# If particles are excitations what are their fields?

It was clear to me that all particles are merely excitation in fields. But it caused a few questions

1. If electron is excitation in electron field, then how does it produce it's own field?
2. For whatever reason electron even after being an excitation produces an electric field, but why doesn't photon (or other such particles if there are) create any such field?
3. All these particles have mass and exert gravity how can they produce gravitational fields?

The major question that partly summarises all three of them has to be " when particles are excitation in some field how do fields arise from them? "

• Why couldn't fields interact? – user259412 Jun 12 '14 at 8:26
• The same thing can be asked about light waves. You probably won’t get a final answer. Usually we get answers like photon’s are excitations of a field, and a field is a wave function and a wave can interact in a field Or a wave function etc. etc. etc. There’s no definite description of anything it just goes around and around. If waves or fields are not made of photons radiating from a common source then how else can they be described? Give one answer. – Bill Alsept Jan 29 '18 at 7:51

An electron can be seen as a localized excitation of the electron field that exists prior to electrons and permeates all of spacetime!! The electron field has nothing to do with the electric field the electron creates!

Now this electron field $\psi$ is coupled to the electromagnetic four-potential field $A_\mu %will this make the edit go through?$ via an interaction $$\mathcal L_\mathrm{int} = -ie \bar \psi \gamma^\mu \psi A_\mu.$$ Thus, all excitations in the electron field create appropirate excitations in the four-potential field as well. This is the electric field in case of an electron at rest.

Photons can also react back on the electron field, this is called pair production, but it is rare, since a single photon cannot induce this process due to conservation laws.

A photon does not couple to the photon field itself, since it is the gauge particle of an abelian gauge group. Gluons on the other hand are from a non-abelian gauge interaction, carry color charges and therefore backreact on the gluon field itself, through interactions like $$\mathcal L_\mathrm{int}^{3g} = if^{abc} G^\mu_a G^\nu_b \big(\partial_\mu G_{\nu c} - \partial_\nu G_{\mu c}\big)$$

Let's start looking at your statements:

It was clear to me that all particles are merely excitation in fields.

This is not exactly true, even though often said. In the model of Quantum Field Theory, we describe an electron as an excitation of the electron field. It's important to stress, that this is just a description, i.e. it's hard to say whether it really "is" like that (whatever that may mean).

If electron is excitation in electron field, then how does it produce it's own field?

It doesn't. In QFT, the an electron field is postulated to exists everywhere (in our flat space-time). This is part of the theory base and as such, the theory can't explain it.

For whatever reason electron even after being an excitation produces an electric field, but why doesn't photon (or other such particles if there are) create any such field?

This statement is not completely true. In QFT, there is no electr(omagnet)ic field. In general physics, an electric field is the mediator of the electromagnetic interaction. In QFT that role is taken by the photon, i.e. an excitation of the photon field. These two fields (photon and electromagnetic) are descriptions in different theories but basically of the same effects. (Not precisely the same, but this is a bit harder to explain.)

This also answers your question as to why the photon (in qft) doesn't create such a field: because they live in different theories.

The fact that the photon doesn't couple to itself is basically an experimental fact. By now, we have a few nice descriptions and theoretical explanations of that, e.g. it's the gauge boson of an abelian gauge group or it simply doesn't have an interaction term in the quantum electrodynamics lagrangian. (The first implies the second reason, but they were found the other way round afaik.)

All these particles have mass and exert gravity how can they produce gravitational fields?

As far as I know, there is no consistent and tested theory that describes gravity and qft together. (more precise: ... results in them in a certain limit/under certain reasonable assumptions.) As Neuneck mentions in his comments, there are reasonable theories about qft in curved background. In qft, mass is just a parameter of the theory nothing more. It's just our interpretation from other parts of physics, that gives it the meaning the mass we know.

Gravity is in this sense closer to normal electrodynamics. The gravitational field is our space-time and is bend through gravity-charges=masses. It's like putting an electron in empty space: First the electrmagnetic field is flat and 0 everywhere (casually said it's not there, but this is imprecise and can create confusions) and afterwards, yxou have a sink in the field and it's bent (casually said, there is a field).

when particles are excitation in some field how do fields arise from them?

With the answer above as background: They don't (in qft). It depends on the theory, what you mean with the word field. It's mostly a confusion because one doesn't always precisely define what one means with a certain word (like electron or field) in physics.

• I disagree with your statement that there is no consistent framework to describe gravity and QFT together. There are lots of meaningful ways to do finite temeperature QFT in curved backgrounds, taking into account classical gravity. The fact that we can't describe gravity as a QFT itself does not limit it's applicability as a classical part of a more general theory framework. – Neuneck Jun 13 '14 at 20:37
• @Neuneck: Thanks, I adjusted my answer a bit in that paragraph. My point was that none of these theories is (succesfully) tested (yet). Most extensions of the standard model and gravity, may it be QFT in curved background, String Theory, Loop Quantum Gravity or Causal Dynamical Triangulations have their meaningful ways of doing physics, but I tried to answer in the least biased but still correct way. I hope it's fine now. – Tim Jun 13 '14 at 21:02

For the case of an electron, it is described by a Dirac spinor field $\psi$ with Lagrangian,

$$\mathcal{L}= \bar{\psi}(i\gamma^\mu\partial_\mu -m)\psi$$

We say the electron arises as an excitation of the quantum field $\psi$. On a technical level, if we expand the field using Fourier analysis, roughly like,

$$\psi \sim \int \frac{d^4p}{(2\pi)^4} \, \left( b_pe^{ipx}+c^\dagger_p e^{-ipx}\right)$$

neglecting many factors. The Fourier coefficients $(b,b^\dagger)$ and $(c,c^\dagger)$ act on the vacuum state of the theory to either create or destroy positrons or electrons; these states live in a Fock space.

In the model of quantum electrodynamics, the field $\psi$ couples to an electromagnetic field $A_\mu$ described by the Maxwell Lagrangian; the interaction term is,

$$\mathcal{L}_{\mathrm{int}} = e \bar{\psi}\gamma^\mu A_\mu \psi$$

The field $\psi$ transforms under a representation of $U(1)$, and we associate a $U(1)$ charge to both the electron $\psi$ and positron $\bar{\psi}$. The electron in isolation, without the existence of an $A_\mu$, would not give rise to an electric field.

An acceptable quantum field theory of gravity is not currently available, and as such precisely how quantum fields interact with a gravitational field is somewhat unknown.