If you want to be not only infinitesimal stochastic but self-adjoint, it’s common in quantum mechanics to assume is a positive self-adjoint operator, e.g.

Then the right equation for stochastic time evolution, e.g. the heat equation, is

On the other hand, in my book with Jacob Biamonte I got tired of all these minus signs and used a different convention, namely

which works well when

It’s all just a matter of different conventions; the physics is the same either way.

]]>If we are thinking about as something like a Hamiltonian, that is, not just a time-evolution operator but an energy-like quantity then it seems to me that the master equation should be . My reasoning is that when you add a potential (one that is not balanced by viscous forces) in, for example, a diffusion (or Fokker-Planck) equation, it enters with a minus sign;

,

and this suggests that either the right-hand side is or , where is a Lagrangian-like operator. Personally, I think the former makes more sense.

]]>Thanks Prof. Baez!

]]>I’ve thought about this a bit, but because this ‘square root map’ from probability distributions to wavefunctions is nonlinear, while time evolution in both quantum and stochastic mechanics is linear, I don’t see how to use it to turn stochastic systems into quantum ones.

The *Quantum Techniques for Stochastic Mechanics* book is based on another idea, which is that it might be interesting to copy some ideas from quantum mechanics to stochastic mechanics while *deliberately ignoring* the fact that the usual relation between probability distributions and amplitudes is nonlinear. While this is weird, it seems to work.

Stochastic mechanics is a lot like quantum mechanics. In particular, the usual path integral description of a free point particle in quantum mechanics is a lot like the path integral description of Brownian motion in stochastic mechanics. In quantum mechanics the amplitude for the free particle to take a path is proportional to the exponential of times its kinetic energy integrated over time. In stochastic mechanics the probability for Brownian motion to take a path is proportional to the exponential of times its kinetic energy integrated over time. Starting from this idea, we should be able to take some ideas about Noether’s theorem for the quantum mechanics of a free point particle and translate them into ideas that apply to Brownian motion. It becomes a bit more interesting if we allow the particle to move around on a Riemannian manifold that’s not necessarily Euclidean

There should also be a version relating quantum field theory to stochastic field theory, but if you want to study this I suggest starting with particles instead of fields.

There is already some work on this, but I don’t understand it very well, and not merely because it’s in languages other than English:

• A. Barros and D. Torres, Teorema de Noether no cálculo das variacoes estocástico, available as arXiv:1208.5529.

• J. Cresson and S. Darses, Plongement stochastique des systémes lagrangiens, *C. R. Math. Acad. Sci. Paris* **342** (2006), 333–336. Also available as arXiv:math/0510655.

Here in Torino, we have Jacob Turner from Penn State visiting for a while and in collaboration with his PhD advisor, Jason Morton we’ve been thinking a lot these days about time-reversal symmetry in quantum mechanics. We have thought about how this relates to the more familiar notions of reversing Markov processes. I want to think more about what you’re saying.

Roughly stated, I think of the difference in probability between starting at network node and going to and the opposite as being a sort of ‘probability current’. The reason I think of this is that it actually relates to time-reversal symmetry in quantum mechanics. So if you let be a valid stochastic generator and define for all non-negative times . Then you can define the ‘stochastic probability current’ as

This definition is relevant to this post since the following are all equivalent.

(i)

(ii)

(iii) is a Dirichlet operator (e.g. under the image of the exponential map generates a 1-parameter unitary group)

Also note that implies the standard definition. I was actually going to ask and see if John was interested in having us create a little mini-series on

time-reversal symmetry breaking. (Full disclosure, as John knows, I’m supposed to be making a post about how some of the ideas in the Network Theory Series can be used to study infectious disease propagation—applied to a long chain of rabbits. This is a missing chapter from our course notes/book).

I only understand quantum theory in a vague sort of way, but it seems to me that it would be ‘quantum-ish’ to define the value of an operator O in the stochastic case as where s is the (element-wise) square root of . I don’t know if this helps…

]]>they actively wanted to support her in her rather “crazy” fight for equal rights.

I would like to add that her main motivation for working were probably the subjects she was working on, that is she probably found that it was important to do this work and the concrete environment she was working in was eventually only of secondary importance.

Or in other words – I don’t know how good the access to literature was in these days, but in principle she eventually could have kept way less contact with those people there,

So I regard the fact that she was teaching and actively contributing as a “fight for her rights”. In particular she could have thought that at one point that this apparent imbalance of treatment is noticed and that she would get more support. It is however not clear to me how many collegues were convinced of the importance of her work at that time and if they were, why they wouldn’t try to change the way things were. May be because she was not making enough fuzz and trouble? But in fact in the end she was at least supported in fleeing from the Nazi’s.

Clearly she thought about these issues, e.g. after she visited the Institute for Advanced Studies at Princeton she wrote that she was not welcome at this “men’s university, where nothing female is admitted”.

I find it problematic to sort out typical “female” or “male” features that is on average I think there are differences, but the distribution is often rather individual and similar to racial differences a lot of differences, if not most, are due to cultural differences. But yes as said on average there seem to be differences. So if there are way more males that females in an institution there may be a noticable dfifference in atmosphere, but that depends a bit also on what kind of males/females. I went from an all girls school to almost basically all boys math-physics school and I found the atmossphere quite different. I think I understand what she could have meant.

It’s indeed interesting to wonder what Emmy Noether’s parents thought about her, and what she thought about her own situation.

I couldn’t find any personal recollections of her or her parents. That seems her writings seem to have been mostly of a mathematical nature.

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