# Relationship between a field and its quanta

• wotanub
In summary, the lecturer explained how coherent states in the simple harmonic oscillator can be linked to excitations of the oscillator, and how Maxwell's equations naturally fall out of attempting to link quantum mechanics with special relativity. He also mentioned a book or resource that would be good for further study on the topic.
wotanub
I just listened to a lecture on this, and I'm not sure I quite get the point.

What the lecturer did was examine coherent states in the simple harmonic oscillator then linked excitations of an oscillator to the number of field quanta. The goal of the lecture was to show how Maxwell's equations naturally fall out of attempting to link quantum mechanics with special relativity. We were not trying to go through second quantization.

Does this ring a bell to anyone? Maybe I'm not explaining it exactly right, but I'm looking for a resource (link or book) where I can read about this in detail to get a better grip on it. He was going quite fast and my lecture notes aren't that good because I was trying to pay attention to understand.

I found the answer to my own question. I was looking to an intro to relativistic quantum mechanics.

Usually, when one says "relativistic quantum mechanics" one talks about a naive copy of the non-relativistic wave-mechanics picture a la Schrödinger ("first quantization"). This is, however, a not very good approach, because it simply doesn't work for relativistic quantum theory. The reason simply is that this formalism is for the case of a conserved number of particles, and in relativistic QT particles are always produced and destroyed again, if only the collision energy is high enough.

Historically there was Dirac's approach, now known as the "hole-theoretic formulation of QED", which uses the 1st-order quantization picture but artificially introduces a many-body concept by assuming the so-called "Dirac sea", i.e., one fills up the negative-energy states with electrons and declares this as the vacuum state of the theory, and then holes in this sea appear as positively charged particles of the same mass as electrons. In this way Dirac predicted anti-particles, and indeed the positive charged "holes in the Dirac sea" were indeed found. These are the positrons, i.e., the anti-particle of the electron. This approach leads to the same theory as the QFT approach, namely Quantum Electrodynamics (QED), but it's pretty cumbersome to learn about it, and thus nobody teaches this at the universities anymore.

So the right way to look at relativistic QT is to use relativistic QFT, i.e., "second quantization", which you left out in your study so far!

Also coherent states (of photons, e.g.) are many-body states, and you need QFT for them (although you can also look at the coherent states of the simple harmonic oscillator in non-relativistic QT as well, and it's a good model to learn about them, and it's pretty similar to the full story in QFT). The point is that these are superpositions of states of any photon number, leading to a state with an indefinite particle number. You can only define an average photon number. At large average photon number, a coherent state physically describes in a quantum-theoretical way classical electromagnetic fields like a laser field. The classical fields are a very good approximation for this case. For very low average photon numbers (even less than 1!) you get the utmost possible dimmed light in some sense, and then the coherent state describes something which still on the average behaves very much like a classical electromagnetic wave, but you have to wait a long time, until you have accumulated inough photon events (e.g., on a CCD screen or photo plate). This is, however NOT a single-photon state but still a superposition of all photon-number states. In this case it's mostly the vacuum state (no photons at all) and the single-photon state.

## 1. What is the relationship between a field and its quanta?

The relationship between a field and its quanta is that a field is a physical quantity that exists throughout space, while quanta are discrete packets of energy that make up the field. Quanta are the smallest possible units of a field, and they interact with each other to create the overall field.

## 2. How are fields and quanta related in quantum mechanics?

In quantum mechanics, fields and quanta are intimately related. Fields are described by quantum field theories, which describe how the field changes over time through interactions with quanta. Quanta are the building blocks of the field and are responsible for creating the observable properties of the field.

## 3. How do fields and quanta interact with matter?

Fields and quanta interact with matter through the fundamental forces of nature. For example, the electromagnetic field and its quanta, photons, interact with charged particles in matter, creating electrical and magnetic effects. Similarly, the strong and weak nuclear forces are mediated by the fields and quanta of gluons and W and Z bosons, respectively.

## 4. What is the significance of the relationship between fields and quanta?

The relationship between fields and quanta is significant because it helps us understand the fundamental nature of the universe. It allows us to explain the behavior of matter and energy at the smallest scales and provides a framework for understanding the interactions between particles. This relationship also forms the basis of modern technologies such as transistors, lasers, and MRI machines.

## 5. Can fields and quanta be observed directly?

Fields and quanta cannot be observed directly, but their effects can be measured and observed through experiments and observations. For example, the existence of the Higgs field was confirmed through the observation of the Higgs boson, which is a quantum excitation of the field. In general, fields and quanta can only be indirectly observed through their interactions with matter and other fields.

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