Req: A conceptual QFT example

• pellman
In summary, the conversation discusses the need for a simple example in quantum field theory (QFT) that illustrates what can be calculated, how it is calculated, and its physical significance. The potential well, step-potential barrier, harmonic oscillator, and two-slit experiment are mentioned as examples in quantum mechanics (QM). Free field theory and the simplest interacting theory, \phi^4-theory, are suggested as examples in QFT. The electron-electron scattering is also mentioned as a successful example of a QFT that can calculate measurable quantities. The Standard Model of particle physics is also mentioned as a successful example of QFT. It is noted that the restriction of particle creation and destruction in QM is removed in QFT, and
pellman
I need someone to point me towards an instance of a relatively simple QFT problem which illustrates what can be calculated from QFT, how it is calculated, and its physical significance. (Note the qualification "relative". I realize there are likely no simple problems in QFT.)

If someone were asking the same of me with regard to QM, I would take them to the standard examples in the texts: The potential well, the step-potential barrier, the harmonic oscillator and the ideal hydrogen model. If I were to single one out it would probably be the step-potential barrier, since it illustrates how a boundary value problem is solved and clearly gives two or more distinct relative probability regions while also exhibiting a non-classical behavior: "tunnelling".

I would also bring up the two-slit experiment since one can discuss all the quantum behaviors and easily visualize the significance of both amplitude and phase--without actually solving the math.

What QFT problem(s) can do this for beginner?

Edit: It should clarify the physical significance of a quantum field and (if possible) show how it is related to the physical entities we call particles.

First of all, obviously, free field theory (which can be "derived" from a classical spring system), and the simplest interacting theory, $\phi^4$-theory; this is for example how Zee's "Introduction to QFT" starts.

The simplest "physical" example I have seen is that of electron-electron scattering, which is ugly algebra but it is one of the things that in "just" a few pages actually calculates a quantity which experiments can measure. Peskin & Schroeder's "An introduction to QFT" starts with this in chapter 1 (although the actual calculation is only in chapter 5, or so).

Perhaps the most successful example of a QFT that we know is the Standard Model of particle physics, which has in (almost) every aspect turned out one of the greatest successes in theoretical physics of the last 50 years or so, and which we now actually trust enough to build a ~ 7 billion dollar experiment (the LHC) just to look for a particle which has so far only been predicted

QM is about systems in which the number of particles (of each kind) is fixed, so it cannot describe particle creation and destruction. This restriction of QM is removed by a more general theory, QFT.
Example:
electron-positron annihilation into a pair of photons

Actually Chapter 1 of Zee's book is freely available in PDF from the publishers website: http://press.princeton.edu/chapters/s7573.pdf

It very clearly introduces the need for QFT and makes the step from QM, I think you will find it useful.

If you are asking the question "why do we need this?", that is, what physical observables require more than just QM, I would suggest a) the Lamb shift and b) the electron's anomalous magnetic moment.

1. What is a conceptual example of QFT (Quantum Field Theory)?

A common example used to illustrate QFT is the quantum harmonic oscillator. This is a model in which a particle is confined to a potential well and can only move in a harmonic motion. It is often used to demonstrate the principles of quantization and the creation and annihilation of particles.

2. How does QFT differ from traditional quantum mechanics?

QFT is an extension of traditional quantum mechanics that takes into account the concept of fields. In traditional quantum mechanics, particles are described as discrete objects, while in QFT, they are treated as excitations of a quantum field. This allows for a more comprehensive understanding of particle interactions and the behavior of matter on a fundamental level.

3. What are some practical applications of QFT?

QFT has numerous applications in modern physics, such as in particle physics, condensed matter physics, and quantum computing. It is also used in various technologies, including lasers, transistors, and superconductors.

4. How does QFT relate to the theory of relativity?

QFT and the theory of relativity are both fundamental theories that describe the behavior of matter on a microscopic and macroscopic level, respectively. In QFT, the principles of special relativity are incorporated into the equations to account for the behavior of particles at high speeds.

5. What are some current challenges and open questions in QFT?

Despite its success in explaining many phenomena, there are still several unanswered questions and challenges in QFT. These include the unification of quantum mechanics and general relativity, the nature of dark matter and dark energy, and the understanding of the hierarchy problem in particle physics.

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