In my last posting Digital Physics and the Software Universe, we covered the idea that our Universe might seem to behave like a large network of quantum computers calculating how to behave, if not in an actual literal sense as the realists would have us believe, at least, perhaps, in a positivistic manner that makes for a good model that yields useful predictions about how our Universe seems to behave. Recall that positivism is an enhanced form of empiricism, in which we do not care about how things “really” are; we are only interested in how things are observed to behave. With positivism, physicists only seek out models of reality - not reality itself. In keeping with that observation, I would like to recommend two offerings on Netflix. Now if you do not currently have Netflix, you can easily stream it for free for 30 days on a trial basis, so I do not feel conflicted about making this recommendation. The first recommendation comes from the Black Mirror series on Netflix that features stories from the dark side of IT. This particular Black Mirror feature is an interactive movie entitled Bandersnatch. The second recommendation is the Netflix series entitled Russian Doll. Both Bandersnatch and the Russian Doll are based on the idea that our Universe may be composed of software running on some kind of cosmic quantum computer, and both feature a computer game developer operating under a tight deadline for a new computer game. While Bandersnatch takes place in 1984 and the Russian Doll series takes place in the present day, both convey the time-pressures that developers experience while working on new code under a tight deadline, but with a twist. In both features, the computer game developers slowly come to the realization that they are caught in some kind of "bad" cosmic code themselves that keeps looping in time. In fact, the line "Life is Like a Box of Timelines" comes from the Russian Doll. Both developers then try to debug and fix the bad cosmic code that they are stuck in. As with all code, debugging and fixing bad cosmic code can be very frustrating and time-consuming. It takes many runs of the bad code with trial code fixes to unit test the cosmic code until it finally performs as intended - see How Software Evolves for more on that.
Fortunately, for the last few decades, we have had IDEs (Integrated Development Environments) like Eclipse that allow developers to step through bad code and watch the variables change with time and also to follow the code logic to see how the code branches at various conditional branch-points. In an IDE, you can also set breakpoints at various points in the code that allow the code to run until the breakpoint is reached. When a breakpoint is reached, the code then stops executing and the developer can then examine the state of all the current code variables. At a breakpoint, the developer can even change the value of a variable on the fly in order to run the code down a different logical path at a conditional branch-point than the code would normally run. Bandersnatch allows the viewer to do this very same thing. At certain points in Bandersnatch, a breakpoint is reached and the viewer then gets to choose which path the cosmic code takes. This allows the viewer to interactively run through the Bandersnatch cosmic code nearly an infinite number of different ways. However, sometimes the viewer will run down a dead-end code path that causes the cosmic code to complete and the credits then begin to roll. At that point, you have to rerun the Bandersnatch cosmic code from the very beginning again. Here is a tip. The Bandersnatch cosmic code also seems to run in the background as a daemon process when you stop debugging the code and come back several hours later. So do not be surprised when you come back and start up Bandersnatch again and find yourself in a different section of the cosmic code than you previously left. All developers should be quite familiar with all of these debugging activities and, therefore, should find debugging cosmic code to be fairly straightforward. Now, in Digital Physics and the Software Universe, we saw that some very bright people hold the position that the best model for explaining the very strange way that our Universe behaves at the microscopic level of atoms and particles is to think of the Universe as some kind of cosmic code running in a quantum-mechanical IDE. But for that, you need some understanding of quantum mechanics. For a brief introduction to quantum mechanics see Quantum Software.
The reason why you will find quantum mechanics useful is that both Bandersnatch and the Russian Doll are based on the idea that our Universe may be composed of software running on some kind of network of cosmic quantum computers, and therefore, also bring in some of the more mind-bending ideas to be found in some of the more esoteric interpretations of quantum mechanics. It should be noted that the more esoteric interpretations of quantum mechanics are also now more important because the classic Copenhagen Interpretation of quantum mechanics no longer seems to carry the weight that it once did many long decades ago. For example, in Quantum Computing and the Many-Worlds Interpretation of Quantum Mechanics we covered Hugh Everett's Many-Worlds Interpretation of quantum mechanics and in Is the Universe a Quantum Computer? we covered John Cramer's Transactional Interpretation of quantum mechanics. Reading both of those postings in advance would help to clarify some of the strange scenes found in both of my Netflix recommendations. For example, in order to explain the strange quantum-mechanical effects that we observe in the lab, the Many-Worlds Interpretation of quantum mechanics relies on timelines in multiple parallel Universes, while the Transactional Interpretation of quantum mechanics relies on a Universe that features multiple timelines moving both forwards and backwards in time at the same time. And both Bandersnatch and the Russian Doll have scenes that display such phenomena. Now to make it a bit easier to understand such phenomena, let us briefly review the Copenhagen Interpretation, the Many-Worlds Interpretation, the Transactional Interpretation and the Decoherence Interpretation of quantum mechanics. But in order to do that we need to know a bit about quantum mechanics and how physicists use the wave model to explain certain phenomena.
The Wave Model
The chief characteristic of the wave model is that waves tend to be everywhere, but nowhere in particular, at the same time and simultaneously explore all possible paths. To see a wave in action, drop a small pebble into a still pond of water containing many obstacles and watch the resulting waves spread out and reflect off the obstacles and interfere with each other before eventually reaching a particular destination near the edge of the pond.
In 1801, Thomas Young conducted a series of experiments with waves. First, using water waves in a shallow ripple tank, he demonstrated the concept of interference. When a water wave encounters a barrier with two slits, the ripples passing through the slits interfere with each other on the other side of the barrier (Figure 1). Where two crests intersect, the wave amplitude doubles in height, and where a crest meets a trough, the two waves cancel each other out entirely. Next, Young used a distant light source with two closely spaced slits in an opaque barrier. On the other side of the barrier, he placed a white projection screen. When light from the distant light source passed through the double-slit barrier, Young observed an interference pattern of alternating bright and dark fringes projected onto the screen which demonstrated the wavelike behavior of light.
Figure 1 – The interference pattern from two slits (click to enlarge)
You can easily repeat Young’s experiment with a piece of thin cloth. At night, hold up a single ply of a pillowcase in front of a distant light source, such as a far-off street light or the filament in your neighbor’s decorative front door light that uses a clear light bulb. Instead of a single diffuse spot of light shining through the pillowcase, you will see a pronounced checkerboard interference pattern of spots, because the weave of your pillowcase has both vertical and horizontal slits between the threads.
The Birth of Modern Quantum Mechanics
As we saw in Quantum Software, Erwin Schrödinger first developed the Schrödinger equation in the winter of 1926 to explain the strange behavior of electrons in atoms and the fact that the electrons only radiated light at certain frequencies when excited. The 1-dimensional version of this famous equation is:
-ħ² ∂²Ψ = iħ ∂Ψ
── ── ──
2m ∂x² ∂t
In the above 1-dimensional Schrödinger equation, Ψ is called the wavefunction of a particle and is pronounced like the word “sigh”. In quantum mechanics, the wavefunction Ψ contains all of the information that can ever be known about the particle.
Now if the particle is just quietly sitting around on its own and not interacting with other particles, like an electron that has been sitting quietly in an atom for a billion years, it means the wavefunction Ψ should not be changing with time, and we can use the 1-dimensional time-independent version of the Schrödinger equation that does not have the time variable "t" in the equation:
-ħ² d²ψ(x) + V(x) ψ(x) = E ψ(x)
── ──────
2m dx²
The lower-case wavefunction ψ is still pronounced like the word "sigh", but we use the lower-case ψ to signify that this is a time-independent wavefunction that does not change with time. When the 3-dimensional Schrödinger equation is solved for the hydrogen atom consisting of just one electron trapped by one proton in an electromagnetic well we get a number of quantized wavefunctions as solutions:
Figure 2 – The n=1 and n=2 orbitals or wavefunctions for the hydrogen atom.
The Strange Motion of Quantum Particles in Space and Time
Now for quantum particles like electrons or photons that are on the move we need to use Richard Feynman’s "sum over histories" approach to quantum mechanics. In Feynman's "sum over histories" approach to quantum mechanics, the wavefunction amplitude of an electron or photon is the same in all directions, like when you drop a pebble in a still pond, but the phase angles of the wavefunction will differ depending upon the path that is taken. So to figure out the probability of finding an electron or photon at a particular point, you have to add up the amplitudes and phases of all the possible paths that the electron or photon could have taken to reach the destination point. Although there are an infinite number of possible paths, the key insight is that most of the paths will be out of phase with each other and will cancel out like the destructive interference shown in Figure 1. This produces some rather strange experimental observations. Imagine a very dim source of photons or electrons that can fire one photon or electron at a time. If we fired the particles at a screen with two slits, as in Young’s experiment, we would expect to see a pattern similar to Figure 3 build up over time, based upon the particle model for electrons and photons.
Figure 3 – What common sense and the particle model would predict for a source that fires electrons or photons one at a time
However, what is actually observed is an interference pattern similar to Figure 4, even though the electrons or photons pass through the slits one at a time. According to quantum mechanics, the individual electrons or photons interfere with themselves as they go through both slits at the same time! This means that if your neighbor could turn down the light by his front door to a very low level, so that it only emitted one photon at a time, and your eye could record a long exposure image, you would still see a checkerboard pattern of light spots through your pillowcase, even though the photons went through the fabric mesh one at a time.
Figure 4 – We actually observe an interference pattern as each particle interferes with itself
Now here comes the really strange part. If we put detectors just in front of the slits so that we can record which slit the electron or photon actually passed through, and keep firing one particle at a time, the interference pattern will disappear, and we will see the pattern in Figure 3 instead. If we turn the detectors off, the interference pattern returns, and we see the pattern in Figure 4. For some reason, Nature will not allow us to observe electrons or photons behaving like particles and waves at the same time. It’s some kind of information thing again. But it gets worse. If we put the detectors at some distance behind the slits and turn them on, the interference pattern again disappears, but if we turn the detectors off, the interference pattern returns. Now, this is after the electrons or photons have already passed through the slits! How do they know whether to behave like a wave or a particle in advance, before they know if the detectors are on or off? In fact, experiments have been performed where the decision to turn the detectors on or off is not made until after the individual electrons or photons have already passed through the slits, but even so, if the detectors are turned on, the interference pattern disappears, and if the detectors are turned off, the interference pattern returns! This means that the present can change the past! This is the famous delayed-choice experiment proposed by John Wheeler in 1978 and actually performed by Alain Aspect and his colleagues in 1982. In another experiment, the detectors are placed beyond the observation screen to detect cloned photons that are created in a splitting process. By observing the cloned photons, it is possible to determine which slit an individual twin photon passed through after its twin has already hit the observation screen. When these distant detectors are turned on, the interference pattern once again disappears, and if the detectors are turned off, the interference pattern returns. Again, the decision to turn the detectors on or off can be made after the photons have already hit the observation screen. This means that the future can change the present!
In 1928, Paul Dirac combined quantum mechanics (1926) with the special theory of relativity (1905) and came up with a relativistic reformulation of the Schrödinger equation. Now, strangely, the solutions to Dirac’s equation predicted both the existence of electrons with a negative charge and positive mass energy and also positrons, the antimatter equivalent of electrons, with a positive charge and a negative mass energy. But in 1947 Richard Feynman came up with an alternate interpretation for Dirac’s positrons with negative mass energy. Feynman proposed that positrons were actually normal electrons moving backwards in time! Recall that the full-blown wave function of an object with constant energy can be expressed as a time-independent wavefunction ψ(x) multiplied by a time-varying term:
Ψ(x, t) = e-iEt/ħ ψ(x)
Now the solutions to Dirac’s equation predicted both the existence of electrons with positive mass energy and also positrons, the antimatter equivalent of electrons, with negative mass energy. For a particle with negative mass energy, the above equation looks like:
Ψ(x, t) = e-i(-E)t/ħ ψ(x)
but since:
-i(-E)t/ħ = -iE(-t)/ħ
Feynman realized that an equivalent equation could be written by simply changing the parenthesis yielding:
Ψ(x, t) = e-iE(-t)/ħ ψ(x)
So a positron with negative mass energy –E could mathematically be thought of as a regular old electron with positive mass energy E moving backwards in time! Indeed, today that is the preferred interpretation. All antimatter is simply regular matter moving backwards in time.
Figure 5 – Above is a Feynman diagram showing an electron colliding with a positron, the antimatter version of an electron.
In Figure 5 we see an electron colliding with a positron, the antimatter version of an electron. When the two particles meet they annihilate each other and turn into two γ gamma rays. In the Feynman diagram, space runs along the horizontal axis and time runs along the vertical axis. In the diagram, we see an electron e- with negative charge and positive mass energy on the left and a positron e+ with positive charge and negative mass energy on the right. As time progresses up the vertical time axis, we see the electron e- and the positron e+ approach each other along the horizontal space axis. When the two particles get very close, they annihilate each other and we see two γ gamma rays departing the collision as time proceeds along the vertical time axis. But notice the red arrowheads on the red arrow lines in the Feynman diagram. The red arrowhead for the negative electron e- is moving upwards on the diagram and forward in time, while the positive positron e+ is moving downwards on the diagram and backwards in time! So in the diagram, the positive positron e+ is portrayed as an ordinary negative electron e- moving backwards in time!
The Copenhagen Interpretation of Quantum Mechanics
In 1927, Niels Bohr and Werner Heisenberg proposed a very positivistic interpretation of quantum mechanics now known as the Copenhagen Interpretation. You see, Bohr was working at the University of Copenhagen Institute of Theoretical Physics at the time. The Copenhagen Interpretation contends that absolute reality does not really exist. Instead, there are an infinite number of potential realities, defined by the wavefunction ψ of a quantum system, and when we make a measurement of a quantum system, the wavefunction of the quantum system collapses into a single value that we observe, and thus brings the quantum system into reality (see Quantum Software for more on wavefunctions). This satisfied Max Born’s contention that wavefunctions are just probability waves. The Copenhagen Interpretation suffers from several philosophical problems though. For example, Eugene Wigner pointed out that the devices we use to measure quantum events are also made out of atoms which are quantum objects in themselves, so when an observation is made of a single atom of uranium to see if it has gone through a radioactive decay using a Geiger counter, the atomic quantum particles of the Geiger counter become entangled in a quantum superposition of states with the uranium atom. If the uranium has decayed, then the uranium atom and the Geiger counter are in one quantum state, and if the atom has not decayed, then the uranium atom and the Geiger counter are in a different quantum state. If the Geiger counter is fed into an amplifier, then we have to add in the amplifier too into our quantum superposition of states. If a physicist is patiently listening to the Geiger counter, we have to add him into the chain as well, so that he can write and publish a paper which is read by other physicists and is picked up by Time magazine for a popular presentation to the public. So when does the “measurement” actually take place? We seem to have an infinite regress. Wigner’s contention is that the measurement takes place when a conscious being first becomes aware of the observation. Einstein had a hard time with the Copenhagen Interpretation of quantum mechanics for this very reason because he thought that it verged upon solipsism. Solipsism is a philosophical idea from Ancient Greece. In solipsism, your Mind is the whole thing, and the physical Universe is just a figment of your imagination. So I would like to thank you very much for thinking of me and bringing me into existence! Einstein’s opinion of the Copenhagen Interpretation of quantum mechanics can best be summed up by his statement "Is it enough that a mouse observes that the Moon exists?". Einstein objected to the requirement for a conscious being to bring the Universe into existence because, in Einstein’s view, measurements simply revealed to us the condition of an already existing reality that does not need us around to make measurements in order to exist. But in the Copenhagen Interpretation, the absolute reality of Einstein does not really exist. Additionally, in the Copenhagen Interpretation, objects do not really exist until a measurement is taken, which collapses their associated wavefunctions, but the mathematics of quantum mechanics does not shed any light on how a measurement could collapse a wavefunction.
The collapse of the wavefunction is also a one-way street. According to the mathematics of quantum mechanics a wavefunction changes with time in a deterministic manner, so like all of the other current effective theories of physics, they are reversible in time and can be run backwards. This is also true in the Copenhagen Interpretation, so long as you do not observe the wavefunction and collapse it by the process of observing it. In the Copenhagen Interpretation, once you observe a wavefunction and collapse it, you cannot undo the collapse, so the process of observation becomes nonreversible in time. That means if you fire photons at a target, but do not observe them, it is possible to reverse them all in time and return the Universe back to its original state. That is how all of the other effective theories of physics currently operate. But in the Copenhagen Interpretation, if you do observe the outgoing photons you can never return the Universe back to its original state. This can best be summed up by the old quantum mechanical adage - look particle, don’t look wave. A good way to image this in your mind is to think of a circular tub of water. If you drop a pebble into the exact center of a circular tub of water, a series of circular waves will propagate out from the center. Think of those waves as the wavefunction of an electron changing with time into the future according to the Schrödinger equation. When the circular waves hit the circular walls of the tub they will be reflected back to the center of the tub. Essentially, they can be viewed as moving backwards in time. This can happen in the Copenhagen Interpretation so long as the electron is never observed as its wavefunction moves forward or backward in time. However, if the wavefunction is observed and collapsed, it can never move backwards in time, so observation becomes a one-way street.
The Many-Worlds Interpretation of Quantum Mechanics
In 1956, Hugh Everett working on his Ph.D. under John Wheeler, proposed the Many-Worlds Interpretation of quantum mechanics as an alternative. The Many-Worlds Interpretation admits to an absolute reality but claims that there are an infinite number of absolute realities spread across an infinite number of parallel universes. In the Many-Worlds Interpretation, when electrons or photons encounter a two-slit experiment, they go through one slit or the other, and when they hit the projection screen they interfere with electrons or photons from other universes that went through the other slit! In Everett’s original version of the Many-Worlds Interpretation, the entire Universe splits into two distinct universes whenever a particle is faced with a choice of quantum states, and so all of these universes are constantly branching into an ever-growing number of additional universes. In the Many-Worlds Interpretation of quantum mechanics, the wavefunctions or probability clouds of electrons surrounding an atomic nucleus are the result of overlaying the images of many “real” electrons in many parallel universes. Thus, according to the Many-Worlds Interpretation wavefunctions never collapse. They just deterministically evolve in an abstract mathematical Hilbert space and are reversible in time, like everything else in physics.
Because Einstein detested the Copenhagen interpretation of quantum mechanics so much, he published a paper in 1935 with Boris Podolsky and Nathan Rosen which outlined what is now known as the EPR Paradox. But to understand the EPR Paradox we need a little background in experimental physics. Electrons have a quantum mechanical property called spin. You can think of an electron’s spin like the electron has a little built-in magnet. In fact, it is the spin of the little electron magnets that add up to make the real magnets that you put on your refrigerator. Now in quantum mechanics, the spin of a single electron can be both up and down at the same time because the single electron can be in a mixture of quantum states! But in the classical Universe that we are used to, macroscopic things like a child's top can only have a spin of up or down at any given time. The top can only spin in a clockwise or counterclockwise manner at one time - it cannot do both at the same time. Similarly, in quantum mechanics, a photon or electron can go through both slits of a double slit experiment at the same time, so long as you do not put detectors at the slit locations.
Figure 6 – A macroscopic top can only spin clockwise or counterclockwise at one time.
Figure 7 – But electrons can be in a mixed quantum mechanical state in which they both spin up and spin down at the same time.
Figure 8 – Similarly, tennis balls can only go through one slit in a fence at a time. They cannot go through both slits of a fence at the same time.
Figure 9 – But at the smallest of scales in our quantum mechanical Universe, electrons and photons can go through both slits at the same time, producing an interference pattern.
Figure 10 – You can see this interference pattern of photons if you look at a distant porch light through the mesh of a sheer window curtain or a pillowcase.
When you throw an electron through a distorted magnetic field that is pointing up the electron will pop out in one of two states. It will either be aligned with the magnetic field (called spin-up) or it will be pointing 180o in the opposite direction of the magnetic field (called spin-down). Both the spin-up and spin-down conditions are called an eigenstate. Prior to the observation of the electron’s spin, the electron is in a superposition of states and is not in an eigenstate. Now if the electron in the eigenstate of spin-up is sent through the same magnetic field again, it will be found to pop out in the eigenstate of spin-up again. Similarly, a spin-down electron that is sent through the magnetic field again will also pop out as a spin-down electron. Now here is the strange part. If you rotate the magnetic field by 90o and send spin-up electrons through it, 50% of the electrons will pop out with a spin pointing to the left, and 50% will pop out with a spin pointing to the right. And you cannot predict in advance which way a particular spin-up electron will pop out. It might spin to the left, or it might spin to the right. The same goes for the spin-down electrons – 50% will pop out spinning to the left and 50% will pop out spinning to the right.
Figure 11 - In the Stern-Gerlach experiment we shoot electrons through a distorted magnetic field. Classically, we would expect the electrons to be spinning in random directions and the magnetic field should deflect them in random directions, creating a smeared out spot on the screen. Instead, we see that the act of measuring the spins of the electrons puts them into eigenstates with eigenvalues of spin-up or spin-down and the electrons are either deflected up or down. If we rotate the magnets by 90o, we find that the electrons are deflected to the right or to the left.
The EPR Paradox goes like this. Suppose we prepare many pairs of quantum mechanically “entangled” electrons that conserve angular momentum. Each pair consists of one spin-up electron and one spin-down electron, but we do not know which is which at the onset. Now let the pairs of electrons fly apart and let two observers measure their spins. If observer A measures an electron there will be a 50% probability that he will find a spin-up electron and a 50% chance that he will find a spin-down electron, and the same goes for observer B, 50% of observer’s B electrons will be found to have a spin-up, while 50% will be found with a spin-down. Now the paradox of the EPR paradox, from the perspective of the Copenhagen Interpretation, is that when observer A and observer B come together to compare notes, they find that each time observer A found a spin-up electron, observer B found a spin-down electron, even though the electrons did not know which way they were spinning before the measurements were performed. Somehow when observer A measured the spin of an electron, it instantaneously changed the spin of the electron that observer B measured. Einstein hated this “spooky action at a distance” feature of the Copenhagen Interpretation that made physics nonlocal, meaning that things that were separated by great distances could still instantaneously change each other. He thought that it violated the speed of light speed limit of his special theory of relativity that did not allow information to travel faster than the speed of light. Einstein thought that the EPR paradox was the final nail in the coffin of quantum mechanics. There had to be some “hidden variables” that allowed electrons to know if they “really” were a spin-up or spin-down electron. You see, for Einstein, absolute reality really existed. For Einstein, the apparent probabilistic nature of quantum mechanics was an illusion, like the random() function found in most computer languages. The random() function just points to a table of apparently random numbers that are totally predictable if you look at the table in advance. You normally initiate the random() function with a “seed” from the system clock of the computer you are running on to simulate randomness by starting at different points in the table.
However, in 1964 John S. Bell published a paper in which he proposed an experiment that could actually test the EPR Paradox. In the 1980s and 1990s, a series of experiments were indeed performed that showed that Einstein was actually wrong. Using photons and polarimeters, instead of the spin of electrons, these experiments showed that photons really do not know their quantum states in advance of being measured and that determining the polarization of a photon by observer A can immediately change the polarization of another photon 60 miles away. These experiments demonstrated that the physical Universe is non-local, meaning that Newton’s spooky “action at a distance” is built into our Universe, at least for entangled quantum particles. This might sound like a violation of the special theory of relativity because it seems like we are sending an instantaneous message faster than the speed of light, but that is really not the case. Both observer A and observer B will measure photons with varying polarizations at their observing stations separated by 60 miles. Only when observer A and observer B come together to compare results will they realize that their observations were correlated, so it is impossible to send a message with real information using this experimental scheme. Clearly, our common-sense ideas about space and time are still lacking, and so are our current effective theories.
Hugh Everett solves this problem by letting the electrons be in all possible spin states in a large number of parallel universes. When observers measure the spin of an electron, they really do not measure the spin of the electron. They really measure in which universe they happen to be located in, and since everything in the Many-Worlds Interpretation relies on “correlated” composite wavefunctions, it should come as no surprise that when observer A and observer B come together, they find that their measurements of the electron spins are correlated. In the Many-Worlds Interpretation, Hugh Everett proposes that when a device, like our magnets above, measures the spin of an electron that is in an unknown state, and not in a spin-up or spin-down eigenstate, the device does not put the electron into a spin-up or spin-down eigenstate as the Copenhagen Interpretation maintains. Instead, the device and the electron enter into a correlated composite system state or combined wavefunction with an indeterminate spin of the electron. Hugh Everett explains how this new worldview can be used to explain what we observe in the lab. In fact, he proposes that from the perspective of the measuring magnets and the electron, two independent observational histories will emerge, one with the measuring magnets finding a spin-up electron and one with the measuring magnets finding a spin-down electron, and both of these will be just as “real” as the other. For them, the Universe has essentially split in two, with each set in its own Universe. That is where the “Many-Worlds” in the Many-Worlds Interpretation of quantum mechanics comes from.
While doing research for The Software Universe as an Implementation of the Mathematical Universe Hypothesis I naturally consulted Max Tegmark’s HomePage at:
http://space.mit.edu/home/tegmark/mathematical.html
and I found a link there to Hugh Everett’s original 137-page Jan 1956 draft Ph.D. thesis in which he laid down the foundations for the Many-Worlds Interpretation. This is a rare document indeed because on March 1, 1957, Everett submitted a very compressed version of his theory in his final 36-page doctoral dissertation, "On the Foundations of Quantum Mechanics", after heavy editing by his thesis advisor John Wheeler to make his Ph.D. thesis more palatable to the committee that would be hearing his oral defense and also to not offend Niels Bohr, one of the founding fathers of the Copenhagen Interpretation and still one of its most prominent proponents. But years later John Wheeler really did want to know what Niels Bohr thought of Hugh Everett’s new theory and encouraged Everett to visit Copenhagen in order to meet with Bohr. Everett and his wife did finally travel to Copenhagen in March of 1959 and spent six weeks there. But by all accounts, the meeting between Bohr and Everett was a disaster, with Bohr not even discussing the Many-Worlds Interpretation with Everett.
Below is the link to Hugh Everett’s original 137-page Jan 1956 draft Ph.D. thesis:
http://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf
I have also placed his thesis on Microsoft One Drive at:
https://onedrive.live.com/redir?resid=21488ff1cf19c88b!1437&authkey=!ADIm_WTYLkbx90I&ithint=file%2cpdf
in Quantum Computing and the Many-Worlds Interpretation of Quantum Mechanics, I step through the above document page-by-page and offer up a translation of the mathematics into easily understood terms.
The Transactional Interpretation of Quantum Mechanics
In Is the Universe a Quantum Computer? I covered John Cramer's Transactional Interpretation of quantum mechanics and compared it to TCP/IP transactions on the Internet. In an email exchange with John Cramer, I learned that such a comparison had never been done before. Now in the Copenhagen interpretation of quantum mechanics, the wavefunctions Ψ of particles and photons are not “real” waves, they are only probability waves – just convenient mathematical constructs that don’t “really” exist. But in Cramer’s Transactional Interpretation of quantum mechanics, the wavefunctions Ψ of particles and photons really do exist. For a physics student new to quantum mechanics, this is truly a comforting idea. Before they teach you about quantum mechanics, you go through a lengthy development of wave theory in courses on classical electrodynamics, optics, and differential equations. In all these courses, you only deal with waves that are mathematically real, meaning that these waves have no imaginary parts using the imaginary number i where i2 = -1. But in your first course on quantum mechanics, you are introduced to Schrödinger’s equation:
-ħ² ∂²Ψ = iħ ∂Ψ
── ── ──
2m ∂x² ∂t
and learn that generally, the wavefunction solutions to Schrödinger’s equation contain both real and imaginary parts containing the nasty imaginary number i. Consequently, the conventional wisdom is that the wavefunction solutions to Schrödinger’s equation cannot really exist as real tangible things. They must just be some kind of useful mathematical construct. However, in the same course, you are also taught about Davisson and Germer bouncing electrons off the lattice of a nickel crystal and observing an interference pattern, so something must be waving! I would venture to suggest that nearly all students new to quantum mechanics initially think of wavefunctions as real waves waving in space. Only with great coaxing by their professors do these students “unlearn” this idea with considerable reluctance.
As we saw previously, the imaginary parts of wavefunctions really bothered the founding fathers of quantum mechanics too. Recall that in 1928, Max Born came up with the clever trick of multiplying the wavefunctions Ψ by their complex conjugates Ψ* to get rid of the imaginary parts. To create the complex conjugate of a complex number or function, all you have to do is replace the imaginary number i with –i wherever you see it. According to Born’s conjecture, the probability of things happening in the quantum world are proportional to multiplying the wavefunction by its complex conjugate Ψ*Ψ. Mathematically, this is the same thing as finding the square of the amplitude of the wavefunction. Now earlier in this posting, I mentioned how Richard Feynman pointed out that instead of thinking of positrons having negative mass energy, you could also think of positrons as regular electrons with negative charge moving backwards in time by shifting the position of the “-“ sign in the wavefunction of a positron. But that is just the same thing as using the complex conjugate Ψ* of an electron wavefunction for a positron. So mathematically, we can think of the complex wavefunction of a particle Ψ* as the wavefunction of the particle moving backwards in time. Cramer suggests that Born’s idea of Ψ*Ψ representing the probability of a quantum event is not just a mathematical trick or construct, rather it is the collision of an outgoing “retarded” wave Ψ moving forwards in time with an incoming Ψ* “advanced” wave moving backwards in time. Essentially, John Cramer's Transactional Interpretation of quantum mechanics sees the collision of outgoing “retarded” waves Ψ moving forwards in time with incoming Ψ* “advanced” waves moving backwards in time.
The Transactional Interpretation easily explains all of the apparent paradoxes of quantum mechanics. As we have seen, there is actual experimental evidence that electrons and photons seem to “know” in advance what they will encounter on the other side of a double-slit experiment. This is easily explained by the Transactional Interpretation. The electrons or photons send out retarded waves into the future which interact with whatever lies beyond the slits. If there are detectors that are turned on, the retarded waves interact with them, if there are no detectors, the waves interact with some electrons on a projection screen instead. In either case, an advanced wave is sent backwards in time from the detectors or the projection screen to the point of origin of the electrons or photons so that they “know” how to behave before they get to the two-slit screen.
The Decoherence Interpretation
Quantum decoherence is another popular interpretation of quantum mechanics. In quantum decoherence, there are again a large number of parallel possibilities for the wavefunctions for things like electrons, but many closely related wavefunctions interact with each other and become quantum-mechanically “entangled” with each other in a quantum superposition of states, forming a Universe of their own. This interpretation of quantum mechanics seems to be very popular amongst the quantum mechanical engineers working on quantum computers because they are always fighting with decoherence. Quantum computers are very delicate machines. If you try to measure the output before the run completes, the whole thing will abend. The problem is that the results of a computation are not in our universe alone; we can only obtain a solution to a computation by looking at the interference of the various results in all the parallel universes together. If we try to take a peek before all these parallel runs complete, the whole system of qubits will decohere because we have taken a measurement of the system, and our quantum calculation will crash.
Conclusion
I hope that some of the above quantum-mechanical craziness that we observe in our Universe helps with watching both of the Netflix features Bandersnatch and the Russian Doll. I would also like to thank Netflix for producing these fine quantum-mechanical productions that can help the general public to become more familiar with the phenomena of quantum mechanics and with the potentials of quantum computer software.
Comments are welcome at scj333@sbcglobal.net
To see all posts on softwarephysics in reverse order go to:
https://softwarephysics.blogspot.com/
Regards,
Steve Johnston
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