Reality #5 - Quantum Logic
von Neumann and Birkhof originated the idea of quantum logic. When we talk about logic here we're talking about how entities' attributes combine to form new attributes (AND, OR, NOR, etc.). Classical objects obey Boolean logic. Because quantum theory deals with waveforms, von Neumann and Birkhof created a waveform logic.
Consider a collection of entities that have attribute A. Another collection has attribute B. We can use Boolean logic to create new sets. Using the AND operand we create a new set called C. A + B = C. C is the set of entities that have attribute A and B. A way to create an easier to use system for entities like this is to create lattices. Lattices are graphical descriptions of the relationships between entities. Think for example of a color lattice where AND has a connotation of mixing colored paints and OR has the connotation of mixing colored lights. Colors are the nodes in the lattice and lines connecting the nodes are relationships between the colors. Here's a couple lattices so you can see what they look like. There are rules for Boolean logic in using the lattice. To use AND between two colors you follow the connections in a particular manner.
An example of a non-Boolean lattice is one for the polarization for single photons. It does not follow the rules for lattices. This is not surprising since it's governed by quantum theory. If we limit ourselves to P(0) and P(45) as before with the H/V and D/S simplifications we can create some logic statements. H OR D = All. All polarizations can be created by a superposition of those two polarizations. H AND D = Null. Meaning no wave can have both H polarization and D polarization (or no wave can go through both polarization measurement devices and score a hit).
Boolean logic dictates that certain laws are upheld. For example the distributive property holds that A OR (B AND C) = (A OR B) and (A OR C). I won't show it here but this sort of thing holds for classical systems but breaks down for certain cases of quantum systems. However, quantum lattices can be broken into sublattices (containing only some of the attributes) that do obey this distributive law. And in all cases they represent the attributes we can measure together with perfect accuracy. Therefore you cannot make a Boolean sublattice with both position and momentum attributes present. And so all quantum lattices have this interesting feature of being the addition of sublattices that obey Boolean logic but as a whole do not obey Boolean logic. It's almost like logic (as we know it) breaks down when you start looking at conjugate variables together.
Finkelstein is one of the few scientists pursuing this type of study for the sake of understanding quantum theory. Others do it primarily to study alternative logic structures. Finkelstein actually thinks that quons do have specific values for their attributes. But quons differ from classical objects in the way that they combine their attributes to form other attributes. In other words quantum objects have classical attributes that obey non-classical logic.A common example of this strangeness is with the 3 polarizer experiment. Take two polarizers and cross them perpendicular. Shine light towards them. No light will go through. Any light that goes through the first one is polarized in one direction and therefore has no component polarized in the perpendicular direction. Once that light hits the second polarizer it blocks all light. Now insert a third polarizer in between at 45 degrees between the two perpendicular polarizers. Light starts to shine through again.
It's a little like having a filter for balls smaller than 2 cm and a filter for balls equal or greater than 2 cm right after each other. Pour a bunch of balls through the two filters and of course nothing goes through. Then we add a new filter inbetween that filters for balls that are smaller than 1 cm and larger than 3 cm. All of a sudden balls smaller than 2 cm are falling all the way through. It's nonsensical in classical mechanics. But in quantum this happens. It just follows another logic.
Reality #6 - Realism
It's both strange and fitting that Einstein was a realist when it came to quantum. He himself had suffered resistance when his special relativity work first came out. And it was invariably from the 'old guard' of science. Now he resisted quantum theory as an incomplete theory. Adding to the oddness of the situation, Einstein started the whole thing by declaring light waves behaved like particles. He never did come to terms with quantum theory. The discussions with Bohr went on until he died. Einstein playing the aggressor to Bohr's defensive stance on quantum theory. To his last days Bohr repudiated all of Einstein's attempts to Einstein's satisfaction. Except for one - The EPR paradox. We'll talk about this later.
It's understandable though. Scientists want to understand the world around them. Quantum is saying, 'too bad'. The world around us has an element of randomness to it that simply cannot be explained. The world can be contained but never fully predicted. This of course changed a general worldview that was common. Namely if you had a powerful enough computer and you knew the laws of the universe you could predict the future by putting in your starting conditions and running it forward. Quantum theory says it will never work because random elements are introduced all the time. That is life.The only way out was to create a hidden variable. An attribute of quons that directs them to behave according to quantum theory. If I decide to measure where a quon is then it directs the momentum of the quon to suddenly shoot off at a higher value. Before we measure the quon, it is at a particular point with a particular momentum. This way the outcome of quantum theory is maintained but underneath quantum is not a complete theory. The only downfall is that this maneuvering the quon must do must be communicated at a speed greater than light.
Reality #7 - Conscious Reality
Again von Neumann started this line of thinking. The concept comes from the excruciating problem of not being able to find a place to collapse the wave function. If we know the quon behaves as a wave before we measure it and like a particle after it hits our measuring device, when does this change occur? von Neumann being the upstanding logician admitted he couldn't find a place. There was nothing inherently different about a measurement device like a phosphor screen and a photon splitting device like a polarizer. He kept following the chain through the experiment and concluded that our consciousness is the only inherently different thing. Consciousness isn't mechanical; isn't made up of atoms and things.
Schrodinger thought this was just crazy and dreamt up his cat experiment. But the fact remains that there is just no good place to put the collapse. It's certainly a viable explanation from that standpoint. von Neumann's argument is this - in order for an all quantum world to work a special process (the quantum jump) must be present in all measurement acts and nowhere else. But how in a world of quantumstuff, where no privileged processes exist, can this occur? The only process that exists outside matter's monopoly is the awareness of the observer. A coupling even with a measuring device is not yet a measurement. A measurement is achieved only when the position of the pointer has been observed.It seems clear to me that none of these explanations are entirely satisfactory. I personally am okay with the Heisenberg Uncertainty Principle. I think of it much in the way that space and time became relative after Einstein. We are always moving at the speed of light. There is no way to stop that reality. We're either moving the speed of light through space or we're moving the speed of light through time or somewhere in between. There is no other possibility. The two sets of dimensions are linked. In a similar manner conjugate attributes are linked. Quons demand to take up a certain amount of 'space' or 'travel the speed of light' in conjugate attribute space. It's very similar in my mind. Bohr made the same point to Einstein - why couldn't he see the parallel nature of his relativity and quantums uncertainty? Einstein replied, "a good joke should never be told twice."
It's an interesting point that I have not brought up that the product of two conjugate attributes is called action. Take position and momentum. The product is (mass)(length)^2/(time). All conjugate attributes have these dimensions. Action is similar to space-time. It's a true unifying perspective of the world.
The wave collapse on the other hand is more troublesome. There are times when the many-world's hypothesis comes in handy by doing away with this onerous aspect of the theory. But the theory itself is so hard to accept otherwise. I definitely think the realist standpoint is a way of clinging on to the past. I remember first comprehending the non-mechanical realities this creates for the universe and it shook me hard in grad school. It's a damn tough thing to get over for a scientist. I do think we will come up with an answer to this though. But only through the creation of a new theory which subsumes quantum theory. The best minds have tackled this issue and I doubt anyone will come up with something at this point without additional theory to guide the way.
At any rate. We now move onto the battle between Bohr and Einstein. Einstein would constantly bang on quantum. He just hated the idea of quantum being complete. I used to sympathize greatly with him on this point. There were many times when Bohr became worried; that Einstein had found a flaw in the system. But each time Bohr would eventually come up with a reason why Einstein's logic was flawed. The strongest reasoning however outlasted both of them. It is known as the EPR paradox. And it eventually led to an experiment which put scientists in every camp I talked about above in a completely uncomfortable position. While some expected to 'win' the quantum vs. reality debate, many came out of this thinking no one won. It's that horrible.
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