You thought bosons were strange? Well, wait, now comes the really strange quantum stuff - fermions.
At first sight, fermions are innocently looking and differing from bosons by the fact that they have half-integer spin. In the standard model, all quarks and leptons are fermions, and have spin one half. No elementary particle is known (though some hypothesized) which are fermions and have a larger spin than one half. But again, some particles made up from several elementary particles may look from afar like having a larger half-integer spin. E.g. the Delta, a heavier cousin of the proton and made up also from three quarks, has spin three halves.
In contrast to bosons, fermions dislike being at the same place. In fact, they can never take the same position, much like the classical balls. But there is a difference to the classical balls. For fermions, this not only applies to position, but also to all quantum numbers and energies. As a consequence, there can never be two fermions being having the same energy. This is the famous Pauli exclusion principle.
This principle has very fundamental consequences: It is responsible for the stability of all matter. If your desk would be made out of bosons, only electromagnetic repulsion would prevent it from collapsing to a pile of bosons. But because it is made out of fermions - all the quarks and electrons - it could never collapse to a single point. Because the fermions can just not get so near to each other. That is the fundamental reason which prevents a white dwarf or a neutron star from collapsing.
Very similar, it also prevents the electrons in an atom, which are attracted by the nucleus by electric forces, from collapsing into the lowest energy level, or into the nucleus outright. All of chemistry works the way it works because the electrons, since they are fermions, cannot all go into the lowest energy level. Otherwise, our chemistry, and thus our biology, would be very different, indeed.
But this is not the only strange thing about fermions. Fermions are also very strange in many other respects. As a consequence of the Pauli principle they obey again a different statistics, the so-called Fermi-Dirac statistics. The consequence of this are at the heart of why there are electric insulators.
But fermions are also strange in the sense that when you turn your coordinate system by 360 degrees, i.e. once fully around, everything is unchanged. Only the fermions do not play along: You have to turn your coordinate system twice around so that they look again the same (or, more precisely, their wave-function explained next time, looks the same). That is so mind-boggling that it is hard to believe it is true, and one cannot really intuitively understand this. It is a very deep combination of our space-time structure and quantum physics. There is no classical objects which behaves like this.
The mathematical consequences of these properties are little less strange. Fermions are the only objects which we cannot describe by ordinary numbers. Theoretical physicists had to invent a whole new type of numbers (well, actually borrow them from your friendly mathematician next door) to describe fermions - so-called Grassmann numbers. These are really strange. If you multiply an ordinary number with itself, you get a new number. If you multiply a Grassmann number with itself, you always get zero. That is the mathematical realization of the Pauli principle. This feature makes fermions very hard to handle in actual calculations, and they have been a bane especially to numerical simulations.
Nonetheless, they are there, and we are bound to live with them, as we are bound to live with bosons. Though - you can always combine two fermions to make something which looks from afar like a boson. But you can never combine two bosons such that they look from afar like a fermion. This fact has been found to be exploited very often by nature, as already described last time. And it lies at the heart of some ideas, so-called technicolor scenarios, to get rid of the Higgs with all its annoying properties: In such proposed extensions of the standard model, the Higgs is just a combinations of two new particles, so-called techniquarks.
Thursday, January 12, 2012
Wednesday, January 11, 2012
Bosons
The first type of particles are bosons bosons. Those are these having integer spin. In the standard model, there is the Higgs particle, which has spin zero, and the photons, the W and Z bosons, and the gluons, which all have spin one.
Particles with spin zero are also called scalar particles. Since their spin is zero, the properties of such particles are the simplest when changing to a different coordinate system: They just look the same.
Particles with spin one are also called vector particles. Such vector particles are described like photons. The name vector stems from the fact that under a coordinate transformation the fields describing a vector particle changes in the same way as a line which connects the origin of a coordinate system and an event. The latter line is also called a vector, and hence the name for particles of spin one.
There is actually also a hypothetical particle with spin two, the graviton. Such a particle is also called a tensor particle. Tensors are generalizations of vectors when it comes to coordinate transformations, and fields of spin two particles transform in the same way as such tensors. In general, tensors are rectangular collections of numbers, where the columns transform like a vector under coordinate transformation.
Elementary particles with higher spin are not known. However, particles made up from elementary particles add their spin together (though not necessarily in the sense 1+1=2 - it can also be subtracted, 1-1=0, and everything in between), and can thus have higher spins.
Furthermore, to each such type of bosons, there exists a so-called pseudo bosons, i. e. a pseudo scalar, a pseudo vector (sometimes for historical reasons also called an axial vector), and a pseudo tensor. The difference between a boson and a pseudo boson is what happenes if you reflect the world in a mirror (a parity transformation). Ordinary bosons just become bosons once more. In contrast, the fields of pseudo bosons are multiplied by minus one.
Ok, after all this classification and name stuff, what is special about bosons? The most striking feature is that you can pile them upon each other. That is different from the small balls one often uses to imagine elementary particles: We can stack such balls next to each other, but never ever can two of these balls be at the same place. But bosons can. That is very hard to get in line with our ideas of how things work, and it shows just how quantum bosons are: they behave in a way which is just unexpected.
This is, of course, only true, if the bosons do not repel each other by some force. For example, if you would have two electrically same-name charged bosons, you would have a hard time to bring them together. But if they have oppositely named charges then they would just love to sit at exactly the same place.
In fact, if bosons do not repel each other because of a force acting between them, they have a tendency to lump together - two bosons rather prefer to be at the same place than being apart. This phenomenon is again a pure quantum effect: If you would have two balls, which are not talking to each other, they ignore each other very consequently. The reason for this different behavior is encoded in what physicists call statistics. In case of the boson this statistics is called Bose-Einstein statistics, in contrast to the classical statistics of the balls. Statistics describes how particles distribute themselves. Classical statistics is essentially randomly distributed, but bosons with Bose-Einstein statistics are not entirely randomly distributed but tend to get together.
This property also pertains to a different thing: The energies the particles have. While classical particles have just their energy, independent of every other particle, as long as they do not interact, bosons tend to have the same energy.
The extreme case of getting together is occurring when a sizable fraction of all available bosons are involved, and all of them have the lowest possible energy. That is what is called a Bose-Einstein condensate. This type of stuff is a state of matter similar to being liquid or being solid. But it only occurs under rather extreme conditions, in particular at very low temperatures. On Earth, there is no naturally occurring case of such a condensate. But it was possible to create such condensates in the laboratory using atoms.
In particle physics, such condensates play a central role. The Higgs effect was associated with a condensate of Higgs particles: It is just such a Bose-Einstein condensate. The same applies to the mass generation from the strong force, though in this case it is not the quarks that form a condensate. Since they are fermions, as will be discussed next, this is not directly possible. But states made up from two quarks (or a quark and an anti-quark) can condense. Since spin adds, such states have either spin zero or one, and thus behave like a boson, if one is not looking too closely. And these effective bosons are, loosely speaking, condensing to a Bose-Einstein condensate in this case.
These are only some examples, but such condensates play very often a role, from superconductors to the interiors of neutron stars. Thus bosons, with their strange properties, are very important to physics, and especially particle physics.
Particles with spin zero are also called scalar particles. Since their spin is zero, the properties of such particles are the simplest when changing to a different coordinate system: They just look the same.
Particles with spin one are also called vector particles. Such vector particles are described like photons. The name vector stems from the fact that under a coordinate transformation the fields describing a vector particle changes in the same way as a line which connects the origin of a coordinate system and an event. The latter line is also called a vector, and hence the name for particles of spin one.
There is actually also a hypothetical particle with spin two, the graviton. Such a particle is also called a tensor particle. Tensors are generalizations of vectors when it comes to coordinate transformations, and fields of spin two particles transform in the same way as such tensors. In general, tensors are rectangular collections of numbers, where the columns transform like a vector under coordinate transformation.
Elementary particles with higher spin are not known. However, particles made up from elementary particles add their spin together (though not necessarily in the sense 1+1=2 - it can also be subtracted, 1-1=0, and everything in between), and can thus have higher spins.
Furthermore, to each such type of bosons, there exists a so-called pseudo bosons, i. e. a pseudo scalar, a pseudo vector (sometimes for historical reasons also called an axial vector), and a pseudo tensor. The difference between a boson and a pseudo boson is what happenes if you reflect the world in a mirror (a parity transformation). Ordinary bosons just become bosons once more. In contrast, the fields of pseudo bosons are multiplied by minus one.
Ok, after all this classification and name stuff, what is special about bosons? The most striking feature is that you can pile them upon each other. That is different from the small balls one often uses to imagine elementary particles: We can stack such balls next to each other, but never ever can two of these balls be at the same place. But bosons can. That is very hard to get in line with our ideas of how things work, and it shows just how quantum bosons are: they behave in a way which is just unexpected.
This is, of course, only true, if the bosons do not repel each other by some force. For example, if you would have two electrically same-name charged bosons, you would have a hard time to bring them together. But if they have oppositely named charges then they would just love to sit at exactly the same place.
In fact, if bosons do not repel each other because of a force acting between them, they have a tendency to lump together - two bosons rather prefer to be at the same place than being apart. This phenomenon is again a pure quantum effect: If you would have two balls, which are not talking to each other, they ignore each other very consequently. The reason for this different behavior is encoded in what physicists call statistics. In case of the boson this statistics is called Bose-Einstein statistics, in contrast to the classical statistics of the balls. Statistics describes how particles distribute themselves. Classical statistics is essentially randomly distributed, but bosons with Bose-Einstein statistics are not entirely randomly distributed but tend to get together.
This property also pertains to a different thing: The energies the particles have. While classical particles have just their energy, independent of every other particle, as long as they do not interact, bosons tend to have the same energy.
The extreme case of getting together is occurring when a sizable fraction of all available bosons are involved, and all of them have the lowest possible energy. That is what is called a Bose-Einstein condensate. This type of stuff is a state of matter similar to being liquid or being solid. But it only occurs under rather extreme conditions, in particular at very low temperatures. On Earth, there is no naturally occurring case of such a condensate. But it was possible to create such condensates in the laboratory using atoms.
In particle physics, such condensates play a central role. The Higgs effect was associated with a condensate of Higgs particles: It is just such a Bose-Einstein condensate. The same applies to the mass generation from the strong force, though in this case it is not the quarks that form a condensate. Since they are fermions, as will be discussed next, this is not directly possible. But states made up from two quarks (or a quark and an anti-quark) can condense. Since spin adds, such states have either spin zero or one, and thus behave like a boson, if one is not looking too closely. And these effective bosons are, loosely speaking, condensing to a Bose-Einstein condensate in this case.
These are only some examples, but such condensates play very often a role, from superconductors to the interiors of neutron stars. Thus bosons, with their strange properties, are very important to physics, and especially particle physics.
Monday, January 9, 2012
Spin
One of the most intriguing and most important properties of an elementary particle is its spin. At the same time, spin is one of the conceptually most problematic quantities, and has led to an enormous amount of misunderstandings.
The reason for this is that there is something in classical physics, which is very closely related to the concept of spin. But this relation is in spirit, rather than literally, and this has led to a lot of confusion. This analogue is angular momentum.
So, first, what is angular momentum? Angular momentum is connected with any kind of rotation of a particle around some center. Formally, it is a product involving the radius of the rotation and the speed along the path of the object. In classical physics, without friction, it is conserved, and it is what keeps the planets' orbits in their respective plane. It is likely also responsible for the fact that all the orbits are more or less in the same plane, or that the milky way has the over-all form of a discus (neglecting the spiral arms). In essence, it is just a reformulation of the ordinary speed, mixed with the mass of a particle. Essentially a kinematic quantity, despite its importance.
If an object just rotates, e.g. a ball, then each of the elements of the ball rotates. This can be described by giving the ball as such an angular momentum. Since the geometry of the ball is known and fixed, it is possible to defer from this total angular momentum the angular momentum of every piece of the ball.
In the world of particles, this angular momentum is reappearing whenever there is something having some kind of relative motion. E. g. in an atom, it is possible to assign the electrons an angular momentum, which is then often called orbital angular momentum (a somewhat complicated name). However, the electrons are not actually small spheres orbiting around the nucleus, bur rather smeared out over the whole of the atom. What this precisely means, I will discuss later. The important thing is that this smeared out something has a kind of orbital movement (the whole object 'rotates' in a certain sense), and can therefore be assigned such an orbital angular momentum.
It is a remarkable observation in quantum physics that angular momentum cannot take any value it likes. It is quantized. The reason for this quantization is the inherent relation between angular momentum and speed, and then speed and energy. Because energy is quantized this implies that angular momentum is quantized.
As orbital angular momentum depends on the momentum, and thus on the speed, its numerical value changes when we as the observer are changing our movement. This does not change the path of the rotating objects, just our perception of it, of course. Therefore, this change of values is closely tied to our change of our coordinate system, when we move.
Now enter spin: It was very early on recognized in quantum physics that elementary particles have a property which changes in the same way as the angular momentum of the ball when we change our coordinate system. This was an intrinsic property of the particles, unchangeable. However, the elementary particles are point-like, at least to the extent we can resolve them. Thus, they cannot rotate in any way, as they do not have any extension. In fact, if this would be an ordinary angular momentum, and the elementary particles would have a small extension, then within our experimental knowledge about the upper limit of this extension, their surface would need to rotate much faster than the speed of light.
Thus, this property got its own name: Spin. This is still inspired by the similarity to (orbital) angular momentum under a change of coordinate system, but by keeping strictly the difference in name, it can always be distinguished from it. However, from time to time its is useful to refer to them both together, and in this case they are called total angular momentum, which is in principle somewhat a misnomer.
Now spin is also quantized, and there exist both half-integer and integer values for it (when choosing appropriate units). This is different from ordinary angular momentum for two reasons. First, there is no simple explanation for the quantization like for angular momentum. There is indeed a complicated explanation, which shows that for the space-time structure which we have, these are the only two possibilities consistent with this type of change under a change of the coordinate system. Second, angular momentum, when measured in the same units, can have only integer values.
The latter is an intriguing difference. It has a very important consequence: Particles having integer spin behave very different from those having a half-integer spin. Therefore, these two types of particles received different names: The former are called bosons, and the latter are called fermions. This distinction is of fundamental importance to particle physics, and therefore the next two entries will discuss both types of particles in more detail. Also, none of these types behave in the same way as an ordinary small ball. But it turns out that if one takes the classical (long-distance) limit, both behave in the same way, and like small balls: Classically fermions and bosons can not be distinguished, their existence is a pure quantum effect, which is intricately linked to the structure of space and time. That is one of the reasons why some people believe that the quantum effect of spin and gravity may be related at a deeper level, but we are very far from understanding whether this suspicion is correct.
The reason for this is that there is something in classical physics, which is very closely related to the concept of spin. But this relation is in spirit, rather than literally, and this has led to a lot of confusion. This analogue is angular momentum.
So, first, what is angular momentum? Angular momentum is connected with any kind of rotation of a particle around some center. Formally, it is a product involving the radius of the rotation and the speed along the path of the object. In classical physics, without friction, it is conserved, and it is what keeps the planets' orbits in their respective plane. It is likely also responsible for the fact that all the orbits are more or less in the same plane, or that the milky way has the over-all form of a discus (neglecting the spiral arms). In essence, it is just a reformulation of the ordinary speed, mixed with the mass of a particle. Essentially a kinematic quantity, despite its importance.
If an object just rotates, e.g. a ball, then each of the elements of the ball rotates. This can be described by giving the ball as such an angular momentum. Since the geometry of the ball is known and fixed, it is possible to defer from this total angular momentum the angular momentum of every piece of the ball.
In the world of particles, this angular momentum is reappearing whenever there is something having some kind of relative motion. E. g. in an atom, it is possible to assign the electrons an angular momentum, which is then often called orbital angular momentum (a somewhat complicated name). However, the electrons are not actually small spheres orbiting around the nucleus, bur rather smeared out over the whole of the atom. What this precisely means, I will discuss later. The important thing is that this smeared out something has a kind of orbital movement (the whole object 'rotates' in a certain sense), and can therefore be assigned such an orbital angular momentum.
It is a remarkable observation in quantum physics that angular momentum cannot take any value it likes. It is quantized. The reason for this quantization is the inherent relation between angular momentum and speed, and then speed and energy. Because energy is quantized this implies that angular momentum is quantized.
As orbital angular momentum depends on the momentum, and thus on the speed, its numerical value changes when we as the observer are changing our movement. This does not change the path of the rotating objects, just our perception of it, of course. Therefore, this change of values is closely tied to our change of our coordinate system, when we move.
Now enter spin: It was very early on recognized in quantum physics that elementary particles have a property which changes in the same way as the angular momentum of the ball when we change our coordinate system. This was an intrinsic property of the particles, unchangeable. However, the elementary particles are point-like, at least to the extent we can resolve them. Thus, they cannot rotate in any way, as they do not have any extension. In fact, if this would be an ordinary angular momentum, and the elementary particles would have a small extension, then within our experimental knowledge about the upper limit of this extension, their surface would need to rotate much faster than the speed of light.
Thus, this property got its own name: Spin. This is still inspired by the similarity to (orbital) angular momentum under a change of coordinate system, but by keeping strictly the difference in name, it can always be distinguished from it. However, from time to time its is useful to refer to them both together, and in this case they are called total angular momentum, which is in principle somewhat a misnomer.
Now spin is also quantized, and there exist both half-integer and integer values for it (when choosing appropriate units). This is different from ordinary angular momentum for two reasons. First, there is no simple explanation for the quantization like for angular momentum. There is indeed a complicated explanation, which shows that for the space-time structure which we have, these are the only two possibilities consistent with this type of change under a change of the coordinate system. Second, angular momentum, when measured in the same units, can have only integer values.
The latter is an intriguing difference. It has a very important consequence: Particles having integer spin behave very different from those having a half-integer spin. Therefore, these two types of particles received different names: The former are called bosons, and the latter are called fermions. This distinction is of fundamental importance to particle physics, and therefore the next two entries will discuss both types of particles in more detail. Also, none of these types behave in the same way as an ordinary small ball. But it turns out that if one takes the classical (long-distance) limit, both behave in the same way, and like small balls: Classically fermions and bosons can not be distinguished, their existence is a pure quantum effect, which is intricately linked to the structure of space and time. That is one of the reasons why some people believe that the quantum effect of spin and gravity may be related at a deeper level, but we are very far from understanding whether this suspicion is correct.
Tuesday, November 29, 2011
Chiral - or why left and right is not always just a mirror image of each other
One of the things we observe in everyday life is that things have a distinct left and right. The simplest case is just the hands of a human: Obviously, the left hand and the right hand are different from each other. That is a very general thing in nature that things can be 'like a left hand' or 'like a right hand'. Of course, they do not need to be so. A ball has obviously no distinct left or right. But things can have. This fact is known in science as chirality, originating from a Greek word for hand.
Left and right are actually not that different. If you take a mirror, and look at a left hand in the mirror, it looks light a right hand. Such a process, which turns something behaving like a left hand into something like a right hand, is called a parity transformation in particle physics.
So far, so good, and some fancy names. Why should this matter? Indeed, it does matter quite a bit. In biology, molecules can also be chiral. And then it turns out that a certain handedness is nutritious for us, while the opposite handedness is at best useless and at worst toxic. Our body has a preference for a certain hand, it is chiral. The fact that the left-handed version of the molecule and the right-handed version of the molecule have different consequences implies that looking through the mirror is not always just a mirror image, but can be something entirely different. Parity is not just a change of perspective: The mirror image in this case is broken, and therefore one tends to say that parity, the property that something becomes just the mirror image without further changes, is broken.
So, what has this to do with particle physics? Well, also some elementary particles have a handedness. This handedness is an intrinsic property of such particles, such as a color for a billiard ball. This is especially important for the quarks and leptons of the standard model. Of each of them two exists: A left-handed one and a right-handed one.
When it comes to the strong interactions or to electromagnetism, this actually does not matter. For these two forces, both types of particles look exactly the same, and thus neither of these forces can actually distinguish between between left and right. These forces are also said to be parity invariant.
This changes when it comes to the weak interactions. The weak interactions are very special, and they distinguish between both types of particles. In fact, they are very extreme in this respect: The only act on the left-handed particles, but completely ignore the right-handed particles. It is said that the weak force is parity violating, or simply it is said that the weak interaction is chiral.
The consequences of this is quite profound, though not obvious. Take for example an atom with a nucleus which is unstable, and decays by emitting so-called beta radiation, i.e. electrons. If you suspend such an atom in a magnetic field, it turns out that the electrons emitted move in a preferential directions. This occurs, because the weak interactions are chiral. If they would not be, this would not happen. Nonetheless, this example shows that it requires something of sophistication to observe this.
Still, this chirality in the standard model is quite important. From a mathematical point of view, it is very restricting for the structure of the standard model. It has also quite important implications for each and every of our attempts to extend the standard model. Furthermore, in actual calculations it is quite a nuisance.
However, after all, we do not know why the weak interaction, but not the other two, are chiral. It is something we observe, and it is one of the bigger mysteries in particle physics. Therefore, looking for modifications of chiral properties is also a big chance to find something new. Since we have either perfect parity or not at all in the standard model, anything else would be new. Also, because we are so completely baffled by it, we think that whatever kind of observation is unexpected in context with a parity violation will very quickly leads us to a glimpse of whatever there is beyond the standard model.
Left and right are actually not that different. If you take a mirror, and look at a left hand in the mirror, it looks light a right hand. Such a process, which turns something behaving like a left hand into something like a right hand, is called a parity transformation in particle physics.
So far, so good, and some fancy names. Why should this matter? Indeed, it does matter quite a bit. In biology, molecules can also be chiral. And then it turns out that a certain handedness is nutritious for us, while the opposite handedness is at best useless and at worst toxic. Our body has a preference for a certain hand, it is chiral. The fact that the left-handed version of the molecule and the right-handed version of the molecule have different consequences implies that looking through the mirror is not always just a mirror image, but can be something entirely different. Parity is not just a change of perspective: The mirror image in this case is broken, and therefore one tends to say that parity, the property that something becomes just the mirror image without further changes, is broken.
So, what has this to do with particle physics? Well, also some elementary particles have a handedness. This handedness is an intrinsic property of such particles, such as a color for a billiard ball. This is especially important for the quarks and leptons of the standard model. Of each of them two exists: A left-handed one and a right-handed one.
When it comes to the strong interactions or to electromagnetism, this actually does not matter. For these two forces, both types of particles look exactly the same, and thus neither of these forces can actually distinguish between between left and right. These forces are also said to be parity invariant.
This changes when it comes to the weak interactions. The weak interactions are very special, and they distinguish between both types of particles. In fact, they are very extreme in this respect: The only act on the left-handed particles, but completely ignore the right-handed particles. It is said that the weak force is parity violating, or simply it is said that the weak interaction is chiral.
The consequences of this is quite profound, though not obvious. Take for example an atom with a nucleus which is unstable, and decays by emitting so-called beta radiation, i.e. electrons. If you suspend such an atom in a magnetic field, it turns out that the electrons emitted move in a preferential directions. This occurs, because the weak interactions are chiral. If they would not be, this would not happen. Nonetheless, this example shows that it requires something of sophistication to observe this.
Still, this chirality in the standard model is quite important. From a mathematical point of view, it is very restricting for the structure of the standard model. It has also quite important implications for each and every of our attempts to extend the standard model. Furthermore, in actual calculations it is quite a nuisance.
However, after all, we do not know why the weak interaction, but not the other two, are chiral. It is something we observe, and it is one of the bigger mysteries in particle physics. Therefore, looking for modifications of chiral properties is also a big chance to find something new. Since we have either perfect parity or not at all in the standard model, anything else would be new. Also, because we are so completely baffled by it, we think that whatever kind of observation is unexpected in context with a parity violation will very quickly leads us to a glimpse of whatever there is beyond the standard model.
Wednesday, October 26, 2011
Always the opposite: Anti-matter
The last time, I made a brief remark about anti-particles. It is about time to illustrate this rather obscure notion.
What is meant, when we talk about anti-particles? Well, just from the experimental point of view, it is found that for every particle there exists another particle, which has (within experimental certainty) exactly the same mass. It has also the same properties when it comes to the way it spins, the so-called spin. This spin is also something I will explain sometimes else, what this mysterious property is.
However, considering everything else, it is exactly the opposite: If the particle has a negative electric charge, the anti-particle has a positive electric charge. If the particle has color red, the anti-particle has an opposite charge, which is called for the lack of a better name anti-red. And so on. The only exception to this rule are those particles which have, except for mass and spin, no other properties. An example is the photon, which is completely uncharged. In this case, the particle is its own anti-particle.
Now, these are rather surprising objects, but we have very good experimental proof that they exist. In fact, we know anti-matter so well that some experiments, like the old LEP at CERN, use matter and anti-matter routinely as a starting point: At LEP electrons and their anti-particles, the positrons, have been collided.
Matter and anti-matter show a very spectacular effect: Because one plus minus one is zero, it is possible for matter and anti-matter to annihilate each other when they are colliding into something else. For example photons. Or other particles. That happens very easily. Hence you may ask, why we do not annihilate whenever we touch something. The answer is surprisingly simple: Because everything around us is made from matter. If we want to use anti-matter or study it, we have to create it artificially. That is not simple, and we can only create very tiny amounts efficiently. Large amounts become rapidly very expensive, mostly because it is not simple to keep it away from matter, so that is not annihilating with it.
That seems a simple enough answer, but the real question baffling physicist is: Why is this so? If they are so equal, why do we not have the same amount of both (and thus vanish in a big photon cloud)? That is another of the questions we do not yet have a real answer to. Irritatingly, the problem is actually not that we do not know how this can be realized. In fact, in the standard model of particle physics, there is a very, very slight preference for matter over antimatter when it comes to the weak force. This implies that though matter and anti-matter are essentially equal, the forces make a difference between them. However, this effect is by far too small to explain why there is so a fantastically little amount of antimatter around us.
Ok, so you might say: Let us forget for the moment about the experimental evidence, and ask, do we really need anti-mater. Could this simple explanation just be a misinterpretation of the experiments, and what we think is anti-matter is really something else? Well, if this should be the case, we would have to rethink our complete view of how the standard model is described theoretically as well. Indeed, the mathematical structure of the standard model requires the existence of an anti-particle for each particle to work properly. If we would remove the anti-particles from the theory, the consequence would be dramatic. It would even be possible to obtain effects without cause or causes without having effects. This is not what we observe, but what we observe is described by the standard model with particles and anti-particles. Thus we take the experimental results as evidence for anti-particles, and everything fits together when we calculate something.
Of course, this means that we essentially double the number of particles. Up to exceptions like the photon, all particles are now accompanied by their anti-particles. And to every charge comes an anti-charge. However, this also provides new options for new phenomena. The last time last time, this gave us the option of a condensate of quarks and anti-quarks. Also, their are bound states of quarks and anti-quarks, the so-called mesons. The most famous and lightest of them are the so-called pions, of which there are three: One is uncharged, and there is one positively charged and one negatively charged. Th neutral one is again its own anti-particle. The reason is that it is made up out of a quark and the corresponding anti-quark. Thus replacing constituent particle by constituent anti-particle gives again the same bound state. The charged ones are each others anti-particle, because they contain an up and an anti-down quark and an anti-up and a down quark, respectively. Exchange particles by anti-particles yields an exchange of both bound states. So, one can have a lot of fun with building things out of particles and anti-particles.
You can also take a hydrogen atom, and exchanges its nucleus, a proton, by the anti-particle of the electron, a positron. Because the positron has the same electric charge as the proton, you get even something looking very much like an atom. This is called positronium, known for a very long time. Recently, it has also been possible to create true anti-atoms, made from an anti-nucleus and positrons. These are very important to test, whether we really have understood everything about anti-mater. If we have, they should behave in the same way as ordinary atoms. And whether this is the case the experimentalists right now try to find out.
What is meant, when we talk about anti-particles? Well, just from the experimental point of view, it is found that for every particle there exists another particle, which has (within experimental certainty) exactly the same mass. It has also the same properties when it comes to the way it spins, the so-called spin. This spin is also something I will explain sometimes else, what this mysterious property is.
However, considering everything else, it is exactly the opposite: If the particle has a negative electric charge, the anti-particle has a positive electric charge. If the particle has color red, the anti-particle has an opposite charge, which is called for the lack of a better name anti-red. And so on. The only exception to this rule are those particles which have, except for mass and spin, no other properties. An example is the photon, which is completely uncharged. In this case, the particle is its own anti-particle.
Now, these are rather surprising objects, but we have very good experimental proof that they exist. In fact, we know anti-matter so well that some experiments, like the old LEP at CERN, use matter and anti-matter routinely as a starting point: At LEP electrons and their anti-particles, the positrons, have been collided.
Matter and anti-matter show a very spectacular effect: Because one plus minus one is zero, it is possible for matter and anti-matter to annihilate each other when they are colliding into something else. For example photons. Or other particles. That happens very easily. Hence you may ask, why we do not annihilate whenever we touch something. The answer is surprisingly simple: Because everything around us is made from matter. If we want to use anti-matter or study it, we have to create it artificially. That is not simple, and we can only create very tiny amounts efficiently. Large amounts become rapidly very expensive, mostly because it is not simple to keep it away from matter, so that is not annihilating with it.
That seems a simple enough answer, but the real question baffling physicist is: Why is this so? If they are so equal, why do we not have the same amount of both (and thus vanish in a big photon cloud)? That is another of the questions we do not yet have a real answer to. Irritatingly, the problem is actually not that we do not know how this can be realized. In fact, in the standard model of particle physics, there is a very, very slight preference for matter over antimatter when it comes to the weak force. This implies that though matter and anti-matter are essentially equal, the forces make a difference between them. However, this effect is by far too small to explain why there is so a fantastically little amount of antimatter around us.
Ok, so you might say: Let us forget for the moment about the experimental evidence, and ask, do we really need anti-mater. Could this simple explanation just be a misinterpretation of the experiments, and what we think is anti-matter is really something else? Well, if this should be the case, we would have to rethink our complete view of how the standard model is described theoretically as well. Indeed, the mathematical structure of the standard model requires the existence of an anti-particle for each particle to work properly. If we would remove the anti-particles from the theory, the consequence would be dramatic. It would even be possible to obtain effects without cause or causes without having effects. This is not what we observe, but what we observe is described by the standard model with particles and anti-particles. Thus we take the experimental results as evidence for anti-particles, and everything fits together when we calculate something.
Of course, this means that we essentially double the number of particles. Up to exceptions like the photon, all particles are now accompanied by their anti-particles. And to every charge comes an anti-charge. However, this also provides new options for new phenomena. The last time last time, this gave us the option of a condensate of quarks and anti-quarks. Also, their are bound states of quarks and anti-quarks, the so-called mesons. The most famous and lightest of them are the so-called pions, of which there are three: One is uncharged, and there is one positively charged and one negatively charged. Th neutral one is again its own anti-particle. The reason is that it is made up out of a quark and the corresponding anti-quark. Thus replacing constituent particle by constituent anti-particle gives again the same bound state. The charged ones are each others anti-particle, because they contain an up and an anti-down quark and an anti-up and a down quark, respectively. Exchange particles by anti-particles yields an exchange of both bound states. So, one can have a lot of fun with building things out of particles and anti-particles.
You can also take a hydrogen atom, and exchanges its nucleus, a proton, by the anti-particle of the electron, a positron. Because the positron has the same electric charge as the proton, you get even something looking very much like an atom. This is called positronium, known for a very long time. Recently, it has also been possible to create true anti-atoms, made from an anti-nucleus and positrons. These are very important to test, whether we really have understood everything about anti-mater. If we have, they should behave in the same way as ordinary atoms. And whether this is the case the experimentalists right now try to find out.
Monday, October 10, 2011
Mass from the strong force
Quite some time ago, I have discussed the Higgs effect, and how it gives the matter particles in the standard model their mass. However, if we look around us, it turns out that most mass we see is actually not due to the Higgs effect.
If we knock on a table, or look at us, then most of this is made up out of atoms. As you might remember, atoms are made up out of nuclei and electrons. The electrons actually get their mass from the Higgs, so that is alright. But they make up less than 0.05% of the mass of the atoms. Thus, one can forget about them for this purpose. Then there are the nuclei. They are made up out of protons and neutrons. These in turn consist out of quarks. But the quarks are rather light, and make up not more than one percent of the mass of the protons and neutrons, and thus of the nuclei. So where does all the remaining mass comes from?
Well, this comes this time from the strong nuclear force, QCD, which has been presented here and here. I have already indicated there that it is a pretty strong force. It is this strength which, indirectly, creates all the mass we are yet missing.
How it works is actually quite complicated in detail, but when being a bit fuzzy about the details, it can be illustrated quite nicely. Looking through such fuzzy glasses, it actually looks like a repetition of the Higgs effect. Remember, the Higgs effect worked by letting the Higgs particles condense. The interaction with this condensate slows particle down, and therefore they behave as having a mass.
Now, how does this proceed in the case of the strong force? The first observation is that the strong force is attractive between quarks, i.e. quarks are attracted to each other. As a consequence, the quarks can form all these nice things like protons. However the force is also attractive between the quarks and their so-called anti-particles. What an anti-particle is I will discuss next time. This time, it is just sufficient to say that it behaves like a quark, but has opposite charges and the same mass.
The strong force now pulls also the quarks and the antiquarks together. The combination of two such particles is then neutral, as all the charges are opposite. It behaves therefore very similar to a Higgs particle. In very much the same way, but this time because of gluons rather than due to the Higgs interacting with itself, this creates also a condensate. That is just like for the Higgs particles. Thus, the universe is filled with quarks and anti-quarks, bound together, and condensed by the strong force. The strong interaction of quarks with this condensate, and the corresponding slowing down, is what provides the remainder mass for the protons and neutrons, and thus for the nuclei. Again, because all the charges cancel, we do not see this condensate, as photons due not directly interact with it.
Putting in numbers, the contribution of the strong force to the mass of the nuclei is much larger than the one due to the Higgs effect. However, the calculation of this was rather challenging. Hence, most of the mass stored in the atoms in the universe is due to the strong force.
It is also said that the strong force provides all the luminous mass in the universe. Here, luminous means actually not only all stars which emit light by themselves, but also everything which reflects light, like planets, interstellar gas clouds, and asteroids. Actually, the latter also emit a kind of light by themselves, but our eyes are not sensitive for the wave-lengths they emit, and therefore we do not see it. The distinction is necessary, because we have indirect evidence that there is also more than just this type of mass in the universe. In fact, we expect that there is about five times more non-luminous, or dark, matter in the universe, than luminous matter. What the origin of mass is in this case is not known.
There is another thing you may wonder about. The quarks have all very different masses due to the Higgs effect. Is the contribution due to the strong interaction also very different for the different quarks? The answer to this is actually no, the contribution from the strong force is about the same for all quarks. Thus, it makes up about 99% of the mass of the light quarks, but less than half a percent for the heaviest one. Thus, while the Higgs makes a difference between the different quark (and lepton) species, the strong force does not. Why this is the case is also yet unknown, and one of the bigger mysteries. Since the different quarks and leptons are also called different flavors of quarks and leptons, it is said that the strong force is flavor-blind, it makes no difference between different flavors. On the other hand, the Higgs makes a different between different flavors.
Finally, it should be noted that the generation of mass can be traced back to a symmetry, though I will not detail this now. This is the so-called chiral symmetry. A thing is called chiral, if it makes a difference between left and right. The associated symmetry in the standard model is a local symmetry. The Higgs effect breaks, loosely spoken, this symmetry to a global one. The strong force then breaks this global symmetry. It is possible, but mathematically involved, to show that these breakings correspond to the existence of mass. Hence, both the Higgs effect and the strong force produce mass. But the above explanations are somewhat more illuminating, I think, though mathematically both views are essentially equivalent. Thus, it is said that mass is created by chiral symmetry breaking, a notion I will right now not dwell on anymore.
If we knock on a table, or look at us, then most of this is made up out of atoms. As you might remember, atoms are made up out of nuclei and electrons. The electrons actually get their mass from the Higgs, so that is alright. But they make up less than 0.05% of the mass of the atoms. Thus, one can forget about them for this purpose. Then there are the nuclei. They are made up out of protons and neutrons. These in turn consist out of quarks. But the quarks are rather light, and make up not more than one percent of the mass of the protons and neutrons, and thus of the nuclei. So where does all the remaining mass comes from?
Well, this comes this time from the strong nuclear force, QCD, which has been presented here and here. I have already indicated there that it is a pretty strong force. It is this strength which, indirectly, creates all the mass we are yet missing.
How it works is actually quite complicated in detail, but when being a bit fuzzy about the details, it can be illustrated quite nicely. Looking through such fuzzy glasses, it actually looks like a repetition of the Higgs effect. Remember, the Higgs effect worked by letting the Higgs particles condense. The interaction with this condensate slows particle down, and therefore they behave as having a mass.
Now, how does this proceed in the case of the strong force? The first observation is that the strong force is attractive between quarks, i.e. quarks are attracted to each other. As a consequence, the quarks can form all these nice things like protons. However the force is also attractive between the quarks and their so-called anti-particles. What an anti-particle is I will discuss next time. This time, it is just sufficient to say that it behaves like a quark, but has opposite charges and the same mass.
The strong force now pulls also the quarks and the antiquarks together. The combination of two such particles is then neutral, as all the charges are opposite. It behaves therefore very similar to a Higgs particle. In very much the same way, but this time because of gluons rather than due to the Higgs interacting with itself, this creates also a condensate. That is just like for the Higgs particles. Thus, the universe is filled with quarks and anti-quarks, bound together, and condensed by the strong force. The strong interaction of quarks with this condensate, and the corresponding slowing down, is what provides the remainder mass for the protons and neutrons, and thus for the nuclei. Again, because all the charges cancel, we do not see this condensate, as photons due not directly interact with it.
Putting in numbers, the contribution of the strong force to the mass of the nuclei is much larger than the one due to the Higgs effect. However, the calculation of this was rather challenging. Hence, most of the mass stored in the atoms in the universe is due to the strong force.
It is also said that the strong force provides all the luminous mass in the universe. Here, luminous means actually not only all stars which emit light by themselves, but also everything which reflects light, like planets, interstellar gas clouds, and asteroids. Actually, the latter also emit a kind of light by themselves, but our eyes are not sensitive for the wave-lengths they emit, and therefore we do not see it. The distinction is necessary, because we have indirect evidence that there is also more than just this type of mass in the universe. In fact, we expect that there is about five times more non-luminous, or dark, matter in the universe, than luminous matter. What the origin of mass is in this case is not known.
There is another thing you may wonder about. The quarks have all very different masses due to the Higgs effect. Is the contribution due to the strong interaction also very different for the different quarks? The answer to this is actually no, the contribution from the strong force is about the same for all quarks. Thus, it makes up about 99% of the mass of the light quarks, but less than half a percent for the heaviest one. Thus, while the Higgs makes a difference between the different quark (and lepton) species, the strong force does not. Why this is the case is also yet unknown, and one of the bigger mysteries. Since the different quarks and leptons are also called different flavors of quarks and leptons, it is said that the strong force is flavor-blind, it makes no difference between different flavors. On the other hand, the Higgs makes a different between different flavors.
Finally, it should be noted that the generation of mass can be traced back to a symmetry, though I will not detail this now. This is the so-called chiral symmetry. A thing is called chiral, if it makes a difference between left and right. The associated symmetry in the standard model is a local symmetry. The Higgs effect breaks, loosely spoken, this symmetry to a global one. The strong force then breaks this global symmetry. It is possible, but mathematically involved, to show that these breakings correspond to the existence of mass. Hence, both the Higgs effect and the strong force produce mass. But the above explanations are somewhat more illuminating, I think, though mathematically both views are essentially equivalent. Thus, it is said that mass is created by chiral symmetry breaking, a notion I will right now not dwell on anymore.
Friday, September 16, 2011
What is strong and what is weak?
Given the previous discussion that things we measure depend on the energy at which we measure them, one could reasonably ask whether names like a strong force or a weak force are justified, or whether the strength of a force may not also depend on the energy.
Actually, it is precisely like this. An this is something which is at the same time a blessing and a bane to theoretical calculations.
When we measure the strength of an interaction, what we do is actually rather indirect. We start out with two particles. These we accelerate by some means, e.g. a particle accelerator like the LHC at CERN. Then we aim them at each other and let them collide. Afterwards, we measure what is left of them, and whether something new has been created in the process.
A practical complication is actually quantum physics. The latter tells us that when we perform an experiment more than once, we cannot perform it such that it will produce precisely the same result. However, this does not mean that physics is arbitrary. Lets make the experiment many times. Then we get an average result. Furthermore, if we build a second experiment to make the same collision, on the average it will yield the same result. In this sense it is reproducible. However, for many things we have to make very many collisions such that we get a precise enough average. That is one of the reasons why we not can just make one experiment at LHC and have immediately the answer whether the Higgs is there or not. We have to perform our averages in this case very precisely.
Well, lets return from this detour. Just keep in mind that when I talk here about an experimental result, it is actually more than the simple picture I will draw.
So, we measured what is left after the two particles collide. Then we may either get the result that the two particles after the collision are essentially unscathed, maybe a bit deflected, but that is it. Or, we may end up with new particles, and a serious redistribution of energy among them. Obviously, in the first case the interaction is not so very strong between the particles, wile it is rather strong in the later case. When looking at the details, it is not that simple, but this should illustrate the point that we can learn something about the strength of an interaction when we let particles collide.
We did this, and what we found is that the strength of the forces is indeed dependent on the energy. These findings are rather remarkably, and where of the following nature:
When we did this with the electromagnetic force, we found that the electromagnetic interaction becomes stronger and stronger the higher the energy. However, the increase is very, very slow. But we were able to measure it very accurately, and it also agrees with the theory to an extremely good precision. The increase of the strength of the electromagnetic force is such that by increasing the energy by a factor of 100, we get only an effect of a few percent. Thus, no man-made experiment is just yet able to test how strong the electromagnetic force is really becoming before we run into the problem that the standard model is just a a low-energy approximation.
The opposite turns out to be the case with the strong nuclear force. At low energies it is actually very strong, the strongest force we know so far, making all the nuclear physics, and keeping our protons and neutrons together in the nuclei, and the quarks even more deeply hidden. However, when increasing the energy, the strength of this force quickly decreases. If we would be able to look at arbitrary large energies, it would become arbitrarily small. This feature is called asymptotic freedom. As a consequence, the strong force becomes the simpler to deal with in theory the higher the energies. Therefore, our best tests of our theory of the strong force comes actually from high energies rather then from nuclear physics, as one would expect.
The weak force is actually very similar to the strong force, and it becomes weaker the higher the energy. The difference is that it starts out already rather weak, at least superficially, and thus its consequences are not nearly as spectacular as the ones of the strong force.
And there is the last remaining set of forces in the standard model. These are the ones associated with the Higgs, which provide the mass to all particles. This is a force we could not yet measure reliably in experiment, since we were not yet able to find the Higgs. Thus, we can make only a statement based on indirect evidence, and theoretical expectations. If they are true, then we expect that the interactions due to the Higgs will, like the electromagnetic interactions, also strengthen with energy. However, it turns out that the question of when these forces become strong depends very much on the mass of the Higgs. If the Higgs is as light as currently still permitted by the experiments, the corresponding interactions will rise very slowly, just like for electromagnetism. Then this rise can be ignored also for most purposes. If the Higgs would be only ten times heavier then the situation would be completely different. We would then expect to see relatively strong interactions between the Higgs and the other particles in the standard model at the LHC. That we did not see them yet does not mean they are not there - it is still hard to produce a Higgs which could interact strongly. Thus such potentially strong interactions are very rare even at such an experiment like the LHC.
Thus, the names of the forces are what they are just because at the low energies, where we first encountered them historically, they were weak and strong. And then these names just stuck, as they always do.
Thursday, September 1, 2011
Why mass depends on energy
One of the fascinating topics one is confronted with in particle physics is the fact that quantities depend on our perspective. For example, masses and the strengths, with which particles interact, depend on the energy we use to probe them. That may sound strange at first. However, it is not so surprising that such quantities may depend on our way of looking at them.
Take an apple. It has a certain mass, say 100 gram. However, if you put yourself inside the apple, the amount of apple you see before your (if you look towards the pit) is less than the whole apple, and its mass is thus less. In a very cartoon idea, you could also say that this depends on the energy: If you ride a (rather slow) bullet, its penetration depth will depend on its initial energy, and thus the higher the energy, the less of the apple mass remains in front of you.
Of course, things can get not only less, but also more. Talk a light bulb (or LED, if you like it more modern). Put veils upon it. The amount of light you perceive depends then on the number of veils between you and the bulb. The more veils are already behind you, and the less before you, the brighter the bulb. Of course, the idea with the bullet applies here, too. Just avoid hitting the bulb.
Well, for particles you do not have veils or outer shells of an apple. So what is going on there? What is going on is that the vacuum around a particle is actually a rather thick soup than really empty. This seems surprising at first - after all, we have been thinking that the space in, e.g., an atom is essentially empty. The reason for this apparent contradiction is quantum physics. In quantum physics, we got used to the fact that we cannot really say what is going on, and everything becomes fuzzy. In particular, we cannot say whether a portion of the vacuum is really empty or contains particles, as long as we do not make a very precise measurement. In the head of theoretical physicists, this observation of nature has formed the picture of so-called virtual particles.
Virtual particles are particles which appear and disappear all the time. They may either be emitted by a source, like another particle, or may even pop out randomly (though necessarily at least in pairs) from the vacuum. They only exist for a very brief glimpse, and are then reabsorbed by the source, or annihilate again into nothing. Only, if we look close enough, i.e. short distances and thus high energies, we can check whether such particles are there or not.
In particular, if we want to look at one particle, or the interaction of two or more particles, they are surrounded by a cloud of such virtual particles. Only if we get close enough to them, and that means at high energies, we can dive through this cloud. However, to really see the pure and unaltered particle (or process), we would need to resolve it with a wave-length of its size, but this is zero. Hence, this would require infinite energy, because we want to measure them at zero distance. So we cannot.
However, the more energy we invest, the deeper we get into the cloud, and the more we see of the particle's or process' properties. Measuring these accurately, we can even extrapolate to the real properties of the particles. Then we find that, e.g., the masses of particles become less and less the closer we get and thus most of their mass is made up by this cloud of virtual particles. Also, we find that some of the interactions become less, and others become stronger. Since these quantities change with energy, a physicists also calls them running quantities. Running means here that they change comparatively quickly with a change of energy. We also know the concept of walking quantities which change slowly, but we do not know an example of such theories (yet) in nature.
When thinking about such things, we should always keep in mind that the standard model of particle physics is, as discussed previously, just a low-energy effective description. Hence, when we try to extrapolate, we use our theory as input, meaning that our extrapolation necessary will fail at some higher energy, and we do not even know precisely when. So this running is telling us just how things change in a certain range of energies we can test. However, this is actually useful: By looking for deviations from what we think should happen, we can find something new.
Take an apple. It has a certain mass, say 100 gram. However, if you put yourself inside the apple, the amount of apple you see before your (if you look towards the pit) is less than the whole apple, and its mass is thus less. In a very cartoon idea, you could also say that this depends on the energy: If you ride a (rather slow) bullet, its penetration depth will depend on its initial energy, and thus the higher the energy, the less of the apple mass remains in front of you.
Of course, things can get not only less, but also more. Talk a light bulb (or LED, if you like it more modern). Put veils upon it. The amount of light you perceive depends then on the number of veils between you and the bulb. The more veils are already behind you, and the less before you, the brighter the bulb. Of course, the idea with the bullet applies here, too. Just avoid hitting the bulb.
Well, for particles you do not have veils or outer shells of an apple. So what is going on there? What is going on is that the vacuum around a particle is actually a rather thick soup than really empty. This seems surprising at first - after all, we have been thinking that the space in, e.g., an atom is essentially empty. The reason for this apparent contradiction is quantum physics. In quantum physics, we got used to the fact that we cannot really say what is going on, and everything becomes fuzzy. In particular, we cannot say whether a portion of the vacuum is really empty or contains particles, as long as we do not make a very precise measurement. In the head of theoretical physicists, this observation of nature has formed the picture of so-called virtual particles.
Virtual particles are particles which appear and disappear all the time. They may either be emitted by a source, like another particle, or may even pop out randomly (though necessarily at least in pairs) from the vacuum. They only exist for a very brief glimpse, and are then reabsorbed by the source, or annihilate again into nothing. Only, if we look close enough, i.e. short distances and thus high energies, we can check whether such particles are there or not.
In particular, if we want to look at one particle, or the interaction of two or more particles, they are surrounded by a cloud of such virtual particles. Only if we get close enough to them, and that means at high energies, we can dive through this cloud. However, to really see the pure and unaltered particle (or process), we would need to resolve it with a wave-length of its size, but this is zero. Hence, this would require infinite energy, because we want to measure them at zero distance. So we cannot.
However, the more energy we invest, the deeper we get into the cloud, and the more we see of the particle's or process' properties. Measuring these accurately, we can even extrapolate to the real properties of the particles. Then we find that, e.g., the masses of particles become less and less the closer we get and thus most of their mass is made up by this cloud of virtual particles. Also, we find that some of the interactions become less, and others become stronger. Since these quantities change with energy, a physicists also calls them running quantities. Running means here that they change comparatively quickly with a change of energy. We also know the concept of walking quantities which change slowly, but we do not know an example of such theories (yet) in nature.
When thinking about such things, we should always keep in mind that the standard model of particle physics is, as discussed previously, just a low-energy effective description. Hence, when we try to extrapolate, we use our theory as input, meaning that our extrapolation necessary will fail at some higher energy, and we do not even know precisely when. So this running is telling us just how things change in a certain range of energies we can test. However, this is actually useful: By looking for deviations from what we think should happen, we can find something new.
Monday, August 15, 2011
The limits of the standard model II
The last entry focused on the low-energy, or long-distance limit, of the standard model. This time, lets have a look at the opposite limit, the one of very short distances, or, as discussed previously, the one of very high energies.
If we go to smaller and smaller distances, we try to look deeper and deeper into something. Just like with the ocean: First, we just see the essentially plain waters. When we go nearer, we see the large movements of very large waves. When we go closer, we see that on the large waves there are small waves, and even deeper, we see ripples on all of the smaller waves. However, if we would go even closer, we would see that the water is made up of water molecules, out of discrete things. Wow. We just made a jump from one description - a continuous amount of water - to another one - that of water molecules. This means, when we look at shorter and shorter distances, we can learn how the things work in the interior, and in detail. Therefore, by looking at smaller and smaller distances, or higher and higher energies, we learn something about the nature of things.
In terms of theories, physicists like to speak of the description in terms of the water molecules as the 'underlying theory'. The description in terms of water as a fluid is called the 'low-energy effective theory', i. e., a theory which describes the relevant features of the underlying theory if we are looking on distances where we cannot distinguish the individual constituents of the underlying theory anymore.
In doing so, we actually notice something: As we have discussed, molecules are made up of atoms and the atoms are made up out of even smaller particles, and these smaller particles are described by another theory, the standard model. Hence, the theory of water molecules is out of a sudden no longer an underlying theory, but a low-energy effective theory for the standard model. Thus, a theory can be both, an underlying theory, and a low-energy effective theory. It is just a question of whether we look at it from larger or smaller distances than the characteristic scale of it.
The characteristic scale, which I have introduced here without warning, is actually a rather sloppy term: It means essentially distances in which the typical behavior of the objects in a theory show themselves. In the case of the waves, this scale is of the order of kilometers down to micrometers, the water-theory with water as molecules then takes over until one reaches the domain of femtometers, where the standard model comes into play. To not give a scale range, one usually uses one intermediate scale to indicate such a characteristic scale. For the theory of water molecules this is typically some Angstrom (about 0.0000000001 meters). For the standard model, it depends on the sector: about 0.000000000000001 meter for the strong interactions, and about 0.000000000000000001 meter for the weak sector, about one-thousand times smaller.
But wait: How can we be sure that the standard model is the underlying theory? The answer is we cannot. In fact, we firmly believe that it is not. The reason for thinking so is the following: On the one hand, it lacks gravity. We think, that the last theory in such a hierarchy of effective theories should include both quantum physics with the standard model as well as gravity. We just do not see right now any logical possibility that these two should be and remain separate.
On the other hand, there is a technical problem, which shows that the standard model is incomplete. If you do calculations in the standard model, then it turns out that for doing everything mathematically consistently you have to consider arbitrarily large energies. However, if you do this, the results are infinite, and thus at first glance meaningless. If you, however, assume that the standard model is just a low-energy effective theory, it is possible to remove these infinities by defining a small number of parameters appropriately. This is called renormalization, and the proof that this is possible for the standard model, at least to some extent, has been awarded with a noble prize. In a way, the standard model is telling us: "Hey, I m not the final answer, but you can parametrize your ignorance such that I still make sense, if you just do not poke me with too large energies."
Ok, all well and fine. But what is the underlying theory to the standard model? We do not know right now. And to figure this out, we have to look at physics a ever shorter distance scales and thus ever higher energies to get an answer to this. That is the reason we built and use the LHC and its predecessors. There is also the possibility to indirectly interfere the very high energy behavior by making very precise measurements. You could imagine this in the following way: You would also figure out that water is made out of molecules when you would weigh water very carefully. Then you would notice that it is not possible to have an arbitrary weight of water, but only discrete portions. And similarly we try to infer the high-energy behavior by very precise measurements.
Anyway, this is the current goal: To see what is the underlying theory of the standard model. This process of identifying the next underlying theory has been driven physics since centuries. Will it ever terminate? That is a good question,a and one we cannot answer (yet). The only thing sure right now is that it did not terminate with the standard model. And that we do not even yet fully understand the standard model, though this is necessary to answer whether something we observe is genuinely a signal of the underlying theory, or just a feature of the standard model. A difficult question indeed.
If we go to smaller and smaller distances, we try to look deeper and deeper into something. Just like with the ocean: First, we just see the essentially plain waters. When we go nearer, we see the large movements of very large waves. When we go closer, we see that on the large waves there are small waves, and even deeper, we see ripples on all of the smaller waves. However, if we would go even closer, we would see that the water is made up of water molecules, out of discrete things. Wow. We just made a jump from one description - a continuous amount of water - to another one - that of water molecules. This means, when we look at shorter and shorter distances, we can learn how the things work in the interior, and in detail. Therefore, by looking at smaller and smaller distances, or higher and higher energies, we learn something about the nature of things.
In terms of theories, physicists like to speak of the description in terms of the water molecules as the 'underlying theory'. The description in terms of water as a fluid is called the 'low-energy effective theory', i. e., a theory which describes the relevant features of the underlying theory if we are looking on distances where we cannot distinguish the individual constituents of the underlying theory anymore.
In doing so, we actually notice something: As we have discussed, molecules are made up of atoms and the atoms are made up out of even smaller particles, and these smaller particles are described by another theory, the standard model. Hence, the theory of water molecules is out of a sudden no longer an underlying theory, but a low-energy effective theory for the standard model. Thus, a theory can be both, an underlying theory, and a low-energy effective theory. It is just a question of whether we look at it from larger or smaller distances than the characteristic scale of it.
The characteristic scale, which I have introduced here without warning, is actually a rather sloppy term: It means essentially distances in which the typical behavior of the objects in a theory show themselves. In the case of the waves, this scale is of the order of kilometers down to micrometers, the water-theory with water as molecules then takes over until one reaches the domain of femtometers, where the standard model comes into play. To not give a scale range, one usually uses one intermediate scale to indicate such a characteristic scale. For the theory of water molecules this is typically some Angstrom (about 0.0000000001 meters). For the standard model, it depends on the sector: about 0.000000000000001 meter for the strong interactions, and about 0.000000000000000001 meter for the weak sector, about one-thousand times smaller.
But wait: How can we be sure that the standard model is the underlying theory? The answer is we cannot. In fact, we firmly believe that it is not. The reason for thinking so is the following: On the one hand, it lacks gravity. We think, that the last theory in such a hierarchy of effective theories should include both quantum physics with the standard model as well as gravity. We just do not see right now any logical possibility that these two should be and remain separate.
On the other hand, there is a technical problem, which shows that the standard model is incomplete. If you do calculations in the standard model, then it turns out that for doing everything mathematically consistently you have to consider arbitrarily large energies. However, if you do this, the results are infinite, and thus at first glance meaningless. If you, however, assume that the standard model is just a low-energy effective theory, it is possible to remove these infinities by defining a small number of parameters appropriately. This is called renormalization, and the proof that this is possible for the standard model, at least to some extent, has been awarded with a noble prize. In a way, the standard model is telling us: "Hey, I m not the final answer, but you can parametrize your ignorance such that I still make sense, if you just do not poke me with too large energies."
Ok, all well and fine. But what is the underlying theory to the standard model? We do not know right now. And to figure this out, we have to look at physics a ever shorter distance scales and thus ever higher energies to get an answer to this. That is the reason we built and use the LHC and its predecessors. There is also the possibility to indirectly interfere the very high energy behavior by making very precise measurements. You could imagine this in the following way: You would also figure out that water is made out of molecules when you would weigh water very carefully. Then you would notice that it is not possible to have an arbitrary weight of water, but only discrete portions. And similarly we try to infer the high-energy behavior by very precise measurements.
Anyway, this is the current goal: To see what is the underlying theory of the standard model. This process of identifying the next underlying theory has been driven physics since centuries. Will it ever terminate? That is a good question,a and one we cannot answer (yet). The only thing sure right now is that it did not terminate with the standard model. And that we do not even yet fully understand the standard model, though this is necessary to answer whether something we observe is genuinely a signal of the underlying theory, or just a feature of the standard model. A difficult question indeed.
Tuesday, June 21, 2011
The limits of the standard model I
After the rather technical discussion in the last few entries let us return this time to a more mundane topic: What is the validity of the standard model. For that purpose assume for the sake of the argument that the Higgs particle will eventually be found.
The question can be paraphrased differently: What is the lowest and the highest energy at which the standard model can be used? This question can also be formulated even more differently: An energy can be associated with a distance. That is very similar to what has been discussed previously in the entry on "Fields, waves, particles, and all that". If you have a very large energy, movement is essentially very rapid. In particular, the fields associated with the particle oscillate very quickly, and thus the distance between the crests of its waves is very small. Hence, changes on very small distances can be sensed by the particle, and thus high energies can be associated with small distances. In the opposite extreme, this means that low energies can be associated with large distances.
Let us then start with the more simple of both limits, the lowest energy. Since the standard model is a quantum theory, this can be also posed as the question when do we no longer observe things, which are distinctively quantum. A quantum theory means associating particles with a field. Thus take again the picture of waves, and let us go again back to the picture of the ocean. If you hover a short distance above it, you can see the individual waves. If you then zoom out, at some point everything blurs together, and you have the impression that only a - more or less - flat surface is there. At this point you do no longer realize the individual particle (wave), but only all of the particles (waves) together, in the form of the ocean as such. Similarly, if you zoom out of the standard model up to, say, the level of your desk, you do not note anymore the particles, but only the surface of the desk.
This is not yet telling you that the standard model is not applicable anymore, just that your are no longer able to distinguish its parts. It is therefore actually a very complicate question, whether the standard model is only valid up to a certain distance scale, because it becomes so hard to see its content. People have tried very hard to see the consequences of the standard model at ever larger distances, but, depending on the part of the standard model you look at, it becomes very hard to make a statement. Once leaving the size of a few times a nuclei, it is essentially only the electromagnetic force we can still test. For that part of the standard model we know that it works at least on the order of our own galaxy, and we have evidence, though far less rigid, that its seems to work rather well even at much larger cosmic distances. Still, answering the question to which distance we can observe the standard model is thus tricky and a persisting challenge. Perhaps even our understanding of the universe would be altered, if we someday would figure out that the standard model is not a suitable description at long distances.
Thus, to the best of our current knowledge, the standard model works (though we have a hard time seeing it) at the largest distance scales, and thus at the lowest energies, we can observe and test. However, it is a technical problem to check whether this is actually true or not: We need very sensitive experiments to check this, and the observation of true quantum effects is up to now limited to very small sizes, like in a Bose-Einstein condensate of atoms. The size of the latter is currently at best below some centimeters. Only some very specific quantum effects can be observed using photons at larger distances, like when using a fiber or making the famous double-slit experiment. But photons are only a very restricted part of the standard model.
The situation will change at high energies. There is also a technical problem, but in addition also a conceptual problem.
The question can be paraphrased differently: What is the lowest and the highest energy at which the standard model can be used? This question can also be formulated even more differently: An energy can be associated with a distance. That is very similar to what has been discussed previously in the entry on "Fields, waves, particles, and all that". If you have a very large energy, movement is essentially very rapid. In particular, the fields associated with the particle oscillate very quickly, and thus the distance between the crests of its waves is very small. Hence, changes on very small distances can be sensed by the particle, and thus high energies can be associated with small distances. In the opposite extreme, this means that low energies can be associated with large distances.
Let us then start with the more simple of both limits, the lowest energy. Since the standard model is a quantum theory, this can be also posed as the question when do we no longer observe things, which are distinctively quantum. A quantum theory means associating particles with a field. Thus take again the picture of waves, and let us go again back to the picture of the ocean. If you hover a short distance above it, you can see the individual waves. If you then zoom out, at some point everything blurs together, and you have the impression that only a - more or less - flat surface is there. At this point you do no longer realize the individual particle (wave), but only all of the particles (waves) together, in the form of the ocean as such. Similarly, if you zoom out of the standard model up to, say, the level of your desk, you do not note anymore the particles, but only the surface of the desk.
This is not yet telling you that the standard model is not applicable anymore, just that your are no longer able to distinguish its parts. It is therefore actually a very complicate question, whether the standard model is only valid up to a certain distance scale, because it becomes so hard to see its content. People have tried very hard to see the consequences of the standard model at ever larger distances, but, depending on the part of the standard model you look at, it becomes very hard to make a statement. Once leaving the size of a few times a nuclei, it is essentially only the electromagnetic force we can still test. For that part of the standard model we know that it works at least on the order of our own galaxy, and we have evidence, though far less rigid, that its seems to work rather well even at much larger cosmic distances. Still, answering the question to which distance we can observe the standard model is thus tricky and a persisting challenge. Perhaps even our understanding of the universe would be altered, if we someday would figure out that the standard model is not a suitable description at long distances.
Thus, to the best of our current knowledge, the standard model works (though we have a hard time seeing it) at the largest distance scales, and thus at the lowest energies, we can observe and test. However, it is a technical problem to check whether this is actually true or not: We need very sensitive experiments to check this, and the observation of true quantum effects is up to now limited to very small sizes, like in a Bose-Einstein condensate of atoms. The size of the latter is currently at best below some centimeters. Only some very specific quantum effects can be observed using photons at larger distances, like when using a fiber or making the famous double-slit experiment. But photons are only a very restricted part of the standard model.
The situation will change at high energies. There is also a technical problem, but in addition also a conceptual problem.
Friday, May 6, 2011
Internal and external space(s)
I have repeatedly discussed symmetries, and often made examples where one imagines some object, and how it looks from different perspectives. It seems surprising at first that something like a symmetry, which is looking like something belonging to the deepest properties of a system, should be so readily visible as an ordinary object. How so?
The reason for this is rather mundane, though far from obvious: There is not such a big difference between symmetries and the world around us. As a physicist, I refer to this fact as an internal and an external space.
An external space is just the world around us - length, width, height, time. It is the arena, in which physics takes place. At the same time, it exhibits symmetries. You can rotate things, and if they are symmetric, they look the same. You can choose a coordinate system, and describe things, but what happens is independent of the coordinate system. That is also a kind of symmetry: Physics is independent of the coordinate system, looking from any coordinate system everything happens in the same way. This is called a space-time symmetry. Physicists have also a more complicated name for it: They call it a diffeomorphism invariance.
Now, how is all of this related to the symmetry, say, of electromagnetism? Well, go back to the four numbers describing electromagnetism, and forget for a while that they change at different places. Then the four numbers can also be taken to describe four directions, four new coordinates, with which I can describe things. Since these coordinates are not the usual ones, it is said that these coordinates describe an internal space. Now, in these new coordinates, we can also choose a coordinate system, and physics is again the same, irrespective of our choice of coordinates. However, with this coordinate system we do not measure lengths or times, but we measure electromagnetism.
If you then combine the internal and external space, you have the total space. Each point is now characterized by eight numbers: The four conventional coordinates, and the four internal coordinates of the photon field.
The fact that we can change the internal coordinate system freely is the reason why we have four numbers, though physics only depends on two numbers: The symmetry permits to make a coordinate system choice, and this does not matter. If there would be no symmetry, there would be just one coordinate system permitted, and we could not change it.
However, even if there is a symmetry, we are not permitted to make any coordinate system choice. For example, we could in the real world, the external space, not make a choice of coordinates such that time were finite, or would make a loop. Similarly, in the internal space, one cannot make always an arbitrary choice. In fact, in the internal space of electromagnetism only coordinate systems where all coordinates do make a loop are permitted. That is one of the big differences between space and time and electromagnetism. Indeed, all the symmetries of the standard model have symmetries, which have only coordinate systems, which have loops. In fact, how one can choose a coordinate system is very hard to understand for the strong and weak force, and we actually only know for sure how to make a choice close to the point where we look at at some instance. How to make a descent choice far away from where we are right now looking is a complicated problem, and actually one of my research topics.
However, for this tourist guide, the most important point to remember is that symmetries and coordinate systems are closely related, and that the coordinate systems of the internal spaces are not so much different from that of the external space.
The reason for this is rather mundane, though far from obvious: There is not such a big difference between symmetries and the world around us. As a physicist, I refer to this fact as an internal and an external space.
An external space is just the world around us - length, width, height, time. It is the arena, in which physics takes place. At the same time, it exhibits symmetries. You can rotate things, and if they are symmetric, they look the same. You can choose a coordinate system, and describe things, but what happens is independent of the coordinate system. That is also a kind of symmetry: Physics is independent of the coordinate system, looking from any coordinate system everything happens in the same way. This is called a space-time symmetry. Physicists have also a more complicated name for it: They call it a diffeomorphism invariance.
Now, how is all of this related to the symmetry, say, of electromagnetism? Well, go back to the four numbers describing electromagnetism, and forget for a while that they change at different places. Then the four numbers can also be taken to describe four directions, four new coordinates, with which I can describe things. Since these coordinates are not the usual ones, it is said that these coordinates describe an internal space. Now, in these new coordinates, we can also choose a coordinate system, and physics is again the same, irrespective of our choice of coordinates. However, with this coordinate system we do not measure lengths or times, but we measure electromagnetism.
If you then combine the internal and external space, you have the total space. Each point is now characterized by eight numbers: The four conventional coordinates, and the four internal coordinates of the photon field.
The fact that we can change the internal coordinate system freely is the reason why we have four numbers, though physics only depends on two numbers: The symmetry permits to make a coordinate system choice, and this does not matter. If there would be no symmetry, there would be just one coordinate system permitted, and we could not change it.
However, even if there is a symmetry, we are not permitted to make any coordinate system choice. For example, we could in the real world, the external space, not make a choice of coordinates such that time were finite, or would make a loop. Similarly, in the internal space, one cannot make always an arbitrary choice. In fact, in the internal space of electromagnetism only coordinate systems where all coordinates do make a loop are permitted. That is one of the big differences between space and time and electromagnetism. Indeed, all the symmetries of the standard model have symmetries, which have only coordinate systems, which have loops. In fact, how one can choose a coordinate system is very hard to understand for the strong and weak force, and we actually only know for sure how to make a choice close to the point where we look at at some instance. How to make a descent choice far away from where we are right now looking is a complicated problem, and actually one of my research topics.
However, for this tourist guide, the most important point to remember is that symmetries and coordinate systems are closely related, and that the coordinate systems of the internal spaces are not so much different from that of the external space.
Thursday, February 17, 2011
Pointing in space and time or why one needs four numbers for a photon
In the previous discussion it was described how photons are described by fields, and that the fields are somehow like the surface of an ocean. The truth is, unfortunately a bit more complex. This can already be seen from the magnetic field. If you have a magnet, you cannot only feel its field in the same plane as where the magnet is, but also above and below it. Thus, the field is something which not only is like the surface of an ocean, but which is more like the ocean itself, it is above and below and all around. Well, this is not yet a problem, since one can imagine that, say, a subsurface explosion also can make a wave which has volume, and the analogy is only a bit more harder to imagine because of the third direction.
But things become still a bit more messy. Take the magnet and take a pretty hot flame, and place it under the magnet, not too close. If you now measure the magnetic field at some point in the space surrounding the magnet, you will notice that the magnetic field decreases over time. That is because when you heat a magnet sufficiently (a couple of hundred degrees), it will loose its magnetic properties. Thus, the field is not static, it changes with time, and can even vanish. Of course, you could have noticed the same feature by just moving the magnet far away, but then you could bring it back again. Thus, a field is something that tells something about a direction and a strength at some point in space and time.
But these seems a bit odd. To identify a position, you need four numbers, four coordinates. But the direction of the magnetic field you can enumerate with just three, two for the direction, one for its strength. There is nothing like a time direction to the magnetic field. Indeed, electric and magnetic fields are peculiar in this sense. As said before, they can be derived from a quantity which had four numbers, as the four coordinates just needed to characterize the evolution of the magnetic field. It is about time to tell what the four numbers are.
Indeed, it turns out that a field which describes a particle has four components, each of which depends on the space-time point one is looking at. So what is this fourth number? In a sense it is the direction of the field in time. That sounds a bit peculiar, and in fact it is. The reason for this is the arena in which physics takes place.
If one goes back to ones experience of reality, then there is the space with its three dimensions, and there is time, which appears to be just flowing along in the background. But in fact space and time are connected, and are not two independent entities. That has been an observation which has actually been made very early on in physics. However, it took a while to note that the structure is peculiar, but this will be discussed at a different time.
Again, it helps to make an analogy. Take a flat cylinder. Put in the cylinder a disc, which fits perfectly in it. Now, if you elevate the disc at a constant rate than everything on the disc can move freely on the disc, but there is a constant change in height, just as time changes constantly. In our world, the disc has one dimension more, and the changing height is the changing time, but otherwise it is the same concept. Somebody on the disc could even measure time by measuring height, because it is lifted constantly.
Now, of course, it is possible to give a direction which is entirely on the disc. But for us, which can see the cylinder as a whole, we can also give a direction which points upwards or downwards from the disc. In contrast to someone living on the disc, we need one quantity more to specify a direction. But if someone on the disc is very clever, he will notice that his space is larger, and then she can invent, at least as a mathematical concept, a direction off the disc, which will agree with our idea of direction. However, since she only knows the disc she has no intuition of what means 'off the disc', but has a mathematical grasp of it.
And so it is the case for us with time. We can mathematical describe our cylinder (though it actually looks very much different from a cylinder), and we can describe a direction off our three-dimensional world by giving it a direction in both time and space. Then, we notice that the field that describes a particle is actually requiring to have such an additional direction, and this is the reason why the photon field has four numbers at every space-time point: a magnitude and a direction in space and time. And the electric and magnetic field with only a direction in space are something like shadows of this object in time and space in a purely spatial world, in which we can move freely.
Of course, these four numbers are not independent, but this is because of the symmetry. Without the symmetry, they would be. The symmetry is something additional, and has nothing to do with space and time.
But things become still a bit more messy. Take the magnet and take a pretty hot flame, and place it under the magnet, not too close. If you now measure the magnetic field at some point in the space surrounding the magnet, you will notice that the magnetic field decreases over time. That is because when you heat a magnet sufficiently (a couple of hundred degrees), it will loose its magnetic properties. Thus, the field is not static, it changes with time, and can even vanish. Of course, you could have noticed the same feature by just moving the magnet far away, but then you could bring it back again. Thus, a field is something that tells something about a direction and a strength at some point in space and time.
But these seems a bit odd. To identify a position, you need four numbers, four coordinates. But the direction of the magnetic field you can enumerate with just three, two for the direction, one for its strength. There is nothing like a time direction to the magnetic field. Indeed, electric and magnetic fields are peculiar in this sense. As said before, they can be derived from a quantity which had four numbers, as the four coordinates just needed to characterize the evolution of the magnetic field. It is about time to tell what the four numbers are.
Indeed, it turns out that a field which describes a particle has four components, each of which depends on the space-time point one is looking at. So what is this fourth number? In a sense it is the direction of the field in time. That sounds a bit peculiar, and in fact it is. The reason for this is the arena in which physics takes place.
If one goes back to ones experience of reality, then there is the space with its three dimensions, and there is time, which appears to be just flowing along in the background. But in fact space and time are connected, and are not two independent entities. That has been an observation which has actually been made very early on in physics. However, it took a while to note that the structure is peculiar, but this will be discussed at a different time.
Again, it helps to make an analogy. Take a flat cylinder. Put in the cylinder a disc, which fits perfectly in it. Now, if you elevate the disc at a constant rate than everything on the disc can move freely on the disc, but there is a constant change in height, just as time changes constantly. In our world, the disc has one dimension more, and the changing height is the changing time, but otherwise it is the same concept. Somebody on the disc could even measure time by measuring height, because it is lifted constantly.
Now, of course, it is possible to give a direction which is entirely on the disc. But for us, which can see the cylinder as a whole, we can also give a direction which points upwards or downwards from the disc. In contrast to someone living on the disc, we need one quantity more to specify a direction. But if someone on the disc is very clever, he will notice that his space is larger, and then she can invent, at least as a mathematical concept, a direction off the disc, which will agree with our idea of direction. However, since she only knows the disc she has no intuition of what means 'off the disc', but has a mathematical grasp of it.
And so it is the case for us with time. We can mathematical describe our cylinder (though it actually looks very much different from a cylinder), and we can describe a direction off our three-dimensional world by giving it a direction in both time and space. Then, we notice that the field that describes a particle is actually requiring to have such an additional direction, and this is the reason why the photon field has four numbers at every space-time point: a magnitude and a direction in space and time. And the electric and magnetic field with only a direction in space are something like shadows of this object in time and space in a purely spatial world, in which we can move freely.
Of course, these four numbers are not independent, but this is because of the symmetry. Without the symmetry, they would be. The symmetry is something additional, and has nothing to do with space and time.
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