Quantum Physics World

Wednesday, March 29, 2006

Algebra and Richard Cohen

In a recent column in the Washington Post, Richard Cohen laments the demise of a high school student named Gabriella who couldn't pass algebra after 7 or so tries. Cohen tells poor Gabriella that he hates math and has never used algebra.

Cohen misses the point. Whether or not you use something later in life is not always the basis of what is important to take in school. It is important to have a broadly educated citizenry who have been exposed to the wide range of human knowledge. This includes among other things history, the arts, science, and yes mathematics.

Furthermore, it shouldn't be lost on people that we live in a highly technical society. The importance of science and technology is increasing, yet the populace as a whole is woefully ignorant in these topics. People are being asked to make decisions about nuclear power, global warming, biotechnology, stem cells, space travel and the like. These are all scientific issues and having a reasonable understanding of them is important. The population should have their own understanding of the issues to a certain degree rather than having to rely on experts for everything. This can only be done through the education system, and while these are conceptually based topics ultimately there is a mathematical underpinning. I am not suggesting that people should be going out and doing their own calculations of say uranium half-lives, but doing some calculations like that in school will allow someone to make more reasoned judgements on many issues-like storing nuclear wastes at Yucca Mountain.

Cohen says that algebra isn't a high or the highest form of human reasoning, and that writing is. Frankly Mr. Cohen I beg to differ. Mathematics is the highest form of human reasoning and is the basic underpinning of our modern society. It transcends the sciences, being at the root of the human genome project, the design of lasers, electric power, radio and cell phones and the internet. In short the entire modern world is fundamentally mathematically based. Writing was already highly developed long before calculus came into existence.

Cohen also claims that a computer or calculator can do math while they can't write. Cohen's understanding of how computers are used in mathematics is naive. In higher mathematics, doing a solution like the quadratic equation is trivial. Its the understanding of the solutions and properties of equations that require higher reasoning. The computer is used to churn out solutions to equations that are too hard for the human mind to solve directly (no analytic or closed form solution). In the end a human being has to analyze and interpret the results-something a computer can't do.

I am sorry that Gabriella couldn't pass the course, but that's too bad. We do need higher standards in this country. Otherwise China and India will come to dominate world affairs in the coming century.

Tuesday, March 28, 2006

One of the themes touched on in "Equations of Eternity" by David Darling is the unreasonable effectiveness of mathematics in describing the physical world. Time and again, as Darling points out, mathematicians have worked on some obscure theoretical idea or area that seems to have nothing to do with reality. Then years later physicists stumble on it and discover that it describes some physical process in absolute detail, down to the last dotted i and crossed t.
A great example of the connection between mathematics and the physical world is the discovery by Maxwell of well, Maxwell's equations. During the 19th century the frontiers of science were being pushed by people studying electromagnetic phenomena. Years earlier Coloumb had figured out how to describe the electric force between two charges. In the early to mid-1800's physics had moved quite a bit beyond that to consider electric currents and magnetic fields. It was here that mathematical insight would prove to be an unusually effective tool-revealing properties of nature hidden to the senses.
In about the 1830's Ampere worked out a "law" that relates the magnetic field to a flow of current. Ampere's law has a very precise mathematical form which was worked out from careful experimental observation. In short Ampere figured out that the curl of the magnetic field B is related to the current density J via



While this "law" was worked out by careful measurement and experimental observation-it violates a tenet of vector calculus. We know from vector calculus that the divergence of the curl of any vector is zero:






Maxwell basically applied this fact to Ampere's law and discovered that Ampere's law could not be true-based strictly on mathematical reasoning. This in turn led to the discovery of radio. Let's get a basic idea of what he did. And so, taking the divergence of the left hand side of Ampere's law, we should get zero



But on the right side, we have






But this isn't zero. In fact, using the conservation of current density together with Gauss's law, it can be shown that this is related to the electric field. That is






So, to make the divergence of Ampere's law vanish, Maxwell added this term. The missing term is the "displacement current" and it leads to a coupling of the electric and magnetic fields-giving us traveling electromagnetic waves or radiation. To sum up Maxwell changed Ampere's law to






These "laws" of vector calculus are abstract mathematical laws--supposedly laws of pure thought. At first sight one might not expect that they would hold precedence over experimental observation. But it turns out they do. Maxwell used the laws to determine what form Ampere's law should really have, and in the process discovered something that was unknown at the time-radio waves.
This is just one small example of the interplay between math and physics. Later we'll explore connections between abstract mathematics and quantum theory which describes every last detail of atomic behavior.

Sunday, March 26, 2006

Quantum mechanics has raised issues about the role of the observer since it's inception. This has led to crazy debates, even prompting some to ask if things exist when a conscious observer isn't looking, or whether the conscious mind has a real fundamental role to play in actually creating reality.
Thoughts like these are explored in "Equations of Eternity" by David Darling, one of the best written popular science books I have come across. Darling has an amazing skill for shedding new light on obvious facts about the brain and the operation of the universe. The first part of the book is an exploration of how the brain works, how its developed through evolution and how it manages to conjure up what our minds view as reality. The first half of the book is very deep and is sure to cause many sleepless nights where you ponder the concepts Darling has put forth.
In the second half of the book he moves on into a discussion of "the unreasonable effectiveness of mathematics", focusing on what quantum mechanics and theoretical physics have to say about the role that conscious observers might play in the universe, and the fundamental role that mathematics plays in the description of the universe. All in all its a fascinating read, highly recommended.

Thursday, March 23, 2006

Readers of Quantum Mechanics Demystified should be pleased to learn about some new books on quantum information science that have just been released for the popular audience. One sure to cause a stir is Programming the Universe : A Quantum Computer Scientist Takes On the Cosmos by Seth Lloyd. In my opinion Lloyd is a bit off the edge. I have heard that he thinks the universe is some kind of computer running a program. The analogy is nice, but I believe that is just a product of the age. Kind of like the 18th century view that the universe was a machine or clock of some kind. If Lloyd really holds that view its too much for me, but I will probably be reading his book anyway since its sure to be full of insights.
Another book that recently came out that caught my attention is
Decoding the Universe: How the New Science of Information Is Explaining Everything in the Cosmos, from Our Brains to Black Holes by Charles Seife. I don't know anything about the author, but the title alone is enough to intrigue me. My feeling is information has some central role to play in the universe, so I intend to pick up this book to stimulate some thinking on this issue.

Wednesday, March 22, 2006

The Holy Grail of physics is the unification of quantum physics and relativity, a Herculean task trying to wed together two spheres as different as night and day. On one hand we have the world of the very large. This is the world of stars, planets and galaxies-the world governed by Einstein’s relativity. On the other hand we have the world of the very small-the world of atoms, neutrons, and quarks-governed by the quantum. Each of these two realms not only describes different types of objects or different sized objects-they require different types of mathematics. Even worse-the world of stars and galaxies seems to be governed by a classical, deterministic physics which fits neatly into a beautiful geometric theory, while the world of elementary particles is governed by probability, randomness, and mysterious mathematical worlds called Hilbert spaces-the world of the quantum dice.

At first glance these two theories can hardly be thought to be describing objects that belong to one and the same universe-but they do exactly that. Stars are made of atoms that obey the laws of quantum physics. Out of chance, chaos and ghostly entanglement-the orderly structure of a galaxy somehow emerges-and if we look at the ghostly quantum particles closely-atoms and elementary particles do fall in a gravitational field. Therefore there must be a path forward to a unified new physics.

Before embarking on the path of unification, it is important to make sure that one has a complete and thorough understanding of those two pillars of physics that are already well established-quantum physics and general relativity. This understanding is necessary before moving on to explore efforts at unification such as string theory and loop quantum gravity.

Let’s take a stab at quantum theory. In my book Quantum Mechanics Demystified, I lay out the mathematical framework of quantum theory. But what is the conceptual framework-the basic building blocks that one wants to come away with before trying to put together a unified theory that describes the universe?. These are the ten keys to quantum physics. They may not be the only ones-I am simply making suggestions of key concepts. You may wish to add your own. Before we start-a brief note on notation. We will denote the state of a particle or system with a bold capital letter, such as F or G, while a scalar (a plain old number) will be denoted by an small italic letter such as a or b.

1. QUANTUM STATES CAN EXIST IN SUPERPOSITION

One principle that plays a central, absolutely vital role in quantum theory is the notion of superposition. Imagine if you will that a particle can be in two mutually exclusive states that we denote F and G. These two states could represent going through one slit or the other in the two slit experiment, or they could be two different energy states of an electron in an atom, for example. The principle of superposition tells us that the state formed by their linear combination-i.e. their sum-is also a valid quantum state. That is a system can be in the state described by H = a F + b G. A quantum superposition is a special sort of beast-when we look at the system, that is when we make a measurement, we never find it in some strange mixture of the states F and G. Rather, it is always in one state or the other. That is when the system is prepared in state H measurement will sometimes find the system in state F, while at other times, measurement will find the system in state G. Note that the numbers a and b can be complex. In key #2, we interpret the meaning of the state H and explore it further.

2. THE BORN RULE

The Born Rule tells us the probability of finding a quantum system in this state or that using a simple recipe. If a quantum system is in the state described by H = a F + b G, then the probability that the system is found in state F when a measurement is made is found by squaring a while the probability that the system is found in state G when a measurement is made is found by squaring b. It is important not to confuse the fact that the Born rule tells us how to extract probabilities from a quantum state with the notion that the state is a mere statistical mixture. A superposition state like H leads to interference effects-like the fringes seen on the screen in the double slit experiment-something a statistical mixture can’t do.

3. IDENTICAL PREPARATION OF STATES DOES NOT RESULT IN IDENTICAL MEAUREMENT RESULTS

A key concept in classical science is that if you set up an experiment in exactly the same way several times- you will get repeatable results. The probabilistic nature of quantum theory-which is inherently fundamental and is not due to a lack of precision in our measurement devices-means that in many cases, if we prepare several systems in a given state, we will get different measurement results when the experiment is run. Once again, suppose that we prepare the system in the state H = a F + b G. If we run the experiment 4 times, we might make measurements and find the results F, F, G, F. The next day if we again prepare systems in state H = a F + b G , we might instead get F, G, G, G. Now if we do the experiment a large number of times, then the relative fractions of F and G will tend to the probabilities given by squaring the coefficients a and b.

4. THE UNCERTAINTY PRINCIPLE

The Heisenberg uncertainty principle tells us that we cannot know the values of two complimentary observables with absolute precision. The quintessential example given in most textbooks is the uncertainty relation between position and momentum. In short, the uncertainty principle tells us that the more precision we use in measurement of position, the less we can know about momentum and vice versa. If we wish, we can know the value of one variable to any precision we like-but at the expense of complete uncertainty in the other variable. For example, if we choose to measure a particle’s momentum with great accuracy, then we sacrifice knowledge of the particles position. The uncertainty principle signs the death warrant of classical, deterministic physics.

5. THE SCHRODINGER EQUATION AND UNITARY EVOLUTION

Quantum states to evolve in time in a deterministic manner that is governed by the Schrodinger equation. Or in a more modern sense-the time evolution of quantum states is described by unitary evolution.

6. LEARN LINEAR ALGEBRA

The fifth key to quantum theory is that the world of quantum mechanics lies in the mathematical realm of vector spaces. This means that the mathematics of quantum physics is the mathematics of linear algebra. If you want to master quantum theory then you need to know linear algebra. Learn how to manipulate matrices, how to calculate determinants, and how to find eigenvectors and eigenvalues. Learn about abstract vector spaces. Finally, learn about special types of matrices, in particular Hermitian and Unitary matrices.

6. PHYSICAL OBSERVABLES ARE REPRESENTED BY OPERATORS

This follows on the heels of point 6. In quantum theory, physical observables like momentum and energy are represented by operators. When considering the wave function approach of the Schrodinger equation, operators are instructions to do something to a function-compute the derivative say. In the mathematically equivalent matrix mechanics derived by Heisenberg, operators are represented by matrices.

7. THE ONLY POSSIBLE RESULTS OF A MEASUREMENT ARE THE EIGENVALUES OF AN OPERATOR

Following key #7, since physical observables are represented by mathematical operators, the next logical question to ask is what are the possible measurement results as predicted by the theory? It turns out that these are the eigenvalues of the operators used to represent physical observables.

8. WHAT IS THE MEANING OF THE WAVEFUNCTION?

An important issue in quantum theory is the following. Is the wavefunction a real, physical entity? Or does it just represent our state of knowledge? The meaning of the wavefunction is an important issue to resolve before we can have a “theory of everything”.

9. DECOHERENCE

Quantum systems interact with their environment. In doing so, the strange quantumness of their nature is lost. That is superposition and the interference that comes along with it is washed out by interactions with the environment. It is believed by many that this is how the classical world of our senses arises out of the quantum morass.

10. ENTANGLEMENT

Quantum physics is filled with mysteries and perhaps the greatest mystery of all is that of entanglement-the spooky action at a distance type correlation originally put forward by Einstein and his colleagues back in 1935. If two particles A and B are entangled, then their properties become correlated. In a certain state, if you measure the spin of A and find it to be spin-up, then the spin of B is spin-down. Or if you measure A and find it to be spin-down, then B is spin-up. If that is too abstract to wrap your mind around, for a loose analogy imagine that A and B are entangled dice correlated such that they always roll the same number. Roll A and find a 3, then rolling B is guaranteed to give a 3.

Spooky action at a distance tells us that we can leave A here on earth and carry B all the way to the other side of the galaxy-and their measurement results will remain correlated. Of course this isn’t completely worked out, even assuming the particles could be protected from the environment, relativistic effects and gravity might hamper such a scenario, but spatial separation, i.e. distance alone doesn’t seem to have an effect on entanglement.

The type of correlation that results in entanglement is a bit spooky in itself. It’s not really mere correlation. Without diving into the mathematics, suffice it to say that basically the two particles loose the essence of their individual identities. In an entangled system, the system as a whole assumes an identity. It is as if the whole becomes greater than the sum of its parts. To understand the entangled system you need to understand the whole and cannot understand it from the individual components alone. This amazing phenomenon-which has been confirmed in the laboratory-is of central importance in quantum computation and quantum cryptography.

Quantum physics can be mathematically daunting. Before you go on to tackle string theory and quantum gravity-make sure you have these central ideas down.

ABOUT THE AUTHOR:

David McMahon is a physicist consulting with Sandia National Labs. He has written books on quantum mechanics, relativity, and mathematics including “Quantum Mechanics Demystified “ and “Relativity Demystified” (more info at http://www.davidmcmahonbooks.com/work2.htm)

Tuesday, March 21, 2006

Does spooky action at a distance allow faster than light communication?

In my book Quantum Mechanics Demystified, I teach the reader the nuts and bolts of quantum mechanics, in the form of the mathematics necessary to calculate using the theory. While this is a challenging and necessary step for mastering quantum physics, the conceptual results that are born out of this mathematical framework is what makes quantum physics truly interesting. Sometimes, the resutls of quantum theory seem almost mystical, almost to the point of validating new age type thinking. Nowhere is this more true than in the phenomenon of entanglement, an interesting subject put on the map by Albert Einstein in one of his last significant scientific papers.

It is often said that scientists do their best work while young. With Albert Einstein this certainly seems to have been the case. Before the age of 40 he developed special relativity, laid the groundwork for quantum theory by explaining the photoelectric effect and in his greatest achievement, developed his elegant theory of gravity, general relativity. However, it was a paper he wrote with two colleagues in 1935-when Einstein was nearly 56 years old-which stands out as his most cited scientific paper. In fact, it may well turn out to be one of the most significant scientific papers of all time.

This is of course the “EPR” paper, written with his colleagues Boris Podolsky and Nathan Rosen. Following a decade of vehement arguments with the great Neils Bohr about the meaning of quantum theory, this paper stands out as Einstein’s “parting shot” in the debate-his last ditch effort to prove that quantum mechanics could not be a fundamental theory. The paper-titled “Can quantum mechanical description of reality be considered complete?”-uses quantum mechanics to demonstrate that particles which interact in someway become entangled, in a loose sense meaning that their properties become correlated. As we’ll see in a moment, this is not an ordinary correlation in any sense of the word. It implies that there exists a strange connection between the particles that persists even when they are separated by great distances. In some sense, this connection is instantaneous, putting it in direct conflict with the special theory of relativity. It was this strange connection that led Einstein to the phrase “spooky action at a distance”.

Quantum Entanglement

The EPR paper is based on the following thought experiment. Two particles interact and then separate. Furthermore, we imagine that they separate such that they are a great distance apart at a time when measurements on the particles can be made. EPR focused on two properties in particular-the position and momentum of each particle.

These properties or variables were chosen because of the Heisenberg uncertainty principle. The uncertainty principle tells us that the position and momentum of a particle are complimentary, meaning that the more you know about one variable, the less you know about the other. If you have complete knowledge of a particles position, then the particles momentum is completely uncertain. Or if instead you have complete knowledge of the particles momentum, then its position becomes completely uncertain. Intermediate ranges of accuracy are possible, the lesson to take home is that you cannot measure one variable without introducing some uncertainty into the value of the corresponding complimentary variable. The amount of uncertainty is quantified precisely by the uncertainty principle. The uncertainty of quantum mechanics never sat well with Einstein, he felt the theory, which is statistical in nature, is statistical because there exist some unknown or “hidden” variables in the microscopic world we are not yet aware of.

We now imagine that two particles interact and then move off in different directions. Because they have interacted, they become entangled. When two particles are entangled, the state of each particle alone has no real meaning-the state of the system can only be described in terms of the whole. In terms of elementary quantum mechanics, there is a wavefunction which describes the two particles together as a single unit. The wavefunction, being a superposition of different possibilities, exists in a ghostly combination of possible states. The Copenhagen interpretation tells us that the properties of the particle, position or momentum, don’t exist in definite values until a measurement is made.

When a measurement is made, and we can choose to make a measurement on one particle or the other, the wavefunction “collapses” and each particle is found to be in a definite state. The measurement results obtained for entangled particles are correlated. So if we make a measurement result on particle A and find its momentum to be a certain value, we know-without making a measurement on particle B-what its momentum is with absolute certainty. As EPR put it, by making a measurement of momentum on particle A, using momentum conservation tells us that pA + pB is an element of physical reality. In other words the wavefunction has collapsed and the variables have definite values-the ghostly superposition of possibilities is gone. The crucial point is that even though no measurement has been made on the distant particle B, the observer at the location of particle A has learned the value of B’s momentum. Somehow the wavefunction has collapsed instantaneously across a spatial distance-presumably in violation of the speed of light limit set by relativity.

The situation can be made even more interesting by noting that we can choose instead to measure the position of particle A. Again, using conservation principles, we will learn the value of the position of particle B, and the quantity qA - qB assumes physical reality.

Notice that the observer at position A can choose, by making different measurements that he or she desires, which properties of particle B assume definite values-or assume physical reality in the terminology of EPR. They can make this choice at a later time without any prior agreement with an observer in possession of particle B. This is another aspect of spooky action at a distance. The observer at A makes a measurement choice-presumably chosen using the free will of the mind-and forces particle B into a definite value instantaneously.

The interpretation of these results is still in debate, some believe that the wavefunction only represents our state of knowledge about the system. However it seems that it would be difficult for anyone who believes this to examine diffraction images from electron scattering and deny that the wavefunction is a real physical entity.

In summary, it appears that the position or momentum of each member of the EPR pair is determined by measurements performed on the other, distant member of the EPR pair. The effect seems to be instantaneous, leading Einstein and his colleagues to refer to the phenomenon as “spooky action at a distance”. The effect is non-local and appears to be instantaneous, but can anything useful come out of it? Can we exploit this to communicate faster than the speed of light? It turns out that as things are currently understood, the answer is no.

Teleportation

In recent years, it was shown that quantum entanglement could be exploited to transmit the state of a quantum particle from one place to another without having that state propagate through the space that separates the two locations. This certainly sounds magical enough-perhaps like something out of Star Trek-and is the reason that the investigators who discovered this phenomenon denoted it by the term teleportation. As we’ll see in a moment, teleportation demonstrates that despite the spooky action at a distance, special relativity is saved because the ability to communicate is limited in an unexpected way. A fundamental observation that should be made this is true even though teleportation is described using non-relativistic quantum mechanics-a theory where as long as no electromagnetic fields are involved, there is no ultimate speed limit.

We imagine two parties who wish to communicate with each other. In the quantum computing literature they are identified by the overused corny labels of Alice and Bob. It works like this. First, Alice and Bob meet. They create an entangled EPR pair. Then each party takes one member of the pair. Alice stays home, while Bob travels off somewhere, perhaps to Las Vegas.

In teleportation, the quantum particles used can have one of two states, so measurement results can be labeled by a 0 or a 1.

Since Alice and Bob each have in their possession one member of an entangled EPR pair, a spooky action at a distance connection exists between them. Alice can exploit this connection to send Bob the state of a quantum particle. The process is quite simple and Alice just follows these steps.

First, Alice gets the particle she wants to send with Bob, and she allows it to interact with her member of the EPR pair. Then she makes measurements on her member of the EPR pair and the particle that she wants to send to Bob. Since she is making measurements of two particles, her possible measurement results are the two-bit combinations 00, 01, 10, and 11.

Since Alice has allowed her half of the EPR pair to interact with another particle, the state of Bob’s half of the EPR pair must have changed. It’s at this point that special relativity peaks its head in-through the back door. Although the state of Bob’s particle has changed, any measurement results he makes on his half of the EPR pair would be completely random. Bob has no information in his possession about the state of the unknown particle Alice wants to send him. Spooky action at a distance has occurred but at this point it’s completely useless. To get something out of the situation-Alice has to call Bob-on an ordinary telephone say-and tell him her measurement results.

If Alice gets the measurement result 00, Bob doesn’t have to do anything-he now has the state of the particle Alice wanted to send him in his possession. However, that only happens 25% of the time, since Alice can get measurement results 00, 01, 10, and 11. If Alice gets measurement results 01, 10, or 11, Bob must make some measurements of his own on his half of the EPR pair in order to obtain the state of the particle Alice wants to send. We won’t get into the technical details, but in each case a different set of operations must be performed by Bob. Alice has to communicate which set of operations to use-based on the measurement result she obtained in the past-using a classical communications channel. Therefore the “instantaneous” nature of the interaction cannot be exploited until a classical communications channel is used.

The interesting thing about teleportation in my view is that it seems to say that special relativity has a major role to play in the transfer of information. In a way this is a fitting cap off to Einstein’s intellectual legacy. Einstein and Bohr both come out winners. Quantum mechanics stands on its own using the standard theory without hidden variables, yet what you can do with it is constrained by Einstein’s special theory of relativity.

ABOUT THE AUTHOR: David McMahon is a physicist who consults at Sandia National Laboratories and is the author of several math and physics books, including “Quantum Mechanics Demystified”. Information on his books can be found at http://www.davidmcmahonbooks.com/

Monday, March 20, 2006

Road to Reality Book Review

In The Road to Reality, Roger Penrose promises to take us on a whilwind tour of math and physics. From this perspective, the book does not disappoint. When I got my copy of the book I was shocked to find that Penrose has virtually laid out the entire mathematics curriculum in the format of a popular book. After a semi-philosophical review of the ancient Greeks, he embarks on a detailed chapter by chapter review of virtually every math topic. Chapters include calculus, complex variables, Fourier series, and topics from differential geometry. If this wasn't enough to give you a headache, that is only the first half of the book. Now that we have the math under our belts, he tackles every imaginable topic in physics. The topics he chooses to include involve much of the standard fair, quantum mechanics is covered in detail as are cosmology, relativity and string theory. But there are a lot of topics readers probably haven't come across in a popular book before, such as Lagrangian dynamics. What is more the physics covered in this book is covered in dramatic detail.

I found his text on quantum mechanics particularly interesting. He puts forward his idea that quantum mechanics needs to be modified, contrasting the deterministic time evolution of the wavefunction with the "collapse" process of measurement, suggesting that these divergent phenomenon speak to a fundamental problem in quantum theory.

I would not really classify this book as a "popular" science book. Rather, it seems to mark out a genre of its own, treading a middle ground I have not seen in a book before. I would call it "semi-popular" instead. The book is a top seller, yet it has many equations throughout the book, contracting the advice once given to a famous physics writer (was it Stephen Hawking?) that every added equation would halve book sales. Add to this the fact the book actually includes exercises for the reader! The book is written with enough detail to make it useful for professional scientists and students, as well as to "laymen" without formal physics training who are interested in the topic-provided they are willing to sit down and wade through the serious text.

The approach taken by Penrose with this book is nothing short of bold. Perhaps it will open up a new genre of physics books written for the general audience that don't completely shy away from the mathematical underpinnings of the science. I would also hope that the detailed exposition on mathematics will help increase interest in math topics.

However, I am not completely comfortable with the writing style used by Penrose, and often don't enjoy his books as much as I do other authors. I believe a book of this type written by Hawking or Brian Greene would be far superior.

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