Download How Albert Einstein invented entanglement despite his intention

How Albert Einstein unintentionally invented entanglement
Paul Drechsel
Johannes Gutenberg University of Mainz, D-55099 Mainz. Germany
[email protected]
Abstract:
When Einstein in 1905 invented the momentum of the photon of light, he probably
could not foresee that some years later de Broglie would derive the matter-wave from
it. Einstein accepted this derivation, but what he could not accept until the end of his
life was another quantum phenomenon, namely entanglement, which he called
‘spooky’. However, one could argue that by formulating his Theory of Special
Relativity (STR), Einstein also laid the foundations for the derivation of an
experimentally based formula for entanglement. The key is the fact that something
which is faster than the constant speed of light c must be imaginary. Exactly this
imaginarity, inserted in the core Lorentz factor  of STR, leads straight to the formula
for entanglement of photons. This imaginarity is in accordance with the fact that
quantum physics is based on the imaginary Schrödinger equation. The consequence is
that entanglement is faster than the constant speed of light c – a fact which was proven
experimentally by Nicolas Gisin et al. in 2008.
Introduction
It is common knowledge that Albert Einstein at the beginning of the last century invented
both the Special (1905) – and the General Theory of Relativity (1915). It is also common
knowledge that in 1905, he invented the photon as an elementary particles of light. Less
commonly known is Einstein’s lifelong fight against the concept of entanglement, which he
considered to be an instantaneous ‘action at a distance’. For Einstein such an instantaneousity
was impossible because it violated the limitation of the constant speed of light c, fundamental
for his theories of relativity. Therefore he called quantum entanglement derogatively a
‘spooky action at a distance’. Einstein assumed that any quantum theory which allowed for
such ‘spooky things’ had to be incomplete. In an article published in 1935 with Roman
Podolsky and Nathan Rosen, he postulated ‘hidden variables’ which have been known ever
since as the EPR-Paradox. It took three decades before John Bell in 1965 offered his now
famous inequality that made experiments of entanglement possible. It took another two
decades before Alain Aspect et al. in the year 1982 could carry out such an experiment and
convincingly prove that entanglement is a quantum physical fact.
Why, then, do I assume that Albert Einstein also induced entanglement? To answer this
question I will proceed in a similar way like de Broglie when he used Einstein’s theory of the
momentum of light particles to derive the wave property of matter from it. I will begin with
Einstein’s notation of the coordinates of the four-dimensional Minkowski space (cf. Einstein
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1961), which introduces the fourth dimension as the product of the constant speed of light c
and time t: ct. Referring to space coordinate differences dx, dy, dz and the time-difference dt
of this four-dimensional space he offers the standard squares of distances of Minkowski space
as:
ds 2  dx2  dy 2  dz 2  cdt 2
In order to be in conformity with the usual positive definite Euclidian space he wants to get
rid of this Minus sign ‘−cdt2‘.On page 63 Einstein says that “we must replace the usual timeco-ordinate t by an imaginary magnitude
1.ct proportional to it.” “If we replace x, y, z,
1ct , by x1 ,x2 ,x3 ,x4 , we also obtain the result that
ds 2  dx12  dy22  dz32  dx42
is independent of the choice of the body of reference.” (1961:102)
Essential for the Special Theory of Relativity (STR) are the Lorentz transformations. Einstein
states: “We can characterize the Lorentz transformation still more simply if we introduce the
imaginary i  1ct in place of t, as time-variable. If, in accordance with this, we insert
x1  x; x2  y; x 3  z; x4  1.ct
…” (1961:139)
It is my intention to demonstrate how a violation of the constant speed of c, which, as Einstein
himself remarked, leads to imaginarity, in a simple way naturally leads to quantum
entanglement. My argumentation is purely formal. It refers to complex number imaginarity
which in my mind is underrated by the community of physicists. George W. Mackey (1984),
E. C. G. Stueckelberg (1960) and others have demonstrated that quantum mechanics and the
Schrödinger equation cannot be based on real numbers. Evidently quantum mechanics –
including photons – must be based on the imaginary Schrödinger equation!
The Lorentz transformations are based on the Lorentz factor  : (v = velocity of an object)

1
v2
c2
The Lorentz transformation can then be formulated like this: (1961:37)
v
t 2 x
x  vt
c
x' 
; y'  y; z'  z;t' 
2
v
v2
1 2
1 2
c
c
1
This is well known, but with regard to the Lorentz factor  of special interest. As long as
v  c there is no problem, but the problems arise when v = c or beyond as v  c . Einstein
postulates the constant speed of light c: “For the velocity v = c we should have
1  v 2 / c 2  0 , and for still greater velocities the square –root becomes imaginary. From
this we conclude that in the theory of relativity the velocity c plays the part of a limiting
velocity, which can neither be reached nor exceeded by any real body.” (1961:41)
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The following point is crucial for my argumentation: Any velocity v faster than c must be
imaginary!
In 2008 Nicolas Gisin et al. carried out an experiment to measure the velocity of entanglement
and to determine its dependence on the principles of relativity. The experiment showed that
entanglement was at a minimum ten thousand times faster than the constant speed of light c.
Gisin’s colleague Cyril Branciard even said in an interview that he assumes it to be infinitely
faster than the constant speed of light c. The experiment also confirmed that this superfast
entanglement does not dependent on the principles of relativity. Referring to the postulate of
Einstein there is only one conclusion possible: entanglement must be imaginary!
To support this possibility formally we need to take a look at the definition of the pivotal
Lorentz factor  and examine it closely. Perhaps then we can find a way to relate it to
imaginarity and discuss the results. The definition of the Lorentz factor is based on the Law of
Pythagoras as, for example, illustrated in the following figure (Jay Orear (1991:151)).
Figure: 1
Figure (a) represents two light-clocks with two ideal reflecting mirrors.  is the time a lightbeam takes to travel from the lower to the upper mirror. Every time a light beam reaches the
upper mirror, this clock “ticks”. The time between two “ticks” is   D / c . In figure (b) the
experiment starts with the two nearby light-clocks A and B. The light-clock B should moves
with the velocity  to the right. We get the triangle cT , c and  T . Now the hypotenuse cT
can appear as faster than c. To avoid that c could be faster than the presumed constant speed
of light c one can refer to the Theorem of Pythagoras and solve it for T.
 cT 
2
  T    c 
2
2
Solving this equation for T one gets:
T
1
1
2

c2
Without the  this formula is the Lorentz-factor  already presented above.
Behind the Lorentz factor is the Pythagorean triangle
3
T
c
v
v
Figure: 2
To clarify my argument let me offer this Pythagorean triangle as a sequence of three abstract
triangles of a unit square:
a)
X
b)
c)
2
c
2
1c
v
1c
Figure: 3
i1
1
Triangle a): This Pythagorean triangle represents the usual classical basis for the Lorentz
factor. In this case we have ‘real’ Einstein photons for both the x- and y-axis with hypotenuses
X, which means both photons should obey the constant speed of light c; therefore v could also
be c: v  c . This leads to a Lorentz factor of zero, which makes no sense because it is not
defined classically.
Triangle b): By introducing formally such a definition that v = c let me assume that we have
a unit square. The constant speed of light c of these real Einstein photons should appear as the
natural units 1 and X as the hypotenuses 2 .
Triangle c): Now I enlarge the definition and take imaginary photons into account. It has to
be defined in the way a complex number is defined. A complex number consist of a real axis x
and an imaginary axis iy. It defines a 2-dimensional square or an imaginary plane wave.
Referring to the logic of the definition of a complex number, we should have a real part
photon referring to the x-axis with real value 1 and an imaginary part photon referring to the
y-axis i1. This also results in hypotenuses 2 .
Now my invention. If we insert this imaginary c  i1 and v  1 into the Lorentz factor, we get:
T
1

1

1
1

11
2
12
12
1

i2
12
At first glance this result might seem plain nonsense, but let us examine the formula for
entanglement in the context of this equation.
1
Alain Aspect et al. (1982) present the following curve-fitting for the data (marks) of their
experiment for entanglement:
4
Figure: 4
With regard to positive y-axis, there is a value of 2 and a maximum of the curve at
2 2  2.8284... . This value is known as the maximum violation of Bell’s inequality (1964)
with a limit of 2 for the domain of classical physics. This is seen as a fundamental break of
quantum physics with classical physics. The formula for this curve is:
3 cos 2  cos 6
This can also be expressed as
cos 2  cos 2  cos 2  cos 6
For an angle of 22.5 degree we would get:
1 / 2 1 / 2 1 / 2 1 / 2  4 2  2 2
Above we have seen that an imaginary Lorentz factor with normalized c results in 2 2 . What
could be the relationship between an imaginary Lorentz factor and this 2 2 ? For an answer
let me come back again to this Lorentz factor, now with a negative −1 and negative
imaginarity –ic:
Figure: 5
The inserted values of the sides of this lower left square into the Lorentz factor results into
1
1
1
x


1
1
2
1 2
1
i
1
Surprisingly we get the same result as before in the positive case. What, then, could these four
times 1 / 2 be? For an answer I offer the following projections:
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Figure: 6
In order to eliminate the minus signs of the left side of zero, one should see every value as an
absolute value. In addition, these four projections of 1 / 2 in addition are exactly the values
we need to calculate entanglement.
This procedure demonstrates that the positive and the negative cases of imaginary photons
cannot be distinguished from each other, which means that in the case of imaginary photons
we have to take both positive and negative cases into considerations! As projections, both
cases refer to a diagonal 2 times 2 which is 2 2 . This value is the violation of Bell’s
inequality; also called the Tsirelson bound. 1
The plus and minus side of the diagonal of the basic Gaussian imaginary square can be
interpreted as two photons moving in opposite directions. Because the diagonal 2 2 as the
hypotenuse refers to the opposite sides of this larger triangle with the absolute value of 2 ,
the projection of it results into:
2
1

2 2
2
Furthermore this diagonal could be expanded infinitely, shortly expressed by n (even). It is
remarkable that this infinite extension also results in the known projection of the diagonal:
n
1
.

n 2
2
This means, in the imaginary case, something of an infinite speed of photons has to be
presupposed, which can no longer be any speed at all! There appears something which is
instantaneous or expresses “instantaneousity” as an “absolute rest”.
To summarize, there is a correspondence between entanglement and the Lorentz factor
depending on whether v  c , leading to imaginarity and entanglement, or v  c , leading to
classical relativity:
1
The maximal value can be 4.
6
Figure: 7
It seems there is an intrinsic relationship between imaginarity and entanglement, but one can
ask the question of which kind? It can be a surprise, because this relationship appears
relatively simple. It consists as

i   1/ 2  i1/ 2

Usually one does not take into account this + and , but it is the case, and this four times of
1/ 2 , which taken as absolute values and without imaginarity i, leads to
1
1
1
1
4




2 2
2
2
2
2
2
+𝑖1√2
We get the value for entanglement! Now the complete figure:
=
−𝑖1/√2
i
+1/√2
−1/√2
Figure: 8
The diagonal of this magic square represents
i as entanglement with the maximum value
2 2.
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Conclusion:
Discussing the possibility of something faster than the constant speed of light c, one should
take into account imaginarity as introduced by Einstein. But one should also take into account
Einstein’s assumption that anything faster than the constant speed of light has to be imaginary.
This leads to the conclusion that there have to be imaginary photons. If, however, we insert an
imaginary photon into the Lorentz factor, we get the value for entanglement. Therefore one
could argue that Albert Einstein, although he regarded entanglement as a ‘spooky action at a
distance’, paved the way for its subsequent derivation from the principles of relativity.
Because Einstein already remarked that something faster than c has to be imaginary, it could
be argued that he already invented entanglement, but unintentionally!
Entanglement as such does not violate STR or the constant speed of light c, because it has
nothing to do with it. But because of the correspondence displayed in figure 7 it can be argued
that entanglement is the quantum complement of Relativity and its two fundamental classical
properties, the constant speed of light c and gravity.
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Bell, John S.: On the Einstein-Podolsky-Rosen Paradoxon. In: Physics 1, 1964, p.195-200.
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