Things that seem difficult to do have easy math–and things that seem extremely easy to do are often accompanied by the most strenuous math. Let’s take the universe as an example. If you were to make a key assumption, that the equations for the entire universe are “easy” to do, however, you must realize that the assumption relies on an average, meaning that all parts of the universe are the same. Once the assumption has been made, the calculation to do the whole universe turns out to be no more than five lines. The reason so is because you are calculating just one number and observing how that one number changes over time. So, what would exemplify “difficulty” in math? Well, take a match, light that might, blow the fire out and then look at the smoke billowing from the head of the match. If you were to try and put numbers to the smoke, you would probably suffer an aneurysm because that would be a painfully difficult equation to solve.
Try to imagine that you’re standing by a pond of water. You pick up a rock and drop it in the pond. The rock creates a force. Where the force is high, the water gets pushed away faster and where the force is low, the water really doesn’t move all too much. Putting numbers to all of this will leave you with a sheet of paper full of Greek letters. Putting numbers to “things” that aren’t all too mathematical [in appearance] is where partial differential equations come in. Simplification is good but most mathematicians like to make things as abstract as possible which is a great thing to do in mathematics. However, as I had just stated, simplification is good especially when you’re delving in topics concerning physics, you tend to think in terms of physical objects. With that said, it’s good to get yourself familiar with tensor calculus.
In reference to the universe, what would it look like if the fine structure constant were to suddenly change to 1 or 1/1000 in some part of the universe? If it causes some effect that would either kill me or set-create a situation in which I’d never come into being in the first place, then in those universes where parts of the universe do change physical constants, I’m dead. If the fine structure constant changes by
…then I don’t die. But, if you were to look at how much the fine structure constant has to change before I die, you’d come up with some limits as to where and how the first-shell crossing can change, which will be interesting. If the laws of physics were to change–sufficiently–it would be impossible for intelligent life to exist. Exactly what are the limits of changes in the laws of physics that would be consistent with the development of life? Queries of this nature would take the argument out of the unobservable multiverse and concentrate it in a singular universe. If you come across the possibility of having considerably large changes in the laws of physics and still have intelligent life then your anthropic arguments are not going to work and realize that this is going to kill anthropic arguments for parallel universes. Is our universe classical? I sure hope not. If it were, the future would be determined and the ability to think of alternative futures–and to act on them–would be utterly useless since there is only one future that has been pre-determined. With that said, our universe has to be just random enough so that you can have some choice in your actions but not so random as to make predicting the outcome of the future inconceivable.
Take into consideration of the reason for me to be thinking along these lines and that reason is that if you take the rules of quantum field theory and turn it sideways you end up with the rules of statistical mechanics. But, statistics of what exactly? Suppose I’m wrong and someone else with a diametrically opposing viewpoint is right and the changes in the laws of physics aren’t occurring at Planck’s timescale. You would have a huge number of universes, generated by the MWI but the first-shell crossing and Planck’s constant would be the same–in all of them. There is a need for the masses to comprehend the reality that our universe isn’t all that big. To draw emphasis to the point I’m making, there’s some benevolent, hyperintelligent being named “Chris” and as “Chris” take a liking towards a certain individual, so much as to the point that when this certain individual dies, “Chris” decides to do something about it. So “Chris” takes some matter and randomly rearranges it. Mathematically, you can calculate how long it takes before the matter ends up with me. This the “possibility” opened up by the Many Worlds Interpretation [MWI]. Now, if you have only one universe the stars will begin to burn out and the universe will suffer heat death. But, we’re going to assume that MWI is right and that you have multiple universes in which in each one, “Chris” rearranges atoms. Once again, mathematically, this can be shown in those universes and what they’ll show is that eventually, I’ll come into existence. “Chris” has his hand on the lever of the “human machine” and he’s systematically going through all of the organic molecules and when the processing is done, I’m going to come out of the machine.
Keeping things in the term of “classical”, you have to bear in mind how superconductors function. Superconductors are a good example in which collective action of large numbers of atoms makes that particular system less “classical”. Also, ferromagnetism is another example where large numbers of atoms makes a particular system less “classical”. If you’re into condensed matter physics, then you’re probably aware of several other examples that contain large numbers of atoms that operate within a system less “classical”. One other thing about MWI that you have to consider is the fact if you assume that there is only one universe, then most things that are deemed possible most likely will not occur. With MWI, anything that is possible will occur and I would only need to show that creating a parallel universe, replete with theme parks filled with fun rides for those that were “good”, isn’t physically impossible–and it will happen. Obviously, “Chris” is Christian-themed, akin to the biblical “God”, and needless to say, me being churned out of a “human machine” and being brought into existence in other worlds exemplifies as an equivalent to being “raised from the dead”. Some may see that as a knock against religiosity but it does the job explaining MWI in laymen’s terms for the masses. I say that because the apparent barriers to “raising the dead” are thermodynamic and there are no physics barriers against what can be deemed as “plausible immortality“. With MWI, there are thermodynamic implications.
If I were to give someone a 1024-bit number, would it be mathematically impossible for that individual to factor that number before our universe suffers heat death? I ask this because, with MWI, it’s trivial to show that in some universe, it is quite possible that someone will factor that number. In fact, this is why we have research in quantum cryptography today.
There is a quantum phenomenon known as shot noise that occurs in resistors. You can imagine a situation in which a resistor that is nearing tolerance fails or doesn’t fail due to the current flow. With that said, you can picture, in your mind, a situation in which a resistor fails or doesn’t fail because a cosmic ray does or doesn’t hit a critical item. To give you a more realistic perspective, say I’m near a piece of uranium that did or did not emit an alpha particle that did or did not knock out a critical part of my DNA that did or did not cause another part of body to become cancerous.
In reference to quantum superpositions, if you were to accept MWI, the wave function is always in a state of superposition. See, in Copenhagen Interpretation, wave functions collapse and the superpositions disappears. With MWI, the superposition does not “disappear” after the measurement has been completed. Remember, in MWI, you are dealing with many worlds and it’s just that the outcome of the measurement are in different universes which aren’t noticeable. Superposition isn’t the only quantum effect that takes place and although the electrons in my computer or those in projected lasers are collapsed, the quantum effects in both are important. It matters greatly because there is one missing part in MWI and it’s that I won’t feel the universe splitting. Quantum suicide and “immortality” are ideas that try to figure out what’s going on; they might be wrong, but it’s not an issue of much irrelevancy and you just can’t avoid a problem by telling yourself that the problem is of no relevance.
Some would think that if I were to advocate what MWI implies that it would directly conflict with my adoption to atheistic principles and/or views, especially if I were to grow anxious about meeting “Chris”. Once again, MWI is just that–an interpretation of “what could be”. But, “what could be” is what MWI addresses.
If I were “Chris”, I would change my name to Desmond, and as Desmond, I’d venture to make a universe that’s intuitively interesting but to do that, it would require a keen sense in things mathematical–to the tune of creating one’s own esoteric corner of mathematics. If I’m creating my own universe, nuclear statistical equilibrium would be the call of the day since nuclear statistical equilibrium is pretty independent of temperature, density and mass for normal temperatures. Equilibrium distributions will change once nuclear temperatures and densities have been reached. Anything less than that and my newly formed universe becomes 100% iron–and I wouldn’t want that. Why, you ask?
Answer: Because stars wouldn’t exist.
At this point, this is the result of the thought experiment. You use math to specify a density, temperature, and electron faction, you’ll be able to calculate the “equilibrium state” of matter.
Suppose you have the electron density
…it would showcase “pure” charge density with the intention of forgetting about the spins of electrons. Okay, that would be valid but clarification comes into play when you work with momenta. Instead of the number of particle (i), you want to work with momenta
This creates a general wave function
Let’s say, I don’t want a generalized wave function….I need something exclusive to electrons, so I’ll take a many electron wave function, square it and integrate all over it with the exception of one of the coordinates. This approach is exampled in Marder’s Condensed Matter Physics.
So, what’s the point of doing that? Well, if I to create a universe that would mathematically span out into multiple universes, or a multiverse, the need to exemplify this in a mathematical equation would be just, without a care which coordinate is singled out since the wave function can be either symmetric or asymmetric in all of its coordinates. Thusly, the squared wave function is symmetric. It’s important that you realize that by mentioning coordinates, it doesn’t imply that I’m going to infer scalars into the fray. Symmetric or asymmetric, that number is going to change, however, that does not define it as a scalar. I can have a can of Coke that is 16 fluid ounces in one coordinate system and 473.18 mL in another coordinate system. Guess what, that does not make it scalar. Volume is an observable, not a scalar. One thing about fluid measurement is that it does not respond to a quantum mechanical operator so it’s plausible to argue that fluids cannot be “observed” in a theoretic field sense, but you have to be careful on how “observed” is defined. With that same can of Coke I just referred to, I can fill it up with water and dump the water into a bucket that has a known volume. This does not fit in with field theory. My definition of “scalar”, notwithstanding, is used in a very narrow and mathematical sense and “observation” in a broad sense. A benefit to doing this is when you open up a book on differential geometry, you’ll see a mathematically precise definition of “scalar” but you won’t see a similar mathematically precise definition for “observation”. Conversely, you can tell what’s a scalar quantity, at times, just by looking at the units. Fine-structure constant and
…are scalar with respect to everything. Charge and rest mass are scalar quantities. Defining scalar as a quantity that does not change when coordinate systems are changed comes off as a different definition to those that don’t see it that way. By this definition, I measure something in my coordinate system; you measure something in your coordinate system and we both end up with the same number. You can measure something that has characteristics, such as an electric charge [when you vary space and time coordinates]; and you can measure something that doesn’t have any characteristic, such as volume.
For the sake of argument, you can say that we have reached the boundary of where the Platonic world is colliding with the real world…