Warped Passages - Lisa Randall [232]
11. A wavefunction is actually a complex-valued function. This is the source of many of quantum mechanics’ strange properties. When you add two complex functions together and then square the sum, you generally get a different result from when you first square and then add. That results in interference phenomena. For example, in the double-slit experiment the probability that is recorded on a screen results from the interference of the waves that describe the electron’s two possible paths.
12. More precisely, it’s the product of Planck’s constant and the absolute value of the commutator of the two quantities divided by 2.
13. Special relativity tells us that a stationary object with rest mass m0 carries energy E = m0c2. More generally, an object that is moving with velocity v (with β = v/c and γ = 1/√(1 − β2)) will carry energy E = γm0c2. The rest mass is also known as the invariant mass (independent of reference frame). That is because, according to the transformation laws of special relativity, the quantity E2 − p2c2 = m0c4 is the same in any reference frame. Notice that you always need an energy at least equal to m0c2 to produce an object of mass m0. Also notice that when an object has low mass compared with its energy (really, energy/c2), the energy and momentum are related approximately by E = pc. That is why, at high energy, energy and momentum are roughly interchangeable.
14. Maxwell’s Equations (in c.g.s. units) are
where E is the electric field, B is the magnetic field, is the charge, and J is the current. These are first-order differential equations; by combining two of them you can derive a second-order differential equation involving only the electric or magnetic field. This equation takes the form of a wave equation—that is, its solutions are sinusoidal waves.
15. Actually, according to special relativity’s underlying principles, there could have been a fourth polarization as well, one that would oscillate in the time direction. But that one doesn’t exist either, and the same internal symmetry that eliminates the third (longitudinal) polarization eliminates “time polarization” as well. Since it plays no role in the discussion in this or the following chapter, we won’t consider it any further.
16. The true symmetries associated with all the forces are actually more subtle and rotate fields, which are complex quantities, into each other. The symmetries don’t merely interchange fields, they turn one field into a linear superposition of the others. The force associated with electromagnetism rotates a single complex field, whereas the weak force rotates two complex fields into each other, and the strong force rotates three.
17. To make a Higgs model work, at least one of the Higgs fields must be forced to take a nonzero value. This would be true if the minimum energy configuration occurs when the value of at least one of the Higgs fields is nonzero. One way this can happen is illustrated in Figure M2, which shows the so-called Mexican hat potential, a plot of the energy the system would take for any combination of values of the two Higgs fields, where the two lower axes are the absolute values of the two Higgs fields and the height of the three-dimensional surface represents the energy of that particular configuration. This particular potential takes the form λ(H12 + H22 − ν2)2, where λ determines how bowed up the potential is and ν determines the value that H12 + H22 will take when the potential is at its minimum. The key feature of this potential is that when both fields have zero value,