Why Does Energy Have Mass? Exploring Quantum Field Theory

[Hasan Delifer]

Hey,
first of all I apologize if this isn't the right thread because I was confused of where to put this.
Recently I've learned about Quantum Field Theory and the Higgs Field. I kind of understand why the Higgs Field gives particles mass. From what I have learned it's because the field is interacting with certain particles, like affecting the spin of Electrons, and therefore slowing it down so it can't travel at light speed, this gives the Electron it's mass.
I have also learned, that Protons and Neutrons on the other hand get most of their mass from energy and that the quarks mass only accounts for 1% of the Protons mass. What bothers me is that I don't understand why energy itself has mass. I know that it has something to do with General Relativity, but I just don't understand why energy would have mass.

I'm sorry if there are some mistakes in the question itself as I am no expert, but I'm looking forward for your answers. :)

 

[Orodruin]

By definition, mass in SR is just rest energy (divided by c^2). If you go to a reference frame where the momentum of your system is zero, then the energy in that frame is the system’s mass by definition. That this in the classical limit is the inertia of the system follows from the mass-energy equivalence. To derive it you need to look at what happens to the system when you act on it with a force.

It is not accurate to say that energy has mass. Energy and mass are both properties, not objects or substances.

I know that it has something to do with General Relativity

No it doesn’t.

 

[Hasan Delifer]

By definition, mass in SR is just rest energy (divided by c^2). If you go to a reference frame where the momentum of your system is zero, then the energy in that frame is the system’s mass by definition. That this in the classical limit is the inertia of the system follows from the mass-energy equivalence. To derive it you need to look at what happens to the system when you act on it with a force.

It is not accurate to say that energy has mass. Energy and mass are both properties, not objects or substances.No it doesn’t.

Hey, thanks for your answers.
How can most of the mass in a proton be energy then? Why does the energy have or provide mass in this case?

 

[Ibix]

In relativistic terms, it turns out that energy of an object squared minus its momentum squared is a constant. And if you take the non-relativistic limit you find that the constant is the square of its mass. That's all there is to it (well - it can be derived rigorously rather than just stated like that, and you need some factors of c to make the units work out).

My understanding of the standard model is fairly limited, and you would probably be better asking in the QM forum (although there are plenty of people who frequent both so you may get an answer here).

 

[dRic2]

I'm pretty dumb when it comes to Relativity, but I read this classic:
http://www.feynmanlectures.caltech.edu/I_15.html#Ch15-S8 (chapter 15, paragraphs 8 nd 9)
and I found it very enlightening (of course, you say... it is Feynman's! ). Not sure if it will help you, but it was of great help to me to figure out something I've always had trouble understanding.

 

[vanhees71]

But here Feynman commits a serious sin, introducing the relativistic mass. I'm very surprised, he did this, because usually he provides the utmost clear view on the fundamental concepts. In this case for some reason he failed. Already Einstein discouraged the use of a "relativistic mass". Today, in current research we only use the concept of invariant mass in relativistic physics with the good reason that everything gets simpler with the use of covariant quantities, i.e., using Minkowski-space vectors and tensors (four-vectors and four-tensors). Accordingly mass is the invariant mass
$$p_{\mu} p^{\mu}=m^2 c^2,$$
where for a point particle of invariant mass $m$ you have
$$p^{\mu}=(E/c,\vec{p}),$$
and then by definition
$$E=c \sqrt{m^2 c^2+\vec{p}^2}.$$
Note that in this choice of the energy the rest energy $E_0=m c^2$ is always included, and this is just convenient, because then $p^{\mu}$ behaves as the components of a four-vector under Lorentz transformations, while $m$ is a scalar quantity which doesn't change.

The relation to the non-covariant velocity follows like this. This quantity is defined as
$$\vec{v}=\frac{\mathrm{d} \vec{x}}{\mathrm{d} t}.$$
It's clear that this is NOT the spatial part of a four vector. It transforms quite complicated under Lorentz transformations. Much better is to use proper time, which is a scalar measure of time for massive particles, it's defined as
$$\mathrm{d} \tau= \frac{1}{c} \sqrt{\mathrm{d} x^{\mu} \mathrm{d} x_{\mu}}.$$
Using the coordinate time $t$ as parameter of the worldline of the particle you get the relation
$$\frac{\mathrm{d} \tau}{\mathrm{d} t}=\sqrt{1-\vec{\beta}^2}, \quad \vec{\beta}=\frac{\vec{v}}{c}.$$
Now momentum is defined as a four-vector
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
with $m$ being again the invariant mass.

It's easy to see that this definition leads by definition to the correct energy-momentum relation, i.e.,
$$p_{\mu} p^{\mu}=m^2 c^2.$$
It's also easy to get the momentum in terms of the non-covariant velocity $\vec{v}$:
$$p^{\mu} = m \frac{\mathrm{d} t}{\mathrm{d} \tau} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} t} = m \gamma \begin{pmatrix}c \\ \vec{v} \end{pmatrix}, \quad \gamma=\frac{1}{\sqrt{1-\vec{\beta}^2}}.$$
This leads to
$$p^0=\frac{E}{c}=m c \gamma, \quad \vec{p}=m \vec{v} \gamma.$$

 

[timmdeeg]

How can most of the mass in a proton be energy then? Why does the energy have or provide mass in this case?

I'm not really sure, there might be an analogy to Einstein's light box. The energy of photons in the box increase it's weight and thus it's mass. Or, another example, the kinetic energy of molecules in a box and their intramolecular vibrations contribute to it's mass. In a sense the proton can be considered a box.

 

[vanhees71]

I'd rather say the photons increase the inertia of the box (compared to the box without the photons or better said the electromagnetic field in it).

"Weight" is a different word for gravitational interaction, and according to Einstein's General Theory of Relativity the energy-momentum tensor of matter and radiation are the sources of the gravitational field, not only the part of it due to rest mass. That's only true in the non-relativistic approximation, where Newton's theory of gravitational interactions becomes a good approximation.

A lot of confusion is avoided in relativity if you distinguish strictly between mass (being a scalar) and energy (being the time component of the energy-momentum four-vector, if you consider a closed system and total energy and momentum).

 

[timmdeeg]

I'd rather say the photons increase the inertia of the box (compared to the box without the photons or better said the electromagnetic field in it).

My wording was influenced by the article: "The box would first be weighed exactly. Then, at a precise moment, the shutter would open, allowing a photon to escape. The box would then be re-weighed."
To avoid ambiguity it might be better to talk about invariant mass.

 

[vanhees71]

There is no other mass than invariant in today's physics. Einstein himself adviced against the use of any other idea about mass. In his 1905 article, although getting the special theory of relativity already right in principle, it was not fully understood. Fortunately there are tons of good textbooks explaining it in a way without all the troubles Einstein et al had to go through until getting everything really consistent. The most important development was Minkowski's mathematical formulation of 1907/08 introducting four-vectors and all that. If you want to go further and learn general relativity, it's the more important to learn special relativity first in the modern way (as far as a more than 110 years old concept is "modern" ;-)), i.e., in terms of Minkowski's 4D spacetime formulation, using covariant notation for all quantities.

 

[Ian J Miller]

My view is that "why" questions are the most dangerous. You can only deduce from something else and sooner or later, as you go up the tree, you come to "the start", which usually involves an induction from observation. As I understand it, Einstein started with the principles that physics should be the same for everyone, light had a constant speed, and the equivalence principle. None of that really involves mass independently. The equations he got from that are well-known, but he had a clear bias towards geometry. There is nothing wrong with that, but it therefore follows that his explanations for what is going on favour contraction of space and time dilation. I think it is wrong to criticize Feynman for bringing out relativistic mass, but ultimately it is a lot easier to use the equations based on the geometric interpretation. The alternative is to go through the whole theoretical structure working on mass, and we have no real idea whether we can do that, and more importantly, why is it worth doing? It really is much better to stick with what we have because it works, but we do not really know it could not be formulated differently.