Dec 16, 2021 04:05 AM
(This post was last modified: Dec 16, 2021 11:22 PM by C C.)
Is the universe actually a fractal?
www.forbes.com/sites/startswithabang/2021/01/06/is-the-universe-actually-a-fractal/
EXCERPT: . . . But our Universe fundamentally differs from this [fractal] scenario in three important ways.
1.) We don’t just have one type of matter, but two: normal and dark matter. While dark matter behaves in this self-similar fashion, normal matter is limited. It collides, forms bound structures, heats up, and even triggers nuclear fusion. Once you reach the small scales on which this occurs, self-similarity ends. The feedback interactions between the normal matter and the dark matter will alter the density profiles of the halos in ways that are not easy to figure out. In fact, this remains an open area of study in dark matter research today.
2.) Matter is joined by radiation, an incredibly important component of the Universe. Radiation, because it has an energy that depends on its wavelength, was actually more important in the early Universe. When the Universe expands, it gets less dense; the number of particles (normal matter, dark matter, and photons) remains the same, while the volume increases. But as the Universe expands, the wavelength of the radiation in it also redshifts, becoming lower in energy. The radiation was more important early on, and gets less important as time goes on.
This means, for the first few hundred thousand years of the Universe (and especially in the first ~10,000 or so), the matter overdensities struggle to grow, as the radiation works to effectively wash them out. There’s a lower limit to the scales at which the Universe is self-similar even at early times: your smallest scale structures are going to have at least ~100,000 solar masses in them, which is approximately the masses of globular clusters and the smallest known dwarf galaxies. Below that, the only structures you get are formed from messy collisions and interactions between various normal matter-based structures.
3.) Our Universe is also made extensively of dark energy, which dominates the energy content of the Universe today. If the Universe kept expanding while gravitating, and if the expansion itself wasn’t accelerating, there would be no upper limit to how large these cosmically self-similar structures could be. But because dark energy exists, it basically sets an upper limit to the size of these structures in the Universe: roughly a few billion light-years across.
That might sound enormous, but in an observable Universe that extends for ~46 billion light-years in all directions, even a structure that was 10 billion light-years in all three dimensions — a value much larger than the largest known structure in the Universe, by the way — would occupy only ~1% of the volume of the Universe. We simply don’t have structures that large and never will.
When you take all of this together, it helps us realize a true but perhaps counterintuitive fact about the Universe: on both the smallest and largest cosmic scales, the Universe is not fractal-like at all, and that only the intermediate scales have any chance at exhibiting fractal-like behavior... (MORE - missing details)
Quantum theory needs complex numbers
https://www.icfo.eu/newsroom/news/article/5232
RELEASE: Physicists construct theories to describe nature. Let us explain it through an analogy with something that we can do in our everyday life, like going on a hike in the mountains. To avoid getting lost, we generally use a map. The map is a representation of the mountain, with its houses, rivers, paths, etc. By using it, it is rather easy to find our way to the top of the mountain. But the map is not the mountain. The map constitutes the theory we use to represent the mountain's reality.
Physical theories are expressed in terms of mathematical objects, such as equations, integrals or derivatives. During history, physics theories evolved, making use of more elaborate mathematical concepts to describe more complicated physics phenomena. Introduced in the early 20th century to represent the microscopic world, the advent of quantum theory was a game changer. Among the many drastic changes it brought, it was the first theory phrased in terms of complex numbers.
Invented by mathematicians centuries ago, complex numbers are made of a real and imaginary part. It was Descartes, the famous philosopher considered as the father of rational sciences, who coined the term "imaginary," to strongly contrast it with what he called "real" numbers. Despite their fundamental role in mathematics, complex numbers were not expected to have a similar role in physics because of this imaginary part. And in fact, before quantum theory, Newton's mechanics or Maxwell's electromagnetism used real numbers to describe, say, how objects move, as well as how electro-magnetic fields propagate. The theories sometimes employ complex numbers to simplify some calculations, but their axioms only make use of real numbers.
Schrödinger's bewilderment. Quantum theory radically challenged this state of affairsbecause its building postulates were phrased in terms of complex numbers. The new theory, even if very useful for predicting the results of experiments, and for instance perfectly explains the hydrogen atom energy levels, went against the intuition in favor of real numbers. Looking for a description of electrons, Schrödinger was the first to introduce complex numbers in quantum theory through his famous equation. However, he could not conceive that complex numbers could actually be necessary in physics at that fundamental level.
It was as though he had found a map to represent the mountains but this map was actually made out of abstract and non-intuitive drawings. Such was his bewilderment that he wrote a letter to Lorentz on June 6, 1926, stating "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ? is surely fundamentally a real function." Several decades later, in 1960, Prof. E.C.G. Stueckelberg, from the University of Geneva, demonstrated that all predictions of quantum theory for single-particle experiments could equally be derived using only real numbers. Since then, the consensus was that complex numbers in quantum theory were only a convenient tool.
However, in a recent study published in Nature, ICFO researchers Marc-Olivier Renou and ICREA Prof. at ICFO Antonio Acín, in collaboration with Prof. Nicolas Gisin from the University of Geneva and the Schaffhausen Institute of Technology, Armin Tavakoli from the Vienna University of Technology, and David Trillo, Mirjam Weilenmann, and Thinh P. Le, led by Prof. Miguel Navascués, from the Institute of Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences in Vienna have proven that if the quantum postulates were phrased in terms of real numbers, instead of complex, then some predictions about quantum networks would necessarily differ. Indeed, the team of researchers came up with a concrete experimental proposal involving three parties connected by two sources of particles where the prediction by standard complex quantum theory cannot be expressed by its real counterpart.
Two sources and three nodes. To do this, they thought of a specific scenario that involves two independent sources (S and R), placed between three measurement nodes (A, B and C) in an elementary quantum network. The source S emits two particles, say photons, one to A, and the second to B. The two photons are prepared in an entangled state, say in polarization. That is, they have correlated polarization in a way which is allowed by (both complex and real) quantum theory but impossible classically. The source R does exactly the same, emits two other photons prepared in an entangled state and sends them to B and to C, respectively. The key point in this study was to find the appropriate way to measure these four photons in the nodes A, B, C in order to obtain predictions which cannot be explained when quantum theory is restricted to real numbers.
As ICFO researcher Marc-Olivier Renou comments "When we found this result, the challenge was to see if our thought experiment could be done with current technologies. After discussing with colleagues from Shenzhen-China, we found a way to adapt our protocol to make it feasible with their state-of-the-art devices. And, as expected, the experimental results match the predictions!." This remarkable experiment, realized in collaboration with Zheng-Da Li,Ya-Li Mao,Hu Chen, Lixin Feng, Sheng-Jun Yang, Jingyun Fan from the Southern University of Science and Technology, and Zizhu Wang from the University of Electronic Science and Technology is published at the same time as the Nature paper in Physical Review Letters.
The results published in Nature can be seen as a generalization of Bell's theorem, which provides a quantum experiment which cannot be explained by any local physics formalism. Bell's experiment involves one quantum source S that emits two entangled photons, one to A, and the second to B, prepared in an entangled state. Here, in contrast, one needs two independent sources, the assumed independence is crucial and was carefully designed in the experiment.
The study also shows how outstanding predictions can be when combining the concept of a quantum network with Bell's ideas. For sure, the tools developed to obtain this first result are such that they will allow physicists to achieve a better understanding of quantum theory, and will one day trigger the realization and materialization of so far unfathomable applications for the quantum internet.
PAPER: https://www.nature.com/articles/s41586-021-04160-4
www.forbes.com/sites/startswithabang/2021/01/06/is-the-universe-actually-a-fractal/
EXCERPT: . . . But our Universe fundamentally differs from this [fractal] scenario in three important ways.
1.) We don’t just have one type of matter, but two: normal and dark matter. While dark matter behaves in this self-similar fashion, normal matter is limited. It collides, forms bound structures, heats up, and even triggers nuclear fusion. Once you reach the small scales on which this occurs, self-similarity ends. The feedback interactions between the normal matter and the dark matter will alter the density profiles of the halos in ways that are not easy to figure out. In fact, this remains an open area of study in dark matter research today.
2.) Matter is joined by radiation, an incredibly important component of the Universe. Radiation, because it has an energy that depends on its wavelength, was actually more important in the early Universe. When the Universe expands, it gets less dense; the number of particles (normal matter, dark matter, and photons) remains the same, while the volume increases. But as the Universe expands, the wavelength of the radiation in it also redshifts, becoming lower in energy. The radiation was more important early on, and gets less important as time goes on.
This means, for the first few hundred thousand years of the Universe (and especially in the first ~10,000 or so), the matter overdensities struggle to grow, as the radiation works to effectively wash them out. There’s a lower limit to the scales at which the Universe is self-similar even at early times: your smallest scale structures are going to have at least ~100,000 solar masses in them, which is approximately the masses of globular clusters and the smallest known dwarf galaxies. Below that, the only structures you get are formed from messy collisions and interactions between various normal matter-based structures.
3.) Our Universe is also made extensively of dark energy, which dominates the energy content of the Universe today. If the Universe kept expanding while gravitating, and if the expansion itself wasn’t accelerating, there would be no upper limit to how large these cosmically self-similar structures could be. But because dark energy exists, it basically sets an upper limit to the size of these structures in the Universe: roughly a few billion light-years across.
That might sound enormous, but in an observable Universe that extends for ~46 billion light-years in all directions, even a structure that was 10 billion light-years in all three dimensions — a value much larger than the largest known structure in the Universe, by the way — would occupy only ~1% of the volume of the Universe. We simply don’t have structures that large and never will.
When you take all of this together, it helps us realize a true but perhaps counterintuitive fact about the Universe: on both the smallest and largest cosmic scales, the Universe is not fractal-like at all, and that only the intermediate scales have any chance at exhibiting fractal-like behavior... (MORE - missing details)
Quantum theory needs complex numbers
https://www.icfo.eu/newsroom/news/article/5232
RELEASE: Physicists construct theories to describe nature. Let us explain it through an analogy with something that we can do in our everyday life, like going on a hike in the mountains. To avoid getting lost, we generally use a map. The map is a representation of the mountain, with its houses, rivers, paths, etc. By using it, it is rather easy to find our way to the top of the mountain. But the map is not the mountain. The map constitutes the theory we use to represent the mountain's reality.
Physical theories are expressed in terms of mathematical objects, such as equations, integrals or derivatives. During history, physics theories evolved, making use of more elaborate mathematical concepts to describe more complicated physics phenomena. Introduced in the early 20th century to represent the microscopic world, the advent of quantum theory was a game changer. Among the many drastic changes it brought, it was the first theory phrased in terms of complex numbers.
Invented by mathematicians centuries ago, complex numbers are made of a real and imaginary part. It was Descartes, the famous philosopher considered as the father of rational sciences, who coined the term "imaginary," to strongly contrast it with what he called "real" numbers. Despite their fundamental role in mathematics, complex numbers were not expected to have a similar role in physics because of this imaginary part. And in fact, before quantum theory, Newton's mechanics or Maxwell's electromagnetism used real numbers to describe, say, how objects move, as well as how electro-magnetic fields propagate. The theories sometimes employ complex numbers to simplify some calculations, but their axioms only make use of real numbers.
Schrödinger's bewilderment. Quantum theory radically challenged this state of affairsbecause its building postulates were phrased in terms of complex numbers. The new theory, even if very useful for predicting the results of experiments, and for instance perfectly explains the hydrogen atom energy levels, went against the intuition in favor of real numbers. Looking for a description of electrons, Schrödinger was the first to introduce complex numbers in quantum theory through his famous equation. However, he could not conceive that complex numbers could actually be necessary in physics at that fundamental level.
It was as though he had found a map to represent the mountains but this map was actually made out of abstract and non-intuitive drawings. Such was his bewilderment that he wrote a letter to Lorentz on June 6, 1926, stating "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ? is surely fundamentally a real function." Several decades later, in 1960, Prof. E.C.G. Stueckelberg, from the University of Geneva, demonstrated that all predictions of quantum theory for single-particle experiments could equally be derived using only real numbers. Since then, the consensus was that complex numbers in quantum theory were only a convenient tool.
However, in a recent study published in Nature, ICFO researchers Marc-Olivier Renou and ICREA Prof. at ICFO Antonio Acín, in collaboration with Prof. Nicolas Gisin from the University of Geneva and the Schaffhausen Institute of Technology, Armin Tavakoli from the Vienna University of Technology, and David Trillo, Mirjam Weilenmann, and Thinh P. Le, led by Prof. Miguel Navascués, from the Institute of Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences in Vienna have proven that if the quantum postulates were phrased in terms of real numbers, instead of complex, then some predictions about quantum networks would necessarily differ. Indeed, the team of researchers came up with a concrete experimental proposal involving three parties connected by two sources of particles where the prediction by standard complex quantum theory cannot be expressed by its real counterpart.
Two sources and three nodes. To do this, they thought of a specific scenario that involves two independent sources (S and R), placed between three measurement nodes (A, B and C) in an elementary quantum network. The source S emits two particles, say photons, one to A, and the second to B. The two photons are prepared in an entangled state, say in polarization. That is, they have correlated polarization in a way which is allowed by (both complex and real) quantum theory but impossible classically. The source R does exactly the same, emits two other photons prepared in an entangled state and sends them to B and to C, respectively. The key point in this study was to find the appropriate way to measure these four photons in the nodes A, B, C in order to obtain predictions which cannot be explained when quantum theory is restricted to real numbers.
As ICFO researcher Marc-Olivier Renou comments "When we found this result, the challenge was to see if our thought experiment could be done with current technologies. After discussing with colleagues from Shenzhen-China, we found a way to adapt our protocol to make it feasible with their state-of-the-art devices. And, as expected, the experimental results match the predictions!." This remarkable experiment, realized in collaboration with Zheng-Da Li,Ya-Li Mao,Hu Chen, Lixin Feng, Sheng-Jun Yang, Jingyun Fan from the Southern University of Science and Technology, and Zizhu Wang from the University of Electronic Science and Technology is published at the same time as the Nature paper in Physical Review Letters.
The results published in Nature can be seen as a generalization of Bell's theorem, which provides a quantum experiment which cannot be explained by any local physics formalism. Bell's experiment involves one quantum source S that emits two entangled photons, one to A, and the second to B, prepared in an entangled state. Here, in contrast, one needs two independent sources, the assumed independence is crucial and was carefully designed in the experiment.
The study also shows how outstanding predictions can be when combining the concept of a quantum network with Bell's ideas. For sure, the tools developed to obtain this first result are such that they will allow physicists to achieve a better understanding of quantum theory, and will one day trigger the realization and materialization of so far unfathomable applications for the quantum internet.
PAPER: https://www.nature.com/articles/s41586-021-04160-4