The Greatest Misconception in Science: Speeds Greater Than the Speed of Light in an Expanding Universe
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The Einstein-Podolsky-Rosen Paradox Is Not So Paradoxical

By Jupiter Scientific staff scientist Dr. Stuart Samuel

Some Features of Quantum Mechanics

Quantum mechanics, which governs Nature at the smallest scales such as those inside an atom, can seem strange especially to those who have only experienced the world at large scales. Since few of us have seen an atom or a nucleus up close, it is not surprising that Nature behaves different at such tiny distances. Why should the way an apple drops from a tree have anything to do with the way an electron orbits around a nucleus? The macroscopic world is governed by classical mechanics and classical mechanics is very different from quantum mechanics. Whereas an apple is a solid object, an electron is a wave. Indeed, according to quantum mechanics, everything is a wave. Only when the momentum is sizable and the wave is relatively well localized or involves dense oscillations does the entity behave like a particle or an object; and, in such a case, quantum mechanics produces the results of classical mechanics, a property known as the correspondence principle. Momentum is sizable if mass or speed or both are big. Since everyday objects are so much heavier than atoms, waves of everyday things behave like particles or objects, and they are well described by classical mechanics. High-speed electrons have high momentum and they too behave like particles in this case.

Because quantum mechanics and the world of the small are so different from our everyday experiences, surprising behavior occurs. For example, an electron in an atom does not have a definite location. Instead, its position can only be described probabilistically. Such electrons have higher probabilities of being in certain places and lower probabilities in being in other places. An electron bound to a nucleus has a high probability of being in the vicinity of the nucleus and a low probability of being far away from it. At any given moment, one cannot say exactly where the electron is. This is related to Heisenberg's uncertainty principle Heisenberg's uncertainty principle. From the point of view of the “wave” of the electron, this is understandable; when you look at a water wave, for example, is it possible to stay that it is located at one particular point?

A second example of the unusual nature of quantum mechanics is that the energy of an electron in an atom (and in many other situations) is only allowed to possess certain specific values; a property that physicists call energy quantization. The quantization of these energy values leads to the specific spectral lines associated with atoms. Third, because waves are involved in quantum mechanics, there can be constructive and destructive interference (meaning that wave components can combine to create larger or smaller amplitudes) and this does not happen in classical mechanics for objects; It does happen classically for waves such as water, sound and light waves. A fourth unusual feature of quantum mechanics is tunneling: although an electron may not have enough energy to pass through a region (a kind of “energy hill or barrier”), there is a certain, typically very low probability of doing so. If this phenomenon were to be blown up to macroscopic distances, it would be like seeing a marble in the bottom of one cup suddenly disappear and appear spontaneously in a neighboring cup. At tiny scales, these seemingly magical things are part of quantum mechanics.

Not only is energy sometimes quantized but other entities also are. Momentum and angular momentum are two examples. Angular momentum is the amount of rotation a system undergoes. Electrons rotate and this rotation is call spin. They are like little tops. The spin of electrons and other elementary particles contributes to the total angular momentum of a system. However, the amount of spin that an electron can undergo is not arbitrary. If one picks an axis, then an electron can spin about this axis by a fundamental amount, equal to ħ/2, in one direction or the opposite direction. Hence, the spin of the electron is quantized: it can only take on two values. Here, ħ (“h bar”) is a fundamental parameter of quantum mechanics known as Planck's constant; it determines the quantization spacing of energy levels and other quantized entities and the degree of uncertainty in Heisenberg's uncertainty principle. The waves associated with situations in which entities such as energy, momentum, angular momentum, et cetera are quantized are called eigenstates. A state in quantum mechanics is the situation corresponding to a particular quantum wave.

The miniscule value of ħ of 1.055 x 10-34 Joule-seconds means that quantum mechanical effects are minute. The spacing of energy levels is very “tight” and seems almost continuous (the situation in classical mechanics) if not viewed in detail. The other contribution to total angular moment, namely orbital angular momentum, is also quantized. The only permitted values are integral multiples of ħ, i.e., nħ where n is an integer. When n=0, the orbital angular momentum is zero and there is only one wave configuration associated with this state corresponding to a wave with perfect spherical symmetry: the wave has the same amplitude in all directions about a point. Orbital angular momentum is always measured about a point, and the word “orbital” is appropriate because it corresponds to the angular momentum of thing going around (or orbiting) the point. For example, take the point to be the center of the Sun. Then the planets contribute to the orbital angular momentum of the solar system because they revolve around the Sun.

Photons, the quantized versions of electromagnetic waves, also have quantized spin, which in this case is known as helicity. Unlike the case of electrons, the spin is quantized in the direction of the motion of the photon. Like electrons, however, there are only two possibilities: clockwise or counterclockwise rotation. Classically, this corresponds to the polarization of electromagnetic waves. The amount of spinning for a photon happens to be twice that of the electron, that is, ħ.

Fundamental to quantum mechanics is the notion a wavefunction Ψ that depends on position and time and often on other variables. It represents the “wave” in the discussion above. Among things, it encodes the quantum-mechanical uncertainty of an entity: Since waves exist over an extended region, there is no notion of the wave being located at any particular point. The wavefunction is a function of the positions (the x, y and z coordinates) of each entity (electron, atom, nucleus, molecule, etc.) in the Universe. Positions where the wavefunction is zero are regions where the entity cannot be found. The entity is more likely to be found in regions where |Ψ|2 (absolute value of the wavefunction squared) is large. Indeed, the probability of finding an entity at x at time t is proportional to |Ψ(x,t)|2.

Quantum mechanics provides a fundamental wave equation describing how the wavefunction Ψ at time t changes during a small time interval Δt:

                  Ψ(t + Δt) = Ψ(t) − iHΨ(t)Δt/ħ       Equation (1)

which can be written as

                  ħdΨ(t)/dt = -iHΨ(t)

where dt = Δt and ΔΨ(t) = Ψ(t + Δt) − Ψ(t). In this equation, H is called the Hamiltonian, ħ is the above-mentioned Planck's constant and i is a special constant. For those that are not mathematically inclined, Equation (1) says that the wavefunction changes smoothly with time in a very definite way determined by the Hamiltonian acting on it. Because quantum mechanical probabilities are determined by the square of the wavefunction, Equation (1) also means that probable locations for an entity evolve smoothly and in a way specified by quantum mechanics. In the non-relativisitic situation, Equation (1) is known as the Schrödinger equation; for relativistic quarks (the constituents of protons and neutrons) and electrons, it is the Dirac equation, and for photons it is the quantized version of Maxwell's equations (the third column of this table).

When HΨ(t) = EΨ(t), where E is a constant, Ψ is called an eigenstate of energy and E is the quantized value of energy. In other words, an eigenstate of energy is just a particular wavefunction involving a definite value of energy. As mentioned above, eigenstates exist for operators other than the Hamiltonian such as the ones for momentum and angular momentum; these operators are the quantum mechanical analogs of the classical notions of momentum and angular momentum. An eigenstate for momentum is just a wavefunction with a specific value for momentum; an eigenstate for angular momentum is a wavefunction with a specific value of angular moment.

This concludes the introduction to quantum mechanics. The reader is now in a position to understand the Einstein-Podolsky-Rosen paradox.

The Setup for an Einstein-Podolsky-Rosen Experiment

The Einstein-Podolsky-Rosen experiment is considered to be a gedanken experiment because it is so difficult to execute in practice. In a gedanken experiment a scientist thinks of an experiment that could be performed in practice given sufficient care and resources. Albert Einstein was the master of the gedanken experiment. He used pure thought to imagine how certain experiments might progress, produce interesting consequences and provide insight into theories.

The Einstein-Podolsky-Rosen (EPR) “paradox” arises in a number of different situations. A simple and commonly considered case is the following. Two detectors capable of measuring spin are placed on opposite sites of and some distance away from a source that produces what is known as entangled spin pairs. The pairs may be atoms, nuclei or particles. In what follows, we take take the pair to be a positron (an electron's anti-particle) and an electron because it will be easy to distinguish the two in the discussion below. One possible source for this situation is a motionless neutral spin-zero particle that is unstable and capable of decaying into an electron-positron pair.

Figure 1: The Experimental Setup for an Einstein-Podolsky-Rosen Experiment

When the spinless entity decays, the electron and positron take off in opposite directions with the same speed because linear moment is conserved. If one is lucky, the two particles head toward the two detectors. Otherwise, many particles repeatedly decay in central region until one of them produces an electron-positron pair moving in the “correct” direction. Alternatively, a multitude of detectors can be employed so that the full spherical surface surrounding the central region is covered.

When the source particle decays, physicists often argue that the spin of the electron heading to one detector is the opposite of the spin of the positron heading to the opposite detector. This is due to the conservation of angular momentum. Because the motionless particle is spinless, the total angular momentum before the decay takes place is zero. After the decay, the spins of the electron and positron should combine to give zero angular momentum. This requires them to spin in opposite directions. Choosing the z-axis (up direction) to quantize the spin, there is a 50% chance that the electron heading to the left in Figure 1 spins clockwise (or down) and a 50% chance that the electron spins counterclockwise (or up).

It is very important for the region between the neutral particle and the detectors to be free of interfering effects. If the electron or positron interact with an atom, a magnetic field, or some other force or entity, it can change its spin.

Suppose you, an experimentalist, are located at the detector on the left. Then if you measure the electron's spin to be up, then you instantly know that the spin of the positron heading to the right must be down. If you measure the electron's spin to be down, then the spin of the positron must be up. In principle, this can be verified experimentally: After detecting the spin of the electron at the left detector, you can proceed to the other detector to see whether the spin at the right detector is the opposite. If such an experiment were to be done carefully, then physicists argure that the spins would be found to be oppositely aligned. If D is the distance between the two detectors, then the verification process must last longer than D/c where c is the speed of light because nothing can travel faster than the speed of light. However, the measurement of the spin of the electron at the left instantly provides knowledge of the spin of the positron on the right.

Actually, given the way the experiment has been described above, the reader probably does not find anything paradoxical about the Einstein-Podolsky-Rosen experiment. However, one aspect of quantum mechanics has been left out. If performed in a pristine environment such as a perfect vacuum, then just before the left detector measures the electron, there is still a 50% chance that its spin is up and a 50% chance that its spin is down. It is not the case that the electron heading to the left is emitted with its spin up in 50% of the decays and is emitted with it spin down in 50% of the time. In a single decay, the 50-50 probability situation persists right up until the time of the measurement. Theorists understand quantum mechanics so well that they know this to be true. Hence, just before the measurement, the two spins exist in this uncertain probabilistic situation: simultaneously being 50% up-down and 50% down-up, where up-down means the left-moving electron has its spin up and the right-moving positron has its spin down, and down-up means the opposite. Some physicists say that the measurement of the electron's spin causes the positron to assume a particular spin state. This is like “action at a distant.” Indeed, Einstein called it “spooky action at a distance.” Something done at the left detector instantaneously causes something to happen to the positron on the right. This bothered Einstein because it seemed to violate special relativity where effects cannot happen instantly. Instead, an action at one point can only affect something a distance D away at a time greater than D/c since effects cannot propogate faster than the speed of light. This bothers a lot of scientists and is the origin for the word “paradox” in the Einstein-Podolsky-Rosen paradox.

In brief, there are two aspects of the Einstein-Podolsky-Rosen paradox. The first is the instantaneous transfer of knowledge about a distance object (which in the above discussion is the spin status of the positron). The second aspect is that one act (the process of measuring the spin of the electron on the left) causes something to occur instantaneously to a far away object (the spin of the positron on the right). In both cases, faster-than-the-speed-of-light propagation seems to be occurring.

In the example of the Einstein-Podolsky-Rosen paradox discussed above, it is possible to replace the electron-positron pair by any two entities that possess non-zero spin as long as the spins get “entangled.” In general, entanglement is the property of one entity being correlated with the property of another entity. In the case of spin 1/2 objects, entanglement means that the final spin state is given by (|↑>L|↓>R − |↓>L|↑>R)/√2, where |↑>L|↓>R means that entity moving to the left has spin up while the entity moving to the right has spin down, and |↓>L|↑>R means that the entity moving to the left has spin down while the entity moving to the right has spin up. This spin state is an admixture of these two possibilities with a 50% chance of either one happening. Entanglement of the spins for the electron-positron case might be accomplished using neutral pions, which sometime decay into electron-positron pairs. To actually conduct an Einstein-Podolsky-Rosen experiment using neutral pions, however, is very difficult for a number of reasons that I will not discuss here.

Two Other Gedanken Experiments

Insight into the Einstein-Podolsky-Rosen paradox can be obtained by considering two other similar gedanken experiments.

Gedanken Experiment 1: A Classical Einstein-Podolsky-Rosen Experiment

The setup for this experiment is similar to the above except that instead of having the decay of a spinless particle into an electron and a positron, sister-brother twins at the site of the source flip a coin. If the coin comes up heads, the sister agrees to hold her thump up and while the brother agrees to hold his thumb down. If the coin comes up tails, the twins hold their thumbs in the opposite directions. The the sister proceeds to the left while the brother proceeds to the right. When the observer at the left sees the sister approach him, he instantly knows the direction of the thumb of the twin brother moving to right.

Gedanken Experiment 2: A Unknown Einstein-Podolsky-Rosen Experiment

An experimentalist determines the spin of an electron with a detector. Unbeknownst to the experimentalist, this electron is one that originated from the decay of a neutral pion and had its spin entangled with a positron. The positron necessarily is moving in the opposite direction with its spin oppositely oriented. However, the experimentalist does not know the source of the detected electron and therefore is unaware of positron.

Discussion of the Two Other Gedanken Experiments

Gedanken Experiment 1 shows that transfer of knowledge at a rate faster than the speed of light can occur even in classical mechanics. Gedanken Experiment 2 shows that if the experimental situation has not been arranged in advance, then transfer of knowledge at a rate faster than the speed of light does not occur in quantum mechanics.

The key point is the following: To have instantaneous faster-than-the-speed-of-light knowledge transfer, a “setup agreement” must occur earlier and the events of the experiment (the decay of the spinless particle and subsequent detection of the electron and its properties, or flipping a coin to decide the orientation of thumbs of the twins and the subsequent observation of one twin's thumb) must be casually connected to the setup-agreement event. An event B is casually connected to A if it is possible for A to communicate with B without having the communication travel faster than the speed of light. Scientists then say that B is inside the future light cone of A. Here is a diagram.

Figure 2: Light Cone Diagram for EPR Paradox

Hence, there is no violation of special relativity in the transfer of knowledge in the Einstein-Podolsky-Rosen experiment. Indeed, one can use the idea of a “setup event” to possible prevent faraway intelligent life from invading Earth. See “How to Defend Earth from Distant Aliens”.

As mentioned above, there is a difference between the classical and quantum mechanical versions of the Einstein-Podolsky-Rosen experiment. In the classical case, the decision as to which thumb is up is made when the two twins are together. In the quantum mechanical case, the uncertainty in the direction of the spins of the electron and positron persists up until the spins are measured. More precisely, just before the spin of the left electron is measured, there is still an equal probability of its spin being up or down.

Another difference between the classical and quantum mechanical versions is that in the quantum case the selection of an axis direction to determine spin-up or spin-down is arbitrary. In Figure 1, the spin is measured in the z direction. However, the x or y (or any) direction can be used and the electron spins would still point in opposite directions. For the x-direction, this follows mathematically because (|↑>zL|↓>zR − |↓>zL|↑>zR)/√2 = (|↑>xL|↓>xR − |↓>xL|↑>xR)/√2 where the z and x subscripts indicate the axis being used to measure the spins. (For the mathematically inclined, substitute the results |↑>z = (|↑>x + |↓>x)/√2 and |↓>z = (|↑>x − |↓>x)/√2 into the left-hand side of the equation to get the right-hand side.) In other words, the entanglement of the spins of the electron and positron do not depend on the spin-quantization axis.

Although knowledge of the spin state of the positron is instantaneously gained, there is no way to use the Einstein-Podolsky-Rosen experiment to send a message faster than the speed of light. Proofs of this result exist. In this sense, faster-than-the-speed-of-light communication is not violated.

One Perhaps Disturbing Aspect of the Einstein-Podolsky-Rosen Experiment

The previous section makes it clear that, although faster-than-the-speed-of-light knowledge takes place in the Einstein-Podolsky-Rosen experiment, there is nothing paradoxical about this. However, there is another aspect of the Einstein-Podolsky-Rosen experiment that bothers some scientists.

When the spin of the electron at the left is measured and determined to be up, one knows that the spin of the positron at the right is down. When the spin of the electron at the left is measured to be down, one knows that the spin of the positron is up. According to several interpretations of quantum mechanics, the measuring of the electron at the left causes the spin of the positron to be selected. So measuring something at one location (at the left detector) is causing something to happen at a location (at the right detector) that seems to violate causality. Here, violating causality means causing something to happen at a distant location instantaneously. If one measures the spin of the electron on the left to be up and immediately sends a light signal with a “message” in it to tell the positron on the right that its spin should be down, then by the time that message arrives at the positron, the spin of that positron is already down. So it appears that one effect (the measuring of the spin of the electron at the left) is causing another effect (the quantization direction and value of the spin of the positron at the right) with a “communication speed” that exceeds the speed of light.

A Possible Resolution of the Einstein-Podolsky-Rosen Paradox

The content of this section is based on ideas of the author (Dr. Stuart Samuel) and has not been evaluated by the general physics community. Among things, the author assumes that quantum mechanics as determined by the “Schrödinger” equation (Equation (1)) governs everything including the devices needed to measure spins in the Einstein-Podolsky-Rosen experiment; some physicists believe this while other physicists believe that “measurement” involves something “in addition” to Equation (1). To get a feeling of the diversity of opinions about aspects of quantum mechanics, see the New York Times article “ Quantum Trickery: Testing Einstein's Strangest Theory.”

This section is rather technical requiring knowledge of quantum mechanics at least at the level of an undergraduate physics major to understand the arguments in detail. I shall try to provide simple descriptions of what is going on with the hope that a more general audience will be able to follow. With this in mind, let me just summarize the result. The last step of the Einstein-Podolsky-Rosen paradox involves a measurement causing a transition of the spin part of the wavefunction to an untangled state. While one can image theoretically doing this, I claim that there is no way to do this in a real experiment. In short, the EPR experiment is not a valid gedanken experiment and hence there is no paradox.

There is an assumption among many physicists that measurement in quantum mechanics causes the “collapse” of the wavefunction. An extreme example of this is the following: If a wavefunction for an electron spreads over a wide region and a measurement determines the electron to be in a smaller region then the wavefunction must collapse to a wavefunction that covers only the smaller region. In the situation with spins in the Einstein-Podolsky-Rosen experiment, the analogous measurement effect is to cause the spin state to collapse from the entangled state (|↑>L|↓>R − |↓>L|↑>R)/√2 to an untangled one: either |↑>L|↓>R or |↓>L|↑>R, each with 50% probability. This is the last step in the Einstein-Podolsky-Rosen experiment: If (|↑>L|↓>R − |↓>L|↑>R)/√2 → |↑>L|↓>R. then the measurement of the left spin has determined it to be up and has caused the right spin to be down. On the other hand, if (|↑>L|↓>R − |↓>L|↑>R)/√2 → |↓>L|↑>R, then the measurement of the left spin has determined it to be down and has caused the right spin to be up. In either case, one has “spooky” action at a distance.

In order for a gedanken experiment to be valid, it must in principle be able to happen physically in a real experiment. If the gedanken experiment involves several steps, then each step must be possible in practice. It is my belief that the last step, the measurement of spin causing the collapse of the spin state to one of the above two untangled spin states cannot experimentally happen.

One standard way of measuring the spin of an entity (an atom, a nucleus, a particle, etc.) uses a gradient magnetic field. When a spin 1/2 object passes in the region of a magnetic field, it is deflected in the direction of or in the opposite direction of the magnetic field depending on whether the spin is oriented with or opposite to the magnetic field.1 The direction of the magnetic field establishes the quantization axis for the spin. This method was first used by Otto Stern and Walther Gerlach in an important experiment that established the quantization of spin values.

The Stern-Gerlach measuring method, however, does not cause a collapse of the spin state. Suppose the magnetic field and the spin-quantization axis are along the z axis (the up-down direction). If the spin state of the spin 1/2 entity when it enters the magnetic field region of the Stern-Gerlach experiment is (a|↑> + b|↓>)/N (where N2 = a2 + b2), meaning that the entity has a probability of a2/N2 of having its spin up and a probability of b2/N2 of having its spin down, then the spatial part of its wavefunction gets split into two pieces: one piece Ψ+ moving up and one piece moving Ψ down. In other words, if Ψ(x,y,z,t)(a|↑> + b|↓>)/N is the wavefunction before ending the gradient magnetic field region, then (aΨ+(x,y,z,t)|↑> + bΨ(x,y,z,t)|↓>)/N is the wavefunction exiting the magnetic region. The Stern-Gerlach measuring method does not cause the spin of the entity to collapse to a particular spin state. We know this from quantum mechanics. Quantum mechanics evolves “independent” states such as a Ψ(x,y,z,t)|↑> /N and bΨ(x,y,z,t)|↓>/N independently. If the spin is purely up, then it will move up and the Stern-Gerlach method allows us to observe that the electron's spin is up; Similarly, if the electron spin is purely down. In short, Stern-Gerlach allows up to observe spin but not to measure it in the sense of causing a wavefunction collapse.

The question then arises: If the Stern-Gerlach method does not measure spin 1/2 but merely observes it, then is there another Spin 1/2 Measurement that produces the last step of the Einstein-Podolsky-Rosen paradox, namely, causing (a|↑> + b|↓>)/N to transition to either |↑> with a probability of a2/N2 or to |↓> with a probability b2/N2?

I now argue that such a Spin ½ Measurement would violate two fundamental principles: conservation of angular moment and what-is-know-as unitarity in quantum mechanics.

Conservation of angular moment is a fundamental property of nature that has never been observed to be violated. In classical mechanics all three components (corresponding to rotations about the x, y or z axes) of angular momentum are measurable, but in quantum mechanics only total angular momentum and a component about one axis can be measured. This is due to Heisenberg's uncertainty principle. Most people think of Heisenberg's uncertainty principle as saying that momentum and position cannot be simultaneously measured. However, the uncertainty principle affects a number of other observables including any two components of angular momentum.

Angular momentum has two contributions: orbital angular momentum due the movements of objects around a selected point and spin, which is the fundamental rotation of elementary particles. Total angular momentum is the sum of these components.

I now show that if a Spin 1/2 Measurement is possible, then conservation of angular momentum can violated in a gedanken experiment: As a specific example, assume that the initial situation before the creation of the electron-positron pair is an eigenstate of total angular momentum J of 0 and z-component Jz of 0 as measured about the central location of the neutral pion. We assume that after the creation of the electron and positron that the spins are entangled. If the entangled spins cannot be created then the Einstein-Podolsky-Rosen paradox cannot arise in the first place. Spin entanglement means that just after the electron-positron pair is created, the wavefunction for the system is Ψ× (|↑>|↓>+ − |↓>|↑>+)/√2 where Ψ is the wavefunction component for the electron, the positron and everything else except the spins of the electron and positron. The angular momentum of Ψ must be 0 and its z-component must be 0 since (|↑>|↓>+ − |↓>|↑>+) has spin 0 and does not contribute to total angular momentum. If a Spin 1/2 Measurement exists, its use causes a transition to either Ψ'|↑>|↓>+ or Ψ''|↓>|↑>+. The problem with either of these final states is that the spin part is a combination of spin 0 and spin 1 and it is not possible to construct a state with the original values of J and Jz. This follows from addition rules for combining angular momentum in quantum mechanics. To create a state with the original J and Jz for the spin 1 part, either |↑>|↑>+ or |↓>|↓>+ or both must be present. For example, in the first case above, Ψ'|↑>|↓>+ = Ψ'(|↑>|↓>+ − |↓>|↑>+)/2 + Ψ'(|↑>|↓>+ + |↓>|↑>+)/2. For the first term to have J and Jz of 0, Ψ' must be an eigenstate of angular momentum with values of J of 0 and Jz of 0 because (|↑>|↓>+ − |↓>|↑>+)/2 is spin 0. Because (|↑>|↓>+ + |↓>|↑>+)/2 is the Jz=0 component of a spin 1 objects, the quantum mechanical rules for constructing an eigenstate of total angular momentum J of 0 require Ψ' to be spin 1 (already a contradiction) and the presence of terms of the form Ψ'|↑>|↑>+ or Ψ'+|↓>|↓>+ where Ψ' and Ψ'+ have total angular momentum 1 and z-components of -1 and +1 respectively. In short, a Spin 1/2 Measurement must involve spin flipping for an initial state of angular momentum 0. Spin flipping means |↑> becomes |↓> or |↓> becomes |↑>. Hence, a transition from (|↑>|↓>+ − |↓>|↑>+) to either only Ψ'|↑>|↓>+ or only Ψ''|↓>|↑>+ is impossible for an initial state with angular momentum 0. A Spin 1/2 Measurement will also violate conservation of angular moment if the initial state is an eigenstate with total angular moment J and z-component Jz.

The existence of a Spin 1/2 Measurement also violates unitarity in quantum mechanics. Equation (1) allows one to determine Ψ(t + Δt) in terms of Ψ(t). The equation can then be used again to determine Ψ(t + 2Δt) in terms of Ψ(t + Δt), and repeated use of the equation determines the wavefunction for all future times. The solution can be expressed as

      Ψ(tf) = U(tf,t0)Ψ(t0)

The value of the wavefunction at a latter time tf is determined from the wavefunction at an earlier time t0 using the operator U(tf,t0). It is known that U is a unitary operator. Unitary operators have the property of preserving the normalization (roughly corresponds to the size) of states. If the Spin 1/2 Measurement caused a collapse of (a|↑> + b|↓>)/N at time t0 just before the measurement to |↑> at time tf just after the measurement then the implication is that a|↑>/N → |↑> and |↓> → 0. However, this transformation would not perserve the normalization of either of these two states. It is well know that a collapse of a wavefunction violates unitarity. Similarity if the Spin 1/2 Measurement caused (a|↑> + b|↓>)/N to transition to |↓> then preservation of normalization of states under U(tf,t0) would be violated. The assumption that everything is governed by quantum mechanics including measurement then precludes the possibility of a Spin ½ Measurement of the type described above.

In brief, I believe the Einstein-Podolsky-Rosen paradox is not a paradox because it is not a valid gedanken experiment. It requires a measurement process that cannot be achieved in real world.

The basic argument of this section is that wavefunction collapse during a measurement does not happen. If (a|↑> + b|↓>)/N cannot collapse to ↑> or to ↓> then (|↑>|↓>+ − |↓>|↑>+)/√2 cannot collapse to |↑>|↓>+ or to |↓>|↑>+ under a Spin 1/2 Measurement and therefore the observation of the spin of the electron at the left does not cause the spin of the positron on the right to achieve a particular value. While the issue of measurement in quantum mechanics is not well understood, I believe that experiments of the “Einstein-Podolsky-Rosen paradox” simply observe the quantum correlations of entangled spins.2

Difficulties in Setting Up the Entangled Spin State for the e+ e- Case

A requirement of the entangled state is that the the electron and positron head in opposite directions to regions where their spins can be observed. Now in classical mechanics there are lots of states with zero orbital angular momentum as measured about the location of the decaying spin-zero particle. Indeed, any straight trajectory starting from this point has zero orbital angular momentum. In quantum mechanics, however, there is essential only one wavefunction with zero orbital angular momentum: the completely spherically symmetric one. Hence, if both the electron and the positron in the final state separately have zero orbital angular momentum then each of their wavefunctions must consist of an outward moving spherical shell. In such a case, the electron's and positron's spatial wavefunctions would be equally distributed in all directions and be overlapping everywhere. However, for the EPR paradox to work, the two particles must be heading in particular (fairly narrow) angular cones. Such a state is not inconsistent with the decay of a spin-zero particle but it means that all quantized values of angular momentum are involved. In addition, the angular momentum values of the electron must match and be opposite to those of the positron in order to create an overall zero orbital angular momentum state. The same is true for linear momentum: The momentum of the electron and positron must be equal in magnitude and opposite in direction. Thus, for the EPR experimental setup to be realized, not only are the spins entangled but the orbital angular momentum and linear momentum states must also be.

If all values of orbital angular momentum are present, it would seem to be a miracle that only one spin state is present. Two 1/2 spins can combine to give spin zero or spin one. If the spin is one, then the orbital angular momenta of the electron and positron could combine to give angular momentum one. Then the spin and orbital angular momentum can combine to give total angular momentum of zero, which is needed because the initial state has total angular momentum zero. Once there are spin one components in the final state there is no reason why the final spin state should be of the form (|↑>|↓>+ − |↓>|↑>+)/√2. Indeed, as discussed above, if a (|↑>|↓>+ + |↓>|↑>+)/√2 component is present, |↑>|↑>+ and |↓>|↓>+ would necessarily have to be present also. Although there is no violation of a fundamental principle, it would seem extremely difficult to generate the pure spin-zero state in practice.

By the way, the state (|↑>|↓>+ − |↓>|↑>+)/√2. has the property that the first term |↑>|↓>+ is orthogonal to the second term |↓>|↑>+, and by unitarity of the evolution operator U(tf,t0) this will be true throughout the future evolution of the Universe. Although there are unitary transformation that can eliminate the entanglement, an example being |↑>|↓>+ → (|↑>|↓>+ + |↓>|↑>+)/√2, |↓>|↑>+ → (− |↑>|↓>+ + |↓>|↑>+)/√2, which leads to (|↑>|↓>+ − |↓>|↑>+)/√2. → |↑>|↓>+ the position and electron would need to return to the same region for these transformations to physically occur. Hence, the spin entanglement is likely to persist forever.

The EPR experimental would also have to unfold in a perfectly pristine environment. Any magnetic fields might cause spatial separations of the two components Ψ1 and Ψ2 or the flipping of a spin. There are only isolated points in the universe where no magnetic field is present. Hence, there is no place where the perfectly pristine environment exists.

Suppose the electron moving to the left passes by a spin 1/2 object with its spin up. Then the component with the spin down can undergo spin exchange with it : (|↑>|↓>+ − |↓>|↑>+)|↑>e/√2 → |↑>(|↑>e|↓>+ − |↓>e|↑>+)/√2. Here the subscript “e” stands for “environment” because the spin 1/2 object is located in a region surrounding the left-moving electron. After this spin exchange, the spin of the left-moving electron is no longer entangled with the positron moving to the right. Instead, the right-moving positron's spin is entangled with the environmental spin 1/2 object. Such spin-exchange effects can ruin the entanglement of the spins of the electron and positron.

Indeed, if the electron or positron interacts with anything in the environment, the EPR experiment would be in jeopardy. In short, the EPR paradox in the case of charged massive objects of spin-1/2 would seem to exist only as a gedanken experiment in terms of setting up entangled spins at two distant locations.

1For objects with charge such as electrons, there is also a a curved motion in a plane perpendicular to the direction of the magnetic field with the “curving direction” depending on the charge of the object.
2In the case of the Stern–Gerlach experiment, some physicists might say that until the electron hits the screen that its spin is not measured. This involves the measurement problem, which which is unsolved. Nevertheless, I believe I can guess what takes place for the Stern–Gerlach case. If the wavefunction for the electron is of the form Ψ(a|↑> + b|↓>)/N when entering the gradient magnetic field (with spin-quantization and the magnetic field both in the positive z direction) and a pristine environment exists all the way to the screen, then by the time the electron arrives just in front of the screen the wavefunction will have the form (aΨ+|↑> + bΨ|↓>)/N where Ψ+ is mostly non-zero above the z-plane and Ψ is mostly non-zero below the z-plane. When the electron reaches the screen, the environment is no longer pristine, the electron feels the forces produced by molecules in the screen, and the (whole) wavefunction reacts to them evolving according to Equation (1). In addition, the electron interacts with the molecules, which in turn, produce an abundance of photons. There is no reason why the spin of the electron should be purely |↑> or |↓> after the electron hits the screen. Hence, there is no point at which the spin collapses to |↑> or |↓>. What an experimentalist observes are electromagnetic waves coming from a rather focused location on the screen and not the spin of the electron. Now because the forces associated with the screen tend to be localized, the probability that the electron interacts with the molecules at (or near) a particular location x on the screen to produce a plethora of photons is proportional to the probability that the electron is located there and hence proportional to |Ψ(x)|2 = |aΨ+(x)|2/N2 + |bΨ(x)|2/N2. So the chances of producing a flash above the z-plane is approximately equal to |a|2/N2 and the chances of producing a flash below the z-plane is approximately equal to |b|2/N2, but not exactly so. Indeed, because of the nature of solutions of Equation (1), the non-zero region of Ψ+ extends below the z-plane and the non-zero region of Ψ extends above the z-plane, so that there is contribution, albeit quite small, of have a flash above the z-plane due to the bΨ|↓> component of the wavefunction. Also, the spin of the electron actually varies continuous from above the z-plane to below the z-plane. This follows because (a|↑> + b|↓>)/N for any a and b can be written as |↑>n for some spin-axis quantization direction n. Defining n(x) by |↑>n(x) = (aΨ+(x)|↑> + bΨ(x)|↓>)/N, one realizes that n(x) rotates from approximately +e3 to approximately -e3 continuously as one moves from values of x well above the z-plane to values of x well below the z-plane (Here, +e3 is a unit vector in the plus z-direction). This means that flashes are also occurring on the screen for electrons whose spin does not point in either the plus or minus z-direction!
Two questions arise: (i) Why does the flash occur at a particular localized region on the screen for a single electron and (ii) Why are there not several flashes at several places on the screen? Both these questions are related to the measurement problem and how a microscopic quantum state interacts with its environment to create to a macroscopic effect. This is currently a poorly understood problem. My guess is that the screen creates a “sensitivity to conditions.” So, each time the experiment is run, the conditions are not precisely the same and this causes the electron to interact with one region of the screen rather than another. It is like placing a pencil vertically on a desk: a tiny difference causes the pencil to topple over in one particular direction. As to question (ii), when the electron interacts with molecules in the screen either it loses enough energy so as not to be able to interact with molecules elsewhere in the screen to produce a flash or the wavefunction evolves (via Equation (1)) sufficiently so that it is not possible to produce a second flash. The second possibility allows the wavefunction to quickly become localized in the region of the flash (a weak version of wavefunction collapse in which Ψ+(x) and Ψ(x) must evolve to a "common region", and such a thing must happen when an electron with an extended wavefunction is captured in a bound state of an atom or molecule). Given the above, it is conceivable to me that there could be a very tiny probability for a single electron to produce two “weak” flashes in two different screen locations.
The observation of photons emanating from a particular spot on the screen neither determines that the wavefunction is concentrated in a particular localized region nor that the electron's spin is a particular value if the spin is of the form (a|↑> + b|↓>)/N with a ≠ 0 and b ≠ 0. Multiple observations for the same initial experimental setup allows the experimentalist to determine the value of the absolute square of the wavefunction as a function of screen position and the values of |a| and |b| in the spin part of the total wavefunction. This is why I say that Stern–Gerlach experiments allow one to observe spin if the spin is up or down in the z-direction but not to measure spin, where by “measure” I mean in the sense of forcing a spin to assume a particular value.

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