Warped Passages (24 page)

To complete the derivation of our simple uncertainty principle, if you had a sufficiently simple quantum mechanical system—a single photon, for example—its energy would be equal to Planck’s constant,
h
, times frequency. For such an object, the product of the time interval over which you measure energy and the error in energy would always exceed
h
. You could measure its energy as precisely as you like, but your experiment would then have to run for a correspondingly longer time. This is the same uncertainty principle we just derived; the added twist is only the quantization relation that relates energy to frequency.

Two Important Energy Values and What the Uncertainty Principle Tells Us About Them

That almost completes our introduction to the fundamentals of quantum mechanics. This and the following section review two remaining elements of quantum mechanics that we will use later on.

This section, which does not involve any new physical principles, presents one important application of the uncertainty principle and special relativity. It explores the relationships between two important energies and the smallest length scales of the physical processes to which particles with those energies could be sensitive—relationships that particle physicists use all the time. The following section will introduce spin, bosons, and fermions—notions that will appear in the next chapter, about the Standard Model of particle physics, and also later on, when we consider supersymmetry.

The position-momentum uncertainty principle says the product of uncertainties in position and in momentum must exceed Planck’s constant. It tells us that anything—whether it is a light beam, a particle, or any other object or system you can think of that could be sensitive to physical processes occurring over short distances—must involve a large range of momenta (since the momentum must be very uncertain). In particular, any object that is sensitive to those physical processes must involve very high momenta. According to special relativity, when momenta are high, so are energies. Combining these two facts tells us that the only way to explore short distances is with high energies.

Another way of explaining this is to say that we need high energies to explore short distances because only particles whose wavefunctions vary over small scales will be affected by short-distance physical processes. Just as Vermeer could not have executed his paintings with a two-inch-wide brush, and just as you can’t see fine detail with blurry vision, particles cannot be sensitive to short-distance physical processes unless their wavefunction varies over only small scales. But according to de Broglie, particles whose wavefunction involves short wavelengths also have high momenta. De Broglie said that the wavelength of a particle-wave is inversely proportional to its momentum. Therefore de Broglie would also have us conclude that you need high momenta, and hence high energies, to be sensitive to the physics of short distances.

This has important ramifications for particle physics. Only high-energy particles feel the effects of short-distance physical processes. We’ll see in two specific cases just how high I mean.

Particle physicists often measure energy in multiples of an
electronvolt
, which is abbreviated as eV, and pronounced by saying the letters “e-V.” An electronvolt is the energy required to move an electron against a potential difference, such as could be provided by a very weak battery, of 1 volt. I’ll also use the related units
gigaelectronvolt
, or GeV (pronounced “G-e-V”) and teraelectronvolt, or TeV; a GeV is 1 billion eV and a TeV is 1 trillion eV (or 1,000 GeV).

Particle physicists often find it convenient to use these units to measure not just energy, but also mass. We can do this because the special relativity relations between mass, momentum, and energy tell
us that the three quantities are related through the speed of light, which is the constant
c
= 299,792,458 meters/second.
¹³
We can therefore use the speed of light to convert a given energy into mass or momentum. For example, Einstein’s famous formula
E
=
mc
²
means that there is a definite mass associated with any particular energy. Since everyone knows that the conversion factor is
c
²
, we can incorporate it and express masses in units of eV. The proton mass in these units is 1 billion eV—that is, 1 GeV.

Converting units in this way is analogous to what you do every day when you tell someone, for example, that “The train station is ten minutes away.” You are assuming a particular conversion factor. The distance might be half a mile, corresponding to ten minutes at walking speed, or it might be ten miles, which is ten minutes at highway speeds. There is an agreed-upon conversion factor between you and your conversation partner.

These special relativity relationships, in conjunction with the uncertainty principle, determine the minimum spatial size of the physical processes that a wave or a particle of a particular energy or mass could experience or detect. We will now apply these relations to two very important energies for particle physics that will appear frequently in later chapters (see Figure 46).

The first energy, also known as the
weak scale energy
, is 250 GeV. Physical processes at this energy determine key properties of the weak force and of elementary particles, most notably how they acquire mass. Physicists (including myself) expect that when we explore this energy, we will see new effects predicted by as yet unknown physical theories and learn a good deal more about the underlying structure of matter. Fortunately, experiments are about to explore the weak scale energy and should soon be able to tell us what we want to know.

Sometimes I will also refer to the
weak scale mass
, which is related to the weak scale energy through the speed of light. In more conventional mass units, the weak scale mass is 10
^-21
grams. But as I just explained, particle physicists are content to talk about mass in units of GeV.

The associated
weak scale length
is 10
^-16
cm, or one ten thousand trillionth of a centimeter. It is the range of the weak force—the
maximum distance over which particles can influence each other through this force.

Because uncertainty tells us that small distances are probed only with high energy, the weak scale length is also the minimum length that something with 250 GeV of energy can be sensitive to—that is, it is the smallest scale on which physical processes can affect it. If any smaller distances could be explored with that energy, the distance uncertainty would have to be less than 10
^-16
cm, and the distance-momentum uncertainty relation would be violated. The currently operating Fermilab accelerator and the future Large Hadron Collider (LHC), to be built at CERN in Geneva within the decade, will explore physical processes down to that scale, and many of the models I will discuss should have visible consequences at this energy.

The second important energy, known as the
Planck scale energy
,
M
_Pl
, is 10
¹⁹
GeV. This energy is very relevant to any theory of gravity. For example, the gravitational constant, which enters Newton’s gravitational force law, is inversely proportional to the square of the Planck scale energy. Gravitational attraction between two masses is small because the Planck scale energy is large.

Moreover, the Planck scale energy is the largest energy for which a classical theory of gravity can apply; beyond the Planck scale energy, a quantum theory of gravity, which consistently describes both quantum mechanics and gravity, is essential. Later on, when we discuss string theory, we will also see that in the old string theory models the tension of a string is very likely determined by the Planck scale energy.

Quantum mechanics and the uncertainty principle tell us that when particles achieve this energy, they are sensitive to physical processes at distances as short as the
Planck scale length
,
^*
which is 10
^-33
cm. This is an extremely small distance—far less than anything measurable. But to describe physical processes that occur over distances this small a theory of quantum gravity is required, and that theory might be string theory. For this reason, the Planck scale length, along with the Planck scale energy, are important scales that will reappear in later chapters.

Bosons and Fermions

Quantum mechanics makes an important distinction among particles, dividing the world of particles into
bosons
and
fermions
. Those particles could be fundamental particles such as the electron and quarks, or composite entities such as a proton or the atomic nucleus. Any object is either a boson or a fermion.

Whether such an object is a boson or a fermion depends on a property called
intrinsic spin
. The name is very suggestive, but the “spin” of particles does not correspond to any actual motion in space. But if a particle has intrinsic spin, it interacts as if it were rotating, even though in reality it is not.

For example, the interaction between an electron and a magnetic field depends on the electron’s classical rotation—its actual rotation in space. But the electron’s interaction with the magnetic field also depends on the electron’s intrinsic spin. Unlike the classical spin that arises from actual motion in physical space,
^*
intrinsic spin is a property of a particle. It is fixed and has a specific value now and for ever. For example, the photon is a boson and has spin-1. That is a property of the photon; it is as fundamental as the fact that the photon travels at the speed of light.

In quantum mechanics, spin is quantized. Quantum spin can take the value 0 (i.e., no spin at all), or 1, or 2, or any integer number units of spin. I’ll call this spin-0 (pronounced “spin-zero”), spin-1, spin-2, and so on. Objects called bosons, named after the Indian physicist Satyendra Nath Bose, have intrinsic spin—the quantum mechanical spin that is independent of rotation—and that is also an integer: bosons can have intrinsic spin equal to 0, 1, 2, and so on.

Fermion spin is quantized in units that no one would have thought possible before the advent of quantum mechanics. Fermions, named after the Italian physicist Enrico Fermi, have half-integer values such as ½ or 3/2. Whereas a spin-1 object returns to its initial configuration after it is rotated a single time, a spin -½ particle would do so only after it were rotated twice. Despite the apparent weirdness of the
half-integer values of fermions’ spins, protons, neutrons, and electrons are all fermions with spin -½. Essentially all familiar matter is composed of spin -½ particles.

The fermionic nature of most fundamental particles determines many properties of the matter around us. The
Pauli exclusion principle
, in particular, states that two fermions of the same type will never be found in the same place. The exclusion principle is what gives the atom the structure upon which chemistry is based. Because electrons with the same spin can’t be in the same place, they have to be in different orbits.

That is why I could make the analogy with different floors of a tall building earlier on. The different floors represented the different possible quantized electron orbits that the Pauli exclusion principle tells us get occupied when a nucleus is surrounded by many electrons. The exclusion principle is also the reason you can’t poke your hand through a table or fall into the center of the Earth. Tables and your hand take the solid structure they do only because the uncertainty principle gives rise to atomic, molecular, and crystalline structure in matter. The electrons in your hand, which are the same as the electrons in a table, have no place to go when you hit a table. No two identical fermions can be in the same place at the same time, so matter can’t just collapse.

Bosons act in exactly the opposite fashion to fermions. They can and will be found in the same place. Bosons are like crocodiles—they prefer to pile up on top of one another. If you shine light where there is already light, it behaves very differently from your hand karate-chopping a table. Light, which is composed of bosonic photons, passes right through light. Two light beams can shine in exactly the same place. In fact, lasers are based on this fact: bosons occupying the same state allow lasers to produce their strong, coherent beams. Superfluids and superconductors are also made of bosons.

An extreme example of bosonic properties is the Bose-Einstein condensate, in which many identical particles act together as a single particle—something that fermions, which have to be in different places, could never do. Bose-Einstein condensates are possible only because the bosons of which they are composed, unlike fermions, can have identical properties. In 2001, Eric Cornell, Wolfgang Ketterle,
and Carl Wieman received the Nobel Prize for Physics for their discovery of the Bose-Einstein condensate.

Later on I won’t need these detailed properties of the way that fermions and bosons behave. The only facts I will use from this section are that fundamental particles have intrinsic spin and can act as if they were spinning in one direction or another, and that all particles can be characterized by whether they are bosons or fermions.

What to Remember

• Quantum mechanics tells us that both matter and light consist of discrete units known as
  quanta
  . For example, light, which seems continuous, is actually composed of discrete quanta called photons.
  
• Quanta are the basis of particle physics. The Standard Model of particle physics, which explains known matter and forces, tells us that all matter and forces can ultimately be interpreted in terms of particles and their interactions.
  
• Quantum mechanics also tells us that every particle has an associated wave, known as the particle’s
  wavefunction
  . The square of this wave is the probability that the particle will be found in a particular location. For convenience, I will sometimes talk about a
  probability wave
  , the square of the more commonly used wavefunction. The values of this probability wave will give probabilities directly. Such a wave will appear later on when we discuss the
  graviton
  , the particle that communicates the force of gravity. The probability wave will also be important when discussing
  Kaluza-Klein
  (
  KK
  )
  modes
  , which are particles that have momentum along the extra dimensions—that is, directed perpendicular to the usual dimensions.
  
• Another major distinction between classical physics and quantum mechanics is that quantum mechanics tells us that you cannot precisely determine a particle’s path—you can never know the precise path a particle took as it traveled from its starting point to its destination. This tells us that we have to consider all the paths that a particle can take when it communicates a force. Because quantum paths can involve any interacting particles, quantum mechanical effects can influence masses and interaction strengths.
  
• Quantum mechanics divides particles into
  bosons
  and
  fermions
  . The existence of two distinct categories of particles is critical to the structure of the Standard Model and also to a proposed extension of the Standard Model known as
  supersymmetry
  .
  
• The
  uncertainty principle
  of quantum mechanics, coupled with the relations of special relativity, tell us that, using physical constants, we can relate a particle’s mass, energy, and momentum to the minimum size of the region in which a particle of that energy can experience forces or interactions.
  
• Two of our most frequent applications of these relations involve the two energies known as the
  weak scale energy
  and the
  Planck scale energy
  . The weak scale energy is 250 GeV (gigaelectronvolts) and the Planck energy is much bigger—ten million trillion GeV.
  
• Only forces with a range smaller than ten million billionths (10
  ^-17
  ) of a centimeter will produce measurable effects on a particle with weak scale energy. This is a very tiny distance, but it is relevant to the physical processes in a nucleus and to the mechanism by which particles acquire mass.
  
• Tiny as it is, the
  weak scale length
  is far greater than the
  Planck scale length
  , which is one million billion billion billionth (10
  ^-33
  ) of a centimeter. That is the size of the region where forces influence particles that have the Planck scale energy. The Planck scale energy determines the strength of gravity; it is the energy that particles would have to have for gravity to be a strong force.