of questions is not yet = in force=20 and in our case I can tell myself: the subsystem is already in some = state, I=20 just don't know which."=20

"Wait. Unfortunately no. There is no `I just don't know.' For as to = the total=20 system, maximal knowledge is at hand..."=20

*The insufficiency of the psi-function as model replacement rests =
solely=20
on the fact that one doesn't always have it.* If one does have it, =
then by=20
all means let it serve as description of the state. But sometimes one =
does not=20
have it, in cases where one might reasonably expect to. And in that =
case, one=20
dare not postulate that it "is actually a particular one, one just =
doesn't know=20
it"; the above-chosen standpoint forbids this. "It" is namely a sum of=20
knowledge; and knowledge, that no one knows, is none. ----=20

We continue. That a portion of the knowledge should float in the form =
of=20
disjunctive conditional statements *between* the two systems can=20
certainly not happen if we bring up the two from opposite ends of the =
world and=20
juxtapose them without interaction. For then indeed the two "know" =
nothing about=20
each other. A measurement on one cannot possibly furnish any grasp of =
what is to=20
be expected of the other. Any "entanglement of predictions" that takes =
place can=20
obviously only go back to the fact that the two bodies at some earlier =
time=20
formed in a true sense *one* system, that is were interacting, =
and have=20
left behind *traces* on each other. If two separated bodies, each =
by=20
itself known maximally, enter a situation in which they influence each =
other,=20
and separate again, then there occurs regularly that which I have just =
called=20
*entanglement* of our knowledge of the two bodies. the combined=20
expectation-catalog consists initially of a logical sum of the =
individual=20
catalogs; during the process it develops causally in accord with known =
law=20
(there is no question whatever of measurement here). The knowledge =
remains=20
maximal, but at its end, if the two bodies have again separated, it is =
not again=20
split into a logical sum of knowledges about the individual bodies. What =
still=20
remains *of that* may have becomes less than maximal, even very =
strongly=20
so. --One notes the great difference over against the classical model =
theory,=20
where of course from known initial states and with known interaction the =
individual end states would be exctaly known.=20

Now how do things stand, after automatically completed measurement? =
We=20
possess, afterwards same as before, a maximal expectation-catalog for =
the total=20
system. The recorded measurement result is of course not included =
therein. As to=20
the instrument the catalog is far from complete, telling us nothing at =
all about=20
where the recording pen left its trace. (Remember that poisoned cat!) =
What this=20
amounts to is that our knowledge has evaporated into conditional =
statements:=20
*if* the mark is at line 1, *then* things are thus and so =
for the=20
measured object, *if* it is at line 2, then such and such, if at =
3, then=20
a third, etc. Now has the psi-function of the measured *object* =
made a=20
leap? Has it developed further in accord with natural law (in accord =
with the=20
partial differential equation)? No to both questions. It is no more. It =
has=20
become snarled up, in accord with the causal law of the =
*combined*=20
psi-function, with that of the measuring instrument. *The =
expectation-catalog=20
of the object has split into a conditional disjunction if=20
expectation-catalogs* - like a Baedeker that one has taken apart in =
the=20
proper manner. Along with each section there is given also the =
probability that=20
it proves correct - transcribed from the original expectation-catalog of =
the=20
object. But which one proves right - which section of the Baedeker =
should guide=20
the ongoing journey - that can be determined only by actual inspection =
of the=20
record.=20

And what if we *don't* look? Let's say it was photographically =
recorded and by bad luck light reaches the film before it was developed. =
Or we=20
inadvertently put in black paper instead of film. Then indeed have we =
not only=20
not learned anything new from the miscarried measurement, but we have =
suffered=20
loss of knowledge. This is not surprising. It is only natural that =
outside=20
interference will almost always spoil the knowledge that one has of a =
system.=20
The interference, if it is to allow the knowledge to be gained back =
afterwards,=20
must be circumspect indeed.=20

What have we won by this analysis? *First*, the insight into =
the=20
disjunctive splitting of the expectation-catalog, which still takes =
place quite=20
continuously and is brought about through embedment in a combined =
catalog for=20
instrument and object. From this amalgamation the object can again be =
separated=20
out only by the living subject actually taking cognizance of the result =
of the=20
measurement. Some time or other this must happen if that which has gone =
on is=20
actually to be called a measurement - however dear to our hearts is was =
to=20
prepare the process throughout as objectively as possible. And that is =
the=20
*second* insight we have won: *not until this inspection*, =
which=20
determines the disjunction, does anything discontinuous, or leaping, =
take place.=20
One is inclined to call this a *mental* action, for the object is =
already=20
out of touch, is no longer physically affected: what befalls it is =
already past.=20
But it would not be quite right to say that the psi-function of the =
object which=20
changes *otherwise* according to a partial differential equation, =
independent of the observer, should *now* change leap-fashion =
because of=20
a mental act. For it had disappeared, it was no more. Whatever is not, =
no more=20
can it change. It is born anew, is reconstituted, is separated out from =
the=20
entangled knowledge that one has, through an act of perception, which as =
a=20
matter of fact is not a physical effect on the measured object. From the =
form in=20
which the psi-function was last known, to the new in which it reappears, =
runs no=20
continuous road - it ran indeed through annihilation. Contrasting the =
two forms,=20
the thing looks like a leap. In truth something of importance happens in =
between, namely the influence of the two bodies on each other, during =
which the=20
object possessed no private expectation-catalog nor had any claim =
thereunto,=20
because it was not independent.=20

For in the first place the knowledge of the total system remains =
always=20
maximal, being in no way damaged by good and exact measurements. In the =
second=20
place: conditional statements of the form "if for A ... then for B B. =
For it is=20
*not* conditional and to it nothing at all can be added on =
relevant to B.=20
Thirdly: conditional statements in the inverse sense (if for B ... then =
for A=20
...) can be transformed into statements about A alone, because all =
probabilities=20
for B are already known unconditionally. The entanglement is thus =
completely put=20
aside, and since the knowledge of the total system has remaind maximal, =
it can=20
only mean that along with the maximal catalog of B came the same thing =
for A.=20

And it cannot happen the other way around, that A becomes maximally =
known=20
indirectly, through measurements on B, before B is. For then all =
conclusions=20
work in the reversed direction - that is, B is too. The systems become=20
simultaneously maximally known, as asserted. Incidentally, this would =
also be=20
true if one did not limit the measurement to just one of the two =
systems. But=20
the interesting point is precisely this, that one *can* limit it =
to one=20
of the two; that thereby one reaches his goal.=20

*Which* measurements on B and in what sequence they are =
undertaken, is=20
left entirely to the arbitrary choice of the experimenter. He need not =
pick out=20
specific variables, in order to be able to use the conditional =
statements. He is=20
free to formulate a plan that would lead him to maximal knowledge of B, =
even if=20
he should know nothing at all about B. And it can do no harm if he =
carries=20
through this plan to the end. If he asks himself after each measurement =
whether=20
he has perhaps already reached his goal, he does only to spare himself =
from=20
further, superfluous labor.=20

What sort of A-catalog comes forth in this indirect way depends =
obviously on=20
the measured values that are found in B (before the entanglement is =
entirely=20
resolved: not on more, on any later ones, in case the measuring goes on=20
superfluously). Suppose now that in this way I derived an A-catalog in a =
particular case. then I can look back and consider whether I might =
perhaps have=20
found a *different one* if I had put into action a =
*different*=20
measuring plan for B. But since after all I neither have actually =
touched the=20
system A, nor in the imagined other case would have touched it, the =
statements=20
of the other catalog, whatever it might be, must *also* be =
correct. They=20
must therefore be entirely contained within the first, since the first =
is=20
maximal. But so is the second. So it must be identical with the first.=20

Strangely enough, the mathematical structure of the theory by no =
means=20
satisfies this requirement automatically. Even worse, examples can be =
set up=20
where the requireement is necessarily violated. It is true that in any=20
experiment one can actually carry out only *one> group of =
measurements (always on B), for once that has happened the entanglement =
is=20
resolved and one leans nothing more about A from further measurements on =
B. But=20
there are cases of entanglement in which two definite programs =
are=20
specifiable, fo which each 1) must lead to resolution of the =
entanglement, and=20
2) must lead to an A-catalog to which the other can not =
possibly lead -=20
whatsoever measured values may turn up in one case or the other. It is =
simply=20
like this, that the two series of A-catalogs, that can possibly =
arise=20
from the one or the other of the programs, are sharply separated and =
have in=20
common not a single term.=20
*

*These are especially pointed cases, in which the conclusion lies so =
clearly=20
exposed. In general one must reflect more carefully. If two programs of=20
measurement on B are proposed, along with the two-series of A-catalogs =
to which=20
they can lead, then it is by no means sufficient that the two series =
have one or=20
more terms in common in order for one to be able to say: well now, =
surely one of=20
these will always turn up - and so to set forth the requirements as =
"presumably=20
fulfilled". That's not enough. For indeed one knows the =
probability of=20
every measurement on B, considered as measurement on the total system, =
and under=20
many ab-ovo-repetitions each one must occur with the frequency assigned =
to it.=20
Therefore the two series of A-catalogs would have to agree, member by =
member,=20
and furthermore the probabilities in each series would have to be the =
same. And=20
that not merely for these two programs but also for each of the =
infinitely many=20
that one might think up. But this is utterly out of the question. The=20
requirement that the A-catalog that one gets should always be the same,=20
regardless of what measurements on B bring it into being, this =
requirement, is=20
plainly and simply never fulfilled.=20
*

*Now we wish to discuss a simple "pointed" example.=20
*

In the cited paper it is shown that between these two systems an =
entanglement=20
can arise, which at a particular moment, can be compactly shown in the =
two=20
equations: q =3D Q and p =3D -P. That means: *I know*, if a =
measurement of q=20
on the system yields a certain value, that a Q-measurement performed =
immediately=20
thereafter on the second will give the *same* value, and vice =
versa; and=20
*I know*, if a p-measurement on the first system yields a certain =
value,=20
that a P-measurement performed immediately thereafter will give the =
opposite=20
value, and vice versa.=20

A single measurement of *q or p or Q or P* resolves the =
entanglement=20
and makes both systems maximally known. A second measurement on the same =
system=20
modifies only the statements about *it*, but teaches nothing more =
about=20
the other. So one cannot check both equations in a single experiment. =
But one=20
can repeat the experiment *ab ovo* a thousand times; each time =
set up the=20
same entanglement; according to whim check one or the other of the =
equations;=20
and find confirmed that one which one is momentarily pleased to check. =
We assume=20
that all this has been done.=20

If for the thousand-and-first experiment one is then seized by the = desire to=20 give up further checking and then measure q on the first system and P on = the=20 second, and one obtains=20

q =3D 4; P =3D 7;=20

can one then doubt that=20

q =3D 4; p =3D -7=20

would have been a correct prediction for the first system, or=20

Q =3D 4; P =3D 7=20

a correct prediction for the second? Quantum predictions are indeed = not=20 subject to test as to their full content, ever, in a single experiment; = yet they=20 are correct, in that whoever possessed them suffered no disillusion, = whichever=20 half he decided to check.=20

There's no doubt about it. Every measurement is for its system the = first.=20 Measurements on separated systems cannot directly influence each other - = that=20 would be magic. Neither can it be by chance, if from a thousand = experiments it=20 is established that virginal measurements agree.=20

The prediction catalog q =3D 4, p =3D -7 would of course by = hypermaximal.=20

But let us once more make the matter very clear. Let us focus =
attention on=20
the system labelled with small letters p, q and call it for brevity the =
"small"=20
one. then things stand as follows. I can direct *one* of two =
questions to=20
the small system, either that about q or that about p. Before doing so I =
can, if=20
I choose, procure the answer to *one* of these questions b a =
measurement=20
on the fully separated other system (which we shall regard as auxiliary=20
apparatus), or I may intend to take care of this afterwards, My small =
system,=20
like a schoolboy under examination, *cannot possibly know* =
whether I have=20
done this or for which questions, or whether and for which I intend to =
do it=20
later. From arbitrarily many pretrials I know that the pupil will =
correctly=20
answer the first question that I put to him. From that it follows that =
in every=20
case he *knows* the answer to *both* questions. That the =
answering=20
of the first question, that it pleases me to put to him, so tires or =
confuses=20
the pupil that his further answers are worthless, changes nothing at all =
of this=20
conclusion. No school principal would judge otherwise, if this situation =
repeated itself with thousands of pupils of similar provenance, however =
he much=20
he might wonder *what* makes all the scholars so dim-witted or =
obstinate=20
after the answering of the first question. he wuold not come to think =
that his,=20
the teacher's, consulting a textbook first suggests to the pupil the =
correct=20
answer, or even, in the cases when the teacher chooses to consult it =
only after=20
ensuing answers by the pupil, that the pupil's answer has changed the =
text of=20
the notebook in the pupil's favor.=20

Thus my small system holds a quite definite answer to the q-question =
and to=20
the p-question in readiness fpr the case that one or the other is the =
first to=20
be put directly to it. Of this preparedness not an iota can be changed =
if I=20
should perhaps measure the Q on the auxiliary system (in the analogy: if =
the=20
teacher looks up one of the questions in his notebook and thereby inded =
ruins=20
with an inkblot *the* page where the other answer stands). The =
quantum=20
mechanician maintains that after a Q-measurement on the auxiliary system =
my=20
small system has a psi-function in which "q is fully sharp, but p fully=20
indeterminate". And yet, as already mentioned, not an iota is changed of =
the=20
fact that my small system also has ready an answer to the p-question, =
and indeed=20
the same one as before.=20

But the situation is even worse yet. Not only to the q-question and = to the=20 p-question does my clever pupil have a definite answer ready, but rather = also to=20 a thousand others, and indeed without my having the least insight into = the=20 memory technique by which he is able to do it. p and q are not the only=20 variables that I can measure. Any combination of them whatsoever, for = example=20

p^2 + q^2=20

also corresponds to a fully definite measurement according to the = formulation=20 of Q.M. Now it can be shown[8] = that also=20 for this the answer can be obtained by a measurement on the auxiliary = system,=20 namely by measurement of P^2 + Q^2, and indeed the answers are just the = same. By=20 general rules of Q.M. this sum of squares can only take on a value from = the=20 series=20

hbar, 3.hbar, 5.hbar, 7.hbar, ...=20

The answer that ym small system has ready for the (p^2+q^2)-question = (in case=20 this should be the first it must face) must be a number from this = series. --It=20 is very much the same with measurement of=20

p^2 + a^2 . q^2=20

where a is an arbitrary positive constant. In this case the answer = must be,=20 according to Q.M. a number from the following series=20

a.hbar, 3a.hbar, 5a.hbar, 7a.hbar, ...=20

For each numerical value of a one gets a different question, and to = each my=20 small system holds ready an answer from the series (formed with the = a-value in=20 question).=20

Most astonishing is this: these answers cannot possibly be related to = each=20 other in the way given by the formulas! For let q' be the answer held = ready for=20 the q-question, and p' for the p-question, then the relation=20

(p'^2 + a^2 . q'^2) / (a.hbar) =3D an odd integer=20

cannot possibly hold for given numerical values q' and p' and for =
*any=20
positive numer a*. This is by no means an operation with imagined =
numbers,=20
that one cannot really ascertain. One can in fact get two of the =
numbers, e.g.,=20
q' and p', the one by direct, the other by indirect measurement. And =
then one=20
can (pardon the expression) convince himself that the above expression, =
formed=20
with the numbers q' and p' and an arbitrary a, is not an odd integer.=20

The lack of insight into the relationships among the various answers =
held in=20
readiness (into the "memory technique" of the pupil) is a total one, a =
gap not=20
to be filled perhaps by a new kind of algebra of Q.M. The lack is all =
the=20
stranger, since on the other hand one can show: the entanglement is =
already=20
uniquely determined by the requirements q =3D Q and p =3D -P. If we know =
that the=20
coordinates are equal and the momenta equal but opposite, then there =
follows by=20
quantum mechanics a *fully determined* one-to-one arrangement of =
*all=20
possible* measurements on both systems. For *every* =
measurement on=20
the "small" one the numerical result can be procured by a suitably =
arranged=20
measurement on the "large" one, and each measurement on the large =
stipulates the=20
result that a particular measurement on the small would give or has =
given. (Of=20
course in the same sense as always heretofore: only the virgin =
measurement on=20
each system counts.) As soon as we have brought the two systems into the =
situation where they (briefly put) coincide in coordinate and momentum, =
then=20
they (briefly put) coincide also in regard to all other variables.=20

But as to how the numerical values of all these variables of =
*one*=20
system relate to each other we know nothing at all, even though for each =
the=20
system must have a quite specific one in readiness, for if we wish we =
can learn=20
it from the auxiliary system and then find it always confirmed by direct =
measurement.=20

Should one now think that because we are so ignorant about the =
relations=20
among the variable-values held ready in *one* system, that none =
exists,=20
that far-ranging arbitrary combination can occur? That would mean that =
such a=20
system of "*one* degree of freedom" would need not merely =
*two*=20
numbers for adequately describing it, as in classical mechanics, but =
rather many=20
more, perhaps infinitely many. It is then nevertheless strange that two =
systems=20
always agree in *all* variables if they agree in two. Therefore =
one would=20
have to make the second assumption, that this is due to our awkwardness; =
would=20
have to think that as a practical matter we are not competent to bring =
two=20
systems into a situation such that they coincide in reference to two =
variables,=20
without *nolens volens* bringing about coincidence also for all =
other=20
variables, even though that would not in itself be necssary. One wold =
have to=20
make these *two* assumptions in order not to perceive as a great =
dilemma=20
the complete lack of insight into the interrelationship of variable =
values=20
within one system.=20

q_t =3D q + (p/m)t Q_t =3D Q + (P/m)t=20

Let us first talk about the small system. The most natural way of =
describing=20
it classically at time t is in terms of coordinate and momentum *at =
this=20
time*, i.e., in terms of q_t and p. But one may do it differently. =
In place=20
of q_t one could specify q. It too is a "determining part at time t", =
and indeed=20
at every time t, and in fact one that does not change with time. This is =
similar=20
to the way in which I can specify a certain determining part of my own =
preson,=20
namely my *age*, either through the hnumber 48, which changes =
with time=20
and in the system corresponds to specifying q_t, or through the number =
1887,=20
which is usual in documents and corresponds to specifying q. Now =
according to=20
the foregoing:=20

q =3D q_t - (p/m)t=20

Similarly for the second system. So we take as determining parts=20

for the first system q_t - (p/m)t and p. for the second system Q_t - = (P/m)t=20 and P.=20

The advantage is that *among these the same entanglement goes on=20
indefinitely*:=20

q_t - (p/m)t =3D Q_t - (P/m)t p =3D -P=20

or solved:=20

q_t =3D Q_t - (2 t/m)P; p =3D -P.=20

So that what changes with time is just this: the coordinate of the = "small"=20 system is not ascertained simply by a coordinate measurement on the = auxiliary=20 system, but rather by a measurement of the aggregate=20

Q_t - (2 t/m)P.=20

Here however, one must not get the idea that maybe he measures Q-t=20
*and* P, because that just won't go. Rather one must suppose, as =
one=20
always must suppose in Q.M., that there is a direct measurement =
procedure for=20
this aggregate. Except for this change, *everything* that was =
said in=20
Sections 12=20
and 13=20
applies at any point of time; in particular there exists at all times =
the=20
one-to-one entanglement of *all* variables together with its evil =
consequences.=20

It is just this way too, if within each system a force works, except = that=20 then q_t and p are entangled with variables that are more complicated=20 combinations of Q_t and P.=20

I have briefly explained this in order that we may consider the =
following.=20
That the entanglement should change with time makes us after all a bit=20
thoughtful. Must perhaps all measurements, that were under discussion, =
be=20
completed in very short time, actually *instantaneously*, in zero =
time,=20
in order that the unwelcome consequences be vindicated? Can the ghost be =
banished by reference to the fact that measurements take time? No. For =
each=20
single experiment one needs just *one* measurement on each =
system; only=20
the virginal one matters, further ones apart from this would be without =
effect.=20
How long the measurement lasts need not therefore concern us, since we =
have no=20
second one following on. One must merely be able to so arrange the two =
virgin=20
measurements that they yield variable values for the same definite=20
*point* of time, known to us in advance - known in advance, =
because after=20
all we must direct the measurements at a pair of variables that are =
entangled at=20
just this point of time.=20

"Perhaps it is not possible so to direct the measurements?"=20

"Perhaps. I even presume so. Merely: *today's* Q.M. must =
require this.=20
For it is now set up so that its predictions are always made for a=20
*point* of time. Since they are supposed to rlate to measurement =
results,=20
they would be entirely without content if the relevant variables were =
not=20
measurable *for* a definite point of time, whether the =
measurement itself=20
lasts a long or a short while."=20

When we *learn* the result is of course quite immaterial.=20
Theoretically that has as little weight as for instance the fact that =
one needs=20
several months to integrate the differential equations of the weather =
for the=20
next three days. --The drastic analogy with the pupil exmaination misses =
the=20
mark in a few points of the law's letter, but it fits the spirit of the =
law. The=20
expression "the system knows" will perhaps no longer carry the meaning =
that the=20
answer comes forth from an instantaneous situation; it may perhaps =
derive from a=20
succession of situations, that occupies a finite length of time. But =
even if it=20
be so, it need not concern us so long as the system somehow brings forth =
the=20
answer from within itself, with no other help than that we tell it =
(through the=20
experimental arrangement) *which* question we would like to have=20
answered; and so long as the answer itself is uniquely tied to a =
*moment*=20
of time: which for better or for worse must be presumed for every =
measurement to=20
which contemporary Q.M. speaks, for otherwise the quantum mechanical =
predictions=20
would have no content.=20

In our discussion, however, we have stumbled across a possibility. If = the=20 formulation could be so carried out that the quantum mechanical = predictions did=20 not or did not always pertain to a quite sharply defined point of time, = then one=20 would also be freed from requiring this of the measurement results. = thereby,=20 since the entangled variables change with time, setting up the = antinomical=20 assertions would become much more difficult.=20

That prediction for sharply-defined time is a blunder, is probable =
also on=20
other grounds. The numerical value of time is like any other the result =
of=20
observation. Can one make exception just for measurement with a clock? =
Must it=20
not like any other pertain to a variable that in general has no sharp =
value and=20
in any case cannot have it simultaneously with *any* other =
variable? If=20
one predicts the value of *another* for a particular *point of =
time*, must one not fear that both can never be sharply known =
together?=20
Within contemporary Q.M. one can hardly deal with this apprehension. For =
time is=20
always considered a priori as known precisely, although one would have =
to admit=20
that every look-at-the-clock disturbs the clock's motion in =
uncontrollable=20
fashion.=20

Permit to repeat that we do not possess a Q.M. whose statements =
should=20
*not* be valid for sharply fixed points of time. It seems to me =
that this=20
lack manifests itself directly in the former antinomies. Which is not to =
say=20
that it is the only lack which manifests itslef in them.=20

The remarkable theory of measurement, the apparent jumping around of =
the=20
psi-function, and finally the "antinomies of entanglement", all derive =
from the=20
simple manner in which the calculation methods of quantum mechanics =
allow two=20
separated systems conceptually to be combined together into a single =
one; for=20
which the methods seem plainly predestined. When two systems interact, =
their=20
psi-functions, as we have seen, do not come into interaction but rather =
they=20
immediately cease to exist and a single one, for the combined system, =
takes=20
their place. It consists, to mention this briefly, at first simply of =
the=20
*product* of the two individual functions; which, since the one =
function=20
depends on qute different variables from the other, is a function of all =
these=20
variables, or "acts in a space of much higher dimension number" than the =
individual functions. As soon as the systems begin to influence each =
other, the=20
combined function ceases to be a product and moreover does not again =
divide up,=20
after they have again become separated, into factors that can be =
assigned=20
individually to the systems. Thus one disposes provisionally (until the=20
entanglement is resolved by an actual observation) of only a =
*common*=20
description of the two in that space of higher dimension. This is the =
reason=20
that knowledge of the individual systems can decline to the scantiest, =
even to=20
zero, while knowledge of the combined system remains continually =
maximal. Best=20
possible knowledge of a whole does *not* include best possible =
knowledge=20
of its parts - and that is what keeps coming back to haunt us.=20

Whoever reflectes on this must after all be left fairly thoughtful by =
the=20
following fact. the conceptual joining of two or more systems into =
*one*=20
encounters great difficulty as soon as one attempts to introduce the =
principle=20
of special relativity into Q.M. Already seven years ago P.A.M. Dirac =
found a=20
startlingly simple and elegant relativistic solution to the problem of a =
single=20
electron.[10] =
A series=20
of experimental confirmations, marked by the key terms electron spin, =
positive=20
electron, and pair creation, can leave no doubt as to the basic =
correctness of=20
the solution. But in the first place it does nevertheless very strongly=20
transcend the conceptual plan of Q.M. (that which I have attempted to =
picture=20
*here*),[11] =
and in=20
the second place one runs into stubborn resistance as soon as one seeks =
to go=20
forward, according to the prototype of non-relativistic theory, from the =
Dirac=20
solution to the problem of several electrons. (This shows at once that =
the=20
solution lies outside the general plan, in which, as mentioned, the =
combining=20
together of subsystems is extremely simple.) I do not presume to pass =
judgment=20
on the attempts which have been made in this direction.[12] =
That they=20
have reached their goal, I must doubt first of all because the authors =
make no=20
such claim.=20

Matter stand much the same with another system, the electromagnetic =
field.=20
Its laws are "relativity personified", a *non*-relatviistic =
treatment=20
being in general impossible. Yet it was this field, which in terms of =
the=20
classical model of heat radiation provided the first hurdle for quantum =
theory,=20
that was the first system to be "quantized". That this could be =
successfully=20
done with simple means comes about because here one has things a bit =
easier, in=20
that the photons, the "atoms of light", do not in general interact =
directly with=20
each other,[13] =
but only=20
via the charged particles. Today we do not as yet have a truly =
unexceptionabl=20
quantum theory of the electromagnetic field.[14] =
One can=20
go a long way with *building up out of subsystems* (Dirac's =
theory of=20
light[15])=
,=20
yet without qute reaching the goal.=20

The simple procedure provided for this by the non-relativistic theory = is=20 perhaps after all only a convenient calculational trick, but one that = today, as=20 we have seen, has attained influence of unprecedented scope over our = basic=20 attitude toward nature.=20

My warmest thanks to Imperial Chemical Industries, London, for the = leisure to=20 write this article.=20

[1] E. Schr=F6dinger, "Die gegenw=E4rtige = Situation in der=20 Quantenmechanik", Naturwissenschaften 23: pp.807-812; = 823-828;=20 844-849 (1935).=20

[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. = Rev. 47:=20 p.777 (1935).=20

[3] E. Schr=F6dinger, Proc. Cambridge Phil. =
Soc.=20
31: p.555 (1935); *ibid.*, 32: p.446 (1936).=20

[4] h =3D 1.041 x 10^(-27) erg sec. Usually in =
the literature=20
the 2.pi-fold of this (6.542 x 10^(-27) erg sec) is designated as h and =
for=20
*our* h an h with a cross-bar is written. [Transl. Note: In =
conformity=20
with the now universal usage, hbar is used in the translation in place =
of h.]=20

[5] For illustration see Fig. 5 or 6 on p.375 of = the 1927=20 volume of this journal; or Fig. 1, p.734 of the preceding year's volume = (1934),=20 though these are proton tracks.=20

[6] Obviously. We cannot fail to have, for = instance,=20 statements on the relation of the two to each other. For that would be, = at least=20 one of the two, something in addition to its psi-function. And such = there cannot=20 be.=20

[7] A. Einstein, B. Podolsky, and N. Rosen, = Phys.=20 Rev. 47: 777 (1935). The appearance of this work motivated the = present -=20 shall I say lecture or general confession?=20

(Paris, 1931); Cursos de la Universidad Internacional de Verano = en=20 Santander, 1: p.60 (Madrid, Signo, 1935).=20

[10] Proc. Roy. Soc. Lond. A117: = p.610=20 (1928).=20

[11] P.A.M. Dirac, The Principles of = Quantum=20 Mechanics, 1st ed., p.239; 2nd ed. p.252. Oxford: Clarendon = Press, 1930=20 or 1935.=20

[12] Herewith a few of the more important = references: G.=20 Breit, Phys. Rev. 34: p.553 (1929) and 39: p.616 (1932); C. = Mo/ller, Z. Physik 70: p.786 (1931); P.A.M. Dirac, = Proc. Roy.=20 Soc. Lond. A136: p.453 (1932) and Proc. Cambridge Phil = Soc.=20 30: p.150 (1934); R. Peierls, Proc. Roy. Soc. Lond. A146: = p.420=20 (1934); W. Heisenberg, Z. Physik 90: p.209 (1934).=20

[13] But this holds, probably, only = approximately. See M.=20 Born and L. Infled, Proc. Roy. Soc. Lond. A144: p.425 and = A147:=20 p.522 (1934); A150: p.141 (1935). This is the most recent attempt at a = quantum=20 electrodynamics.=20

[14] Here again the most important works, = partially=20 assignable, according to their contents, also according to the = penultimate=20 citation: P. Jordan and W. Pauli, Z. Physik 47: p.151 = (1928); W.=20 Heisenberg and W. Pauli, Z. Physik 56: p.1 (1929); 59: = p.168=20 (1930); P.A.M. Dirac, V.A. Fock, and B. Podolsky, ,cite>Physik. Z.=20 Sowjetunion 6: p.468 (1932); N. Bohr and L. Rosenfeld, = Danske.=20 Videns. Selsk. (math.-phys.) 12: p.8 (1933).=20

[15] An excellent reference: E. Fermi, = Rev. Mod.=20 Phys. 4: p.87 (1932).=20