# Quantum Field Theory: What is a Quantum Field?

Quantum field theory brings together quantum mechanics and relativity. It is the mathematical language in which the standard model of particle physics is written.

What is a quantum field? Most people will be familiar with the concept of classical fields, for example electromagnetic fields. Fields are used to explain how a force can be transmitted over a distance. For instance, how does the magnet in the picture 'push' the iron filings into that pattern without touching them? The answer is that a field exists around the magnet, which exerts a force on the iron filings causing them to line up along the field lines.

In particle physics, we use fields to describe particles. In quantum physics, particles are not thought of as point-like objects. The uncertainty principle of quantum mechanics tells us that we cannot know the exact location of a particle. The location of the particle is described using a probability field, which is strongest at the points where the particle is most likely to be found and weakest where we are unlikely to see the particle.

## Path Integrals

This quantum uncertainty means that we cannot be sure what a particle is doing when we are not looking at it. If we observe a particle at point A, and some time later observe it at point B, we do not know what path it has taken to get from A to B. Quantum mechanics tells us that in order to calculate the probability of observing the particle at B after having seen it at A, we must add up the probabilities of all the paths it could possibly have taken from A to B. Feynman showed that this summing up of paths can be approximated by an integration over all the paths known as a path integral.

This expression means: "add up the “probability” exp(S[q]) of each of the paths q1, q2, etc... to get the probability of finding the particle at B". The classical path is the most likely so it has the largest contribution to the integral. When we consider everyday objects, which are much larger than elementary particles, the contribution from the classical path is overwhelmingly larger than from any of the other paths, which is why we do not see everyday objects behaving in a quantum way.

These integrals are rather complicated and difficult to evaluate. Fortunately, Feynman invented a pictorial way of representing them, known as Feynman diagrams.

Quantum field theory is a powerful mathematical tool which underpins the whole of modern particle physics theory. It is also used in statistical physics, cosmology, and theories of quantum gravity, which aim to find a “theory of everything” that is free from the inconsistencies that currently exist between general relativity and quantum physics. Resources for learning more are listed below.

## Lecture Notes

- David Tong: Quantum Field Theory

A Cambridge University course with lecture notes, covering the canonical quantization of scalar fields, Dirac fields and QED.

## Books

Peskin and Schroeder is the ultimate quantum field theory textbook. I found this book invaluable during my degree.

User-friendly book which keeps the math to a minimum and explains the physics clearly.

A useful book which explains the theory well and puts it in context.

## Other physics hubs by topquark

- Feynman Diagrams: An Introduction

A Feynman diagram is a representation of an interaction between elementary particles. Feynmann diagrams are incredibly useful in calculating the probability of a reaction occurring. - Where is all the antimatter? Mystery of matter-antimatter asymmetry

This article is an introduction to the unsolved problem of matter-antimatter asymmetry in particle physics and cosmology. Why is the amount of matter in the universe so much greater than the amount of antimatter? - The Evidence for Dark Matter

Dark matter is the name given to the matter that holds together the Milky Way and other galaxies. It cannot be seen using telescopes as it does not emit light or other radiation. However, there is significant evidence for its existence.

*topquark works as a researcher in theoretical particle physics and blogs about research at The Particle Pen.*

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## Comments 6 comments

Another interesting hub. What are your ideas on the double slit experiment with a sensor?

Personally I can't see how we can consider the sensor as an observer. To say that the observer collapses the wave function in this experiment seems like nonsense to me considering no human observer observed anything.

Yes. I am sure it was done. ;) but I do not know the details of how. Was the sensor completely passive? Can we rule out any electromagnetic interference from it? What kind of sensor was it exactly? Have we done the experiment with different types?

You say the observer observed the results but did we observe it after the fact or as it was happening?

I also know Feynman said there was no wave particle duality and that in the double split experiment the particle takes all possible paths creating interference with itself which emerges as a wave pattern. But I can not find his ideas on the sensor experiment or wave function collapse, which I know many scientists have doubt actually exists.

Can you enlighten me on any of these questions? I think it would make a great hub, actually.

You have written another very easy to follow hub that explains the concept in a way that anyone can understand. I also think that the resources you provide for further reading are excellent.

Have you considered writing a series for A-Level students as I feel your treatment of the concepts would help so many of them develop a much better understanding and appreciation for the material they will be tested on. Just an idea.

Interesting hub, you made it very easy to understand. However, I would like to ask how practical is it to locate a particle in quantum physic without a perimeter or a field as you mentioned in particle physics?

I would have thought that since there are infinite possible path in which a particle could travel, than to locate which position using the probability field to find a particle is like trying to tap into an infinity amount of path which of-course seems impossible. For that matter the behavior of a particle using the probability field should appear as if this particle as disappeared from field " A" to field "B".

The next question I'm thinking about is that could it be that there are other similar particles that are invisible within the same field, and that the particle we think is the same is actually a different particle with similar characteristic. If this was the case we could not have predicted how a particle moves within a field.

In that sense, we would have had to find out why is the particle visible in this field while it is not in another field. If there is a force acting on the field which causes the particle to suddenly appear, than we would have had to find out what's causing this force. I'm not a physicist but I thought this was a reasonable question to ask. Thanks for making it simple to understand.

I would also like to ask how does the particle behave under different environment, such as temperature. If differences in temperature change the behavior of the particle, than temperature is a force acting on the particle, then what other forces could there be there?

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