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 An experiment creates a new universe

One of the most important things to remember about physics is that Nature looks different depending on the distance or energy scale being looked at. Physical constants like the speed of light and Planck's constant have values that tell us the size or energy where the the observable physics begins to change substantially, and a different way of describing the physics mathematically is called for.

Relativity

The first example is the speed of light

In a physical system where all the velocities that matter are much much less than the speed of light, the system can usually be described very well by ordinary Newtonian physics. When some important velocity in the system begins to approach the speed of light, the Newtonian description doesn't fit so well any more, and the system has to be described in fully relativistic terms using the mathematics of Einstein's theory of special relativity.

Quantum physics

The most important physical constant uncovered in the 20th century is Planck's constant

which tells us about the distance or momentum scale where classical physics stops making sense and a quantum description becomes necessary. (The unit labeled eV is called an electron-volt, it is the amount of work one needs to do to move one electron across one volt of electric potential. This is a standard unit for measuring particle physics energy scales. You'll see more of it on this web site.)
Planck's constant, in the form of the de Broglie wavelength, tells us when the wave/particle duality of quantum physics becomes measurable in a physical object. The de Broglie wavelength of a particle of momentum p is

In quantum particle-wave duality, if the size of an object of momentum p is smaller than its de Broglie wavelength, then quantum wave interference will be measurable enough so that classical physics will fail to be a good description of the object's behavior.
Planck's constant is a very very small number. The de Broglie wavelength of something like a piece of cheese in the fridge would very very tiny. The cheese would have to be subatomic size before quantum cheese effects would take over -- and then it wouldn't be identifiable as cheese any longer.

Atomic physics

Planck's constant combined with the mass and electric charge of an electron gives us another important number in physics called the Bohr radius

which tells us the average size of a hydrogen atom. The Bohr radius describes the distance scale where atomic physics, using nonrelativistic quantum mechanics in a classical electromagnetic field, is the best way to describe the physics.

Elementary particle physics

In elementary particle physics, electromagnetism, the strong and weak nuclear forces are described by a combination of relativity and quantum mechanics, called relativistic quantum field theory. The important numbers for comparing these three forces are called coupling constants, whose size measures the strength of the respective force. For electromagnetism, the coupling constant is called the fine structure constant a and is formed by the electron charge, Planck's constant and the speed of light.

Notice this constant ends up being just a plain number, with no units associated with it. That is what is meant by a dimensionless coupling constant.
The combination of Planck's constant with the speed of light and the electron charge means this coupling constant is telling us about the quantum relativistic physics of electromagnetism, that is, it tells us something about electromagnetism at distance scales where quantum mechanics and special relativity are both important to the physics at the same time.
There are also dimensionless coupling constants for the strong and weak nuclear interactions. The table below compares their relative strengths and ranges.

Force Symbol Strength Range
Strong nuclear force as 1/3 10-15 m
Weak nuclear force aW 1/30 10-16 m
Electromagnetic force aEM 7x10-3 Infinity

The weak nuclear force isn't actually that weak when measured by aW, but it has the shortest range, because the gauge bosons are very heavy and have short lifetimes, so they can't travel very far without decaying into lighter particles. The strong nuclear force binds quarks into neutrons, protons and other hadrons, and binds protons and neutrons into the nuclei of atoms, but because of quark confinement, the strong force has a very small range as well.

Spontaneous symmetry breaking

Physicists originally had a lot of trouble reconciling relativistic quantum field theory with the weak interaction that is responsible, for example, for what is called beta decay of neutron into proton, electron and neutrino. The very short range of weak interaction suggested that the gauge boson the mediated the force must be very heavy. But quantum relativistic gauge field theories only seemed to make sense if the gauge boson had zero mass.
The paradox was resolved by the discovery that a special type of coupling to a particle called the Higgs could make the weak interaction gauge bosons able to become very heavy without destroying the symmetries that make the quantum theory mathematically consistent. This interaction between the gauge bosons and Higgs particles is called spontaneous symmetry breaking, which is a misnomer because the symmetry of the theory is still there, it's just hidden in the interactions of the theory.
Spontaneous symmetry occurs at some energy scale determined by the quantum interactions of the Higgs particle with itself, and this scale in turn determines the mass of the gauge boson. If the scale of symmetry breaking is very large, then it takes a very large particle accelerator to find evidence of the effect. In 1983, this theory was confirmed by the direct observation of the heavy gauge bosons in the powerful particle accelerator at CERN in Geneva. The masses of the three weak interaction gauge bosons are now known to be

Gauge boson Mass
W+ 80 GeV
W- 80 GeV
Z0 91 GeV

Running coupling constants

Assigning a coupling strength to each force is tricky, however, because of the quantum relativistic effect known as the running coupling constant. In quantum relativistic field theory, the calculation of a simple particle interaction, say, an electron interacting with another electron by exchanging a photon, is complicated by an infinite series of possible virtual particle interactions in a cloud of quantum relativistic fluctuations.
That cloud of quantum relativistic fluctuations changes the measured value of the coupling constant and makes it depend on the energy scale at which one is trying to measure it.
The electromagnetic coupling constant increases with energy. The strong and weak nuclear coupling constants decrease with energy. The strong force in particular exhibits a property called asymptotic freedom. The force that binds quarks together into a proton gets stronger at lower energy but becomes negligible at very high energies. That's why, in very high energy scattering experiments, the quarks inside a proton scatter almost like free particles.

A unified theory

The three running coupling constants wind up having the same strength at some very high energy scale, much higher than the weak interaction scale of about 80 GeV. That fact, and some nifty mathematics concerning particle multiplets in group theory convinced physicists that there should be some energy scale where these three forces all have the same strength and where all the different types of particles fit into the same mathematical theory in one unified group. This type of elementary particle theory is called a Grand Unified Theory or GUT for short.
The three gauge groups of the Standard Model of known elementary particle physics are SU(3)xSU(2)xU(1). In a Grand Unified Theory, these three groups all fit into a single unified group with a unified set of gauge bosons whose number is determined by the properties of the unified group. The most studied theories have been SU(5) and SO(10). The same process of spontaneous symmetry breaking that makes the weak bosons massive at a scale of 80 GeV is invoked by physicists to make most of the unified gauge bosons massive at a much higher scale.
When physicists look for a mass scale where the three known running coupling constants run into a single value necessary for a Grand Unified Theory, they find a very high mass scale of

This mass scale is far too high to be reached by particle accelerators in the near or distant future.
But there is a way to test Grand Unified Theories without probing that mass scale directly. The weak interaction was discovered because of beta decay, where a neutron decays into a proton, and electron and a neutrino. In a Grand Unified Theory, something like beta decay can happen to a proton. Even a small rate of proton decay would be disastrous and very noticeable, because the stability of the proton is the basis for the stability of all known matter in the Universe. So far, no evidence of proton decay has been observed in experiments set up to detect it.

The natural constant that describes the measured strength of the gravitational force is called Newton's constant. This is the constant that appears in Newton's law for the gravitation force between two objects (written here in its generalization to higher dimensions, where d is the dimension of spacetime)

Newton's constant is very different from the speed of light and Planck's constant, because the units depend on the number of spacetime dimensions

Gravity feels like a strong force at the macroscopic distance scales where humans experience it, but gravity is a very very weak force from a microscopic point of view. For example, the equivalent of the fine structure constant for an electron and a proton interacting according to Newton's law is

The gravitational radius of an object of mass M is a distance scale made from Newton's constant and the speed of light

When the size of an object approaches its gravitational radius, the object can collapse into a black hole. This gives a natural length scale where we expect the system to be described by the Einstein equations rather than by Newtonian physics.
The natural length scale at which quantum gravity should become important is called the Planck length, made from Newton's constant (in four dimensions), the speed of light and Planck's constant.

In string theory, the set of physical quantum states usually contains a graviton that gives rise to gravitational interactions. Therefore, it has been widely assumed that the natural distance scale of string theory should be the Planck scale. However string theories contain many duality symmetries that connect a string theory at one distance scale to a different string theory at a different scale. So the idea of a distance scale itself is not as firm and reliable in string theory as it it normally is in quantum field theory.
String theory is supposed to contain the physics of the quantum behavior of gravity. This implies a very subtle and rich structure where the idea of distance itself is shifting and obscure.

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Virtual interactions make the electron charge depend on the distance scale at which is it measured.

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