Moyal’s equation for a Wigner function of a quantum system with (Wigner-transformed) Hamiltonian is where the Moyal bracket is a binary operator on the space of functions over phase space. Unfortunately, it is often written down mysteriously as
where the arrows over partial derivatives tell you which way they act, i.e., . This only becomes slightly less weird when you use the equivalent formula , where “” is the Moyal star product given by
The star product has the crucial feature that , where we use a hat to denote the Weyl transform (i.e., the inverse of the Wigner transform taking density matrices to Wigner functions), which takes a scalar function over phase-space to an operator over our Hilbert space. The star product also has some nice integral representations, which can be found in books like Curtright, Fairlie, & ZachosThe complete 88-page PDF is here.a , but none of them help me understand the Moyal equation.
A key problem is that both of these expressions are neglecting the (affine) symplectic symmetry of phase space and the dynamical equations. Although I wouldn’t call it beautiful, we can re-write the star product as
where is a symplectic index using the Einstein summation convention, and where symplectic indices are raised and lowered using the symplectic form just as for Weyl spinors: and , where is the antisymmetric symplectic form with , and where upper (lower) indices denote symplectic vectors (co-vectors).
With this, we can expand the Moyal equation as
where we can see in hideous explicitness that it’s a series in the even powers of and the odd derivates of the Hamiltonian and the Wigner function .… [continue reading]
This post is (a seed of) a bibliography covering the primordial research area that goes by some of the following names:
Although the way this problem tends to be formalized varies with context, I don’t think we have confidence in any of the formalizations. The different versions are very tightly related, so that a solution in one context is likely give, or at least strongly point toward, solutions for the others.
As a time-saving device, I will just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. I hope to update this from time to time, and perhaps turn it into a proper review article of its own one day. If you have a recommendation for this bibliography (either a single citation, or a paper I should quote), please do let me know.
Carroll & Singh
From “Quantum Mereology: Factorizing Hilbert Space into Subsystems with Quasi-Classical Dynamics”, arXiv:2005.12938:
While this question has not frequently been addressed in the literature on quantum foundations and emergence of classicality, a few works have highlighted its importance and made attempts to understand it better. Brun and Hartle  studied the emergence of preferred coarse-grained classical variables in a chain of quantum harmonic oscillators. Efforts to address the closely related question of identifying classical set of histories (also known as the “Set Selection” problem) in the Decoherent Histories formalism [3–7, 10] have also been undertaken.
… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7,8.]
On microscopic scales, sound is air pressure fluctuating in time . Taking the Fourier transform of gives the frequency distribution , but in an eternal way, applying to the entire time interval for .
Yet on macroscopic scales, sound is described as having a frequency distribution as a function of time, i.e., a note has both a pitch and a duration. There are many formalisms for describing this (e.g., wavelets), but a well-known limitation is that the frequency of a note is only well-defined up to an uncertainty that is inversely proportional to its duration .
At the mathematical level, a given wavefunction is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction . The mapping of the analogyI am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal the “musical score” of .a is . Wavefunctions can of course be complex, but we can restrict ourself to a real-valued wavefunction without any trouble; we are not worrying about the dynamics of wavefunctions, so you can pretend the Hamiltonian vanishes if you like.
In order to get the acoustic analog of Planck’s constant , it helps to imagine going back to a time when the pitch of a note was measured with a unit that did not have a known connection to absolute frequency, i.e.,… [continue reading]
Physicists often define a Lindbladian superoperator as one whose action on an operator can be written as
for some operator with positive anti-Hermitian part, , and some set of operators . But how does one efficiently check if a given superoperator is Lindbladian? In this post I give an “elementary” proof of a less well-known characterization of Lindbladians:
Thus, we can efficiently check if an arbitrary superoperator is Lindbladian by diagonalizing and seeing if all the eigenvalues are positive.
A quick note on terminology
The terms superoperator, completely positive (CP), trace preserving (TP), and Lindbladian are defined below in Appendix A in case you aren’t already familiar with them.
Confusingly, the standard practice is to say a superoperator is “positive” when it is positivity preserving: . This condition is logically independent from the property of a superoperator being “positive” in the traditional sense of being a positive operator, i.e., for all operators (matrices) , where
is the Hilbert-Schmidt inner product on the space of matrices. We will refer frequently to this latter condition, so for clarity we call it op-positivity, and denote it with the traditional notation .
It is reasonably well known by physicists that Lindbladian superoperators, Eq. (1), generate CP time evolution of density matrices, i.e., is completely positive when and satisfies Eq.… [continue reading]
Curtright et al. have a monograph on the phase-space formulation of quantum mechanics. I recommend reading their historical introduction.
Wigner’s quasi-probability distribution function in phase-space is a special (Weyl–Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics, nuclear physics, quantum computing, decoherence, and chaos. It is also of importance in signal processing, and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory: The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways.
Here are some quotes. First, the phase-space formulation should be placed on equal footing with the Hilbert-space and path-integral formulations:
When Feynman first unlocked the secrets of the path integral formalism and presented them to the world, he was publicly rebuked: “It was obvious”, Bohr said, “that such trajectories violated the uncertainty principle”.
However, in this case, Bohr was wrong. Today path integrals are universally recognized and widely used as an alternative framework to describe quantum behavior, equivalent to although conceptually distinct from the usual Hilbert space framework, and therefore completely in accord with Heisenberg’s uncertainty principle…
Similarly, many physicists hold the conviction that classical-valued position and momentum variables should not be simultaneously employed in any meaningful formula expressing quantum behavior, simply because this would also seem to violate the uncertainty principle…However, they too are wrong.
… [continue reading]
The implacable Aspelmeyer group in Vienna announced a gnarly achievement in November (recently published):
We report quantum ground state cooling of a levitated nanoparticle in a room temperature environment. Using coherent scattering into an optical cavity we cool the center of mass motion of a
nm diameter silica particle by more than
orders of magnitude to
phonons along the cavity axis, corresponding to a temperature of
μK. We infer a heating rate of
kHz, which results in a coherence time of
μs – or
coherent oscillations – while the particle is optically trapped at a pressure of
mbar. The inferred optomechanical coupling rate of
kHz places the system well into the regime of strong cooperativity (
). We expect that a combination of ultra-high vacuum with free-fall dynamics will allow to further expand the spatio-temporal coherence of such nanoparticles by several orders of magnitude, thereby opening up new opportunities for macroscopic quantum experiments.
Ground-state cooling of nanoparticles in laser traps is a very important milestone on the way to producing large spatial superpositions of matter, and I have a long-standing obsession with the possibility of using such superpositions to probe for the existence of new particles and forces like dark matter. In this post, I put this milestone in a bit of context and then and then toss up a speculative plot for the estimated dark-matter sensitivity of a follow-up to Delić et al.’s device.
One way to organize the quantum states of a single continuous degree of freedom, like the center-of-mass position of a nanoparticle, is by their sensitivity to displacements in phase space.… [continue reading]
Our paper discussed in the previous blog post might prompt this question: Is there still a way to use Landauer’s principle to convert the free energy of a system to its bit erasure capacity? The answer is “yes”, which we can demonstrate with a simple argument.
Summary: The correct measure of bit-erasure capacity N for an isolated system is the negentropy, the difference between the system’s current entropy and the entropy it would have if allowed to thermalize with its current internal energy. The correct measure of erasure capacity for a constant-volume system with free access to a bath at constant temperature is the Helmholtz free energy (divided by , per Landauer’s principle), provided that the additive constant of the free energy is set such that the free energy vanishes when the system thermalizes to temperature . That is,
where and are the internal energy and entropy of the system if it were at temperature . The system’s negentropy lower bounds this capacity, and this bound is saturated when .
Traditionally, the Helmholtz free energy of a system is defined as , where and are the internal energy and entropy of the system and is the constant temperature of an external infinite bath with which the system can exchange energy.Here, there is a factor of Boltzmann’s constant in front of because I am measuring the (absolute) entropy in dimensionless bits rather than in units of energy per temperature. That way we can write things like .a (I will suppress the “Helmholtz” modifier henceforth; when the system’s pressure rather than volume is constant, my conclusion below holds for the Gibbs free energy if the obvious modifications are made.)… [continue reading]
People often say to me “Jess, all this work you do on the foundations of quantum mechanics is fine as far as it goes, but it’s so conventional and safe. When are you finally going to do something unusual and take some career risks?” I’m now pleased to say I have a topic to bring up in such situations: the thermodynamic incentives of powerful civilizations in the far future who seek to perform massive computations. Anders Sandberg, Stuart Armstrong, and Milan M. Ćirković previously argued for a surprising connection between Landauer’s principle and the Fermi paradox, which Charles Bennett, Robin Hanson, and I have now critiqued. Our comment appeared today in the new issue of Foundations of Physics:
In their article [arXiv:1705.03394
], 'That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradox', Sandberg et al. try to explain the Fermi paradox (we see no aliens) by claiming that Landauer's principle implies that a civilization can in principle perform far more (~1030
times more) irreversible logical operations (e.g., error-correcting bit erasures) if it conserves its resources until the distant future when the cosmic background temperature is very low. So perhaps aliens are out there, but quietly waiting. Sandberg et al. implicitly assume, however, that computer-generated entropy can only be disposed of by transferring it to the cosmological background. In fact, while this assumption may apply in the distant future, our universe today contains vast reservoirs and other physical systems in non-maximal entropy states, and computer-generated entropy can be transferred to them at the adiabatic conversion rate of one bit of negentropy to erase one bit of error.
… [continue reading]