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Authors: Lawrence M. Krauss
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(or 1/energy 3 ) in this new system. Making the appropriate conversions to these new units one finds, for example, that a volume of 1 cubic meter (1 meter 3 ) is equivalent to (1/[10 –20 electron Volts 3 ]).
While this is an unusual and new way of thinking, the beauty of it is that with only one fundamental independent dimensional parameter left, we can approximate the results of what may be intrinsically very complicated phenomena simply in terms of a single quantity. In so doing we can perform some magic. For example, say a new elementary particle is discovered that has three times the mass of the proton or, in energy units, about 3 billion electron Volts—3 GeV (Giga electron Volts), for short. If this particle is unstable, what might we expect its lifetime to be before it decays? It may seem impossible to make such an estimate without knowing any of the detailed physical processes involved. However, we can use dimensional analysis to make a guess. The only dimensional quantity in the problem is the rest mass, or equivalently rest energy of the particle. Since the dimensions of time are equivalent to the dimensions of 1/mass in our system, a reasonable estimate of the lifetime would be k/(3 Ge V), where k is some dimensionless number that, in the absence of any other information, we might hope is not too different from 1. We can convert back to our normal units, say, seconds, using our conversion formula (1/1 eV) = 6 × 10 –16 sec. Thus we estimate the lifetime of our new particle to be about k × 10 –25 seconds.
Of course, there really is no magic here. We have not gotten something for nothing. What dimensional analysis has given us is the scale of the problem. It tells us that the “natural lifetime” of unstable particles with this kind of mass is around k × 10 –25 seconds, just like the “natural” lifetime of human beings is of the order of k × 75 years. All the real physics (or, in the latter case, biology) is contained in the unknown quantity k. If it is very small, or very large, there must be something interesting to be learned in order to understand why.
Dimensional analysis has, as a result, told us something very important. If the quantity k differs greatly from 1, we know that the processes involved must be either very strong or very weak, to make the lifetime of such a particle deviate from its natural value as given by dimensional arguments. It would be like seeing a supercow 10 times the size of a normal cow but weighing only 10 ounces. Simple scaling arguments in that case would tell us that such a cow was made of some very exotic material. In fact, many of the most interesting results in physics are those in which naive dimensional scaling arguments break down. What is important to realize is that without these scaling arguments, we might have no idea that anything interesting was happening in the first place!
In 1974, a remarkable and dramatic event took place along these lines. During the 1950s and 1960s, with the development of new techniques to accelerate high-energy beams of particles to collide, first with fixed targets and then with other beams of particles of ever higher energy, a slew of new elementary particles was discovered. As hundreds and hundreds of new particles were found, it seemed as if any hope of simple order in this system had vanished—until the development of the “quark” model in the
early 1960s, largely by Murray Gell-Mann at Caltech, brought order out of chaos. All of the new particles that had been observed could be formed out of relatively simple combinations—fundamental objects that Gell-Mann called quarks. The particles that were created at accelerators could be categorized simply if they were formed from either three quarks or a single quark and its antiparticle. New combinations of the same set of quarks that make up the proton and neutron were predicted to result in unstable particles, comparable in mass to the proton. These were observed, and
While this is an unusual and new way of thinking, the beauty of it is that with only one fundamental independent dimensional parameter left, we can approximate the results of what may be intrinsically very complicated phenomena simply in terms of a single quantity. In so doing we can perform some magic. For example, say a new elementary particle is discovered that has three times the mass of the proton or, in energy units, about 3 billion electron Volts—3 GeV (Giga electron Volts), for short. If this particle is unstable, what might we expect its lifetime to be before it decays? It may seem impossible to make such an estimate without knowing any of the detailed physical processes involved. However, we can use dimensional analysis to make a guess. The only dimensional quantity in the problem is the rest mass, or equivalently rest energy of the particle. Since the dimensions of time are equivalent to the dimensions of 1/mass in our system, a reasonable estimate of the lifetime would be k/(3 Ge V), where k is some dimensionless number that, in the absence of any other information, we might hope is not too different from 1. We can convert back to our normal units, say, seconds, using our conversion formula (1/1 eV) = 6 × 10 –16 sec. Thus we estimate the lifetime of our new particle to be about k × 10 –25 seconds.
Of course, there really is no magic here. We have not gotten something for nothing. What dimensional analysis has given us is the scale of the problem. It tells us that the “natural lifetime” of unstable particles with this kind of mass is around k × 10 –25 seconds, just like the “natural” lifetime of human beings is of the order of k × 75 years. All the real physics (or, in the latter case, biology) is contained in the unknown quantity k. If it is very small, or very large, there must be something interesting to be learned in order to understand why.
Dimensional analysis has, as a result, told us something very important. If the quantity k differs greatly from 1, we know that the processes involved must be either very strong or very weak, to make the lifetime of such a particle deviate from its natural value as given by dimensional arguments. It would be like seeing a supercow 10 times the size of a normal cow but weighing only 10 ounces. Simple scaling arguments in that case would tell us that such a cow was made of some very exotic material. In fact, many of the most interesting results in physics are those in which naive dimensional scaling arguments break down. What is important to realize is that without these scaling arguments, we might have no idea that anything interesting was happening in the first place!
In 1974, a remarkable and dramatic event took place along these lines. During the 1950s and 1960s, with the development of new techniques to accelerate high-energy beams of particles to collide, first with fixed targets and then with other beams of particles of ever higher energy, a slew of new elementary particles was discovered. As hundreds and hundreds of new particles were found, it seemed as if any hope of simple order in this system had vanished—until the development of the “quark” model in the
early 1960s, largely by Murray Gell-Mann at Caltech, brought order out of chaos. All of the new particles that had been observed could be formed out of relatively simple combinations—fundamental objects that Gell-Mann called quarks. The particles that were created at accelerators could be categorized simply if they were formed from either three quarks or a single quark and its antiparticle. New combinations of the same set of quarks that make up the proton and neutron were predicted to result in unstable particles, comparable in mass to the proton. These were observed, and
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