. The human body is more or less bilaterally symmetric, like the triangle shown in the figure. Some studies have actually indicated that more symmetrical faces are generally considered more beautiful.
and it says, With every continuous symmetry of a physical system, there exist certain corresponding quantities that are conserved, meaning that they do not change in time. As we all know, in nature things are constantly changing, and one of our primary goals in our ongoing efforts to understand nature is to be able to predict how those changes occur. Therefore, it is of the deepest relevance that we identify those aspects that do remain unchanged as things change and evolve in time. In the jargon of physics, a physical quantity is said to be conserved if it stays constant as a system changes with time. Noether’s theorem is extremely powerful because it allows us to determine those constants simply by looking at the symmetries possessed by a system! It would be no exaggeration to say that this theorem is one of the pillars of our understanding of the universe, particularly in our quest for the fundamental building blocks and the origins of the universe.
whereas in sexual reproduction, one half of genetic material is inherited from each parent, in effect creating new genetic material. Breaking the sexual symmetry leads to the violation of genetic conservation, meaning that genetic material gets significantly altered. It is rather amusing to note that even among sexual creatures, there might be some conservation rules if symmetry is imposed! Gay sex is symmetric as regards the absence of sexual dimorphism of the participants, and gay sex does conserve numbers—two people have sex, and at the end of that there are still two people. Straight sex between a man and a woman is asymmetric, and the conservation of numbers is frequently violated—at the end of about nine months, a completely new person can come into existence, sometimes even two or three or more!
defined by quantum field theory is based on two key insights:
1. Space-time (even when seemingly empty) is permeated by certain physical quantities called fields, which can vary from point to point, meaning that they have “infinite degrees of freedom,” since there are an infinite number of points in space-time. Electromagnetism (associated with electricity and magnetism) and gravity are examples of fields. We experience gravity although we cannot see it, and it can vary in strength at different points; for example, when we move up from the earth’s surface, gravity gets weaker. It may help to think of fields as sort of like the Force in Star Wars—they can be intangible and invisible and still permeate space-time and have significant influence on all that exist, but, unlike in the movie, fields have no dependence on life.
2. All elementary particles—basic building blocks of the universe—are simply manifestations of changes or disturbances (called excitations) of certain fields. As astounding as this might sound, we have actually already brushed with the idea earlier when we mentioned in that the electromagnetic force is associated with the photon (particle of light) and the weak nuclear force with two W and one Z bosons. We can now understand the photon as being an excitation of the electromagnetic field and W and Z bosons as that of the weak nuclear field.
). At any point in space-time, a field can take any one of a continuum of values, just the way the temperature can vary. But as the field changes, its potential energy usually changes, as well—and how exactly it changes is of critical importance. An example is shown in , where positions on the flat, circular grid define a map of values the field can take at any one particular point in space-time, and the height of the parabolic surface measures the corresponding potential energy. In this example, the potential energy is minimum when the field is zero (at the center). As the field increases (moves out from the center) the potential energy increases as indicated by the upward curving of the surface. Since the universe prefers lower potential energy, as we saw in , the field is most likely to have the value of zero. This becomes clear if we think of a ball rolling about on the potential energy surface: Its horizontal position marks the value of the field, and how high up on the curved surface determines its potential energy. Such a hypothetical ball will clearly settle at the center illustrating the preference for zero value for the field.
That was an example. In reality, all of space-time is infused with a scalar field that has a somewhat more complicated potential energy surface, in the shape of a Mexican hat or sombrero as shown in . (We left out the flat grid that marks the field, with the understanding that the field is zero at the center and increases outward as before.) Clearly, now the potential energy has a local maximum (the peak of the hat) at the center where the field is zero. The minimum value occurs all around a ring (the valley of the hat) located a certain distance from the center. By the rule that lowest potential energy is preferred, the field is most likely to settle somewhere along that circular valley in the Mexican hat shape. Again, it is easy to visualize this by considering how a hypothetical ball on that surface would behave.
) has its value at the center, the scenario retains that symmetry. But when the field takes a value somewhere in the circle of minima (as in the ball lying at some point in the valley of the hat ), the potential energy surface still remains symmetric, but the lop-sided positioning of the field (like the ball) on one specific side breaks that symmetry, as can be seen. The field is then said to have spontaneously broken the symmetry associated with the shape of the potential energy surface.
, we encountered the standard definition of mass as the amount of matter an object contains, so that heavier objects have more mass. But later in that same chapter, in the context of Newton’s second law, we identified a more general meaning of mass as a measure of inertia of an object—a measure of its resistance to having its state of rest or motion altered by some external forces in the environment. Our everyday experiences confirm that: A massive tractor trailer is much more resistant to being moved about than a significantly less massive bicycle. “Massless” therefore simply means that the entity of interest has no such resistance or inertia.
Going back to , we can see that if the scalar field tries to change or move in the radial direction, it will face resistance like a ball trying to roll up the steep sides. Therefore, such a radial “excitation” or disturbance of the field has mass in the sense we just described, and so will the associated particle (since field excitations are manifest as particles according to the second tenet of quantum field theory mentioned at the beginning of this discussion). However, if the field tries to change along the circular valley, there is no resistance since the potential energy does not curve up that way, so any excitation in that direction, and the associated particle, is massless. The massive particle associated with the field excitation in the radial direction is the famous Higgs boson!
, along which the field (or the ball) is present, as shown in . In the jargon of physics, this is called “fixing the gauge,” meaning that we choose the one slice of interest from the infinitely many possible directions around the circle that the field could have picked (corresponding to the different points in the valley the ball could have rolled into).
Figure 13.4 When the scalar field spontaneously breaks the radial symmetry present in (a), it chooses one point in the circular valley (represented by where the ball settles when it rolls off the tip). Thereafter, instead of the full sombrero-shape, we need consider only a slice of it, taken along the direction where the ball has settled, as shown in (b). This directly affects the vector field coupled to it. When the scalar field is symmetric, the vector field has a flat potential energy surface seen in (c), so the field (represented by a ball again) can change with no resistance, and the associated particles are massless. When the scalar field breaks the symmetry, (d) potential energy of the vector field rolls up in one direction, and the associated particles acquire mass since now there is resistance to field changes (like a ball rolling) in that direction.
so that there is no resistance to changes in the field, meaning all particles associated with the vector field have no mass. When the scalar field breaks the symmetry, it makes the potential energy of the vector field roll up into a u-shape as shown in . Field changes along the valley of the u-shape still face no resistance, so the associated particle remains massless. This particle happens to be the photon—the particle of light. But field changes up the steep walls of the valley face resistance, and the associated particle gains mass. This actually corresponds to three particles that happen to be two W bosons and the Z boson.
This has profound implications, because this process known as the Higgs mechanism gives mass not only to the W and the Z bosons, but also to all the elementary particles like the electrons, protons, and neutrons that we are all made of—they would otherwise have remained as massless fields. Thus, the Higgs mechanism and the Higgs boson is truly and literally the origin of mass and, therefore, of all matter in the universe. Without it, the universe as we know it would not exist!
There is a very interesting analog to this in the evolution of life on this planet and how eternal life was lost. For the first two thirds of the billions of years life has existed on this planet, there was no sex—reproduction was asexual, whereby the offspring are exact genetic copies of the parent (via a few different mechanisms, for example, single-cell organisms might simply split in two). So these creatures never died naturally because their genetic code was preserved. On the other hand, offspring of sexually reproducing creatures are a blend of two distinct parents, so when the parents die, their specific genetic code is lost. Sexual reproduction has numerous evolutionary benefits due to the mixing of DNA from two separate parents, which can correct errors and also speed up advantageous evolutionary changes. However, it came at a price, because along with sex came death: Gradual decay and death are programmed into the cells of sexually reproducing organisms. A fascinating account of this can be found in a book with the evocative title, Sex, and the Origins of Death.
, which is quite analogous to but representing this new interpretation.
Viewing an organism’s life-vector as the vector field, we can define eternal life as the lack of any natural resistance to living forever—just like a ball rolling on a flat potential surface represented in . When sexual symmetry is broken, an absolute resistance emerges that prevents lifespans of sexually reproducing creatures from “rolling on” forever. However, some creatures still remained asexual, so not all living organisms today are doomed to natural death; some immortals are still with us in the form of single-celled organisms and certain others—just as not all the vector field excitations acquired mass, only some did.
. But, all massless particles always travel at the speed of light in vacuum—the maximum speed available. According to time dilation in relativity, time goes slower and slower as any entity approaches the speed of light so that time literally stands still at the speed of light. From that perspective, massless particles exist for an eternity. Massive particles exist in time—time can slow down, but it can never stand still, so they have lost eternity.
Emmy Noether was one of the greatest mathematicians of all times. As one of the few women in the male-dominated mathematical world of the early twentieth century, she faced crippling sexual discrimination. As if that was not bad enough, being Jewish in Nazi Germany brought on persecutions that led her to eventually leave Europe and settle in the United States. It is a testament to her genius that despite such overwhelming odds, she made highly significant contributions to physics and mathematics.
There is also the possibility of changes to the genetic material due to mutations, but it is a slow process, which is part of the reason sexually reproducing creatures have been dominating evolution since sex came on the scene.
Here we should understand “fundamental” in the sense of what has been experimentally established, which is at the level of elementary particles like electrons, protons, and quarks, and so forth—the ultimate building blocks we know of. There exist some very convincing and beautiful ideas about even smaller structures within some of those, such as described by string theory, but none of those have been validated by any experiments.
Here we will frequently refer to space and time as a single continuum called space-time that can be visualized as a four-dimensional (4D) fabric or mesh.
Actually, the vector field has four directions, but that is impossible to draw, so to get the main idea across I have represented it as a flat surface (before symmetry breaking) that has only two independent directions. But in reality, when the symmetry is broken, the vector field “rolls up” in three directions (hence three massive particles) and remains flat in the fourth direction (corresponding to the massless photon).