String Theory For Dummies. Andrew Zimmerman Jones

would no longer be broken (which is what makes it so interesting to study). We cover supersymmetry in Chapters 2 and 10.

      Translational symmetry: Same system, different spot

      If an object has translational symmetry, you can move it and it continues to look the same (for a detailed explanation of this, flip to Chapter 4). Moving objects in space doesn’t change the physical properties of the system.

      Now, didn’t we just say in the last section that the potential energy due to gravity changes depending on where an object is? Yes, we did. Moving an object’s location in space can have an impact on the physical system, but the laws of physics themselves don’t change (so far as we can tell). If the Empire State Building, Earth, and the penny held over the edge (the entire “system” in this example) were all shifted by the same amount in the same direction, there would be no noticeable change to the system.

      Internal symmetry: The system changes, but the outcome stays the same

      In an internal symmetry, some property of the system can undergo a change without changing anything that we may measure in an experiment.

      For example, changing every particle with its antiparticle — changing positive charges to negative and negative charges to positive — leaves the electromagnetic forces involved completely identical. This is a form of internal symmetry called charge conjugation symmetry.

      Spontaneous symmetry breaking: A gradual breakdown

      Physicists believe that the laws of the universe used to be even more symmetric but have gone through a process called spontaneous symmetry breaking, where the symmetry falls apart in the universe we observe.

If everything were perfectly symmetric, the universe would be a very boring place. The slight differences in the universe — the broken symmetries — are what make the natural world so interesting, but when physicists look at the physical laws, they tend to find that the differences are fairly small in comparison to the similarities.

      

To understand spontaneous symmetry breaking, consider a pencil perfectly balanced on its tip. The pencil is in a state of perfect balance, of equilibrium, but it’s unstable. Any tiny disturbance will cause it to fall over. However, no law of physics says which way the pencil will fall. The situation is perfectly symmetrical because all directions are equal.

      As soon as the pencil starts to fall, however, definite laws of physics dictate the direction it will continue to fall. The symmetrical situation spontaneously (and, for all intents and purposes, randomly) begins to collapse into one definite, asymmetrical form. As the system collapses, the other options are no longer available to the system.

      The Standard Model of particle physics, as well as string theory (which includes the Standard Model in an appropriate limit), predicts that some properties of the universe were once highly symmetrical but have undergone spontaneous symmetry breaking into the universe we observe now.

All Shook Up: Waves and Vibrations

      In string theory, the most fundamental objects are tiny strings of energy that vibrate or oscillate in simple, regular patterns. In physics, such systems are called harmonic oscillators. Harmonic oscillators are the simplest (and in many ways most universal) physical system that you will ever encounter.

      

Though the strings of string theory are different, understanding the vibrations of classical objects — like air, water, jump ropes, springs — can help you understand the behavior of these exotic little creatures when you encounter them. These classical objects can carry what are called mechanical waves.

      Catching the wave

Waves (as we usually think of them) move through some sort of medium. Like in the examples we discussed when talking about kinetic energy, tidal waves can move through the water, and sound waves through the air, with those materials acting as the medium for the motion. Similarly, if you flick the end of a jump rope or string, a wave moves along the rope or string. In classical physics, waves transport energy, but not matter, from one region to another. One set of water molecules transfers its energy to the nearby water molecules, which means that the wave moves through the water, even though the actual water molecules don’t really travel all the way from the start of the wave to the end of the wave.

      This is even more obvious if we take the end of a jump rope and shake it, causing a wave to travel along its length. Clearly, the molecules at our end of the jump rope aren’t traveling along it. Each group of jump-rope molecules is nudging the next group of jump-rope molecules, and the end result is the wave motion along its length.

There are two types of mechanical waves, as shown in Figure 5-1:

       Transverse wave: A wave in which the displacement of the medium is perpendicular to the direction of travel of the wave along the medium, like the flicking of a jump rope

       Longitudinal wave: A wave that moves in the same direction in which the wave travels, like a piston pushing on a cylinder of water

      FIGURE 5-1: Waves come in two types: transverse, shown on top, and longitudinal, shown on the bottom.

      The highest point on a transverse wave (or the densest point in a longitudinal wave) is called a crest. The lowest point on a transverse wave (or the least dense point in a longitudinal wave) is called a trough.

      The displacement from the resting point to the crest — in other words, how high the wave gets — is called the amplitude. The distance from one crest to another (or one trough to another) is called the wavelength. These values are shown on the transverse wave in Figure 5-1. The wavelength is shown on the longitudinal wave as well, although the amplitude is hard to show on that type of wave, so it isn’t included.

Another useful thing to consider is the velocity (speed and direction) of the wave. This can be determined by its wavelength and frequency, which is a measure of how many times the wave passes a given point per second. If you know the frequency and the wavelength, you can calculate the velocity. This, in turn, allows you to calculate the energy contained within the wave.

Another trait of many waves is the principle of superposition, which states that when two waves overlap, the total displacement is the sum of the individual displacements, as shown in Figure 5-2. This property is also referred to as wave interference.

      FIGURE 5-2: When two waves overlap, the total displacement is the sum of the two individual displacements.

      Consider waves when two ships cross each other’s path. The waves made by the ships cause the water to become choppier, and as the waves add height to each other, they cause massive swells.