121 Spin- 1 photons spin- ½ electrons Bells inequalities and Feynmans special perspective on quantum mechanics Masud Mansuripur

2025-04-28 0 0 665.68KB 21 页 10玖币
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Spin-1 photons, spin-½ electrons, Bell’s inequalities, and Feynman’s
special perspective on quantum mechanics
Masud Mansuripur
James C. Wyant College of Optical Sciences, The University of Arizona, Tucson
[Published in Proceedings of SPIE 12205, Spintronics XV, 122050B (3 October 2022); doi: 10.1117/12.2633646]
Abstract. The Einstein-Podolsky-Rosen (EPR) paradox that argues for the incompleteness of quantum
mechanics as a description of physical reality has been put to rest by John Bell’s famous theorem,
which inspired numerous experimental tests and brought about further affirmations of quantum reality.
Nevertheless, in his writings and public presentations, Richard Feynman never acknowledged the
significance of Bell’s contribution to the resolution of the EPR paradox. In this paper, we discuss
several variants of the Bell inequalities (including one that was specifically espoused by Feynman), and
explore the ways in which they demolish the arguments in favor of local hidden-variable theories. We
also examine the roots of Feynman’s attitude toward Bell’s theorem in the context of Feynman’s special
perspective on quantum mechanics.
1. Introduction. Quantum weirdness is rooted in the fact that the states of a quantum system have
probability amplitudes, that these amplitudes are complex numbers, and that the evolution,
superposition, and combination of these complex amplitudes give rise to phenomena that often
seem strange, counterintuitive, and outright paradoxical to the worldview that is firmly rooted in our
understanding of classical physics, where probability amplitudes play no role. Richard Feynman has
consistently and persistently emphasized the importance of probability amplitudes in his teachings
and his descriptions of quantum phenomena.1,2 In fact, his path integral formulation of quantum
mechanics is based on the fundamental idea that one must add up all the probability amplitudes that
contribute to an event along different paths — so long as the paths are physically allowed and are, in
principle, indistinguishable from one another.3 The squared magnitude of the sum total of these
complex amplitudes will then yield the ordinary probability of occurrence of the event.1-3
Given Feynman’s profound understanding of physics, in general, and of quantum mechanics, in
particular, one would be inclined to think that he had reached the point where quantum weirdness
no longer disturbed him. Yet, as late as 1982, he would write:4 “we always have had a great deal of
difficulty in understanding the world view that quantum mechanics represents. At least I do, because I’m an old
enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it.” In
the same paper,4 Feynman describes an experiment with a pair of polarization entangled photons
which demonstrates the incompatibility of quantum mechanical predictions with the existence of
local hidden variables. While the argument is very much in line with John Bell’s views on the
subject,5-7 Feynman does not credit Bell, and indeed appears to have quietly known and internalized
the implications of such local hidden-variable theories (LHVTs) for quite some time.8,9 Be it as it
may, the goal of the present paper is not to adjudicate the issue for purposes of parceling out
historical credit, but rather to provide an accessible account of Bell’s inequalities that is informed
by Feynman’s special perspective on quantum mechanics — specifically, his belief in the primacy
of probability amplitudes.1-4
The organization of the paper is as follows. The maximally entangled photon pair discussed by
Feynman in the aforementioned paper4 is the subject of Sec.2. Here, we begin by describing the
polarization state of the photon pair, then present the quantum mechanical explanation of
experiments in which each photon passes through a linear polarizer (also known as polarization
analyzer) before being detected. This is followed by a concise argument as to why any LHVT
cannot possibly explain the observed correlations in certain experiments of the sort. In the particular
arrangement of polarizers chosen by Feynman, the probability of agreement between the signals of
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the two detectors (i.e., both receive a photon or neither does) is ¾ according to the quantum theory,
but cannot exceed 󰆠 if an LHVT happens to be operative. It is in the context of this observation
that Feynman concludes by saying: “I have entertained myself always by squeezing the difficulty of
quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item.
It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than
another. But there you are — it is bigger than any logical argument can produce, if you have this kind of logic.”4
The maximally-entangled photon pair aside, there exist other entangled states that also
demonstrate the incompatibility of quantum mechanics with LHVTs.10-12 In Sec.3, we discuss the
case of a photon pair in the Hardy state, and follow this in Sec.4 by examining the case of a triplet
of photons in the Greenberger-Horne-Zeilinger (GHZ) thought experiment.
In photon-polarization-state studies, the Poincaré sphere serves as a powerful tool by mapping
each and every state of polarization to a unique point on a unit-sphere’s surface. Section 5 provides
a detailed description of the Poincaré sphere in conjunction with the Stokes parameters, first defined
in a basis of linearly-polarized states, then translated to a basis of right- and left-circularly-polarized
states. This provides a segue into Sec.6, where the Bloch sphere, a close relative of the Poincaré
sphere, is briefly introduced to help visualize the polarization states of spin-½ particles such as
electrons and protons. The spin polarization of such particles can be examined with the aid of Stern-
Gerlach analyzers,1 which — much the same as polarizing beam-splitters used in photon
experiments — can be oriented in three-dimensional space for purposes of separating spin-up and
spin-down electrons (relative to an axis of the analyzer) by directing the particles of differing spins
along divergent paths. For particles with nonzero mass (e.g., electrons, protons, neutrons, atoms),
the mathematics of coordinate rotations is more nuanced than that needed for zero-mass particles
(e.g., photons, neutrinos), which lack a rest frame. Thus, we devote Sec.7 to a systematic study of
2 × 2 unitary operators whose action on the spin state of a spin-½ particle at rest can bring about an
arbitrary rotation of the coordinate system.†
In Sec.8, a pair of maximally entangled electrons is sent through two Stern-Gerlach analyzers
with their axes oriented in different directions. For a particular setting of the analyzer angles,
quantum mechanics predicts a probability of ¼ for the electrons emerging from the Stern-Gerlach
devices to have anti-parallel spins. In contrast, LHVTs put the chances of such occurrences at
greater than 󰆟. Once again, the incompatibility of LHVTs with a description of Nature in quantum
mechanical terms is brought into sharp focus.
Up to this point in the paper, we have managed to discuss the predictions of LHVTs on a case
by case basis. Section 9 presents a formal (and more general) formulation of these theories, where
one of Bell’s inequalities (the one due to Clauser, Holt, Horne, and Shimony) is rigorously derived.
The unitary 2 × 2 rotation operator developed in Sec.7 for our subsequent study (in Sec.8) of a
pair of entangled spin-½ particles, has numerous interesting properties and applications in quantum
mechanics that go beyond the immediate concerns of the present paper. As an example, we show
how the rotation operator helps to derive the spin angular momentum operator
=
+
+
for spin-½ particles, then use the latter operator in Sec.10 to examine the singlet and triplet states of
a pair of electrons.1 This leads to a derivation of the 3 × 3 rotation operator for spin-1 particles of
nonzero mass, thus providing a useful comparison with the 2×2 rotation operator for spin-1
photons mentioned toward the end of Sec.8.
† These 2 × 2 unitary operators act within the rest frame of two-state particles to bring about desired coordinate
reorientations. However, they are not applicable to two-state particles with zero-mass, such as single photons, whose
allowed rotations are restricted to those around the particle’s propagation direction.1
3/21
’
2
source
photon 1 photon 2
analyzer 1 analyzer 2
detector 1 detector 2
With the 3 × 3 rotation operator for spin-1 particles at hand, we proceed to derive in Sec.11 the
angular momentum operator =
+
+ for spin-1 particles of nonzero mass, then use the
results to examine the states of a two-particle system consisting of a spin-1 particle and a spin-½
particle . Once again, the methodology used and the insights gained from these last sections are in
keeping with Feynman’s unique approach to quantum mechanics.1 The paper closes with a brief
summary and a few concluding remarks in Sec.12.
2. Maximally entangled photon pair. The state of a maximally entangled pair of photons having
linear polarization along the and axes is4
|=
(||+||). (1)
With reference to Fig.1, if analyzer 1 is rotated through an angle around the -axis, photon 1
may be described as being in a superposition of | and |, in which case the state of photon 2
automatically becomes a superposition of | and | corresponding to | and | associated
with a rotated analyzer 2 through the same angle , as follows:
|=
[(cos |sin |)|+(sin |+cos |)|]
=
[|(sin |+cos |)+|(cos |sin |)]
=
(||+||). (2)
Clearly, when analyzers 1 and 2 are set to the same angle , detectors 1 and 2 both detect a
photon, or both fail to detect a photon, irrespective of the specific value of .
Digression: For a single photon in the pure state |=|+|, one can write |=(cos +sin )|
(sin cos )|. Thus, when = ±90°, |= ±(||), indicating that a 90° rotation of the analyzer
turns the probability || of passage into that of blockage, and vice-versa. Consistency requires that two successive 90°
rotations restore the analyzer to its initial state. Indeed, when = ±180°, |=(|+|), which has the same
probability || of passage and ||= 1 || of blockage as the original setting at =0°.
Fig.1. A source emits a pair of maximally entangled photons, one going to the left, the other to the right,
toward detectors 1 and 2. (This could be due to a 321 transition in a hydrogen atom, resulting in
two successive photoemissions.) Analyzers are placed before each detector, so that a photon polarized along
the -axis passes through to detector 1, whereas a photon polarized along is blocked. Similarly, a photon
polarized along passes through to detector 2, while a photon polarized along is blocked. Each analyzer
can be rotated independently of the other, so that when the transmission axis of the first analyzer is rotated
through the angle , that of the second analyzer is set to . [The  and coordinate systems are
right-handed (with the thumb along the -axis). Both depicted angles and are taken to be positive.]
4/21
Next, assume analyzer 1 is set to while analyzer 2 is set to . The entangled state | of the
photon pair may now be written as
|=
[(cos |sin |)(sin |+cos |)
+(sin |+cos |)(cos |+ sin |)]
=
[sin() ||+cos() ||
+cos() ||sin() ||]. (3)
In this case, the probability that both detectors click, or that neither clicks, is cos(),
whereas the probability of only one detector clicking is sin().
Digression: The state | in Eq.(1) may be written in terms of the right and left circularly-polarized states (i.e., the “up”
and “down” angular momentum states), where |=(|+ i|)2
and |=(|i|)2
, as follows:
|=
[(|+|)(||)+(||)(|+|)]=
(||||). (4)
Note that the angular momentum of each photon in its up and down states is aligned with its own ±-axis, where
is the direction of propagation. Thus, both the || and || states have zero net angular momentum. In spite of
the distinguishability of photons of different color emitted in the 321 transition in a hydrogen atom, there is
no a priori way to know whether the emitted pair is in the angular momentum state || or ||, hence the
necessity of describing the state as a superposition of the two. For our pair of photons, not only is it necessary for the
total , whose operator is , to vanish, but also the total angular momentum , associated with the operator
==()=
+
2, must vanish as well. This is the fundamental reason why the photon pair
must be in the superposition state | given by Eq.(4) — the singlet state.
Given that photons are spin-1 particles, one might be curious as to why their spin angular momenta assume only
the eigenvalues = ±; shouldn’t there be a = 0 eigenvalue as well? Here is how Feynman explains the absence of
the corresponding quantum number = 0: “But light is screwy; it has only two states. It does not have the zero case.
This strange lack is related to the fact that light cannot stand still. For a particle of spin which is standing still, there
must be the 2+ 1 possible states with values of going in steps of 1 from  to +. But it turns out that for something
of spin with zero mass only the states with the components + and  along the direction of motion exist. For
example, light does not have three states, but only two — although a photon is still an object of spin one.”1
2.1. Density matrix. The two-particle pure state | of Eq.(3) has a 4 × 4 density matrix13 (or
density operator) consisting of 16 elements, as follows:
(,)=||= ½[sin() ||||
+ sin()cos() ||||
sin()cos() ||||
+ sin() ||||]. (5)
The trace of  over particle 1 now yields the density matrix of particle 2, namely,
(,)=||||+||||= ½||+ ½||. (6)
This is the 2 × 2 matrix ½, showing that the state of particle 2 is maximally mixed, with |
and | each having probability ½.
5/21
2.2. A local hidden-variable theory. Suppose each photon carries an instruction set as to which
setting of an analyzer it is allowed to pass through, and which setting it is not.4 Considering that the
two (maximally entangled) photons behave in precisely the same way when the two analyzers are
identically oriented (i.e., when =), it is natural to assume that the instructions carried by the
two photons are identical. However, in each photon-pair-emission event, the pair is expected to
receive a new (perhaps randomly selected) instruction set.
Fig.2. The two (maximally entangled) photons of Fig.1 are assumed to abide by identical instruction sets that
specify the settings of their respective analyzer at which they are permitted to pass through to the corresponding
detector. The instruction sets, although always identical for the two photons, change (perhaps randomly) from
one emitted photon pair to the next. In each experiment, the angle of the first analyzer is set to a random
multiple of 30° (relative to the -axis), while the second analyzer is set to =+30° (relative to the -axis).
In each realization, the photons are instructed to pass or not pass through their respective analyzer when the
analyzer is set at =0°, 30° and 60°. Considering that the rotation of an analyzer by 90° must result in the
opposite outcome, the behavior of each photon for all the remaining angles (at integer-multiples of 30°) is
automatically determined by their instructions for =0°, 30° and 60°. In the diagrams, a green dot at a given
angle instructs the photon to pass through, whereas a red dot is an instruction to stop. There are numerous
possible instruction sets, of course, but they can all be neatly divided into the depicted eight categories
depending on the specific instructions they contain for the 0°, 30° and 60° analyzer settings. In the cases
identified with the numerals 3 and 6, one detector always detects a photon and the other does not. For each of the
remaining six cases, however, given that is randomly set to an integer-multiple of 30°, in eight out of twelve
possible experiments (i.e., an average of two-thirds), the outputs of the two detectors will agree with each other.
Consequently, no matter what fraction of the instruction sets falls into each of the eight possible categories, the
fraction of experiments in which the readings of the two detectors agree with each other cannot exceed 󰆠.
Now, in each repetition of our experiment, we randomly set the angle of the first analyzer to
an integer-multiple of 30°, while fixing the second analyzer at =+30°.‡ Considering that the
allowed orientation angles of both analyzers are integer-multiples of 30°, all conceivable instruction
sets can be divided into the eight categories depicted in Fig.2. Here a green dot associated with a
given angle ( or ) indicates that the photon will pass through to the detector, whereas a red dot
‡To avoid the need to decide the values of and prior to the release of each photon pair, the two measurements can be done
independently of each other, where, in each trial, the analyzer angles are set to random multiples of 30°. Afterward, when the
experimenters come together, they compare only the results of those measurements that pertain to cases where =+ 30°. Also,
to ensure that coincidences are properly recorded, it is necessary in practice to replace the analyzers with polarizing beam-splitters,
each followed by two photodetectors, one for detecting -polarized photons, the other for -polarized photons.
0°
30°
60°
90°
1 2 3 4
5 6 7 8
摘要：

1/21Spin-1photons,spin-½electrons,Bell’sinequalities,andFeynman’sspecialperspectiveonquantummechanicsMasudMansuripurJamesC.WyantCollegeofOpticalSciences,TheUniversityofArizona,Tucson[PublishedinProceedingsofSPIE12205,SpintronicsXV,122050B(3October2022);doi:10.1117/12.2633646]Abstract.TheEinstein-Pod...

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