## Symmetries and solutions

Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).a  . Consider the space of all possible trajectories for a system, a real-valued Lagrangian functional on that space, the “directions” at each point, and the corresponding functional gradient in each direction. Classical solutions are exactly those trajectories such that the Lagrangian is stationary for perturbations in any direction , and continuous symmetries are exactly those directions such that the Lagrangian is stationary for any trajectory . That is,

(1)

There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry , being a tangent direction on the manifold of trajectories, can vary with the tangent point it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.

### Footnotes

(↵ returns to text)

1. You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).

## How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

[Other parts in this series: 1,2,3,4,5,6,7,8.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well.[continue reading]

## Reeh–Schlieder property in a separable Hilbert space

As has been discussed here before, the Reeh–Schlieder theorem is an initially confusing property of the vacuum in quantum field theory. It is difficult to find an illuminating discussion of it in the literature, whether in the context of algebraic QFT (from which it originated) or the more modern QFT grounded in RG and effective theories. I expect this to change once more field theorists get trained in quantum information.

The Reeh–Schlieder theorem states that the vacuum is cyclic with respect to the algebra of observables localized in some subset of Minkowski space. (For a single field , the algebra is defined to be generated by all finite smearings for with support in .) Here, “cyclic” means that the subspace is dense in , i.e., any state can be arbitrarily well approximated by a state of the form with . This is initially surprising because could be a state with particle excitations localized (essentially) to a region far from and that looks (essentially) like the vacuum everywhere else. The resolution derives from the fact the vacuum is highly entangled, such that the every region is entangled with every other region by an exponentially small amount.

One mistake that’s easy to make is to be fooled into thinking that this property can only be found in systems, like a field theory, with an infinite number of degrees of freedom. So let me exhibitMost likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code…a   a quantum state with the Reeh–Schlieder property that lives in the tensor product of a finite number of separable Hilbert spaces:

As emphasized above, a separable Hilbert space is one that has a countable orthonormal basis, and is therefore isomorphic to , the space of square-normalizable functions.… [continue reading]

## Legendre transform

The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I’ve mentioned before that the preface of Sussman and Wisdom’s Structure and Interpretation of Classical Mechanics is a good starting point for thinking about these issues. As a pointed example, in this blog post I’ll look at how badly the Legendre transform is taught in standard textbooks,I was pleased to note as this essay went to press that my choice of Landau, Goldstein, and Arnold were confirmed as the “standard” suggestions by the top Google results.a   and compare it to how it could be taught. In a subsequent post, I’ll used this as a springboard for complaining about the way we record and transmit physics knowledge.

Before we begin: turn away from the screen and see if you can remember what the Legendre transform accomplishes mathematically in classical mechanics.If not, can you remember the definition? I couldn’t, a month ago.b   I don’t just mean that the Legendre transform converts the Lagrangian into the Hamiltonian and vice versa, but rather: what key mathematical/geometric property does the Legendre transform have, compared to the cornucopia of other function transforms, that allows it to connect these two conceptually distinct formulations of mechanics?

(Analogously, the question “What is useful about the Fourier transform for understanding translationally invariant systems?” can be answered by something like “Translationally invariant operations in the spatial domain correspond to multiplication in the Fourier domain” or “The Fourier transform is a change of basis, within the vector space of functions, using translationally invariant basis elements, i.e.,… [continue reading]

## Toward relativistic branches of the wavefunction

I prepared the following extended abstract for the Spacetime and Information Workshop as part of my continuing mission to corrupt physicists while they are still young and impressionable. I reproduce it here for your reading pleasure.

Finding a precise definition of branches in the wavefunction of closed many-body systems is crucial to conceptual clarity in the foundations of quantum mechanics. Toward this goal, we propose amplification, which can be quantified, as the key feature characterizing anthropocentric measurement; this immediately and naturally extends to non-anthropocentric amplification, such as the ubiquitous case of classically chaotic degrees of freedom decohering. Amplification can be formalized as the production of redundant records distributed over spatial disjoint regions, a certain form of multi-partite entanglement in the pure quantum state of a large closed system. If this definition can be made rigorous and shown to be unique, it is then possible to ask many compelling questions about how branches form and evolve.

A recent result shows that branch decompositions are highly constrained just by this requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. However, this result is grounded in a preferred tensor decomposition into independent microscopic subsystems associated with spatial locality. This structure breaks down in a relativistic setting on scales smaller than the Compton wavelength of the relevant field. Indeed, a key insight from algebraic quantum field theory is that finite-energy states are never exact eigenstates of local operators, and hence never have exact records that are spatially disjoint, although they can approximate this arbitrarily well on large scales.… [continue reading]

## Branches and matrix-product states

I’m happy to use this bully pulpit to advertise that the following paper has been deemed “probably not terrible”, i.e., published.

When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops “branches”: a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.

Here’s the figureThe editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better!a   and caption:

Spatially disjoint regions with the same coloring (e.g., the solid blue regions ) denote different records for the same observable (e.g., ).

## Comments on Cotler, Penington, & Ranard

One way to think about the relevance of decoherence theory to measurement in quantum mechanics is that it reduces the preferred basis problem to the preferred subsystem problem; merely specifying the system of interest (by delineating it from its environment or measuring apparatus) is enough, in important special cases, to derive the measurement basis. But this immediately prompts the question: what are the preferred systems? I spent some time in grad school with my advisor trying to see if I could identify a preferred system just by looking at a large many-body Hamiltonian, but never got anything worth writing up.

I’m pleased to report that Cotler, Penington, and Ranard have tackled a closely related problem, and made a lot more progress:

Locality from the Spectrum
Jordan S. Cotler, Geoffrey R. Penington, Daniel H. Ranard
Essential to the description of a quantum system are its local degrees of freedom, which enable the interpretation of subsystems and dynamics in the Hilbert space. While a choice of local tensor factorization of the Hilbert space is often implicit in the writing of a Hamiltonian or Lagrangian, the identification of local tensor factors is not intrinsic to the Hilbert space itself. Instead, the only basis-invariant data of a Hamiltonian is its spectrum, which does not manifestly determine the local structure. This ambiguity is highlighted by the existence of dualities, in which the same energy spectrum may describe two systems with very different local degrees of freedom. We argue that in fact, the energy spectrum alone almost always encodes a unique description of local degrees of freedom when such a description exists, allowing one to explicitly identify local subsystems and how they interact.