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Stream: event: Categorical Probability and Statistics 2020 workshop

Topic: Jun 5: Malte Gerhold's talk

Paolo Perrone (Jun 04 2020 at 15:16):

Hey all,
This is the discussion thread of Malte Gerhold's talk, "Independence and Lévy processes in monoidal categories".
The talk, besides being on Zoom, is livestreamed here: https://youtu.be/fVEucjkYSmw

Paolo Perrone (Jun 04 2020 at 19:24):

Date and time: Friday, 5 Jun, 16h UTC.

Paolo Perrone (Jun 05 2020 at 15:40):

Hi! The talk will start in 20 minutes.

Paolo Perrone (Jun 05 2020 at 15:59):

1 minute!

Tomáš Gonda (Jun 05 2020 at 17:00):

I am also very interested in the notions of independence and conditional independence you mentioned! Do you know of a good introductory reference(s) on the topic? Also, do you have a sense of how the interpretation of these various independence notions differs, say when we instantiate them?

Chris Heunen (Jun 05 2020 at 17:27):

Is there a reference where all the notions of independence (linear algebra, transcendentals over fields, probability distributions) are nicely listed as examples? I'm wondering if they all form matroids, and if the construction "that lifts independence from elements to injective morphisms" is a general construction of matroids.

Malte Gerhold (Jun 05 2020 at 17:30):

Roland Speicher has written nice introductory material about free independence. The survey in the link also touches contitional independence: https://www.math.uni-sb.de/ag/speicher/surveys/speicher/bielefeld.pdf. As for interpretation, the tensor independence is used to model noninteracting quantum systems while free probability is closely related to random matrices. I know that Boolean independence was first used in a theoretical physics paper by von Waldenfels, but I cannot say much about how to interpret it.

Malte Gerhold (Jun 05 2020 at 17:37):

We listed some examples in our paper https://arxiv.org/pdf/1612.05139.pdf. Many examples are probably not expressable via matroids. For example, orthogonal subsets of a Hilbert space do not form a matroid (augmentation property is not fulfilled). I don't think that stochastic independence can be expressed via matroids.

Arthur Parzygnat (Jun 05 2020 at 17:43):

As a reminder of my question, I was curious about the example of creation/annihilation operators of the boson system viewed as a categorical Levy process in your sense. Feel free to point me to a reference if it is easier to do that. :) (Perhaps such an example is the paper you just mentioned above?)

Malte Gerhold (Jun 05 2020 at 17:45):

Just trying to figure out how to write math here :)

Arthur Parzygnat (Jun 05 2020 at 17:47):

Ah double dollar signs!

Malte Gerhold (Jun 05 2020 at 17:47):

Let $\Gamma(H):= \mathbb C \oplus H \oplus H\otimes_s H \oplus\ldots$ denote the symmetric Fock space over $H$ and $a^*(x), a(x)$ the creation and annihilation operators corresponding to $x\in H$. We think of the algebra $A$ generated by all creation and annihilation operators as a nc probability space with respect to the state given by vacuum expectation $\Phi(x):=\langle \Omega, x\Omega\rangle$, where $\Omega=1\oplus 0\oplus0\oplus\ldots$ is the vacuum vector. Now let $H=L^2(\mathbb R_+)$. Then $(a^*(1_{[0,t[}),a(1_{0,t}))$ form a Lévy process; to connect to the definition in the talk, take the *-Homomorphism from the polynomial algebra in two non-commuting indeterminates $x$ and $x^*$ to $A$ mapping $x$ to $a^*(1_{[0,t[})$. Independence of increments comes from the facts that $a^*(1_{[0,t[})-a^*(1_{[0,s[})=a^*(1_[s,t[)$ and the canonical isomorphisms $\Gamma(L^2(\mathbb R_+))\cong \Gamma(L^2([0,s[))\otimes \Gamma(L^2([s,t[))\otimes \Gamma(L^2([t,\infty[))$. Hope that helps a bit. I will try to find a reference soon and post it here.

Alex Simpson (Jun 05 2020 at 19:58):

Thanks for the very nice talk Malte! In my question at the end I mentioned work I did on the monoidal category definition of independence and on an extension of it to conditional independence. This can be found at https://www.sciencedirect.com/science/article/pii/S1571066118300318 . It also references some other examples of notions of "independence" that arise in computer science. With respect to Malte's talk, one mismatch is that I use a monoidal structure with projections (the unit is terminal) rather than injections (the unit is initial) as in Malte's talk. This is of course purely a matter of convention.

Arthur Parzygnat (Jun 05 2020 at 20:22):

Malte Gerhold said:

Let $\Gamma(H):= \mathbb C \oplus H \oplus H\otimes_s H \oplus\ldots$ denote the symmetric Fock space over $H$ and $a^*(x), a(x)$ the creation and annihilation operators corresponding to $x\in H$. We think of the algebra $A$ generated by all creation and annihilation operators as a nc probability space with respect to the state given by vacuum expectation $\Phi(x):=\langle \Omega, x\Omega\rangle$, where $\Omega=1\oplus 0\oplus0\oplus\ldots$ is the vacuum vector. Now let $H=L^2(\mathbb R_+)$. Then $(a^*(1_{[0,t[}),a(1_{0,t}))$ form a Lévy process; to connect to the definition in the talk, take the *-Homomorphism from the polynomial algebra in two non-commuting indeterminates $x$ and $x^*$ to $A$ mapping $x$ to $a^*(1_{[0,t[})$. Independence of increments comes from the facts that $a^*(1_{[0,t[})-a^*(1_{[0,s[})=a^*(1_[s,t[)$ and the canonical isomorphisms $\Gamma(L^2(\mathbb R_+))\cong \Gamma(L^2([0,s[))\otimes \Gamma(L^2([s,t[))\otimes \Gamma(L^2([t,\infty[))$. Hope that helps a bit. I will try to find a reference soon and post it here.

Thank you! I think this does help. But perhaps it added some confusion with the corresponding physics. Does $t$ represent time here or physical space? Based on how you plugged in characteristic functions, I suspect it's the latter (please correct if I'm mistaken). So let me call the variable $x$ instead. So physically what's happening is that $a^*(1_{[0,x)})$ is describing the creation of bosons uniformly distributed on the interval $[0,x)$ in your half space. And as $x$ increases, you are merely increasing the range on which these particles are being created? And then the tensor product decomposition you gave shows how the distribution of particle creation on disjoint regions is independent?

Paolo Perrone (Jun 05 2020 at 21:51):

Hi all! Here's the recording:
https://youtu.be/JJtPfCgybs0

Malte Gerhold (Jun 06 2020 at 15:18):

I am not sure about the physical interpretation details, but I will give it a try :) I guess one interpretation would be to see $a+a^*$ as position and $i(a-a^*)$ as momentum. So in the situation above, $t$ could be time and with the time the variance of the distribution of position and momentum grows. (If you calculate the moments of the position operator in the vacuum state $\langle\Omega,(a+a^*)^n\Omega\rangle$, you will get the moments of a Gaussian distribution.) One can probably think of it as a deformation of classical phase space where the coordinates position and momentum do not commute but fulfill canonical commutation relation. Then the above described process is what happens to 2-dimensional Brownian motion in the 2 dimensional phase space under this deformation.

In https://projecteuclid.org/download/pdf_1/euclid.cmp/1103941122, these processes are described and used as basic processes to build quantum stochastic calculus, however there is no concrete physical interpretation.

In the wikipedia article on quantum stochastic calculus there might be hints how to connect all of this to actual physics. https://en.wikipedia.org/wiki/Quantum_stochastic_calculus

Arthur Parzygnat (Jun 10 2020 at 11:52):

@Malte Gerhold, Thank you for taking the time to write this out. I'm still a little confused about some minor things (like if this is actually describing a single particle or a quantum field), but I think I get the overall idea. Keeping this example in mind when I read your paper will definitely be very helpful and I suspect I'll be able to fix any confusion I have. Thanks again!