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First, a quick comment about something pretty basic that I completely forgot to mention in part 4 on the "Boltzmann's brain" paradox: why do they call it Boltzmann's brain? Seems obvious I should have explained this, but it's because the type of dilemna raised by it is similar to the "brain in a vat" thought experiment that philosophers of metaphysics like to argue about. The basic question is depicted explicitly in the movie The Matrix: how do you know that you aren't just a brain in a vat somewhere, with wires plugged into you, feeding this brain electrical impulses that it interprets as sensory experiences? I think for the most part, the answer is: you don't. I mentioned that the Boltzmann's brain paradox could involve the vacuum fluxuation of a spontaneously generated galaxy, or solar system, or even just a single room. But the ultimate limit of this would be, just a single brain in a vat, with no actual world around it at all. Anyway, just wanted to add that to make part 4 more clear, because you may have been wondering why it was called "Boltzmann's brain". If I ever decided to make these notes into a book, I guess this would be one of the chapters. Now on to the matter at hand...

One of the most puzzling things about quantum mechanics, especially when you first learn it, is why there appear to be two completely different types of rules for how physics works. One is the microscopic set of rules which physicists usually refer to as "unitary time evolution of the wavefunction". In quantum mechanics, what's called the wavefunction is similar to a probability distribution either in regular space or momentum space (not phase space, the combination of the two) for which state a particle (or field, or string) could be in. (Actually, it's more like the square root of a probability, but no matter.) While it is similar to a probability distribution in regular space or momentum space, it can be much more simply represented as a single vector in a much larger space called a Hilbert space. The Hilbert space in quantum mechanics is usually infinite dimensional, so much much bigger than even the 6N dimensional phase space we talked about in part 5. As the system progresses further into the future, this state vector traces out a single path through the Hilbert space, and that path is always exactly reversible according to these microscopic rules. The key word here is "unitary". The vector is moved around in the Hilbert space mathematically by applying a unitary matrix to it. (Heisenberg's formulation of quantum mechanics was orginally called "matrix mechanics" because of this.) Unitary matrices are matrices whose transpose is equal to their complex conjugate (replacing all imaginary components with real components and vice versa, where by imaginary and real I'm talking about the mathematical notion of i, the square root of -1). If this doesn't make any sense, don't worry about it, the only important thing to understand is that this property of unitarity guarantees many nice things about time evolution in quantum mechanics. It makes things nice and smooth, so that a single state always moves to a single state, and if the total probability of the particle being anywhere is initially 100% then it will stay 100% in the future. But most importantly, it guarantees reversibility. Because the time evolution matrix in quantum mechanics is unitary, it means Louiville's theorem holds and you can always reverse time and go backwards exactly to where the system came from.

But wait--if this is the case, then that also means that all quantum mechanical systems are conservative, ie non-dissipative. Does dissipation not happen in quantum mechanics at all? This brings us to the second type of rule in quantum mechanics: the process of measurement. Originally, this rule was called the "collapse of the wavefunction". Because when looked at through Schrodinger's wave mechanics, it appears that when a macroscopic observer makes a measurement in the lab, of a property of a microscopic system (for instance if he asks the question "where is this particle actually located?") the rule is that what used to be a probability distribution over many different states--suddenly that distribution is reduced to, or collapses to, a single state. Before the measurement, we mathematically describe the particle as being in a "superposition" of many different positions at once, but after the measurement it is only found at one of those positions. Mathematically, this is achieved with a projection matrix. A projection matrix is very different from a unitary matrix. Its action in the Hilbert space is that it maps many vectors onto a single vector, instead of mapping one vector onto a single vector. Because of this, the action is irreversible, and does not satisfy Liouville's theorem. The measurement process is therefore dissipative rather than conservative. In other words, the rule for how an observer of a microscopic system makes measurements in quantum mechanics seems completely opposite to the rule for how microscopic systems evolve in time when they are not being observed. One is nice and smooth and reversible, the other is a sudden reduction, an irreversible collapse. Dissipation only seems to happen during observation.

But the way I've explained this highlights the paradox, called the "measurement problem in quantum mechanics". This paradox is what has spawned many different interpretations of quantum mechanics, which philosophers still argue fiercely about today. But the real question is how to reconcile reversibility with irreversibility, and the answer lies in thermodynamics. And once you understand it, you realize that it doesn't really make a whole lot of sense to talk about measurement in this way, as a sudden "collapse" of the wavefunction. And it leads to deeper questions about whether the wave function is real or just a mathematical abstraction--and if there is anything in quantum mechanics which can be said to be real at all. To be continued...

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