SymplecticMan
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Ah, so I'm working at a level of generality that applies to all sorts of dynamical systems, including ones with no well-defined volume. As long as there's a conserved quantity , we can define the entropy as the log of the number of states with that value of . This is a univariate function of , and temperature can be defined as the multiplicative inverse of the derivative .
You still in general to specify which macroscopic variables are being held fixed when taking partial derivatives. Taking a derivative with volume held constant is different from one with pressure held constant, etc. It's not a universal fact that all such derivatives give temperature. The fact that we're talking... (read 431 more words →)
I'm going to open up with a technical point: it is important, not only in general but particularly in thermodynamics, to specify what quantities are being held fixed when taking partial derivatives. For example, you use this relation early on:
.
This is a relationship at constant volume. Specifically, the somewhat standard notation would be
,
where U is the internal energy. The change in internal energy at constant volume is equal to the heat transfer, so it reduces to the relationship you used.
That brings us to the lemma you wanted to use:
.
To get what you wanted, it has to actually be the derivative with constant volume on the right, but then there's a problem: it... (read more)
Note, though, that time reversal is still an anti-unitary operator in quantum mechanics in spite of the hand-waving argument failing when time reversal isn't a good symmetry. Even when time reversal symmetry fails, though, there's still CPT symmetry (and CPT is also anti-unitary).
I argue that counting branches is not well-behaved with the Hilbert space structure and unitary time evolution, and instead assigning a measure to branches (the 'dilution' argument) is the proper way to handle this. (See Wallace's decision-theory 'proof' of the Born rule for more).
The quantum state is a vector in a Hilbert space. Hilbert spaces have an inner product structure. That inner product structure is important for a lot of derivations/proofs of the Born rule, but in particular the inner product induces a norm. Norms let us do a lot of things. One of the more important things is we can define continuous functions. The short version is, for a continuous function,... (read 411 more words →)
I will amend my statement to be more precise:
Everett's proof that the Born rule measure (amplitude squared for orthogonal states) is the only measure that satisfies the desired properties has no dependence on tensor product structure.
Everett's proof that a "typical" observer sees measurements that agree with the Born rule in the long term uses the tensor product structure and the result of the previous proof.
I kind of get why Hermitian operators here makes sense, but then we apply the measurement and the system collapses to one of its eigenfunctions. Why?
If I understand what you mean, this is a consequence of what we defined as a measurement (or what's sometimes called a pre-measurement). Taking the tensor product structure and density matrix formalism as a given, if the interesting subsystem starts in a pure state, the unitary measurement structure implies that the reduced state of the interesting subsystem will generally be a mixed state after measurement. You might find parts of this review informative; it covers pre-measurements and also weak measurements, and in particular talks about how to actually implement measurements with an interaction Hamiltonian.
I don't see how that relates to what I said. I was addressing why an amplitude-only measure that respects unitarity and is additive over branches has to use amplitudes for a mutually orthogonal set of states to make sense. Nothing in Everett's proof of the Born rule relies on a tensor product structure.
Why should (2,1) split into one branch of (2,0) and one branch of (0,1), not into one branch of (1,0) and one branch of (1,1)?
Again, it's because of unitarity.
As Everett argues, we need to work with normalized states to unambiguously define the coefficients, so let's define normalized vectors v1=(1,0) and v2=(1,1)/sqrt(2). (1,0) has an amplitude of 1, (1,1) has an amplitude of sqrt(2), and (2,1) has an amplitude of sqrt(5).
(2,1) = v1 + sqrt(2) v2, so we need M[sqrt(5)] = M[1] + M[sqrt(2)] for the additivity of measures. Now let's do a unitary transformation on (2,1) to get (1,2) = -1 v1 + 2 sqrt(2) v2 which still has an amplitude of... (read more)
I guess I don't understand the question. If we accept that mutually exclusive states are represented by orthogonal vectors, and we want to distinguish mutually exclusive states of some interesting subsystem, then what's unreasonable with defining a "measurement" as something that correlates our apparatus with the orthogonal states of the interesting subsystem, or at least as an ideal form of a measurement?
Material properties such as thermal conductivity can depend on temperature. The actual calculation of thermal conductivity of various materials is very much outside of my area, but Schroeder's "An Introduction to Thermal Physics" has a somewhat similar derivation showing the thermal conductivity of an ideal gas being proportional to √T based off the rms velocity and mean free path (which can be related to average time between collisions).