# I Origin of mass and Noethers theorem

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1. Aug 24, 2016

### AlanKirby

Let me set up the question briefly. Emmy Noether's theorem relates symmetry to conserved quantities, e.g. invariance under translations in time => conservation of energy. A fundamental truth revealed.
Massive guage bosons, leptons and quarks all appear to aquire mass through the spontaneous symmetry breaking of the guage invariance and hadrons aquire there mass primarily through the spontaneous symmetry breaking of the chiral invariance.
Through relativity we see that energy and mass are related by a conversion factor of c (speed of light).

And so, firstly, is it correct to say that energy is a fundamental conserved quantity and that mass is simply an alternate form that it takes as a consequence of the spontaneous symmetry breaking.
Secondly, would it be reasonable to assume that other conserved quantities such as linear momentum (from invariance under translations in space) take on an alternate form as a consequence of spontaneous symmetry breaking, or perhaps due to something else. If not, what makes energy special?

Thanks.

2. Aug 25, 2016

### haushofer

Yes. Energy in general is conserved, not mass. This is a relativistic result, however; basically, the reason is that one of the Casimirs of the Poincaré group is given by $P_{a}P^a$. Non-relativistically, mass is conserved, but there algebraically it is not given by a Casimir (which is per definition quadratic in the generators) but by a central extension of the Galilei group called "Bargmann group".

However, most of the mass of matter is not given by the Higgs-mass but by the kinetic energy of the quarks moving inside hadrons!

No, because the Higgs-mechanism does not break translation invariance, like e.g. mediums or gravitational fields do.

3. Aug 25, 2016

### ChrisVer

not only motion... but the whole dynamics inside hadrons.

yes, it is something we currently trust to a high degree.

mass is a form of energy; it is also affected by the interactions and so on... Higgs provides the bare mass of particles, and that is not their physical mass (which is obtained after you take some loop interactions into account).

In special relativity nothing is special about energy or momentum. In special relativity, those two quantities get "unified" into a single object (the 4-momentum)...
Does the conservation of energy take an alternative form? I didn't get where you mentioned that in all your previous phrases.

4. Aug 25, 2016

### AlanKirby

I didn't mean that the conservation of energy takes an alternate form, but rather that energy and momentum are both conserved quantities, energy can take an alternate form (mass), so can momentum then also take an alternate form? I.e. Is energy a special conserved quantity in this sense, such that other conserved quantities don't also take alternate forms. Thanks.

5. Aug 25, 2016

### Staff: Mentor

What does "alternate form" mean? Mass is one type of energy. Kinetic energy is another one, and potential energy is also one. There is nothing special about mass.

Momentum doesn't have those groups.

6. Aug 25, 2016

### ChrisVer

I guess correcting this phrase of yours will "solve" your confusion?
$E=mc^2$ is not the complete equation (it applies only for stationary objects).
The complete equation is $E^2 - p^2c^2 = m^2 c^4$, where $p$ is the momentum's magnitude.

7. Aug 25, 2016

### AlanKirby

Thank you for the insight, this is helping me to progress. I think I need to properly take on QFT and group theory before I can expect to really make sense of those things that i'm thinking about.

Can I just ask though, what would you consider both energy and mass (seperately) to be? Thanks.

8. Aug 25, 2016

### Staff: Mentor

There is no need to include quantum mechanics at all, you can understand this using classical physics only.
Important concepts in physics?

9. Aug 25, 2016

### AlanKirby

I thought I'd get pulled for saying that. I was just referring to the relation between energy and mass, or rather that they are in a sense the same. I see from the above posts that I'm not quite thinking about that peoperly though. What I'm really trying to understand is what is energy (maybe that's a dead end question), and what is mass (eluded to above as a form of energy in the same sense as kinetic energy). Thanks.

10. Aug 25, 2016

### AlanKirby

Wouldn't I need QFT to understand the origin of the majority of the mass within hadrons?

I understanding that asking "why" repeatedly within physics eventually leads to the answer of "because that's how/what the universe is", and I understand that we learn of energy and mass early on. But I can't stop myself asking what energy actually is, and maybe that isn't a good question. Similarly i'm realising that I actually don't understand mass all that much either. I understand that it sounds like i'm asking something daft or that i've never seen the inside of a lecture theatre but this confusion usually comes out of nowhere and results with an improved understanding. Thanks.

11. Aug 25, 2016

### Staff: Mentor

To understand how much: yes. To understand that there are other contributions: no.

12. Aug 25, 2016

### ohwilleke

One important caveat to this notion. While there are both easy Hamiltonian and Lagrangian equation ways to define conservation of mass-energy in a system, one has to be careful in how you apply the principle to use consistent definitions in a particular application of the concept, because defining energy itself in absolute terms is a much more slippery concept than conservation of mass-energy.

Frequently systems trade off between kinetic energy and potential energy, for example. Yet, the absolute value of the potential energy in a system is pretty much arbitrary; potential energy relative to what? Similarly, the conventional classical definition of kinetic energy (1/2mv^2) (and likewise the usual formula for linear momentum, another conserved quantity) is also arbitrary because velocity is defined relative to a reference frame and there is no preferred reference frame for an object's velocity (even in Newtonian mechanics). Throw in special relativity and definitions of energy in gauge fields and it gets even trickier.

Part of what makes general relativity distinctive mathematically is its ability to give answers that are consistent in all reference frames related to motion and in all consistent definitions of potential energy.

I also personally prefer the term "mass-energy" to "energy" as the fundamental conserved quantity, rather than presuming that one form of mass-energy or another is inherently more fundamental. Otherwise you create a linguistic ambiguity between "energy" referred to in the sense of things that aren't mass, and "energy" referred to in the sense of the conserved quantity "mass-energy".

Last edited: Aug 25, 2016
13. Aug 25, 2016

### ohwilleke

It is true that only about 1% of the mass of a proton or neutron comes from the rest Higgs-mass of the quarks (its really a bit more complicated than that because if you do the QCD estimates of hadron mass assuming that all quarks are massless, the reduction in mass is, if I recall correctly, more like 8-9% due to the way that quark mass impacts the gauge fields in the hadron in a way that impacts the total mass of the system more than the rest mass of the quarks themselves).

This is less true of heavier hadrons. As a rule of thumb, the heavier the quarks, the greater the percentage of the mass of the hadron is Higgs mass. Heavier quarks give rise to a strong force field that has more energy than the strong force field between lighter quarks, but the energy of the strong force field rises much more slowly than the mass of the constituent quarks. For example, the vast majority in the electrically neutral B meson (total mass 5.280 GeV), a bit less than 4.2 GeV of the mass (almost 80%) comes from the Higgs-mass of the quarks, a bit more than 1 GeV of mass comes from the gluon field, and something less than 0.1 GeV comes from the kinetic energy of the quarks, electromagnetic fields, etc.

More importantly, the lion's share of the remaining mass of a hadron comes not from the kinetic energy of the quarks, but instead from the strong force gauge field that binds the quarks together (with the electromagnetic field between the quarks, virtual weak force loops, and kinetic energy of the quarks making orders of magnitude smaller contributions that are comparable to each other). In other words, most of the mass of a proton or neutron and most of the non-quark mass in other hadrons comes from the energy of the gluons that bind them.

Also, FWIW in a proton or neutron, most of the energy of the gluon fields is concentrated in the middle 1/3rd of the volume of the proton or neutron (the definition of which is a wee bit tricky since 99.99....% of the volume of a hadron is empty space and the constituent point particles are in constant motion) which makes up just 1/9th of the total volume of the proton or neutron.

14. Aug 26, 2016

### haushofer

Yes, I said "kinetic" energy while that should be "binding energy". Just curious: to which extent is it possible now to calculate the proton mass from the quark masses and QCD? Do you have any nice reference where I can find more about the examples you give?

This is actually one of those things I see popping up every now and then in popular context but have never looked upon more carefully.

15. Aug 26, 2016

### vanhees71

16. Aug 28, 2016

### haushofer

Thanks (again)!

17. Aug 30, 2016

### ohwilleke

The most recent first principles calculation of the proton mass from QCD had a precision of about 1%. I'll see if I can find a reference. Another approach to calculate a theoretically expected mass is to use the Coleman-Glashow relation which argues generally that certain combinations of hadron masses should be equivalent to each other based on symmetry considerations that follow from QCD. So, if you measure the masses of the other hadrons that should be equivalent, you can back out a phenomenologically predicted mass for the hadron whose mass you care about.

One of the biggest sources of uncertainty in that calculation is uncertainty in the QCD coupling constant (roughly 0.5% in the world average measurement). The uncertainty from omitting higher order loops from the calculation (because the infinite series approximation terms in QCD converge much more slowly than in QED) is another big one, and is more fundamental - one of the reasons that the QCD coupling constant and quark masses are not known with great precision is that the theoretical error due to omitted loops is so significant in QCD.

In contrast, the precision with which the proton mass has been measured experimentally has about eight significant digits (a million times more precise than the theoretical calculation), so theory lags far behind experiment here.

The disparity between theory and experiment, however, is far less extreme for heavy hadrons containing higher generation quarks, with experimental measurements often only precise to three or four significant digits, and theoretical calculations approaching three significant digit precision.

Last edited: Aug 30, 2016
18. Aug 31, 2016

### haushofer

Thanks. I recently read "Exploring fundamental particles" by Wolfenstein and Silva to boost my knowledge of phenomenology, and there it states that "it is not actually possible to calculate the mass of the proton in QCD" (ch6.5), but 1% off seems to me pretty accurate. The book was released in 2011, so I guess a lot can happen in a few years ;) If you can find that reference, I'd be happy to see it. I'll do a search on my own too.

19. Aug 31, 2016

### Staff: Mentor

20. Aug 31, 2016

### ohwilleke

The arVix post-print of the study mfb refers to is at http://arxiv.org/abs/0906.3599 The link to the published result is http://science.sciencemag.org/content/322/5905/1224.full The lead author explains the result with conference slides at http://www.durr.itp.unibe.ch/talk_09_psi.pdf The conference slides also mention that if quarks were massless that the proton mass would be about 890 MeV rather than about 940 MeV, a 5% difference despite the fact that the quark masses themselves are only about 1% of the total mass of the proton and neutron.

Electromagnetic corrections are discussed by the same author are discussed at http://arxiv.org/abs/1011.4189 and a first principles calculation of the proton-neutron mass difference by the same author is at http://arxiv.org/abs/1406.4088 An effort to back out the light quark masses from the known hadron masses by the same author is http://arxiv.org/abs/1011.2403

A survey of efforts to directly calculate the proton mass from QCD can be found at http://arxiv.org/abs/0906.0126 Some related points are discussed at http://arxiv.org/abs/0810.4234

A study predicting the masses of the Delta and Omega baryons with the proton and neutron masses as inputs is here: http://arxiv.org/pdf/1109.0199.pdf with experiment measuring a mass about one order of magnitude more precise than theory.

A nice overview of mass generation in QCD is found at http://arxiv.org/abs/1501.06581

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