Herb— R F Streater in his 2007 book ‘lost causes in and beyond physics’ (see amazon) discusses and dismisses frieden’s papers (physics from fisher information).

I happen to think Frieden was on to something, and he kept publishing ( Streater was top notch but he may have got that wrong. Even Oppenheimer said Feynman didnt make any sense.)

. The same issue applies in population genetics ( is ‘fisher’s fundamental theorem of natural selection’ a ‘maximum entropy principle’ or a newtonian ‘potential energy minimization’ problem? elliot sober (philosopher of of science at U Wisconsin) among others have numerous papers on this.

The same issue occurs in General equilibrium theory in economics (Arrow-Hahn-Debreu-Mantel-Sonnenschein) . Do markets maximize entropy or minimize potential energy to achieve ‘pareto optimality’?

My impression is you can do it both ways. (see the book by ingrao and israel ‘economic equilibirium in the history of science’).

More recent papers derive general relativity from a max ent principle.

]]>What is the philosophy behind the general idea: (quantity) = (conjugate1) – (conjugate2), e.g., the Lagrangian dS = Tdt -Vdt.

Roy Frieden has introduced the Extreme physical information principle, but it’s a controversial idea. https://en.wikipedia.org/wiki/Extreme_physical_information

]]>In all relativistic theories of classical point particles you can write the action as

where is an arbitrary parameter, not necessarily arclength. (Indeed, for particles moving at the speed of light the arclength is zero, so it’s no good as a parameter!)

You can see this done e.g. starting in Section 3.5.2 of this book:

• John Baez, Blair Smith and Derek Wise, *Lectures on Classical Mechanics*.

It’s just a draft but it’s fairly readable.

]]>Ah, I think I’ve got it! Thanks a lot.

I suppose the awkward bit would be finding an expression for the action that is also agnostic with respect to the choice of time coordinate. A physical theory with action does seem to give a special role to , while it’s much more elegant to parametrise by arc length as you did and never have to mention time at all.

I wonder if it would turn into a field theory at that point, though.

]]>I think what I don’t understand is what role the HPF plays in this recipe for mechanics. Is it:

1. Use some minimisation procedure to find path

2. Evaluate HPF for that path

3. Decide which represents time, label as and hence derive Hamilton’s equations

?

Yes, that’s about right.

In which case I think I was looking for step 1 somewhere in your post, and getting confused.

I explained it here:

Suppose we have a particle on the line. Consider smooth paths where it starts at some fixed position at some fixed time and ends at the point at the time . Nature will choose a path with least action—or at least one that’s a stationary point of the action. Let’s assume there’s a unique such path, and that it depends smoothly on and . For this to be true, we may need to restrict and to a subset of the plane, but that’s okay: go ahead and pick such a subset.

Given and in this set, nature will pick the path that’s a stationary point of action; the action of this path is called

Hamilton’s principal functionand denoted (Beware: this is not the same as entropy!)

Of course this is just a special case of a more general procedure. For example we could consider a spacetime manifold and fix a point

Suppose we can define a quantity called the ‘action’ for any path . Suppose there is a unique action-minimizing path from to any point . Then given , let be the action of the action-minimizing path from to the point Then

is **Hamilton’s principal function**, and starting from here we can derive Hamilton’s equations as explained in my blog article.

We can work in arbitrary coordinates and write the coordinates of as We don’t need to choose one coordinate and call it ‘time’: the formalism doesn’t really require that. Hamilton’s equations are really just the trivial statement that

Here I’m assuming that everything is as differentiable as we want it to be, and just to be careful I’m assuming there’s a unique action-minimizing path. The second assumption is false when there are ‘caustics’.

]]>Sorry, I meant integrating .

I think what I don’t understand is what role the HPF plays in this recipe for mechanics. Is it:

1. Use some minimisation procedure to find path

2. Evaluate HPF for that path

3. Decide which represents time, label as , and hence derive Hamilton’s equations

?

In which case I think I was looking for step 1 somewhere in your post, and getting confused.

Incidentally, I find methods like this that put time and space coordinates on an equal footing to be very attractive — as far as I’m concerned, they only differ because they show up in different places in a Lorentzian metric. However, looking it up it seems that this is called ‘non-autonomous mechanics’ and is in general fairly tricky, e.g. I found http://arxiv.org/pdf/math-ph/0604063.pdf

(Though the text you posted http://math.ucr.edu/home/baez/toby/Hamiltonian.pdf also seems to apply to this case, though it seems to assume you’ve already picked a hypersurface, and doesn’t contain any details of a minimisation procedure)

]]>ejlflop wrote:

Given that the integral of Hamilton’s principal function over a path…

I’m confused. Hamilton’s principal function is itself defined as an integral over a path; it’s not something we typically integrate over a path.

]]>Given that the integral of Hamilton’s principal function over a path seems only to depend on the difference , what is the result of ‘minimising S’? It surely can’t be a path. Or perhaps I’m missing something crucial about how to integrate ?

]]>Is the contrast between having QM derive from stat mech and having it from classical mechanics, not really stiffer or steeper than presented here? If you start from the stat mech side isn’t there a sense by which the other view misleads on the materiality or definiteness of the wave function outside any context of identically prepared systems?

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