SymplecticMan

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 $\sqrt{T}$  based off the rms velocity and mean free path (which can be related to average time between collisions).

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 $Q$, we can define the entropy $S\left(Q\right)$ as the log of the number of states with that value of $Q$. This is a univariate function of $Q$, and temperature can be defined as the multiplicative inverse of the derivative $\frac{dS}{dQ}$.

You still in general to specify which macroscopic variables are being held fixed when taking partial derivatives. Taking a derivative with volume... (read more)

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:

$\frac{dS}{dQ}=\frac{1}{T}$.

This is a relationship at constant volume. Specifically, the somewhat standard notation would be

${\frac{\partial U}{\partial S}|}_V=T$,

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 us... (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 o... (read more)

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. Y... (read more)

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 t... (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?

I don't know if it would make things clearer, but questions about why eigenvectors of Hermitian operators are important can basically be recast as one question of why orthogonal states correspond to mutually exclusive 'outcomes'. From that starting point, projection-valued measures let you associate real numbers to various orthogonal outcomes, and that's how you make the operator with the corresponding eigenvectors.

As for why orthogonal states are important in the first place, the natural thing to point to is the unitary dynamics (though there are also various more sophisticated arguments).

Everett argued in his thesis that the unitary dynamics motivated this:

...we demand that the measure assigned to a trajectory at one time shall equal the sum of the measures of its separate branches at a later time.

He made the analogy with Liouville's theorem in classical dynamics, where symplectic dynamics motivated the Lebesgue measure on phase space.

The earlier post has problems of its own: it works with an action with nonstandard units (in particular, mass is missing), its sign is backwards from the typical definition, and it doesn't address how vector potentials should be treated. The Lagrangian doesn't have to be positive, so interpreting it as any sort of temporal velocity will already be troublesome, but the Lagrangian is also not unique. It simply does not make sense in general to interpret a Lagrangian as a temporal velocity, so importing that notion into field theory also does not make sense.

T... (read more)

You say this theory of your predicts "that localized quantum fields will maximize proper time". It's not clear how it's a prediction of this theory, since the statement of the prediction is the first time you mention proper time in this article. I looked through the follow-up link to see if the prediction came from there, but the link doesn't mention entropy (nor does this post mention action). And I don't believe that "The physics establishment sidesteps this quandary by defining time to progress in the direction of increasing entropy" is a fair assessmen... (read more)

Demanding that the time reversal operator leaves Q unchanged but reverses the sign of P (which is how time reversal in classical mechanics works) means that the time reversal operator has to be implemented by an anti-unitary operator. More hand-wavingly, since the Schrodinger equation gives  $e^{-iHt}$ as the forward time evolution of a state $\psi$, $e^{+iHt}$ (flipping the sign of time) should give the backward time evolution. But that's just the normal time evolution of ${\psi }^{\ast }$ as you can see if you just conjugate the Schrodinger equ... (read more)

That's a good point; $⟨A|B⟩=0$ is a strong precise notation of "mutually exclusive" in quantum mechanics. (...)

I'd be remiss at this point not to mention Gleason's theorem: once you accept that notion of mutually exclusive events, the Born rule comes (almost) automatically. There's a relatively large camp that accepts Gleason's theorem as a good proof of why the Born rule must be the correct rule, but there's of course another camp that's looking for more solid proofs. Just on a personal note, I rea... (read more)

I haven't closely read the details on the hypothetical experiments yet, but I want to comment on the technical details of the quantum mechanics at the beginning.

In quantum mechanics, probabilities of mutually exclusive events still add: $P(A\vee B)=P\left(A\right)+P\left(B\right)$. However, things like "particle goes through slit 1 then hits spot x on screen" and "particle goes through slit 2 then hits spot x on screen" aren't such mutually exclusive events.

This may seem like I'm nit-picking, but I'd like to make the point by example. Le... (read more)