Can boundaries and Hamiltonians get along?

by J. Fernando Barbero G., Benito A. Juárez-Aubry, Juan Margalef-Bentabol and
Eduardo J. S. Villaseñor.

Boundaries are ubiquitous in physics, if anything because most material objects tend to have one… In the context of gravity they play a number of interesting roles: from the definition of conserved quantities in asymptotically flat spacetimes to holography (no lasers here, sorry) or the modeling of black holes. A natural question in the context of the canonical quantization of gravitational theories is how to obtain their Hamiltonian description – in particular the constraints – in the presence of boundaries.


Eduardo and Fernando trying to get inspiration for the new group logo

“Juan, can you hear us?”

“Yes, the connection seems to be working better these days.”

“Hi Benito. Good to see you! Can you also see us?”

“Sorry for the delay, it is early in the morning here. Yeah, I can see you perfectly. Hi, Eduardo and Fernando! Hi, Juan!”

“Hi, there!”

“So, as I told you in my last email, we have to write this CQG+ Insight piece about our traces paper. You know, it should be informative, informal and infused with deep physical insights, so, any suggestions? — Yes, Fernando.”

“Well, you have mentioned several things in the abstract but many other questions come to mind, for instance: To what extent can boundary conditions be interpreted as constraints? Are they first class or second class? How comfortably can physical degrees of freedom live on boundaries?”

“Also, we should mention that the model that inspires the paper – the string with the masses – can be used as a toy model to understand how detectors work in quantum field theories in curved spacetimes. Ah! and also in quantum information and all that stuff.”

“Yes, Benito, you are right. Juan, you were saying…?”

“Just that we should also insist on the importance of geometric methods and, also, emphasize that physical degrees of freedom may or may not live on boundaries.”

“So, what about this (I have written some draft paragraphs that, hopefully, incorporate all your suggestions):”

In order to answer these questions our group has been studying for some time now a number of simple systems in bounded regions. Our goal has been to obtain a detailed description of the corresponding phase spaces, their geometry, the identification of the relevant constraints and their geometric properties. This is not as easy as it sounds because the configuration spaces of field theories are infinite dimensional function spaces (actually infinite dimensional manifolds), which tend to be quite formidable beasts to tame. As one can easily find out when applying the usual methods to this task (i.e. the good, old and trustworthy Dirac algorithm) one quickly runs into trouble as Poisson bracket deltas and boundaries do not get along very well.

“Well, it sounds good”

“Thanks, Fernando. Let’s go ahead.”

A neat way to avoid the difficulties that arise in the straightforward  implementation of Dirac’s ideas is to resort to purely geometric methods, in particular the approach devised by Gotay, Nester and Hinds (imaginatively labeled as the GNH method) some time ago. In any case it is fair to say that a geometric interpretation of the original ideas due to Dirac can be made to work (you can find this in Gotay’s thesis) and, also, the covariant phase space methods provide an efficient way to obtain information (if one is willing to sidestep the detailed description of the spaces of solutions to the field equations).

In the paper we expand a simple model – that we have studied in the past – consisting of a string tied to a pair of masses, themselves connected to springs. One of its (relatively) surprising features is that no classical degrees of freedom are associated with the masses because their positions are given by the configuration of the string (continuity!). When one Fock-quantizes the system this reflects on the fact that the resulting Hilbert space cannot be factorized in a natural way as a tensor product of Hilbert spaces associated with the string and the masses.

“Don’t forget to mention the traces!”

“Don’t be impatient, Juan, there is still another paragraph! Here it goes!”

In order to describe the quantum dynamics of the masses one can use the so called trace operators (not to be confused with the usual traces of matrices or operators) and lift them from the one-particle Hilbert space to the full Fock space of the system. The details on this can be found in our paper.


Juan summoning a hyperbolic dodecahedron.

“So, anything else?”

“Well, we could add that we are implementing ideas developed by Marsden and collaborators in the geometric description of elasticity. Maybe say something about the role of the measures that we use to write the fundamental elliptic operators that define the system and, also, say that our models resemble QFT’s in curved space times but, instead of generalizing the flat Laplacian by using non-trivial metrics, we introduce non-trivial measures.”

“Yes, Juan, you are right.”

“So, what do you think? Don’t you find it too boring?”

“Yeah, a little bit but, this is us…”

“Ok, then, I take notice of your enthusiastic approval. We should wrap it up now. One last thing: Any ideas to catch people’s attention?”

“Try writing it in Spenserian stanzas.

“Why not tell the readers to look up Tupper’s formula and just send a bunch of very long numbers to CQG?”

“And what about a 800 words-long palindromic text?”

“Ok! Ok! Benito, Juan, Fernando, thank you for your very useful and easy to implement suggestions, but I was after something a little bit more complicated… By the way, we could claim that our title is a good exception to Hinchliffe’s rule, I am sure this will blow everybody’s minds!


The Teotihuacán pyramids in México paying a visit to Benito.

So, I guess I have enough material. I will try now to put it in writing in a readable form and send you a draft in a couple of hours. Ah! don’t forget to email some pictures of you. According to the instructions, they should be striking but, please, behave! (some of them will be, I’m pretty sure…) Bye!”

“Yes, bye, bye!”


Note for the readers: In case anyone is wondering… No, we do not use English when we talk to each other.

Don’t waste a second! You can read the paper right away by following this link
Boundary Hilbert spaces and trace operators, Class. Quantum Grav. 34 (2017) 095005
Other related papers can be found in our group webpage.

About the authors:

The members of our intercontinental collaboration are: J. Fernando Barbero G., Research Scientist at the Instituto de Estructura de la Materia-CSIC (Spanish Research Council) and Eduardo J. S. Villaseñor, Associate Professor at the Universidad Carlos III de Madrid. Both co-chair the Grupo de Teorías de Campos y Física Estadística, (Universidad Carlos III de Madrid, Unidad Asociada al Instituto de Estructura de la Materia, CSIC).

Benito A. Juárez-Aubry is currently a postdoc at the Departamento de Gravitación y
Teoría de Campos, Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de

Finally, Juan Margalef-Bentabol is close to getting his Ph.D. at the Universidad Carlos
III de Madrid.

Read the full article in Classical and Quantum Gravity:
Boundary Hilbert spaces and trace operators,
Barbero, J. Fernando G., et al 2017. Class. Quantum Grav. 34, 095005.

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