Tag Archives: math

Astronomy, Physics

Quantum Gravity…Oh my!

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So here’s the big problem.

Right now, physicists have a detailed mathematical model for how the fundamental forces in nature work: electromagnetism, and the strong and weak nuclear forces. Added to this is a detailed list of the fundamental particles in nature like the electron, the quarks, photons, neutrinos and others. Called the Standard Model, it has been extensively verified and found to be an amazingly accurate way to describe nearly everything we see in the physical world. It explains why some particles have mass and others do not. It describes exactly how forces are generated by particles and transmitted across space. Experimenters at the CERN Large Hadron Collider are literally pulling out their hair to find errors or deficiencies in the Standard Model that go against the calculated predictions, but have been unable to turn up anything yet. They call this the search for New Physics.

Along side this accurate model for the physical forces and particles in our universe, we have general relativity and its description of gravitational fields and spacetime. GR provides no explanation for how this field is generated by matter and energy. It also provides no description for the quantum structure of matter and forces in the Standard Model. GR and the Standard Model speak two very different languages, and describe two very different physical arenas. For decades, physicists have tried to find a way to bring these two great theories together, and the results have been promising but untestable. This description of gravitational fields that involves the same principles as the Standard Model has come to be called Quantum Gravity.

The many ideas that have been proposed for Quantum Gravity are all deeply mathematical, and only touch upon our experimental world very lightly. You may have tried to read books on this subject written by the practitioners, but like me you will have become frustrated by the math and language this community has developed over the years to describe what they have discovered.

The problem faced by Quantum Gravity is that gravitational fields only seem to display their quantum features at the so-called Planck Scale of 10^-33 centimeters and  10^-43 seconds. I cant write this blog using scientific notation, so I am using the shorthand that 10^3 means 1000 and 10^8 means 100 million. Similarly, 10^-3 means 0.001 and so on. Anyway, the Planck scale  also corresponds to an energy of 10^19 GeV or 10 billion billion GeV, which is an energy 1000 trillion times higher than current particle accelerators can reach.

There is no known technology that can reach the scales where these effects can be measured in order to test these theories. Even the concept of measurement itself breaks down! This happens because the very particles (photons) you try to use to study physics at the Planck scale carry so much energy  they turn into quantum black holes and are unable to tell you what they saw or detected!

One approach to QG is called Loop Quantum Gravity.  Like relativity, it assumes that the gravitational field is all there is, and that space and time become grainy or ‘quantized’ near the Planck Scale. The space and time we know and can experience in-the-large is formed from individual pieces that come together in huge numbers to form the appearance of a nearly-continuous and smooth gravitational field.

The problem is that you cannot visualize what is going on at this scale because it is represented in the mathematics, not by nuggets of space and time, but by more abstract mathematical objects called loops and spin networks. The artist rendition above is just that.

So here, as for Feynman Diagrams, we have a mathematical picture that represents a process, but the picture is symbolic and not photographic. The biggest problem, however, is that although it is a quantum theory for gravity that works, Loop Quantum Gravity does not include any of the Standard Model particles. It represents a quantum theory for a gravitational field (a universe of space and time) with no matter in it!

In other words, it describes the cake but not the frosting.

The second approach is string theory. This theory assumes there is already some kind of background space and time through which another mathematical construct called a string, moves. Strings that form closed loops can vibrate, and each pattern of vibrations represents a different type of fundamental particle. To make string theory work, the strings have to exist in 10 dimensions, and most of these are wrapped up into closed balls of geometry called Calabi-Yau spaces. Each of these spaces has its own geometry within which the strings vibrate. This means there can be millions of different ‘solutions’ to the string theory equations: each a separate universe with its own specific type of Calabi-Yau subspace that leads to a specific set of fundamental particles and forces. The problem is that string theory violates general relativity by requiring a background space!

In other words, it describes the frosting but not the cake!

One solution proposed by physicist Lee Smolin is that Loop Quantum Gravity is the foundation for creating the strings in string theory. If you looked at one of these strings at high magnification, its macaroni-like surface would turn into a bunch of loops knitted together, perhaps like a Medieval chainmail suit of armor. The problem is that Loop Quantum Gravity does not require a gravitational field with more than four dimensions ( 3 of space and one of time) while strings require ten or even eleven. Something is still not right, and right now, no one really knows how to fix this. Lacking actual hard data, we don’t even know if either of these theories is closer to reality!

What this hybrid solution tries to do is find aspects of the cake that can be re-interpreted as particles in the frosting!

This work is still going on, but there are a few things that have been learned along the way about the nature of space itself. At our scale, it looks like a continuous gravitational field criss-crossed by the worldlines of atoms, stars and galaxies. This is how it looks even at the atomic scale, because now you get to add-in the worldlines of innumerable ‘virtual particles’ that make up the various forces in the Standard Model.  But as we zoom down to the Planck Scale, space and spacetime stop being smooth like a piece of paper, and start to break up into something else, which we think reveals the grainy nature of gravity as a field composed of innumerable gravitons buzzing about.

But what these fragmentary elements of space and time ‘look’ like is impossible to say. All we have are mathematical tools to describe them, and like our attempts at describing the electron, they lead to a world of pure abstraction that cannot be directly observed.

If you want to learn a bit more about the nature of space, consider reading my short booklet ‘Exploring Quantum Space‘ available at amazon.com. It describes the amazing history of our learning about space from ancient Greek ‘common sense’ ideas, to the highlights of mind-numbing modern quantum theory.

Check back here on Thursday, December 22 for the last blog in this series!

Astronomy, Physics

Relativity and Space

Psychologists and physicists often use a similar term to describe one of the most fundamental characteristics of humans and matter: The Story. Here, for example, is the timeline story for key events in the movie The Hunger Games.

Oliver Sacks, in his book ‘The Man Who Mistook His Wife for a Hat’ describes the case of Jimmy G who was afflicted with Korsakov’s Syndrome. He could not remember events more than a few minutes in the past, and so he had to re-invent his world every few minutes to account for new events. As Sacks notes ‘If we wish to know about a man, we ask ‘what is his story – his real, inmost story? – for each of us is a biography, a story..[and a] singular narrative, which is constructed, continually, unconsciously, by, through, and in us – through our perceptions, our feelings, our thoughts, our actions..and our narratives…we must constantly recollect ourselves’.

Physicist Lee Smolin, in his book ‘Three Roads to Quantum Gravity’ , describes the essential foundation of relativity as the ‘story’ about processes and not the things-as-objects.   “A marble is not an inert thing, it is a process…There are only relatively fast processes and relatively slow processes. And whether it is a short story or a long story, the only kind of explanation of a process  that is truly adequate is a story.”

In both cases, we cannot define an object, be it a human, a table, or an electron by merely describing its properties at one instant in time. We can only define an object in terms of a process consisting of innumerable events, which create the story that defines it. This is very obvious when we are talking about humans, but it also applies to every object in the universe.

In relativity, the history or ‘story’ of a process such as a football or a galaxy, consists of a series of events that are tied together by cause-and-effect to create the process that you see at any particular moment. These events include the interactions of one process with others that cumulatively create what you see as the history of the process at a particular moment. In relativity, we call this history of a process its worldline.

This is a worldline map (Credit Aaron Koblin / BBC)of airlines traveling to and from the United States. The lines give the history of each flight on the 2-d surface of Earth. Each worldline consists of a huge number of ‘hidden’ events contributed by each passenger! By carefully studying these worldlines you could mathematically deduce that Earth is a sphere.

What Einstein said is that only worldlines matter, because that is the only thing we have access to. Even better than that, we are only able to see that part of a processes that can be communicated to us by using light, which is the fastest signal we can ever use to transfer information. When we are ‘looking’ at something, like a car or a star, what we are actually doing is looking back along its history carried to us as information traveling by photons of light.

In an earlier essay, I mentioned how we do not see objects in space, but only the end points of a light ray’s history as, for example, it leaves the surface of an object (Event 1) arrives at dust mote along the way and was re-emitted (Event 2) to arrive at our retina, and cause a rod or a cone cell to fire (Event 3). Because these events are strictly determined by cause-and-effect, and travel times are limited by the speed of light, we can organize these events in a strict history for the object we viewed (which was in fact a ‘process’ in and of itself!).

So, what does this say about space? Space  is irrelevant, because we can completely define our story only in terms of the ‘geometry’ of these history worldlines and the causal connections between events on these worldlines, without any mention of space as a ‘background’ through which things move.

This leads to another problem.

Einstein’s new relativistic theory of gravity makes use of a convenient mathematical tool called 4-dimensional spacetime. Basically we live in a world with three dimensions of space and one dimension of time, making a 4-dimensional thing called spacetime. Without knowing, you live and work in 4-dimensions because there is nothing about you that does not ‘move’ in time as well as space from second to second. All physical process take place in 4 dimensions, so all theories of physics and how things work are necessarily statements about 4-dimensional things.

It is common to refer to gravity as a curvature in the geometry of this spacetime ‘fabric’, but we can just as easily talk about the curvature of worldlines defining gravity and not even bother with the idea of spacetime at all! Remember, when you look at an object, you are ‘just’ looking back through its history revealed by the network of photons of light.

So we have used a mathematical tool, namely spacetime, to make visualizing the curvature of worldlines easier to describe, but we now make the mistake of thinking that spacetime is real because we have now used the mathematical tool to represent the object itself. This is similar to what we did with the idea of Feynman Diagrams in the previous blog! As Lee Smolin says ‘When we imagine we are seeing into an infinite three-dimensional space, we are actually falling for a fallacy in which we substitute what we actually see [a history of events] for an intellectual construct [space]. This is not only a mystical vision, it is wrong.”

But what about infinity?

In my next essay I will discuss why infinity is probably not a real concept in the physical world.

 

Check back here on Friday, December 16 for the next installment!

Physicist Lee Smolin’s book ‘The Three Roads to Quantum Gravity’ discusses many of these ideas in more detail.

Astronomy, Brain Research, Physics

Is Space Real?

I take a walk to the store and can’t help but feel I am moving through something that is more than the atmosphere that rushes by my face as I go. The air itself is contained within the boundaries of the space through which I pass. If I were an astronaut in the vacuum of outer space, I would still have the sense that my motion was through a pre-existing, empty framework of 3-dimensions. Even if I were blind and confined to a wheelchair, I could still have the impression through muscular exertion that I was moving through space to get from my kitchen to my living room ‘over there’. But what is space as a physical thing? Of all the phenomena, forces and particles we study, each is something concrete though generally invisible: a field; a wave; a particle. But space, itself, seems to be none of these. WTF!

Spider web covered with dew drops

Way back in the early 1700s, Sir Isaac Newton proposed that space was an ineffable, eternal framework through which matter passed. It had an absolute and immutable nature. Its geometry pre-existed the matter that occupied it and was not the least bit affected by matter. A clever set of experiments in the 20th century finally demonstrated rather conclusively that there is no pre-existing Newtonian space or geometry ‘beneath’ our physical world. There is no absolute framework of coordinates within which our world is embedded. What had happened was that Albert Einstein developed a new way of thinking about space that essentially denied its existence!

Albert Einstein’s relativity revolution completely overturned our technical understanding of space and showed that the entire concept of dimensional space was something of a myth. In his famous quote he stressed that We entirely shun the vague word ‘space’ of which we must honestly acknowledge we cannot form the slightest conception. In the relativistic world we live in, space has no independent existence. “…[prior-geometry] is built on the a priori, Euclidean [space], the belief in which amounts to something like a superstition“. So what could possibly be a better way of thinking about space than the enormously compelling idea that each of us carries around in our brains, that space is some kind of stage upon which we move?

To understand what Einstein was getting at, you have to completely do away with the idea that space ‘is there’ and we move upon it or through it. Instead, relativity is all about the geometry created by the histories (worldlines) of particles as they move through time. The only real ‘thing’ is the collection of events along each particle’s history. If enough particles are involved, the histories are so numerous they seem like a continuous space. But it is the properties of the events along each history that determine the over-all geometry of the whole shebang and the property we call ‘dimension’, not the other way around.

This figure is an example where the wires (analogous to worldlines) are defining the shape and contours of a dimensional shape. There is nothing about the background (black) space that determines how they bend and curve. In fact, with a bit of mathematics you could specify everything you need to know about the surface of this shape and from the mathematics tell what the shape is, and how many dimensions are required to specify it!

Princeton University physicist Robert Dicke expressed it this way, “The collision between two particles can be used as a definition of a point in [space]…If particles were present in large numbers…collisions could be so numerous as to define an almost continuous trajectory…The empty background of space, of which ones knowledge is only subjective, imposes no dynamical conditions on matter.”

What this means is that so long as a point in space is not occupied by some physical event such as the interaction point of a photon and an electron, it has no effect on a physical process ( a worldline) and is not even observable. It is a mathematical ‘ghost’ that has no effect on matter at all. The interstitial space between the events is simply not there so far as the physical world based upon worldlines is concerned. It is not detectable even by the most sophisticated technology, or any inventions to come. It does not even supply something as basic as the ‘dimension’ for the physical world!

We should also be mindful of another comment by Einstein that “…time and space are modes by which we think and not conditions in which we live“. They are free creations of the human mind, to use one of Einstein’s own expressions. By the way, the 18th century philosopher Immanuel Kant also called the idea of ‘space’ an example of a priori knowledge that we are born with to sort out the world, but it is not necessarily a real aspect of the world outside our senses.

Like a spider web, individual and numerous events along a worldline define the worldline’s shape, yet like the spider web, this web can be thought of as embedded in a larger domain of mathematically-possible events that could represent physical events…but don’t. The distinction between these two kinds of points is what Einstein’s revolutionary idea of relativity provided physicists, and is the mainstay of all successful physical theories since the 1920s. Without it, your GPS-enabled cell phones would not work!

So what are these events? Simply put, according to Physicist Lee Smolin, they are exchanges of information, which are also the interaction points between one particle’s worldline and another particle’s world line. If you think at the atomic level, each time a particle of light interacts with (collides or is emitted by) an electron it generates an event. These events are so numerous the electron’s worldline looks like a continuous line with no gaps between the events. So the shape of one worldline, what we call its history, is a product of innumerable interactions over time with the worldlines of all other objects (photons etc) to which it can be in cause-and-effect contact.

Even though this new idea of space being a myth has gained enormous validity among physicists over the last century, and I can easily speak the language of relativity to describe it, personally, my mind has a hard time really understanding it all. I also use the mathematical theory of quantum mechanics to make phenomenally accurate predictions, but no Physicist really understands why it works, or what it really means.

Next time I want to examine how the history of a particle is more important than the concept of space in Einstein’s relativity, and how this explains the seeming rigidity of the world you perceive and operate within.

Check back here on Thursday, December 15 for the next installment!

Physics

Seeing with Mathematics

Our brain uses sensory data to sift for patterns in space and time that help us create a mental model of the world through which we can navigate and stay alive. At some point, this model of the external world becomes our basis for thinking symbolically and mathematically about it.

Mathematics is an amazingly detailed, concise and accurate way of examining the world to state the logical relationships we find there, but many physicists and mathematicians have been astonished about why this is the case. The physicist Eugene Wigner wrote an article about this in 1960 titled ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In fact, since the enormous successes of Sir Isaac Newton in mathematically explaining a host of physical phenomena, physicists now accept that mathematics actually serves as a microscope (or telescope!) for describing things and hidden relationships we cannot directly experience. This amazing ability for describing relationships in the world (both real and imagined!) presents us with a new problem.

parabola

Mathematics is a symbolic way of describing patterns our world, and sometimes these symbolically-defined descriptions actually look like the things we are studying. For example, the path of a football is a parabola, but the equation representing its path, y(x), is also that of a parabolic curve drawn on a piece of paper. But what happens when the mathematical description takes you to places where you cannot see or confirm the shape of the object?

Mathematics is a tool for understanding the world and symbolically stating its many logical interconnections, but the tool can sometimes be mistaken for the thing itself. Here is a very important example that comes up again and again when physicists try to ‘popularize’ science.

In the late-1940s, physicist Richard Feynman created a new kind of mathematics for making very precise calculations about how light (photons) and charged particles (such as electrons) behave. His famous ‘Feynman Diagrams’ like the one below, are very suggestive of particles moving in space, colliding, and emitting light. This diagram, with time flowing from left to right, shows a quark colliding with an anti-quark, which generates a photon that eventually produces an electron and anti-electron pair.

feynman_qqgamee1

The problem is that this is not at all a ‘photograph’ of what is actually happening. Instead, this is a tool used for setting up the problem and cranking through the calculation. Nothing more. It is a purely symbolic representation of the actual world! You are not supposed to look at it and say that for the solid lines, ‘particles are like billiard balls moving on a table top’ or that the photon of light they exchange is a ‘wiggly wave traveling through space’. What these objects are in themselves is completely hidden behind this diagram. This is a perfect example of what philosopher Immanuel Kant was talking about back in the 1700s. He said that there is a behind-the-scenes world of noumena where the things-in-themselves (ding-an-sich) exist, but our senses and observations can never really access them directly. The Feynman diagram lets us predict with enormous precision how particles will interact across space and time, but hides completely from view what these particles actually look like.

Another example of how math lets us ‘see’ the world we cannot directly access is the answer to the simple question: What does an electron actually look like?

Since the 1800’s, electricity increasingly runs our civilization, and electricity is merely a measure of the flow of electrons through space inside a wire. Each of us thinks of electrons as tiny, invisible spheres like microscopic marbles that roll through our wires wicked fast, but this is an example of where the human brain has created a cartoon version of reality based upon our ‘common sense’ ideas about microscopic particles of matter. In both physics and mathematics, which are based upon a variety of observations of how electrons behave, it is quite clear that electrons can be thought of as both localized particles and distributed waves that carry the two qualities we call mass and charge. They emit electric fields, but if you try to stuff their properties inside a tiny sphere, that sphere would explode instantly. So it really does not behave like an ordinary kind of particle at all. Also, electrons travel through space as matter waves and so cannot be localized into discrete sphere-like particles. This is seen in the famous Double Slit experiment where electrons produce distinct wave-like interference patterns.

electronwave

So the bottom line is that we have two completely independent, mathematical ways of visualizing what an electron looks like, particles and matter waves, and each can facilitate highly accurate calculations about how electrons interact, but the two images (particle and wave – localized versus distributed in space) are incompatible with each other, and so we cannot form a single, consistent impression of what an electron looks like.

Next time we will have a look at  Einstein and his ideas about relativity, which completely revolutionized our common-sense understanding of space created by the brain over millions of years of evolution.

Check back here on Tuesday, December 13 for the next installment!

Brain Research

Logic and Math

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So here we have a brain whose association cortex works overtime to combine sensory information into a stream of relationships in space and time. In fact, you cannot shut off this process or stop the brain from constantly searching its sensory inputs for patterns, even when there are no patterns to be found!

The time domain is particularly interesting because it is here that we build up the ideas of cause-and-effect and create various rules-of-thumb that help us predict the future, find food, and many other activities. This specific rule-type association largely takes place in the dominant hemisphere of the brain (left side for right-handed people) which also has the speech center. Specifically, the frontal lobes are generally considered to be the logic and reasoning centers of the brain. So when the brain is talking to you, it also has easy access to logical tools of thinking. We also have a minor hemisphere (right-side for right-handed people) that specializes in pattern recognition, but it contains no language centers and is therefore mute. Its insights about the patterns that it finds in sensory data are totally overtaken by the constantly babbling left-hemisphere. All it can muster is that non-verbal feeling of ‘Eureka!’ you get  once in a while.

problemsolving

This sequence of brain scans shows the four stages of math problem solving from left to right: encoding (downloading), planning (strategizing), solving (performing the math), and responding (typing out an answer).

Anyway, brain researchers have done brain imaging studies to explore where math reasoning occurs. They found that, when mathematicians think about advanced concepts, the prefrontal, parietal, and inferior temporal regions of the brain become very active. But this activity didn’t also happen in brain regions linked to words. This means, as many mathematicians can tell you, mathematics is not related to the brain’s language centers. It is an entirely different way of communication. In other words, you can carry-out mathematical thinking without an internal voice speaking. It is an entirely non-verbal activity until you are interested in communicating your results to someone else. I know this myself when I am solving complex equations. Not a single conscious word is involved. Instead, I move my hands, squirm in my chair, and robotically step through the methods of solving the equations without any verbal prattle like ‘Ok…this goes over here and that goes down here and x moves to the other side of the equals sign’ …and so on. I keep telling the public that math is the language of science, but in fact it is not really a language at all. It’s like saying that oil is the language of painting a work of art! In fact, when you are proficient at mathematics, you are behaving like a concert violinist who does not think about each note she plays, but flows along on a trained sequence of steps that she learned. Her sequence of fine motor skills are stored in the cerebellum, but a mathematician’s skills are stored in an entirely different part of the brain.

Researchers have found a group of a few million cells located in a region called the inferior temporal gyrus. These cells seem to respond very strongly when you are doing concrete, numerical calculations like balancing your bank account, filling out your taxes, or plugging-in numbers in a complex equation to get an answer . So the brain does have discrete regions and clumps of neutrons that make mathematical reasoning possible. An entire hemisphere is dedicated to ‘logical analysis’, but specific locations allow you to work with this implicit logic in an entirely symbolic manner that resembles the language centers in the Broca’s and Wernicke areas, which facilitate speech and writing words.

So here we now have all the elements for creating a model of the outside world by using sensory data to find patterns in time, and to use these patterns to eventually deduce general mathematical If A then B logical statements about them that are entirely symbolic, and far more accurate than what the brain’s language centers can provide through its endless chatter.

The basis for all these deductions about the world outside the brain is a concrete understanding of what space and time are all about. We learn about space through our binocular vision, but also because our mobility allows us to move through space. Even with perfect stereo vision, you have no idea what those objects are that you are looking at unless you can literally walk over to them and appreciate their actual sizes, textures and other features. So our deep, personal understanding of 3-dimensional space comes about because we have mobility and stereo vision. But actual vision as a sense may not be that important after all. Echolocation used by bats and dolphins is not a visual decomposition of the world but an auditory one, processed to give back a 3-dimensional model of the world without vision, color or even the perception of black and white being involved.

Evidence from  brain-imaging experiments indicates that, when blind people read Braille using touch, the sensory data is being sent to, and processed in, the visual cortex. Using touch, they get a sense of space and the relative locations of the raised dots that form Braille letters . Although the information is processed in the visual cortex, their impression is not a visual,  but is instead a directly spatial one without the intermediary of vision to get them there!

Next time I will begin the discussion about how astronomers and physicists ‘envision’ space itself.

Check back here on Monday, December 12 for the next installment!

Mathematical Ability Revealed in Brain Scans, 2016, By Mindy Weisberger
http://www.livescience.com/54370-math-brain-network-discovered.html

Other related essays:
Your Brain on Math

How Do Blind People Picture Reality? By Natalie Wolchover
http://www.livescience.com/23709-blind-people-picture-reality.html