The Greatest Misconception in Science: Speeds Greater Than the Speed of Light in an Expanding Universe| Home | Books | Reports | Reviews | Contact Us | Virtual Astronomer | Press Releases |

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

The miniscule value of ħ of 1.055 x 10

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|

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 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 (|↑>

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 (|↑>

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.

When the spin of the electron at the left is

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 (|↑>

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.

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 N

The question then arises: If the Stern-Gerlach method does not

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 J

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(t

The value of the wavefunction at a latter time t

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 (|↑>

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 (|↑>

By the way, the state (|↑>

The EPR experimental would also have to unfold in a perfectly pristine environment. Any magnetic fields might cause spatial separations of the two components Ψ

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 : (|↑>

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.

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

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.

This report was prepared by Jupiter Scientific, an organization devoted to the promotion of science through books, the internet and other means of communication.

This web page may NOT be copied onto other web sites, but other sites may link to this page.

Copyright ©2016 by Jupiter Scientific

To Jupiter Scientific's Information Page