Final conception of Calabi Yau Manifold magnified from an arbitrary point in space.
 

In my article, I discuss how the difficulties encountered by Superstring Theory also point to the existence of multiple universes. In Superstring Theory, we have a theory that has been remarkably successful in its attempts to unify the discrepancies between our foremost physical theories, Einstein’s theory of General Relativity and Quantum Mechanics. But on the other hand, Superstring Theory has failed considerably in its other goal; to predict the values of the fundamental constants from first principles. Superstring Theory itself postulates that elementary particles are not points but higher dimensional strings, where different particles like protons or neutrons are differentiated by the way their string is vibrating, like different notes on a violin. In addition, Superstring Theory finds that mathematically, space itself must have extra dimensions, with a total of as many as 11 or 26, if the inconsistencies of Quantum Mechanics and General Relativity are to be resolved.
Since we only see three spatial dimensions around us, the others must then be tightly curled up at the microscopic level, avoiding easy detection. What is interesting is that the geometry of the way in which the extra dimensions are curled up determines the values of the fundamental constants of physics. Change the topology of these curled up spaces, and you change the speed of light and all of its companion constants. But what Superstring theorists have found is that contrary to expectation, there are literally an infinite number of ways to curl up these spaces that are all mathematically consistent, and thus an infinite number of allowed combinations of the physical parameters. The fact that we can not pin down a unique set of parameters, and predict our own values straight from the theory, gives further credence to the idea that all possible values are just as viable, and may be realized in other universes.
The mathematical term for the kind of curled up extra dimensional spaces allowed by Superstring Theory is what’s called a Calabi Yau Manifold. The difficulty posed in constructing a graphic to illustrate the concept of extra dimensions is this; It is ridiculously impossible to draw a complicated 6-dimensional object like a Calabi Yau Manifold. Thus one must settle for a schematic illustration purely designed to capture the essence of the complexity of the object, without worrying about the impossible task of actually constructing an accurate artistic representation of one of these things.
The idea I thought of was to use an abstract geometric piece that I did a while back, which seemed to have the complexity I was looking for. With the help of my friend Tom, we then wrapped it around a sphere in 3D Studio, then we used the program to construct what’s called a displacement map, where regions of different brightness on the sphere are translated into bumps of corresponding size on the surface of the sphere, creating a rough, complex shape that is our conception of a Calabi Yau Manifold. Finally, we have a graphic where we magnify an arbitrary point in space to reveal the existence of the Calabi Yau Manifold at the microscopic level. Just imagine that every point in space harbors one of these crazy things, which we could only see if we could resolve space itself on the tiniest of scales. The amazing thing is that this actually may be true of our world. The universe never ceases to be ridiculous, it seems.
-Andy Friedman

Image credit: Andy Friedman

Final conception of Calabi Yau Manifold magnified from an arbitrary point in space.

 

In my article, I discuss how the difficulties encountered by Superstring Theory also point to the existence of multiple universes. In Superstring Theory, we have a theory that has been remarkably successful in its attempts to unify the discrepancies between our foremost physical theories, Einstein’s theory of General Relativity and Quantum Mechanics. But on the other hand, Superstring Theory has failed considerably in its other goal; to predict the values of the fundamental constants from first principles. Superstring Theory itself postulates that elementary particles are not points but higher dimensional strings, where different particles like protons or neutrons are differentiated by the way their string is vibrating, like different notes on a violin. In addition, Superstring Theory finds that mathematically, space itself must have extra dimensions, with a total of as many as 11 or 26, if the inconsistencies of Quantum Mechanics and General Relativity are to be resolved.

Since we only see three spatial dimensions around us, the others must then be tightly curled up at the microscopic level, avoiding easy detection. What is interesting is that the geometry of the way in which the extra dimensions are curled up determines the values of the fundamental constants of physics. Change the topology of these curled up spaces, and you change the speed of light and all of its companion constants. But what Superstring theorists have found is that contrary to expectation, there are literally an infinite number of ways to curl up these spaces that are all mathematically consistent, and thus an infinite number of allowed combinations of the physical parameters. The fact that we can not pin down a unique set of parameters, and predict our own values straight from the theory, gives further credence to the idea that all possible values are just as viable, and may be realized in other universes.

The mathematical term for the kind of curled up extra dimensional spaces allowed by Superstring Theory is what’s called a Calabi Yau Manifold. The difficulty posed in constructing a graphic to illustrate the concept of extra dimensions is this; It is ridiculously impossible to draw a complicated 6-dimensional object like a Calabi Yau Manifold. Thus one must settle for a schematic illustration purely designed to capture the essence of the complexity of the object, without worrying about the impossible task of actually constructing an accurate artistic representation of one of these things.

The idea I thought of was to use an abstract geometric piece that I did a while back, which seemed to have the complexity I was looking for. With the help of my friend Tom, we then wrapped it around a sphere in 3D Studio, then we used the program to construct what’s called a displacement map, where regions of different brightness on the sphere are translated into bumps of corresponding size on the surface of the sphere, creating a rough, complex shape that is our conception of a Calabi Yau Manifold. Finally, we have a graphic where we magnify an arbitrary point in space to reveal the existence of the Calabi Yau Manifold at the microscopic level. Just imagine that every point in space harbors one of these crazy things, which we could only see if we could resolve space itself on the tiniest of scales. The amazing thing is that this actually may be true of our world. The universe never ceases to be ridiculous, it seems.

-Andy Friedman

Image credit: Andy Friedman

Neutrino masses: 60 seconds review
   

Neutrino masses are extremely difficult to measure. While we know precisely how much an electron weighs, we have little information on the mass of its neutral partner, the electron neutrino. The same is true of the muon neutrino and tau neutrino.
For a long time scientists thought neutrinos might be massless. Then experiments revealed that the three types of neutrinos can transform into each other, a process known as neutrino oscillation. According to quantum theory, this is possible only if neutrinos have mass.
Cosmological observations and laboratory-based experiments indicate that the masses of the three neutrino types must be extremely small: The electron, the lightest charged elementary particle, is at least a million times heavier than any neutrino.
Physicists think the origins of neutrino masses are closely tied to subatomic processes that took place right after the big bang. Determining which neutrino types are heaviest and lightest—the neutrino mass ordering—is a first step toward revealing these processes.
So far, neutrino oscillation experiments have provided some information on the differences in mass between the different neutrino types. Future experiments, requiring accelerator-produced, high-intensity neutrino beams traveling at least 500 miles through the Earth, will tell us what the neutrino mass ordering is.
[www.symmetrymagazine.org/cms/?pid=1000645]

Neutrino masses: 60 seconds review

 

Neutrino masses are extremely difficult to measure. While we know precisely how much an electron weighs, we have little information on the mass of its neutral partner, the electron neutrino. The same is true of the muon neutrino and tau neutrino.

For a long time scientists thought neutrinos might be massless. Then experiments revealed that the three types of neutrinos can transform into each other, a process known as neutrino oscillation. According to quantum theory, this is possible only if neutrinos have mass.

Cosmological observations and laboratory-based experiments indicate that the masses of the three neutrino types must be extremely small: The electron, the lightest charged elementary particle, is at least a million times heavier than any neutrino.

Physicists think the origins of neutrino masses are closely tied to subatomic processes that took place right after the big bang. Determining which neutrino types are heaviest and lightest—the neutrino mass ordering—is a first step toward revealing these processes.

So far, neutrino oscillation experiments have provided some information on the differences in mass between the different neutrino types. Future experiments, requiring accelerator-produced, high-intensity neutrino beams traveling at least 500 miles through the Earth, will tell us what the neutrino mass ordering is.

[www.symmetrymagazine.org/cms/?pid=1000645]

The mass of neutrinos
   

Physicists had already observed this type of mixing behavior in neutral mesons, but they had no reason to expect it in neutrinos. After all, the Standard Model assumed that neutrinos had no mass. However, oscillation between neutrino flavors, which means that individual neutrinos change their identities, is theoretically possible only if different flavors of neutrinos have different masses. Physicists still do not know the absolute mass scale of neutrinos, but they have measured the mass differences between pairs of neutrino flavors through careful study of their oscillation properties. These differences are very tiny, suggesting that neutrinos may be a million times lighter than the electron. Now theorists face the challenge of explaining why nature should have given neutrinos such miniscule masses.
Experiments to make more accurate measurements of neutrinos’ mass differences and their mixing rates are under way in several countries. Some use nuclear reactors as the sources of neutrino beams. Others rely on neutrinos produced in accelerators by the decay of a secondary beam of mesons produced when high-energy protons smash into a target. The results of both types of studies may make it experimentally feasible for the next generation of projects to look for CP violation in neutrinos.
[www.learner.org/courses/physics/unit/text.html?unit=1&secNum=6]

   
Image credit: The Sudbury Neutrino Detector led to the discovery of neutrino mass. Source: © Lawrence Berkeley National Laboratory. More info

The mass of neutrinos

 

Physicists had already observed this type of mixing behavior in neutral mesons, but they had no reason to expect it in neutrinos. After all, the Standard Model assumed that neutrinos had no mass. However, oscillation between neutrino flavors, which means that individual neutrinos change their identities, is theoretically possible only if different flavors of neutrinos have different masses. Physicists still do not know the absolute mass scale of neutrinos, but they have measured the mass differences between pairs of neutrino flavors through careful study of their oscillation properties. These differences are very tiny, suggesting that neutrinos may be a million times lighter than the electron. Now theorists face the challenge of explaining why nature should have given neutrinos such miniscule masses.

Experiments to make more accurate measurements of neutrinos’ mass differences and their mixing rates are under way in several countries. Some use nuclear reactors as the sources of neutrino beams. Others rely on neutrinos produced in accelerators by the decay of a secondary beam of mesons produced when high-energy protons smash into a target. The results of both types of studies may make it experimentally feasible for the next generation of projects to look for CP violation in neutrinos.

[www.learner.org/courses/physics/unit/text.html?unit=1&secNum=6]

 

Image credit: The Sudbury Neutrino Detector led to the discovery of neutrino mass. Source: © Lawrence Berkeley National Laboratory. More info

Fundamental forces

 

christinetheastrophysicist:

Feynman Diagrams are used to represent the fundamental forces by which elementary particles interact. In all interactions, time flows horizontally to the right.

a)  Quantum Electrodynamics:  An electron converts into an electron, with the emission of a photon (e → e + γ)

b)  Quantum Chromodynamics: A quark converts into a quark, with the emission of a gluon (q → q + g)

c)  Neutral Weak Interactions: Any lepton or quark converts into the corresponding lepton or quark, with the emission of a Z boson (f → f + Z)

d)  Charged Weak Interactions for Leptons: A negative lepton converts into a corresponding neutrino, with the emission of a W- boson (l- → νl + W-)

e)  Charged Weak Interactions for Quarks: A quark with charge -1/3 converts into the corresponding quark with charge +2/3, with the emission of a W- boson (q-1/3 → q+2/3 + W-)

(via hadron94-deactivated20130407)

Introduction to gauge theory - I
 

In physics, gauge invariance (also called gauge symmetry) is the property of a field theory in which different configurations of the underlying fields — which are not themselves directly observable — result in identical observable quantities. A theory with such a property is called a gauge theory. A transformation from one such field configuration to another is called a gauge transformation.
Modern physical theories describe reality in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields for the electron and all other elementary particles. A general feature of these theories is that none of these fundamental fields, which are the fields that change under a gauge transformation, can be directly measured. On the other hand, the observable quantities, namely the ones that can be measured experimentally — charges, energies, velocities, etc. — do not change under a gauge transformation, even though they are derived from the fields that do change. This (and any) kind of invariance under a transformation is called a symmetry.
For example, in classical electromagnetism the electric field, E, and the magnetic field, B, are observable, while the underlying and more fundamental electromagnetic potentials V and A are not. Under a gauge transformation which jointly alters the two potentials, no change occurs either in E or B or in the motion of charged particles. In this example, the gauge transformation was just a mathematical feature without any physical significance, except that gauge invariance is intrinsically connected to the fundamental law of charge conservation.
With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time. 
Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory explaining all of the fundamental interactions except gravity.
[en.wikipedia.org/wiki/Introduction_to_gauge_theory]

Introduction to gauge theory - I

 

In physicsgauge invariance (also called gauge symmetry) is the property of a field theory in which different configurations of the underlying fields — which are not themselves directly observable — result in identical observable quantities. A theory with such a property is called a gauge theory. A transformation from one such field configuration to another is called a gauge transformation.

Modern physical theories describe reality in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields for the electron and all other elementary particles. A general feature of these theories is that none of these fundamental fields, which are the fields that change under a gauge transformation, can be directly measured. On the other hand, the observable quantities, namely the ones that can be measured experimentally — charges, energies, velocities, etc. — do not change under a gauge transformation, even though they are derived from the fields that do change. This (and any) kind of invariance under a transformation is called a symmetry.

For example, in classical electromagnetism the electric fieldE, and the magnetic fieldB, are observable, while the underlying and more fundamental electromagnetic potentials V and A are not. Under a gauge transformation which jointly alters the two potentials, no change occurs either in E or B or in the motion of charged particles. In this example, the gauge transformation was just a mathematical feature without any physical significance, except that gauge invariance is intrinsically connected to the fundamental law of charge conservation.

With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time

Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory explaining all of the fundamental interactions except gravity.

[en.wikipedia.org/wiki/Introduction_to_gauge_theory]

                                      Quantum Field Theory - III

Most branches of physics treat “elementary particles” just as particles - i.e. points or solid little balls of matter and charge. This statement also applies to atomic and nuclear physics. The theoretical foundation of these sciences is Quantum Mechanics. In the mathematical framework of Quantum Mechanics the basic entities are usually represented either by points of matter and charge or by spheres, where matter and charge is uniformly distributed throughout the sphere. The main difference between a classical (or Newtonian) particle and its Quantum Mechanical counterpart is that, while the position and outline of the former can be determined exactly, this is not so for the latter. Nevertheless, as far as the interactions on the atomic and nucleonic levels are concerned, Quantum Mechanics is doing pretty well. However, when it comes to the levels of “elementary particles”, that framework has at least one crucial drawback. In the Quantum Mechanical framework the basic entities have existed in their current shape forever and will continue to exist in this shape forever. Consequently, Quantum Mechanics provides no clues on how or why “particles” come into existence nor how and why they vanish. A natural consequence of this defect is that a large variety of well-known processes, such as e.g. those illustrated above, cannot be treated within the frames of Quantum Mechanics.
In order to be able to consider the immanent qualities and substance of “elementary particles”, physicists have to turn to the framework of Quantum Field Theory. Essentially, Quantum Field Theory is a specialized set of mathematical formulas and rules by means of which one, in a consistent and very detailed manner, can consider questions such as ‘what is the true origin and intrinsic nature of matter, charge, force etc.’ and ‘what is really going on at the most fundamental levels of nature’. Actually, Quantum Field Theory is not a theory as such but is more like a complex and content rich language well equipped for describing and “verbalizing” realms where other languages (such as English or the mathematics of Quantum Mechanics) have to give up.
The overall result of the sequence of processes, illustrated by the image above, is that two entities - a charm-quark and its anti-quark - are split into six other entities. The actual appearances of this sequence are well-confirmed by “particle” physics experiments. The same statement applies to a large variety of similar processes with any other set of “elementary particles” in the initial stage. Thus, these entities obviously are neither elementary nor indivisible, little balls of matter. No, - according to Quantum Field Theory, the “elementary particles” are discrete, vibrational states of certain underlying fields. Consequently, the fundamental constituents of the physical world are not “particles”. The fundamental constituents are fields. - This is the reason for hitherto having put the word ‘elementary’ in quotes. Next  .  .  .  the reason for the extensive use of quotes around the word ‘particle’ will be revealed.
[buddhasociety.com/quantum-physics/quantum-mechanics-3]

                                      Quantum Field Theory - III

Most branches of physics treat “elementary particles” just as particles - i.e. points or solid little balls of matter and charge. This statement also applies to atomic and nuclear physics. The theoretical foundation of these sciences is Quantum Mechanics. In the mathematical framework of Quantum Mechanics the basic entities are usually represented either by points of matter and charge or by spheres, where matter and charge is uniformly distributed throughout the sphere. The main difference between a classical (or Newtonian) particle and its Quantum Mechanical counterpart is that, while the position and outline of the former can be determined exactly, this is not so for the latter. Nevertheless, as far as the interactions on the atomic and nucleonic levels are concerned, Quantum Mechanics is doing pretty well. However, when it comes to the levels of “elementary particles”, that framework has at least one crucial drawback. In the Quantum Mechanical framework the basic entities have existed in their current shape forever and will continue to exist in this shape forever. Consequently, Quantum Mechanics provides no clues on how or why “particles” come into existence nor how and why they vanish. A natural consequence of this defect is that a large variety of well-known processes, such as e.g. those illustrated above, cannot be treated within the frames of Quantum Mechanics.

In order to be able to consider the immanent qualities and substance of “elementary particles”, physicists have to turn to the framework of Quantum Field Theory. Essentially, Quantum Field Theory is a specialized set of mathematical formulas and rules by means of which one, in a consistent and very detailed manner, can consider questions such as ‘what is the true origin and intrinsic nature of matter, charge, force etc.’ and ‘what is really going on at the most fundamental levels of nature’. Actually, Quantum Field Theory is not a theory as such but is more like a complex and content rich language well equipped for describing and “verbalizing” realms where other languages (such as English or the mathematics of Quantum Mechanics) have to give up.

The overall result of the sequence of processes, illustrated by the image above, is that two entities - a charm-quark and its anti-quark - are split into six other entities. The actual appearances of this sequence are well-confirmed by “particle” physics experiments. The same statement applies to a large variety of similar processes with any other set of “elementary particles” in the initial stage. Thus, these entities obviously are neither elementary nor indivisible, little balls of matter. No, - according to Quantum Field Theory, the “elementary particles” are discrete, vibrational states of certain underlying fields. Consequently, the fundamental constituents of the physical world are not “particles”. The fundamental constituents are fields. - This is the reason for hitherto having put the word ‘elementary’ in quotes. Next  .  .  .  the reason for the extensive use of quotes around the word ‘particle’ will be revealed.

[buddhasociety.com/quantum-physics/quantum-mechanics-3]

                                      Quantum Field Theory - II

All matter in the world around us consists of atoms. All atoms consist of electrons, which interact with a nucleus consisting of quarks. In ordinary matter the quarks interact with each other in such a way, that they form the compound bodies neutrons and protons. As seen from the figure above, all the various appearances of matter around us are composed of just three kind of “elementary particles” - electrons, up-quarks and down-quarks. However, in high-energy laboratories, in the interior of stars and at the beginning of this universe, other species of “elementary particles” are encountered.
The interactions between the atomic electrons and the nucleus, as a whole, is the domain of atomic physics while the internal affairs of the nucleus is the domain of nuclear physics. Thus, these two sciences are concerned with the first few steps in the construction of all the material objects we see around us. “Particle” physics, on the other hand, is concerned with what happens on a much smaller scale - about 10-100 million times smaller than the size of an atom. The domain of this branch of physics is the creation and behavior of “elementary particles” - which may combine to form atoms or may not combine to form anything at all.
Fermions
The electrons belong to the class of “elementary particles” called leptons. The leptons and the quarks together constitute the class called fermions. According to the Standard Model all matter is constituted by fermions. Whether the fermions combine to form a table, a star, a human body, a flower or do not combine at all depend on the four elementary forces - the electromagnetic, the gravitational, the weak and the strong forces.

Bosons
According to the Standard Model all force is mediated by exchange of (gauge) bosons. The electromagnetic force is mediated by exchange of photons, the strong force by exchange of gluons while the weak force is mediated by exchange of W and Z bosons. The gravitational force is special in the sense that, on the levels currently pondered by quantum physicists, it is too insignificant to even be considered.

The whole set of presently detected “elementary particles” is listed in the tables above. However, each quark comes in three slightly different variants (three colors), the gluon comes in eight variants and all fermions have an “antiparticle” of the same mass but of opposite charge.
[buddhasociety.com/quantum-physics/elementary-particles-2]

                                      Quantum Field Theory - II

All matter in the world around us consists of atoms. All atoms consist of electrons, which interact with a nucleus consisting of quarks. In ordinary matter the quarks interact with each other in such a way, that they form the compound bodies neutrons and protons. As seen from the figure above, all the various appearances of matter around us are composed of just three kind of “elementary particles” - electrons, up-quarks and down-quarks. However, in high-energy laboratories, in the interior of stars and at the beginning of this universe, other species of “elementary particles” are encountered.

The interactions between the atomic electrons and the nucleus, as a whole, is the domain of atomic physics while the internal affairs of the nucleus is the domain of nuclear physics. Thus, these two sciences are concerned with the first few steps in the construction of all the material objects we see around us. “Particle” physics, on the other hand, is concerned with what happens on a much smaller scale - about 10-100 million times smaller than the size of an atom. The domain of this branch of physics is the creation and behavior of “elementary particles” - which may combine to form atoms or may not combine to form anything at all.

Fermions

The electrons belong to the class of “elementary particles” called leptons. The leptons and the quarks together constitute the class called fermions. According to the Standard Model all matter is constituted by fermions. Whether the fermions combine to form a table, a star, a human body, a flower or do not combine at all depend on the four elementary forces - the electromagnetic, the gravitational, the weak and the strong forces.

Table of fermions

Bosons

According to the Standard Model all force is mediated by exchange of (gauge) bosons. The electromagnetic force is mediated by exchange of photons, the strong force by exchange of gluons while the weak force is mediated by exchange of W and Z bosons. The gravitational force is special in the sense that, on the levels currently pondered by quantum physicists, it is too insignificant to even be considered.

Table of fermions

The whole set of presently detected “elementary particles” is listed in the tables above. However, each quark comes in three slightly different variants (three colors), the gluon comes in eight variants and all fermions have an “antiparticle” of the same mass but of opposite charge.

[buddhasociety.com/quantum-physics/elementary-particles-2]