64 votes

Deriving Lagrangian density for electromagnetic field

Abstract In the following we'll prove that a compatible Lagrangian density for the electromagnetic field in presence of charges and currents is \begin{equation} \mathcal{L}_{em}\:=\:\epsilon_{0}\cdot\...
Frobenius's user avatar
  • 15.5k
45 votes

How do we know that gravity is spacetime and not a field on spacetime?

Practically speaking, what's the difference? There exists a rank-two tensor field on spacetime called the "metric" $g_{\mu \nu}$ which couples to all mass-energy, and things that we intuitively call "...
tparker's user avatar
  • 47.2k
41 votes

Is there just one EM field for the whole universe?

You can define a "wind field" for the Earth by putting a weather vane at every point. You've probably seen drawings of these wind fields in weather reports; you can even define 'wind field lines' in ...
knzhou's user avatar
  • 102k
38 votes
Accepted

Why we don't have macroscopic fields of Higgs bosons or gluons?

There are slightly different answers for each particle type. Macroscopic photon and graviton fields can exist because these forces are long-ranged, which is directly related to the force carriers ...
knzhou's user avatar
  • 102k
36 votes
Accepted

If Energy can be converted into mass, why can it not be converted into charge?

You're making some category errors in the question. Energy can't be converted into mass, mass is a form that energy can take. In other words, when energy is "converted" into mass it never stops being ...
Sean E. Lake's user avatar
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36 votes
Accepted

Can the center of charge and center of mass of an electron differ in quantum mechanics?

Can the center of charge and center of mass of an electron differ in quantum mechanics? They can. Particle physics does allow for electrons (and other point particles) to have their centers of mass ...
Emilio Pisanty's user avatar
31 votes

What are quantum fields mathematically?

The replies which suggest that the answer to "What is a quantum field?" is unclear or even open are wrong. The impression that this could be unclear is owed to the standard textbooks sticking to the ...
Urs Schreiber's user avatar
31 votes
Accepted

Does the gravitational field have a gravitational field?

Yes. One interpretation of the fact that Einstein's equations of general relativity are non-linear is that "gravity gravitates." In other words, the gravitational field itself carries energy,...
Andrew's user avatar
  • 47.6k
29 votes

Differentiating Propagator, Green's function, Correlation function, etc

It has been many years since you asked this question. I assume that over time you have compiled meaning definitions and distinctions for the other terms in your list. However, there are terms not ...
ThomasTuna's user avatar
29 votes
Accepted

What, to a physicist, are instantons and the Donaldson invariants?

1. Instantons 1.1 Instantons as a classical solution An instanton, is pretty much exactly what you say: An (anti-)self-dual configuration of the curvature of a principal bundle. The curvature $F_A$ ...
ACuriousMind's user avatar
  • 124k
28 votes
Accepted

Physical difference between gauge symmetries and global symmetries

The first answer to such a question must always be: A gauge symmetry has no "physical" meaning, it is an artifact of our choice for the coordinates/fields with which we describe the system (cf. Gauge ...
ACuriousMind's user avatar
  • 124k
28 votes
Accepted

Covariance in gauge theories: why should the Lagrangian be gauge invariant?

We do not start from the assumption that the Lagrangian "should" be invariant under gauge transformations. This assumption is often made because global symmetries are seen as more natural than local ...
ACuriousMind's user avatar
  • 124k
28 votes
Accepted

Why is the kinetic energy of a fluid given as an integral?

The kinetic energy of a fluid is the same as normal mechanics, $T=mv^2/2$. However, that's not generally useful as we don't usually have masses but densities, so we instead consider the kinetic energy ...
Kyle Kanos's user avatar
  • 28.1k
27 votes

What are quantum fields mathematically?

The definition of a quantum field depends slightly on the formalism that you adopt, but globally, quantum fields are defined as operator-valued distributions. That is, if you have a quantum field $\...
Slereah's user avatar
  • 16.1k
25 votes
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What are quantum fields mathematically?

There is no mathematically sound formulation of realistic QFT yet so at this point we have no real answer to your question. The QFT that physicists use to make predictions is in the so-called ...
Conifold's user avatar
  • 5,273
24 votes
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Why is Noether's theorem not guaranteed by calculus?

Since the problem here appear to be coordinates, let's just stop using coordinates, and for simplicity consider the theory of a single scalar field on space(time) $M$: Our field is a function $\phi : ...
ACuriousMind's user avatar
  • 124k
22 votes

Zero dimensional field theory

Zero-dimensional quantum field theory is exactly like a standard quantum field theory, except that the background spacetime is exactly one point. Consider, for a moment, a $d$-dimensional QFT defined ...
Bob Knighton's user avatar
  • 8,450
21 votes

Can particles be in a superposition of times as well as positions?

I don't think this question has an exciting answer. Strictly speaking, superposition refers to a wavefunction occupying multiple states at a particular time, so the question you've asked is "why ...
g s's user avatar
  • 13.1k
20 votes
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Invariance under boosts but not rotations?

Every quantum field theory has a symmetry group under which its Lagrangian is invariant. Like every group, it must be closed. The boosts are not closed under composition, so they cannot form a ...
tparker's user avatar
  • 47.2k
20 votes

Difference between field and wavefunction

Ignore spin, polarization, and even Lorentz issues, absorb all superfluous constants, and consider time and one space dimension. A one-particle complex wavefunction $\psi(x,t)=\langle x|\psi \...
Cosmas Zachos's user avatar
20 votes
Accepted

What are Quantum Field Theories?

A field theory is a mathematical model where the "basic ingredients" are fields. Maxwell's theory of electromagnetic fields and continuum mechanics are prominent examples of classical field ...
Tobias Fünke's user avatar
19 votes
Accepted

About the notion of the self-interaction of a field

I'm afraid you're searching for a picture that isn't there, because you think of both fields and charges still classically. In quantum field theory, there are no fields in which localized particles ...
ACuriousMind's user avatar
  • 124k
19 votes

Is energy localised in space?

Energy is a property of a system. It depends on how the system's fields and particles are arranged and on what they're doing. At least in classical physics, energy can be localized in the sense that ...
Chiral Anomaly's user avatar
19 votes
Accepted

Does a non-lagrangian field theory have a stress-energy tensor?

Most theories do not have a conserved energy-momentum tensor, regardless of whether they are Lagrangian or not. For example you need locality and Lorentz invariance. When you have those, you can ...
AccidentalFourierTransform's user avatar
18 votes
Accepted

When is stress-energy tensor defined as variation of action with respect to metric conserved?

Actually, the metric variational definition for the stress-energy tensor (due to Hilbert, as remarked by Qmechanic) is an universal improvement procedure for the canonical stress-energy tensor (and ...
Pedro Lauridsen Ribeiro's user avatar
18 votes
Accepted

Noether's Theorem and scale invariance

First of all, let's see what Noether's Theorem says about your specific case (Klein-Gordon under global rescaling of the fields). Noether's theorem states that To every differentiable symmetry of ...
Giorgio Comitini's user avatar
18 votes
Accepted

Examples of "gauging a global symmetry"

Here is a simple example, one of the first you should try to understand. The theory has a free $U(1)$ scalar field $\phi$ in $d+1$ spacetime dimensions, discussed in the modern notation of ...
Ryan Thorngren's user avatar
17 votes
Accepted

How do fields co-exist physically?

The concept of fields is the concept of assigning numbers to every instance of time and to every point in space. Think of temperature. You can assign a temperature value to every point in space ...
image357's user avatar
  • 3,094
17 votes

Why don't fundamental particles propagate outwards like classical waves?

As stated by several people including @Ruslan, @Marco Ocram, and @John Doty in the comments, this question is related to the following classic paper by Mott: https://royalsocietypublishing.org/doi/10....
Andrew's user avatar
  • 47.6k
17 votes

How to interpret quantum fields?

There is one usual confusion about quantum fields which is, at least in my perspective, perhaps caused by one very familiar example which we all have met before studying QFT. This example is that of ...
Gold's user avatar
  • 35.8k

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