Albert And Curvature.

Posed 17 years ago, the bounded L2 curvature conjecture has finally been proved by a group of three researchers at the Laboratoire Jacques-Louis Lions (CNRS / UPMC / Université Paris Diderot) and Princeton University. It provides a potentially minimal framework in which it is possible to solve the Einstein equations, which in turn could be a critical step toward the proof of major conjectures, such as Penrose’s cosmic censorship conjectures. This work has appeared in Inventiones Mathematicae.

Even though 2015 marked its 100th anniversary, Albert Einstein’s theory of general relativity still holds its share of mysteries. This theory of gravitation stipulates that matter curves spacetime in proportion to the mass of the object. This phenomenon is measured using a mathematical tool called the curvature tensor, on which the bounded L2 curvature conjecture focuses to find possible frameworks for making sense of solutions to Einstein’s equations. Proposed 17 years ago by Sergiu Klainerman, this conjecture has at last been demonstrated by Sergiu Klainerman, Igor Rodnianski and Jérémie Szeftel.

The bounded L2 curvature conjecture stipulates that Einstein’s equations admit a solution if, at the initial time, the space curvature tensor is square-integrable – in other words, if the integral of its square is a finite number. This resolution of the bounded L2 curvature conjecture is important because it is a potential step towards the proof of Penrose’s famous cosmic censorship conjectures, which concerns gravitational singularities: pathological regions of spacetime where the gravitational field becomes infinite, like in the centre of a black hole. The presence of such cases in the solutions to Einstein’s equations could challenge the physical validity of general relativity.

Roger Penrose posits that these singularities are never visible because they are generically hidden behind the event horizon: the region of a black hole from which light cannot escape and become perceptible to observers. Although much remains to be learnt about these phenomena, the equations that govern them are now somewhat less mysterious in light of this new proof.


Quantum Psychology

I’m not irrational, I’m just obeying the laws of quantum physics.

A new trend taking shape in psychological science not only uses quantum physics to explain humans’ sometimes paradoxical thinking, but may also help researchers resolve certain contradictions among the results of previous psychological studies.

According to Zheng Joyce Wang and others who try to model our decision-making processes mathematically, the equations and axioms that most closely match human behaviour may be ones that are rooted in quantum physics.

“We have accumulated so many paradoxical findings in the field of cognition, and especially in decision-making,” said Wang, who is an associate professor of communication and director of the Communication and Psychophysiology Lab at The Ohio State University.

“Whenever something comes up that isn’t consistent with classical theories, we often label it as ‘irrational.’ But from the perspective of quantum cognition, some findings aren’t irrational anymore. They’re consistent with quantum theory and with how people really behave.”

In two review papers in academic journals, Wang and her colleagues spell out their new theoretical approach to psychology. One paper appears in Current Directions in Psychological Science, and the other in Trends in Cognitive Sciences.

Their work suggests that thinking in a quantum-like way, essentially not following a conventional approach based on classical probability theory, enables humans to make important decisions in the face of uncertainty and lets us confront complex questions despite our limited mental resources.

When researchers try to study human behaviour using only classical mathematical models of rationality, some aspects of human behaviour do not compute. From the classical point of view, those behaviours seem irrational, Wang explained.

For instance, scientists have long known that the order in which questions are asked on a survey can change how people respond, an effect previously thought to be due to vaguely labelled effects, such as “carry-over effects” and “anchoring and adjustment,” or noise in the data. Survey organisations normally change the order of questions between respondents, hoping to cancel out this effect. But in the Proceedings of the National Academy of Sciences last year, Wang and collaborators demonstrated that the effect can be precisely predicted and explained by a quantum-like aspect of people’s behaviour.

We usually think of quantum physics as describing the behaviour of sub-atomic particles, not the behaviour of people. But the idea is not so far-fetched, Wang said. She also emphasised that her research program neither assumes nor proposes that our brains are literally quantum computers. Other research groups are working on that idea; Wang and her collaborators are not focusing on the physical aspects of the brain, but rather on how abstract mathematical principles of quantum theory can shed light on human cognition and behaviours.

“In the social and behavioural sciences as a whole, we use probability models a lot,” she said. “For example, we ask what is the probability that a person will act a certain way or make a certain decision? Traditionally, those models are all based on classical probability theory, which arose from the classical physics of Newtonian systems. So it’s really not so exotic for social scientists to think about quantum systems and their mathematical principles, too.”

Quantum physics deals with ambiguity in the physical world. The state of a particular particle, the energy it contains, its location, all are uncertain and have to be calculated in terms of probabilities.

Quantum cognition is what happens when humans have to deal with ambiguity mentally. Sometimes we aren’t certain about how we feel, or we feel ambiguous about which option to choose, or we have to make decisions based on limited information.

“Our brain can’t store everything. We don’t always have clear attitudes about things. But when you ask me a question, like ‘What do you want for dinner?” I have to think about it and come up with or construct a clear answer right there,” Wang said. “That’s quantum cognition.”

“I think the mathematical formalism provided by quantum theory is consistent with what we feel intuitively as psychologists. Quantum theory may not be intuitive at all when it is used to describe the behaviours of a particle, but actually is quite intuitive when it is used to describe our typically uncertain and ambiguous minds.”

She used the example of Schrödinger’s cat, the thought experiment in which a cat inside a box has some probability of being alive or dead. Both possibilities have potential in our minds. In that sense, the cat has a potential to become dead or alive at the same time. The effect is called quantum superposition. When we open the box, both possibilities are no longer superimposed and the cat must be either alive or dead.

With quantum cognition, it’s as if each decision we make is our own unique Schrödinger’s cat.

As we mull over our options, we envision them in our mind’s eye. For a time, all the options co-exist with different degrees of potential that we will choose them: That’s superposition. Then, when we zero in on our preferred option, the other options cease to exist for us.

The task of modelling this process mathematically is difficult in part because each possible outcome adds dimensions to the equation. For instance, a Republican who tried to decide among the candidates for U.S. president in 2016 confronted a high-dimensional problem with almost 20 candidates. Open-ended questions, such as “How do you feel?” have even more possible outcomes and more dimensions.

With the classical approach to psychology, the answers might not make sense, and researchers have to construct new mathematical axioms to explain behaviour in that particular instance. The result: There are many classical psychological models, some of which are in conflict, and none of which apply to every situation.

With the quantum approach, Wang and her colleagues argued, many different and complex aspects of behaviour can be explained with the same limited set of axioms. The same quantum model that explains how question order changes people’s survey answers also explains violations of rationality in the prisoner’s dilemma paradigm, an effect in which people cooperate even when it’s in their best interest not to do so.

“The prisoner’s dilemma and question order are two completely different effects in classical psychology, but they both can be explained by the same quantum model,” Wang said. “The same quantum model has been used to explain many other seemingly unrelated, puzzling findings in psychology. That’s elegant.”


Humble Pi

Old From New.

In 1655 the English mathematician John Wallis published a book in which he derived a formula for pi as the product of an infinite series of ratios. Now researchers from the University of Rochester, in a surprise discovery, have found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.

“We weren’t looking for the Wallis formula for pi. It just fell into our laps,” said Carl Hagen, a particle physicist at the University of Rochester. Having noticed an intriguing trend in the solutions to a problem set he had developed for students in a class on quantum mechanics, Hagen recruited mathematician Tamar Friedmann and they realised this trend was in fact a manifestation of the Wallis formula for pi.

“It was a complete surprise; I jumped up and down when we got the Wallis formula out of equations for the hydrogen atom,” said Friedmann. “The special thing is that it brings out a beautiful connection between physics and math. I find it fascinating that a purely mathematical formula from the 17th century characterises a physical system that was discovered 300 years later.”

The theory of quantum mechanics dates back to the early 20th century and the Wallis formula has been around for hundreds of years, but the connection between the two had remained hidden until now.

“Nature had kept this secret for the last 80 years,” Friedmann said. “I’m glad we revealed it.”

The researchers report their findings in the Journal of Mathematical Physics, from AIP Publishing.


Big Dark And Mysterious

Using the world’s most powerful telescopes, an international team of astronomers has found a massive galaxy that consists almost entirely of dark matter.

The galaxy, Dragonfly 44, is located in the nearby Coma constellation and had been overlooked until last year because of its unusual composition: It is a diffuse “blob” about the size of the Milky Way, but with far fewer stars.

“Very soon after its discovery, we realized this galaxy had to be more than meets the eye. It has so few stars that it would quickly be ripped apart unless something was holding it together,” said Yale University astronomer Pieter van Dokkum, lead author of a paper in the Astrophysical Journal Letters.

Van Dokkum’s team was able to get a good look at Dragonfly 44 thanks to the W.M. Keck Observatory and the Gemini North telescope, both in Hawaii. Astronomers used observations from Keck, taken over six nights, to measure the velocities of stars in the galaxy. They used the 8-meter Gemini North telescope to reveal a halo of spherical clusters of stars around the galaxy’s core, similar to the halo that surrounds our Milky Way galaxy.

Star velocities are an indication of the galaxy’s mass, the researchers noted. The faster the stars move, the more mass its galaxy will have.

“Amazingly, the stars move at velocities that are far greater than expected for such a dim galaxy. It means that Dragonfly 44 has a huge amount of unseen mass,” said co-author Roberto Abraham of the University of Toronto.

Scientists initially spotted Dragonfly 44 with the Dragonfly Telephoto Array, a telescope invented and built by van Dokkum and Abraham.

Dragonfly 44’s mass is estimated to be 1 trillion times the mass of the Sun, or 2 tredecillion kilograms (a 2 followed by 42 zeros), which is similar to the mass of the Milky Way. However, only one-hundredth of 1% of that is in the form of stars and “normal” matter. The other 99.99% is in the form of dark matter, an hypothesized material that remains unseen but may make up more than 90% of the universe.

The researchers note that finding a galaxy composed mainly of dark matter is not new; ultra-faint dwarf galaxies have similar compositions. But those galaxies were roughly 10,000 times less massive than Dragonfly 44.

“We have no idea how galaxies like Dragonfly 44 could have formed,” said Abraham. “The Gemini data show that a relatively large fraction of the stars is in the form of very compact clusters, and that is probably an important clue. But at the moment we’re just guessing.”

Van Dokkum, the Sol Goldman Family Professor of Astronomy and Physics at Yale, added: “Ultimately what we really want to learn is what dark matter is. The race is on to find massive dark galaxies that are even closer to us than Dragonfly 44, so we can look for feeble signals that may reveal a dark matter particle.”

Additional co-authors are Shany Danieli, Allison Merritt, and Lamiya Mowla of Yale, Jean Brodie of the University of California Observatories, Charlie Conroy of Harvard, Aaron Romanowsky of San Jose State University, and Jielai Zhang of the University of Toronto.


Space Doughnuts

In the 1960s, NASA launched six satellites to study Earth’s atmosphere, magnetosphere and the space between Earth and the moon. Using observations from those satellites, Christopher Russell, a UCLA graduate student at the time, detected mysterious plasma waves in the Van Allen radiation belts, the doughnut-shaped rings surrounding Earth that contain high-energy particles trapped by the planet’s magnetic field.

Referred to as equatorial noise or “Russell noise,” in tribute to Russell who is now a professor of space physics and planetary science at UCLA, the waves are among the most frequently observed emissions in the near-Earth space. But until recently, scientists could not explain how these waves are excited.

Now, after nearly a half century, the mystery has been solved by a team co-led by another UCLA scientist.

Yuri Shprits, a research geophysicist in the UCLA College, and his colleagues discovered the structure of these waves when they are very close to the equator. The scientists observed 13 equally spaced lines measured by two European Space Agency Cluster satellites, and found highly structured wave spectrograms that look like a zebra pedestrian crossing.

“It’s truly remarkable how nature managed to draw such clear, very narrow, and periodic lines in space,” said Shprits, who led the study with Michael Balikhin of the University of Sheffield.

The finding represents a major advance because the high-energy particles can be harmful for satellites and humans in space. The research is reported in the journal Nature Communications.

The European Cluster spacecraft observed ring distributions of protons in space that provide the energy for the plasma waves. Modelling of waves based on these observations provided additional evidence that waves are excited by so-called proton ring distributions.

Scientists have been especially interested in equatorial noise because it can accelerate particles in the Van Allen belts to high energies and cause the particles to disappear into the atmosphere. This phenomenon may have important implications for space weather and may play an important role in the acceleration and scattering of electrons and ions by these waves that can cause problems ranging from minor anomalies to the complete failure of critical satellites. Better understanding of space radiation will be instrumental in better protecting astronauts and equipment, Shprits said.

Shprits added that similar wave generation mechanisms may also be taking place in the magnetospheres of the outer planets, close to the sun and in distant corners of the universe.

Russell, who also is the principal investigator of NASA’s Dawn mission, was pleased with the findings. “It is interesting that with Yuri’s work, almost a half century later, scientists are finally making the measurements in space that explain the surprising observations made in 1966 and reported in my 1968 thesis,” he said. “The waves were a real puzzle, and now they make much more sense.”

The wave modelling was done by Lunjin Chen, who received his doctorate at UCLA in 2011 and is now an assistant professor at the University of Texas, Dallas.


Coming And Going

Where Does The Time Go?

New research from Griffith University’s Centre for Quantum Dynamics is broadening perspectives on time and space.

In a paper published in the journal Proceedings of the Royal Society A, Associate Professor Joan Vaccaro challenges the long-held presumption that time evolution, the incessant unfolding of the universe over time, is an elemental part of Nature.

In the paper, entitled Quantum asymmetry between time and space, she suggests there may be a deeper origin due to a difference between the two directions of time: to the future and to the past.

“If you want to know where the universe came from and where it’s going, you need to know about time,” says Associate Professor Vaccaro.

“Experiments on subatomic particles over the past 50 years ago show that Nature doesn’t treat both directions of time equally.

“In particular, subatomic particles called K and B mesons behave slightly differently depending on the direction of time.

“When this subtle behaviour is included in a model of the universe, what we see is the universe changing from being fixed at one moment in time to continuously evolving.

“In other words, the subtle behaviour appears to be responsible for making the universe move forwards in time.

“Understanding how time evolution comes about in this way opens up a whole new view on the fundamental nature of time itself.

“It may even help us to better understand bizarre ideas such as travelling back in time.”

According to the paper, an asymmetry exists between time and space in the sense that physical systems inevitably evolve over time whereas there is no corresponding ubiquitous translation over space.

This asymmetry, long presumed to be elemental, is represented by equations of motion and conservation laws that operate differently over time and space.

However, Associate Professor Vaccaro used a “sum-over-paths formalism” to demonstrate the possibility of a time and space symmetry, meaning the conventional view of time evolution would need to be revisited.

“In the connection between time and space, space is easier to understand because it’s simply there. But time is forever forcing us towards the future,” says Associate Professor Vaccaro.

“Yet while we are indeed moving forward in time, there is also always some movement backwards, a kind of jiggling effect, and it is this movement I want to measure using these K and B mesons.”

Associate Professor Vaccaro says the research provides a solution to the origin of dynamics, an issue that has long perplexed science.


Through The Looking Glass

It’s What We Don’t Know.

Almost every particle has an antimatter counterpart: a particle with the same mass but opposite charge, among other qualities.

This seems to be true of neutrinos, tiny particles that are constantly streaming through us. Judging by the particles released when a neutrino interacts with other matter, scientists can tell when they’ve caught a neutrino versus an anti-neutrino.

But certain characteristics of neutrinos and anti-neutrinos make scientists wonder: Are they one and the same? Are neutrinos their own antiparticles?

This isn’t unheard of. Gluons and even Higgs bosons are thought to be their own antiparticles. But if scientists discover neutrinos are their own antiparticles, it could be a clue as to where they get their tiny masses and whether they played a part in the existence of our matter-dominated universe.

The idea of the antiparticle came about in 1928 when British physicist Paul Dirac developed what became known as the Dirac equation. His work sought to explain what happened when electrons moved at close to the speed of light. But his calculations resulted in a strange requirement; that electrons sometimes have negative energy.

“When Dirac wrote down his equation, that’s when he learned antiparticles exist,” says André de Gouvêa, a theoretical physicist and professor at Northwestern University. “Antiparticles are a consequence of his equation.”

Physicist Carl Anderson discovered the antimatter partner of the electron that Dirac foresaw in 1932. He called it the positron, a particle like an electron but with a positive charge.

Dirac predicted that, in addition to having opposite charges, antimatter partners should have another opposite feature called chirality, (distinguishable from its mirror image)which represents one of the inherent quantum properties a particle has. A particle can have either a right-handed or left-handed chirality.

Dirac’s equation allowed for neutrinos and anti-neutrinos to be different particles and as a result, four types of neutrino were possible: neutrinos with left- and right-handed chirality and anti-neutrinos with left- and right-handed chirality.

But if the neutrinos had no mass, as scientists thought at the time, only left-handed neutrinos and right-handed anti-neutrinos needed to exist.

In 1937, Italian physicist Ettore Majorana debuted another theory: Neutrinos and anti-neutrinos are actually the same thing. The Majorana equation described neutrinos that, if they happened to have mass after all, could turn into anti-neutrinos and then back into neutrinos again.

Whether neutrino masses were zero remained a mystery until 1998, when the Super-Kamiokande and SNO experiments found they do indeed have very small masses, an achievement recognised with the 2015 Nobel Prize for Physics. Since then, experiments have cropped up across Asia, Europe and North America searching for hints that the neutrino is its own antiparticle.

The key to finding this evidence is something called lepton number conservation. Scientists consider it a fundamental law of nature that lepton number is conserved, meaning that the number of leptons and anti-leptons involved in an interaction should remain the same before and after the interaction occurs.

Scientists think that just after the big bang, the universe should have contained equal amounts of matter and antimatter. The two types of particles should have interacted, gradually cancelling one another until nothing but energy was left behind. Somehow, that’s not what happened.

Finding out that lepton number is not conserved would open up a loophole that would allow for the current imbalance between matter and antimatter. And neutrino interactions could be the place to find that loophole.

Scientists are looking for lepton number violation in a process called double beta decay, says SLAC theorist Alexander Friedland, who specializes in the study of neutrinos.

In its common form, double beta decay is a process in which a nucleus decays into a different nucleus and emits two electrons and two anti-neutrinos. This balances leptonic matter and antimatter both before and after the decay process, so it conserves lepton number.

If neutrinos are their own antiparticles, it’s possible that the anti-neutrinos emitted during double beta decay could annihilate one another and disappear, violating lepton number conservation. This is called neutrino-less double beta decay.

Such a process would favour matter over antimatter, creating an imbalance.

“Theoretically it would cause a profound revolution in our understanding of where particles get their mass,” Friedland says. “It would also tell us there has to be some new physics at very, very high energy scales, that there is something new in addition to the Standard Model we know and love.”

It’s possible that neutrinos and anti-neutrinos are different, and that there are two neutrino and anti-neutrino states, as called for in Dirac’s equation. The two missing states could be so elusive that physicists have yet to spot them.

But spotting evidence of neutrino-less double beta decay would be a sign that Majorana had the right idea instead; neutrinos and anti-neutrinos are the same.

“These are very difficult experiments,” de Gouvêa says. “They’re similar to dark matter experiments in the sense they have to be done in very quiet environments with very clean detectors and no radioactivity from anything except the nucleus you’re trying to study.”

Physicists are still evaluating their understanding of the elusive particles.

“There have been so many surprises coming out of neutrino physics,” says Reina Maruyama, a professor at Yale University associated with the CUORE neutrino-less double beta decay experiment. “I think it’s really exciting to think about what we don’t know.”

The report can be found online at: