Gauge Theories2.3 Fields and Quantum Fields (1) To refresh your mind, I sum up the previous posting: quantum mechanics is a simple, logic, objective and truthful description of the microscopic reality. The most important rule we must follow is that ‘reality is the observed reality’, or ’reality is a body of information you could possibly collect on a system.’ To collect information, one must perform measurements. The order of measurements is of paramount importance and its consequence is neatly expressed with commutators. Therefore, any quantum theory ultimately boils down to finding the correct commutators. In this and next postings, I will extend this idea to fields, i.e., quantization of fields.
First, answer to the final question of the previous posting: Because both the kinetic energy p^2/2m and potential energy, -Ze/r, are ‘time dependent’, if one looks at them individually, an electron in an atom does radiate energy. However, the total energy, after average over the state configuration (wave function), has definite, discrete values. This means, the radiation of the electrons inside an atom is self-compensated so that the electrons essentially do not lose energy (this becomes clear if one calculates the average of the nucleus-electron dipole moments over the wave functions of an atom in any stationary states-so called eigen states). Therefore, the peculiar commutation relationship between p and r renders the energies of the electrons definite and guarantees the stability of the atoms. In formal language, as a result of the commutation relationship between position and momentum, the spectrum of the energy operator has a limited, definite minimum. Spectrum reflects our spirit of ‘reality is reality only when it is observed.’ Had there been no such a commutation rule, the atoms would not have existed at all. How lucky we are in this universe where the microscopic world obeys quantum mechanical rules!
Generally speaking, all bound systems have discrete energies, called quantized energies, or energy levels. For many bound systems, there exists a nonzero minimal energy, called zero point energy.
Now turn to the major topic of this posting—fields and how it is quantized.
Fields were introduced into physics by Michael Faraday. We know he was a scientist (physicist and chemist) of rarely high caliber. He was not formally trained and known to lack math skills, but few physicists in history left lasting influence on physics as he. In addition to great specific discoveries such as dynamo, electromagnetic induction, electrolysis, electroplating etc., he proposed the existence of electric field and magnetic field. He created such a great concept based on his experience and physics intuition: two electric charges may attract or repel each other even though there is only a vacuum between them. He hypothesized that there must be something very special between the charges, field as he called it. He also developed operational methods to measure the strength of the field.
However, the question of “what in the world is a field?” has been in the mind of physicists. In Faraday’s picture, field is a ‘special stuff’ generated by electric charge or electric current. For a long time, his idea was accepted without much further scrutiny. Anyhow, nobody could say better than that and nobody could understand what that ‘stuff’ really is. People adopted a kind of pragmatism and described a field as something generated by a ‘source’ or charge (such as electric charge, magnetic moment, mass, ‘color’ etc), added to the void or vacuum or nothingness or spacetime. Fields can be regarded as some medium in spacetime albeit a very, very special kind of medium. However, about 100 years ago, Albert Einstein showed that gravitational field causes a distorted vacuum or spacetime. Crazy enough, he told us that spacetime is the gravitational field itself. We will not mention gravitational field until in the last postings of this series. Before then, we just visualize spacetime as a vast void waiting for ‘things’ to fill in. But here I must caution you that vacuum in physics texts is very different from void or nothing. Vacuum must be understood as some extra stuff in void or spacetime. I hope you will eternally keep this picture in your mind after finishing this posting.
According to today’s fashion, fields are the most fundamental ‘being’. A vacuum is the ground state, namely the lowest energy state, of some being. Notice here, that we have different vacuua because the ground states of different beings can be (and should be as some people insist) different. Actually, from here we can sense something eerie about modern physics, but I do not want to distract you at this moment. We will come back to this subject later.
Therefore, at the most fundamental level of reality, we have fields and we have fields of many kinds. Vacuua are the lowest energy states of these fields and particles are the other states of the fields. In other words, particles are created as a result of field excitation. Take a violin as an example. The vacuua are the strings at rest and particles are created if the strings are pulled. The vibrating strings may affect each other when they vibrate in harmony. According to this picture, we naturally arrive at an important conclusion that interactions between particles are mediated by fields. In fact, fields and particles are the same ‘stuff’.
Particles such as electron, quarks, photon, muons, mesons, are all ‘excited’ states of their corresponding fields. Although we tend to visualize these ‘particles’ as some tiny-whinny point-like billiards roaming around in spacetime, we know we must be careful not to go too far.
I can imagine someone now would jump and protest: “Aren’t particles localized in space and fields distributed throughout the space?” Great question. Now it is good time for us to review and apply quantum mechanics we learnt in the last posting. The most important lesson from quantum mechanics is those commutators. For instance, position and momentum have such a commutation relationship: xp – px = h (ignoring a constant). Translating this into a more operational language, we say that the position and momentum of an object cannot be measured precisely at the same time. For a point-like object, by definition, it has a fixed value of position, i.e., the position has a precise value. This means the momentum is completely indeterminate to keep the commutation relationship valid. This is to say that the position at the next instant is completely uncertain since we have no information about its momentum (hence speed). It could be anywhere in the universe. Or it is nowhere. What is this kind of object? Isn’t it just like a ‘wave’ with no specific position?
On the other hand, if we get a definite momentum of an object upon a measurement, its position would be completely unknown, i.e., it could be anywhere. Here you would not oppose that it is essentially a ‘wave’.
Therefore, you see, a ‘particle’ and a ‘wave’ in quantum mechanics are the same ‘stuff’. They are wave-particle or particle-wave or non-wave-particle or non-particle-wave, as you please. From now on, I hope you will get used to this picture that ‘particles’ and ‘fields’ are the same ‘stuff’. Forget old-time counter-revolutionary, totalitarian slogans such as ‘localized particles’ and ‘distributed fields’ and embrace new-age, revolutionary, democratic buzzwords such as “fields and particles are the equal” and “fields are particles and particles are fields”, ”Vauua are the fields at the lowest state or ground state and particles are the higher-energy states of the fields.” Brainwash yourself for as many times as possible.
In brief: spacetime + ground state of field --> vacuum --> particles are born when a field is excited. Particles are babies born from vacuua. Vacuua are embryos of particles. We have different babies, so we should have different vacuua. You would agree that it is a good idea to use the enlightening symbol |0> to denote a vacuum.
You may ask: “how a field is excited so that particles are created?” It sounds hard but it is sucking easy. To see this, again, the violin metaphor is helpful. You pull the strings and you excite them and you create ‘particles’. What does it really mean to say ‘to pull the strings’? It is simply putting energy into the strings. Physicists use a similar method to give birth to particles. They add energy to a vacuum and the field is excited and particles are born. In formal words to exert terror, they use something called creation operators, denoted as a’ where the superscript ‘ means creation. When a creation operator acts on a vacuum, a particle is created, i.e., a new state of the field is created, simply denoted as a’|0>. Essentially, creation of particles is just pulling strings, or adding energy to a vaccum. The equation |1> = a’|0> is just a shorthand record of “a particle is created after adding some energy to a vacuum.” Kindergarten kids can tell us how to work with quantum fields. To compute how much energy is added, you simply count the number of particles created.
Just like a violin string may stop vibration if it loses energy, the excited field, or particles, may lose energy and go back to the vacuum state. When this happens, we say particles are annihilated or destroyed. By symbols, we have |0> = a|1>. Creation and annihilation of particles, or excitation and de-excitation of fields, are daily life of quantum fields.
In quantum mechanics, we measure all kinds of physical quantities such as position, momentum, angular momentum and energy. In quantum field theories, we do the same, but some simplicity as well as some complexity come in when we deal with fields. As for simplicity, we count the number of particles as kids can tell us. As for complexity, we have infinite number of positions and momenta so we have to deal with infinite number of degrees of freedom. There is another complexity, the so-called relativistic invariance, which has to be postponed to the next posting.
Quantum field theories can be regarded as an application of quantum mechanics to systems with infinite number of degrees of freedom. Therefore, it is no wonder in quantum field theories, the same principle applies: the order doing measurements matters. Creation and annihilation of particles is the central theme in quantum field theories as mentioned earlier. Creation and annihilation operators are what we need to perform measurements. The commutators in a quantum field theory, therefore, may be denoted as [a’,a] = h (ignoring a constant). That, as you may instantly recognize,is just the commutation relationship lifted from quantum mechanics. Yes. quantum field theories steal or herit a lot from quantum mechanics. But why don't we simply call them quantum mechanics?
To answer this question, some new element has to be introduced and we’d better leave it for the next week.
I hope by far you have not been deterred. I promise that you will have an easier time in the future if you have come along without much agony. As long as we maintain some percentages of the curiosity and persistence of our chief scientist Shortriver Scholar, we will be able to climb up to the summit of modern physics.