As I mentioned in my previous post, I have been working for some five years on trying to understand quantum field theory from a quantum-information perspective. This has finally come to a fruition of sorts: I’m pretty sure I have an operationally motivated way to build nontrivial *nongaussian* quantum field states using a variety of tensor network states.

The input to the procedure is any family of tensor network states (or, indeed, any family of states) whose correlation functions diverge in a controllable way as a function of a scale parameter. The procedure then produces a continuum limit with the corresponding quantum field data modelling the quantum fluctuations around the limit.

There are two main ideas behind the procedure: (1) it begins by extending the mean-field formalism of Hepp and Lieb (developed later by Verbeure and coworkers) to identify the emergent continuous large-scale degrees of freedom describing the classical bulk fluctuations (remarkably the continuous degrees of freedom are not prescribed beforehand) — this is a kind of generalised law of large numbers result; and then (2) by exploiting a generalised quantum central limit-theorem type argument the quantum fluctuations around the bulk are then identified and the emergent quantum field operators are subsequently identified. The applicability of this procedure is contingent on the family of input states satisfying certain criteria, which essentially boils down to the ability to tune the correlation length in a controlled way.

A nontrivial result is that several tensor network states naturally satisfy the criteria required by the continuum limit procedure: in particular, for the continuum limit of the matrix product state and projected entangled-pair state classes we recover their recently introduced continuous counterparts and for tree tensor network classes arising from Kadanoff block spin renormalisation and the multi-scale renormalisation ansatz class we obtain continuum descriptions generalising the recently introduced continuous MERA.

For me the most exciting discovery in all of this is that there are simply an enormous number of non-gaussian states which can serve as fixed points of Wilson’s RG and give rise to very reasonable renormalisable QFTs.

## An open science experiment

I gave up on open science a while ago (see this post for details). However, I’ve always wanted to give it another try.

The open-source software (OSS) movement is often held up as a model for how open science should work and it occurred to me recently we could exploit a powerful tool used in OSS to facilitate scientific collaborations, namely, github. Thus today I’d like to announce a new github-based open-science project based on the aforementioned continuum limit construction: I’ve created a github repository for this project and uploaded the latex source of a paper I’ve been working on for some time describing this construction. It is my hope that this initial incomplete draft could serve as the basis for a collaborative project on understanding how to implement Wilsonian renormalisation for tensor network states.

## The rules

I will continue to edit this document and push commits. When the paper is ready, and after any pull requests and merges, the final version will be submitted to the arXiv and, eventually, a scientific journal for publication.

However, anyone who is interested in contributing to this projected paper is most warmly invited to do so *at any time* by forking the repository and issuing pull requests.

If you fork this paper then please add yourself to the author list in your fork (regardless of whether you make any changes). If you make *substantial contributions* to the draft then I will merge your changes into the final version and include you on the final submission version author list.

What is a “substantial contribution”? At the moment I define this to be a nontrivial calculation, e.g., working out the details of an example, sorting out some of the unfinished calculations, or similar. Another form of “substantial contribution” is writing one or more paragraphs of new material. Also, nontrivial figures and diagrams, e.g., graphs of interesting quantities, will be considered. Spelling, grammar, and/or stylistic changes do not count as “substantial contributions” (but are still welcome!). This definition is subject to change as the experiment proceeds.

It is my fondest ambition to be the benevolent dictator for this project: all contributions are welcome, but I retain the right to approve their inclusion in the final “submission” version.

### A (partial) list of what needs to be done:

- Is the current paper a good medium to communicate the idea in its current form? Should we write a different paper?
- More worked examples: e.g., what about MPS whose classical mean-field limit isn’t the empty vacuum? What about more mean-field models?
- Fermions: there is a totally natural generalisation to fermions where the classical limit is described by grassmann numbers. I haven’t worked this out yet. Will fermion doubling be a problem?
- Single-particle systems: the whole procedure admits a reasonably simple “generalisation” to single-particle systems. Elaborate this. What about quantum walks?
- Applications to quantum cellular automata? (This is, by the way, an approach to dealing with fermion doubling.)
- Applications to perturbative QFT: do we recover standard perturbative QFT when we apply it to the perturbation series coming from the standard lattice discretisation of a perturbative QFT?
- The operator product expansion: I gave some notes on this in the appendix, but there needs to be some interesting examples.
- Lorentz invariance is conspicuously absent: this is partially equivalent to the “parent hamiltonian problem”. This problem would be answered by the construction of a Lorentz invariant parent hamiltonian for the continuum limit. However, we currently understand very little of parent hamiltonians for continuous tensor network states.
- Supersymmetry: could there be a way to obtain interesting supersymmetric models exactly as the continuum limits of interesting tensor network states.
- Can the continuum limit procedure actually solve a problem whose answer isn’t known in the literature? To this end, what systems would be interesting to focus on?
- Topological order: I sketched in the draft how I think this should be covered. Is this correct? Do we get a TQFT when we do the continuum limit correctly?
- Topological insulators: can the continuum limit say anything here?

## Organisation

For the moment I’ll organise this by this blog and email; if you are interested in contributing please go ahead and fork the repository and email me/post a comment here.

## Future projects

In the near future I will upload the following three papers in this series to github. The plan is to run these four projects concurrently.

- Continuum limits for (non)abelian gauge theories: in which we obtain a gauge-invariant RG fixed point via a gauge-invariant tensor network family.
- The path integral and tensor networks: in which we define the path integral via a tensor network, obtain the continuum limit via the procedure described above, and study the dynamics of nontrivial quantum fields.
- QCD via the continuum limit of tensor network states: in which we study QCD via continuum limits of tensor networks. This is still in a preliminary sketch stage. My dream would be to study QCD in 3+1 dimensions.

Hi Tobias, How is this project progressing? It certainly looks fascinating.

Dear Gil,

Many thanks for your comment! On the open science/collaborative side this project has moved rather slowly. I have had the good fortune to meet quite a few new exciting collaborators via the github repository who are interested in, and have, contributed to the project. But, the project was already large and unwieldy to begin with, which makes it hard for people to get up to speed. (This is my fault; I should have opened this up way earlier when it was easy to learn everything… Waiting too long to open a project is a classic mistake in the open source community…)

Secondly, this project has been on a sort of hiatus for about a year. This is because I was not happy with the operational interpretation of the construction: while the calculations so far are correct, I wanted a compelling physical story that would connect the continuous limit construction as introduced in this paper to the way physicists understand the continuous limit. For the physicist one understands the continuous limit as the a process where one removes a small scale “ultraviolet” (UR) regulator whiles preserving large scale “infrared” (IR) features. The way that physicists are assured that this is ok is via Wilson’s (momentum-shell) renormalisation group (RG), which needs some adaptation to the present context. Also, the operational story is slightly different in the latter case. This was the motivation for a lot of our recent papers on the RG and on defining quantum states of quantum fields (see, e.g., https://github.com/tobiasosborne/What-is-a-quantum-field-state- and http://arxiv.org/abs/1402.4949).

We now think we understand all of this and I am soon going to come back to this project.

Best wishes!

Sincerely,

Tobias

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