What is quantum field theory? A quantum information theorist’s perspective

October 19, 2015

Daniel Burgarth and Matt Leifer have really been an inspiration for me as they have successfully embraced and exploited new technologies to improve scientific communication and discourse. One of their innovations, with the help of Ravi Kunjwal, has been the Q+ hangouts talk series. This excellent series of talks is delivered online via google hangouts, and has now attracted a wonderful line up of topics and speakers. It has also inspired the TCS+ hangouts series.

Last week I had the honour and pleasure to participate in the Q+ hangout series. I spoke about a topic very close to my heart, namely, quantum field theory, and would like to share the video with you here today

In this talk I described a way to formalise Wilson’s approach to quantum field theory as an effective theory in quantum information theorist friendly terms. If you are interested in seeing the details behind what I discussed in the talk then please do check out the (as yet still incomplete) paper draft of the content at my github repository. This paper contains very many definitions and worked out examples. The purpose of my talk and the paper is to provide a definition for what a quantum field state is in such a way that it: (a) matches the modern Wilsonian definition used by physicists; (b) allows us to answer quantum information theory style questions about QFT; and (c) is mathematically rigourous (or, at least, rigourisable…)

If you feel like contributing to this project then please don’t hesitate to fork the github repository!

An introduction to the continuous limit construction I

May 19, 2014

In this post I’d like to begin to explore what is meant by the continuum limit of a quantum lattice system. This post is meant to serve as the first in a series of intuitive overviews of the ideas involved in the open science project “continuous-limits-of-quantum-lattice-systems” hosted on github.

The continuous limit is a power tool in the condensed-matter theorist’s toolkit: by identifying the appropriate effective field theory modelling the low-energy large-scale physics of a complex quantum system one can bring the fully developed apparatus of (perturbative) field theory and the renormalisation group to bear on a problem, often delivering results unavailable via any other means.

Now I’m pretty sure I’m not alone in feeling confused by much of the available physical literature on this topic. Over the past decade I’ve tried to understand the process whereby a field theory is produced to describe a given quantum lattice system. However, up until recently, this has always seemed like a kind of mysterious black magic to me. I know it has to do with symmetries etc. etc.. But this didn’t really help me! I had so many questions. E.g., how exactly does the state of the effective field theory relate to that of the original lattice system? And, for that matter, how do you know what quantities are “fieldlike” and which don’t admit a field-theoretic representation? That is, what has most puzzled me is the quantitative side of things: ideally what I would like is some kind of map which associates, one to one, lattice quantities with field quantities in an operationally transparent way.

Thus I was very excited when I discovered that there is indeed such a map and, further, is naturally associated with the quantum de Finetti theorem. Here I’d like to explain the idea behind this construction using the quantum information theoretic language of exchangeable states.

Yang-Mills theory and tensor networks

April 30, 2014

In my two previous posts I discussed a quantum-information inspired approach to the study of quantum field theory.

So far everything I’ve talked about applies only to standard bosonic quantum fields. However, there are, apart from fermionic quantum fields, another type of quantum field which requires a more careful approach, namely, gauge fields. Read the rest of this entry »

Returning to open science: continuous limits of quantum lattice systems

April 28, 2014

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. Read the rest of this entry »

Quantum field theory: a quantum-information view

April 28, 2014

I always wanted to be a string theorist.

However, my career took a different turn and I ended up a quantum information theorist. Nevertheless, my fascination with particle physics and quantum gravity has never lessened. I suppose this has had a direct impact on what I chose to work on, namely, complex quantum systems and entanglement. Luckily this was to prove a good choice: I got to be part of an excitating revolution in quantum many body theory where entanglement-inspired thinking led to the development of a dazzling array of new variational classes, namely, tensor network states.

Tensor network states have proved to be a very useful tool in trying to reason about the dynamics of many strongly interacting quantum systems because they provide a wonderfully parsimonious description of the degrees of freedom that are actually relevant for observable physics without needing to keep direct track of the exponentially diverging set of complex numbers required to specify a general quantum state in the hilbert space of N particles.

But my first love was always quantum field theory and string theory so I was delighted when, in late 2009, I realised that one could take a continuum limit of a matrix product state (a tensor network state that I’d spent a lot of time working with) to produce a nontrivial quantum field state. With great excitement I worked out the details of the construction and started writing it up for publication. However, as is often the case with a good idea, it occurs to many people at the same time — such ideas are somehow “in the air”. This was no exception: to my astonishment I discovered a paper submitted some two weeks prior entitled “Continuous matrix product states for quantum fields” which read, almost word for word, like my draft. This was doubly embarrassing because the paper had already appeared on the arXiv and I had somehow missed it and, further, it had been written by my good friends and collaborators Frank Verstraete and Ignacio Cirac!

Thus began a very fruitful time where we worked out many of the details of this new approach to quantum field theory: in the past few years myself and my collaborators have been enthusiastically extending cMPS to a variety of settings and along the way developing a continuous limit of the PEPS and MERA tensor network states.

My hope, all this time, has been that these new continuous tensor network states would immediately lead to progress on the major unsolved problems in quantum field theory, just as they had in quantum many body theory. With the arrogance of the ignorant I was convinced we’d solve the problem of “infinities” and usher in a new era of quantum information-inspired quantum field theory.

It turned out not to be so easy.

During the past five years I’ve thrown myself into the deep end of quantum field theory: I realised early on I’d have to go back to basics and actually really learn the subject properly. Along the way I realised that my original naive enthusiasm was somewhat misguided and misdirected. But I also discovered some truly fascinating things that I’m very excited about. I’d like to share some of these with you today; although my initial conception of quantum field theory was indeed wrongheaded, I discovered that quantum information theorists are really very well-suited to think about all sorts of interesting problems in quantum field theory. Read the rest of this entry »

The variational principle, relativistic quantum field theory, and holographic quantum states

June 13, 2010

The variational principle is the basis of a tremendous number of highly successful calculational tools in physics, which is surprising because it is simple enough to cover in an introductory quantum mechanics course. Remember how it works? Pick some wavefunction that, on physical grounds, you believe captures the essential physics of the ground state and, crucially, make sure the wavefunction depends on one or more free parameters. A classic example is the wavefunction

$\displaystyle \psi_a(x) \propto e^{-ax^2}. \ \ \ \ \ (1)$

Now plug this wavefunction into the expression for the energy: ${E(a) = \langle\psi_a|H|\psi_a\rangle/\langle\psi_a|\psi_a\rangle}$, and vary ${a}$ until you reach a minimum value. There is a theorem which guarantees that the optimal value of the energy will always overestimate the correct value and that the closer ${E(a)}$ gets to the true value the better the approximation the wavefunction ${\psi_a(x)}$ is to actual ground-state wavefunction. That’s it.

Of course life is never simple, especially for interacting systems of many particles. The trouble is that it is very hard to come up with a variational wavefunction for which you can actually calculate ${E(a)}$ and which bears some resemblence to the actual ground-state. You need to be extremely clever to design reasonable variational wavefunctions, and it often requires deep new insights into the physics of a system to come up with a good one. Some classic examples include the BCS wavefunction in superconductivity, the Gutzwiller wavefunction, and the Laughlin wave function for the fractional quantum Hall effect (ok the last one is a bit of a cheat, it doesn’t really have any free parameters).

The dominance of the variational principle does not extend, however, to quantum field theory where it has not met with the same success as in other branches of many-body physics. The reason for this was very clearly explained by Feynman in a lecture given at a conference in Wangarooge in Germany in 1987. In this article I want to revisit Feynman’s lecture in the light of recent developments in quantum information theory and condensed matter physics.

In his lecture Feynman described some of his unsuccessful attempts at getting the principle to work. More importantly, he shared the insight he gained during this process into why he believes the principle will not work. While the lecture strikes a rather negative tone, he does actually offer some suggestions as to what he believes would repair the principle. In reading the article I was struck by the prescience of his suggestions. But I’ll get to this later. Read the rest of this entry »