I had to brush up on my Hamilton-Jacobi mechanics to referee a paper. I’d like to share, from this Physics.StackExchange answer, Qmechanic’ clear catalog of the conceptually distinct functions all called “the action” in classical mechanics, taking care to specify their functional dependence:
At least three different quantities in physics are customary called an action and denoted with the letter .
The (off-shell) action
is a functional of the full position curve/path for all times in the interval . See also this question. (Here the words on-shell and off-shell refer to whether the equations of motion (eom) are satisfied or not.)
If the variational problem with well-posed boundary conditions, e.g. Dirichlet boundary conditions
has a unique extremal/classical path , it makes sense to define an on-shell action
which is a function of the boundary values. See e.g. MTW Section 21.1.
The Hamilton’s principal function in Hamilton-Jacobi equation is a function of the position coordinates integration constants , and time , see e.g. H. Goldstein, Classical Mechanics, chapter 10.
The total time derivative
is equal to the Lagrangian on-shell, as explained here. As a consequence, the Hamilton’s principal function can be interpreted as an action on-shell.
These sorts of distinctions are constantly swept under the rug in classical mechanics courses and textbooks (even good books like Goldstein). This leads to serious confusion on the part of the student and, more insidiously, it leads the student to think that this sort of confusion is normal. Ambiguity is baked into the notation! This is a special case of what I conjecture is a common phenomena in physics:
Original researcher thinks deeply, discovers a theory, and writes it down.
… [continue reading]
(This post is vague, and sheer speculation.)
Following a great conversation with Miles Stoudenmire here at PI, I went back and read a paper I forgot about: “Entanglement and the foundations of statistical mechanics” by Popescu et al.S. Popescu, A. Short, and A. Winter, “Entanglement and the foundations of statistical mechanics” Nature Physics 2, 754 – 758 (2006) [Free PDF]. a . This is one of those papers that has a great simple idea, where you’re not sure if it’s profound or trivial, and whether it’s well known or it’s novel. (They cite references 3-6 as “Significant results along similar lines”; let me know if you’ve read any of these and think they’re more useful.) Anyways, here’s some background on how I think about this.
If a pure quantum state is drawn at random (according to the Haar measure) from a -dimensional vector space , then the entanglement entropy
across a tensor decomposition into system and environment is highly likely to be almost the maximum
for any such choice of decomposition . More precisely, if we fix and let , then the fraction of the Haar volume of states that have entanglement entropy more than an exponentially small (in ) amount away from the maximum is suppressed exponentially (in ). This was known as Page’s conjectureD. Page, Average entropy of a subsystem. b , and was later provedS. Foong and S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem. c J. Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem. d ; it is a straightforward consequence of the concentration of measure phenomenon.… [continue reading]
Lemos et al. have a relatively recent letterG. Lemos, V. Borish, G. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, “Quantum imaging with undetected photons”, Nature 512, 409 (2014) [ arXiv:1401.4318 ]. a in Nature where they describe a method of imaging with undetected photons. (An experiment with the same essential quantum features was performed by Zou et al.X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference”, Phys. Rev. Lett. 67, 318 (1991) [ PDF ]. b way back in 1991, but Lemos et al. have emphasized its implications for imaging.) The idea is conceptually related to decoherence detection, and I want to map one onto the other to flesh out the connection. Their figure 1 gives a schematic of the experiment, and is copied below.
Figure 1 from Lemos et al.: ''Schematic of the experiment. Laser light (green) splits at beam splitter BS1 into modes a and b. Beam a pumps nonlinear crystal NL1, where collinear down-conversion may produce a pair of photons of different wavelengths called signal (yellow) and idler (red). After passing through the object O, the idler reflects at dichroic mirror D2 to align with the idler produced in NL2, such that the final emerging idler f does not contain any information about which crystal produced the photon pair. Therefore, signals c and e combined at beam splitter BS2 interfere. Consequently, signal beams g and h reveal idler transmission properties of object O.''
The first two paragraphs of the letter contain all the meat, encrypted and condensed into an opaque nugget of the kind that Nature loves; it stands as a good example of the lamentable way many quantum experimental articles are written.… [continue reading]
In this post I’m going to give a clean definition of idealized quantum Brownian motion and give a few entry points into the literature surrounding its abstract formulation. A follow-up post will give an interpretation to the components in the corresponding dynamical equation, and some discussion of how the model can be generalized to take into account the ways the idealization may break down in the real world.
I needed to learn this background for a paper I am working on, and I was motivated to compile it here because the idiosyncratic results returned by Google searches, and especially this MathOverflow question (which I’ve answered), made it clear that a bird’s eye view is not easy to find. All of the material below is available in the work of other authors, but not logically developed in the way I would prefer.
Quantum Brownian motion (QBM) is a prototypical and idealized case of a quantum system , consisting of a continuous degree of freedom, that is interacting with a large multi-partite environment , in general leading to varying degrees of dissipation, dispersion, and decoherence of the system. Intuitively, the distinguishing characteristics of QBM is Markovian dynamics induced by the cumulative effect of an environment with many independent, individually weak, and (crucially) “phase-space local” components. We will defined QBM as a particular class of ways that a density matrix may evolve, which may be realized (or approximately realized) by many possible system-environment models. There is a more-or-less precise sense in which QBM is the simplest quantum model capable of reproducing classical Brownian motion in a limit.
In words to be explained: QBM is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the evolution is specified by a quadratic Hamiltonian and linear Lindblad operators.… [continue reading]
For the upteenth time I have read a paper introducing the Wigner function essentially like this:
The Wigner-representation of a quantum state is a real-valued function on phase space definedActually, they usually use a more confusing definition. See my post on the intuitive definition of the Wigner function. a (with ) as
It’s sort of like a probability distribution because the marginals reproduce the probabilities for position and momentum measurements:
But the reason it’s not a real probability distribution is that it can be negative.
The fact that can be negative is obviously a reason you can’t think about it as a true PDF, but the marginals property is a terribly weak justification for thinking about as a “quasi-PDF”. There are all sorts of functions one could write down that would have this same property but wouldn’t encode much information about actual phase space structure, e.g., the Jigner“Jess” + “Wigner” = “Jigner”. Ha! b function , which tells as nothing whatsoever about how position relates to momentum.
Here is the real reason you should think the Wigner function is almost, but not quite, a phase-space PDF for a state :
Consider an arbitrary length scale , which determines a corresponding momentum scale and a corresponding setNot just a set of states, actually, but a Parseval tight frame. They have a characteristic spatial and momentum width and , and are indexed by as it ranges over phase space. c of coherent states .
If a measurement is performed on with the POVM of coherent states , then the probability of obtaining outcome is given by the Husimi Q function representation of :
If can be constructed as a mixture of the coherent states , thenOf course, the P function cannot always be defined, and sometimes it can be defined but only if it takes negative values.
… [continue reading]
In May, Losada and Laura wrote a paperM. Losada and R. Laura, Annals of Physics 344, 263 (2014). a pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained historiesGell-Mann and Hartle usually use the term “fine-grained set of histories” to refer to a set generated by the finest possible partitioning of histories in path integral (i.e. a point in space for every point in time), but this is overly specific. As far as the consistent histories framework is concerned, the key mathematical property that defines a fine-grained set is that it’s an exhaustive and exclusive set where each history is constructed by choosing exactly one projector from a fixed orthogonal resolution of the identity at each time. b ). Let the elementary histories be defined by projective decompositions of the identity at time steps (), so that
where are the class operators. Then Losada and Laura showed that the following two conditions are equivalent
The set is consistent“Medium decoherent” in Gell-Mann and Hartle’s terminology. Also note that Losada and Laura actually work with the obsolete condition of “weak decoherence”, but this turns out to be an unimportance difference. For a summary of these sorts of consistency conditions, see my round-up. c for any state: .
The Heisenberg-picture projectors at all times commute: .
However, this is not as general as one would like because assuming the set of histories is elementary is very restrictive. (It excludes branch-dependent sets, sets with inhomogeneous histories, and many more types of sets that we would like to work with.) Luckily, their proof can be extended a bit.… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7.]
A common mistake made by folks newly exposed to the concept of decoherence is to conflate the Schmidt basis with the pointer basis induced by decoherence.… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7.]
Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding“Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper”, although I’m sure there’s another term for this used by philosophers of science.) a .
Here is everything you need to know:
A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160] b
“Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.
… [continue reading]
The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.Here’s an example of a memory effect: An atom immersed in an electromagnetic field can be in one of two states, excited or ground. If it is in an excited state then, during a time interval, it has a certain probability of decaying to the ground state by emitting a photon. If it is in the ground state then it also has a chance of becoming excited by the ambient field. The situation where the atom is in a space of essentially infinite size would be Markovian, because the emitted photon (which embodies a record of the atom’s previous state of excitement) would travel away from the atom never to interact with it again. It might still become excited because of the ambient field, but its chance of doing so isn’t influenced by its previous state. But if the atom is in a container with reflecting walls, then the photon might be reflected back towards the atom, changing the probability that it becomes excited during a later period. a Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.
Given an arbitrary orthonormal basis of the space of operators on the -dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product ), the Lindblad equation takes the following form:
with .… [continue reading]
[This post was originally “Part 1” of my HTTAQM series. However, it’s old, haphazardly written, and not a good starting point. Therefore, I’ve removed it from that series, which now begins with “Measurements are about bases”. Other parts are here: 1,2,3,4,5,6,7. I hope to re-write this post in the future.]
It’s often remarked that the Aharonov–Bohm (AB) effect says something profound about the “reality” of potentials in quantum mechanics. In one version of the relevant experiment, charged particles are made to travel coherently along two alternate paths, such as in a Mach-Zehnder interferometer. At the experimenter’s discretion, an external electromagnetic potential (either vector or scalar) can be applied so that the two paths are at different potentials yet still experience zero magnetic and electric field. The paths are recombined, and the size of the potential difference determines the phase of the interference pattern. The effect is often interpreted as a demonstration that the electromagnetic potential is physically “real”, rather than just a useful mathematical concept.
The magnetic Aharanov-Bohm effect. The wavepacket of an electron approaches from the left and is split coherently over two paths, L and R. The red solenoid in between contains magnetic flux
. The region outside the solenoid has zero field, but there is a non-zero curl to the vector potential as measured along the two paths. The relative phase between the L and R wavepackets is given by
However, Vaidman recently pointed out that this is a mistaken interpretation which is an artifact of the semi-classical approximation used to describe the AB effect. Although it is true that the superposed test charges experience zero field, it turns out that the source charges creating that macroscopic potential do experience a non-zero field, and that the strength of this field is dependent on which path is taken by the test charges.… [continue reading]
[Added 2015-1-30: The paper is now in print and has appeared in the popular press.]
One criticism I’ve had to address when proselytizing the indisputable charms of using decoherence detection methods to look at low-mass dark matter (DM) is this: I’ve never produced a concrete model that would be tested. My analysis (arXiv:1212.3061) addressed the possibility of using matter interferometry to rule out a large class of dark matter models characterized by a certain range for the DM mass and the nucleon-scattering cross section. However, I never constructed an explicit model as a representative of this class to demonstrate in detail that it was compatible with all existing observational evidence. This is a large and complicated task, and not something I could accomplish on my own.
I tried hard to find an existing model in the literature that met my requirements, but without luck. So I had to argue (with referees and with others) that this was properly beyond the scope of my work, and that the idea was interesting enough to warrant publication without a model. This ultimately was successful, but it was an uphill battle. Among other things, I pointed out that new experimental concepts can inspire theoretical work, so it is important that they be disseminated.
I’m thrilled to say this paid off in spades. Bateman, McHardy, Merle, Morris, and Ulbricht have posted their new pre-print “On the Existence of Low-Mass Dark Matter and its Direct Detection” (arXiv:1405.5536). Here is the abstract:
Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB).
… [continue reading]
I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.I posted the answer on Physics.SE too since they encourage the answering of one’s own question. How lonely is that?!? a
Q: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?
A: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.
The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which , , and are all approximately diagonal in the basis of wavepackets, but is not, where is the unitary operator which maps
(This operator creates a Schrodinger’s cat state by reflecting about .)
For two different coherent states and , we want to require an approximately diagonal operator to satisfy , but we only want to do this if . For , we sensibly expect to be within the eigenspectrum of .
One might consider the negativity of the Wigner-Weyl transformCase has a pleasingly gentle introduction. b of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]
[Edit: Scott Aaronson has posted on his blog with extensive criticism of Integrated Information Theory, which motivated Tegmark’s paper.]
Max Tegmark’s recent paper entitled “Consciousness as a State of Matter” has been making the rounds. See especially Sabine Hossenfelder’s critique on her blog that agrees in several places with what I say below.
Tegmark’s paper didn’t convince me that there’s anything new here with regards to the big questions of consciousness. (In fairness, I haven’t read the work of neuroscientist Giulio Tononi that motivated Tegmark’s claims). However, I was interested in what he has to say about the proper way to define subsystems in a quantum universe (i.e. to “carve reality at its joints”) and how this relates to the quantum-classical transition. There is a sense in which the modern understanding of decoherence simplifies the vague questions “How does (the appearance of) a classical world emerge in a quantum universe? ” to the slightly-less-vague question “what are the preferred subsystems of the universe, and how do they change with time?”. Tegmark describes essentially this as the “quantum factorization problem” on page 3. (My preferred formulation is as the “set-selection problem” by Dowker and Kent. Note that this is a separate problem from the origin of probability in quantum mechanicsThe problem of probability as described by Weinberg: “The difficulty is not that quantum mechanics is probabilistic—that is something we apparently just have to live with. The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.” HT Steve Hsu. a .)
Therefore, my comments are going to focus on the “object-level” calculations of the paper, and I won’t have much to say about the implications for consciousness except at the very end.… [continue reading]
Max Schlosshauer has a new review of decoherence and how it relates to understanding the quantum-classical transition. The abstract is:
I give a pedagogical overview of decoherence and its role in providing a dynamical account of the quantum-to-classical transition. The formalism and concepts of decoherence theory are reviewed, followed by a survey of master equations and decoherence models. I also discuss methods for mitigating decoherence in quantum information processing and describe selected experimental investigations of decoherence processes.
I found it very concise and clear for its impressive breadth, and it has extensive cites to the literature. (As you may suspect, he cites me and my collaborators generously!) I think this will become one of the go-to introductions to decoherence, and I highly recommend it to beginners.
Other introductory material is Schlosshauer’s textbook and RMP (quant-ph/0312059), Zurek’s RMP (quant-ph/0105127) and Physics Today article, and the textbook by Joos et al.… [continue reading]
When two initially uncorrelated quantum systems interact through a weak coupling, no entanglement is generated at first order in the coupling constant. This is a useful and very easy to prove fact that I haven’t seen pointed out anywhere, although I assume someone has. I’d love a citation reference if you have one.
Suppose two systems and evolve under where the Hamiltonian coupling them is of the form
with and as usual. We’ll show that when the systems start out uncorrelated, , they remain unentangled (and therefore, since the global state is pure, uncorrelated) to first order in . First, note that local unitaries cannot change the entanglement, so without loss of generality we can consider the modified unitary
which peels off the unimportant local evolution of and . Then the Baker–Campbell–Hausdorff formula gives
where the first few ‘s are given by
The key feature here is that every commutators in each of the ‘s contains at least one copy of , i.e. all the ‘s are at least first order in . That allows us to write
for some new that is independent of . Then we note just that a general Hamiltonian cannot produce entanglement to first order:
This is potentially a very important (negative) result when considering decoherence detection of very weakly coupled particles. If the coupling is so small that terms beyond first order are negligible (e.g. relic neutrinos), then there is no hope of being sensitive to any decoherence.
Of course, non-entangling (unitary) effect may be important. Another way to say this result is: Two weakly coupled systems act only unitarily on each other to first order in the coupling constant.… [continue reading]