Explicit symmetry breaking terms are classified as "hard" and "soft". If the the explicit breaking is induced by a term whose dimension is less than four it is called "soft" breaking. Does soft breaking has anything to do with low breaking scale? I tried to follow  but couldn't follow the "low energy" interpretation. Moreover, I think the answer is particularly addressed for soft SUSY breaking.  In short, why "soft"?
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For example it is stated that if supersymmetry breaking is soft then stability of gauge hierarchy can be still maintained.
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I was told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation}  {\cal L} = \frac{1}{2} \partial _\mu \phi _i \partial ^\mu \phi _i - \frac{m ^2 }{2} \phi _i \phi _i  - \frac{ \lambda }{ 4} ( \phi _i \phi _i ) ^2  \end{equation}  with $m ^2 =-\mu^2 &gt; 0$, adding a linear term in one of the fields $\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with).   Are there any other effects of linear terms that we should keep in mind or is this the only exception of the "forget about linear terms" rule?
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The Higgs boson and gluons have no electric charge and photons couple to charge, so there is no tree level interaction between them and photons.  But what prevents higher order diagrams from contributing a non-zero mass term to the photon, for instance    where a photon couples to some fermion (say an electron, or a top quark) which can interact with the Higgs field.   Or consider that same diagram but with quarks and a gluon interacting between them? Or any higher diagram with even more loops?  I have heard charge conservation depends on gauge invariance, which in turn depends on photons being massless.  So it appears the photon has no mass, and these diagrams must all cancel somehow.  So I'm hoping there is a very nice symmetry explanation for why they all disappear, but if I say "because of gauge invariance" that would be circular logic, so there must be another symmetry at stake here?     What prevents photons from obtaining a mass from high-order self-energy loop diagrams?
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Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might not be the natural inclusion. (The difference pointwise isomorphic/diagramwise isomorphic.)  Maybe one can realize an arrow $M\oplus N → N$ with a kernel isomorphic to $M$, but not being an (internal) direct summand of $M \oplus N$? I tried something along the lines of $ℚ$ and $ℚ/ℤ$, but I have been unsuccessful so far.  Can anyone provide me with a counterexample?
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Intro:  In completing , he presents an experiment (go to 42:44) which leaves me baffled.    Experiment:  (I recommend watching the video; see link above.)   There is a $2$ kg block with 2 identical strings attached to it: one at the top, the other at the bottom. The top string is attached to a "ceiling", and the bottom to a "floor". Professor Lewin "stretches" the system (by pulling on the bottom string) with the block not accelerating. One string snaps.     Prediction:   Initially, the top string has a tension of approximately $20$ N, to counter the force of gravity. The bottom string has no tension at all. Then, when Lewin pulls the bottom string, it gains some tension $n$ N. To counter act the force exerted by the bottom string, the top string exerts now $20 + n$ N. I assume that the string with more force will give out sooner, leading me to conclude that the top string will break.     Results:  (This was conducted by Lewin, not me; see link above.)   Trial 1: Bottom string breaks. Trial 2: Top string breaks. Trial 3: Bottom string breaks.     Additional Notes:  The results don't seem consistent. If I was right, I'd expect all 3 experiments to be right; conversely, if I was wrong, I'd expect all 3 experiments wrong, with one exception: the results are more-less random and one result isn't preferred over the other.    Question:        Why was my prediction incorrect?   Was there a flaw in my logic?   Why were the results inconsistent?
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I suppose there was a discussion already on .  But I silly wonder whether all the massless mediators of four fundamental forces, i.e.  Graviton: $g_{\mu\nu}$ (gravity)  Photon $\gamma$: $A_\mu$ (electromagnetism)  Gluons: $A_\mu^a$ (strong interactions)  Necessarily travel at the same speed? Is there a no-go theorem or theoretical proof to say that it is impossible to have these three mediators have different speeds?   Or does QCD confinement makes the story of gluons any different from gravitons and photons?  [ps. excluded massive $Z^{0}$ and $W^{\pm}$ bosons (weak interactions)]  Another way to say this: Speed of photon, graviton, gluon all equal to $c$? or  Whether all massless particles necessarily have the same speed?  Note add: Notice however, in condensed matter systems, there can be emergent gauge fields and emergent massless particles (Dirac or Weyl cones), but their speeds need not be the same, unless there is some emergent symmetry...
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I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
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Electricity and magnetism was unified in the 19th century, and unification of electromagnetism with the weak force followed suit, bringing into play the electroweak force.  I've been told that unifying these with the strong force is likely to be far easier than unifying them with gravity, albeit still very hard.   Apparently this is because of the fact that electromagnetic, weak nuclear and weak nuclear force equations are relatively similar, whereas the equations for gravity differ greatly.  What are the fundamental differences between these equations? (If you could write some that would be great, and then explain the technical differences, if necessary.)  What is it about these differences that make them much harder to unify?
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In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously  symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of local gauge symmetry, some people says that we can formulate Higgs mechanism in a gauge invariant way, some people also says that we need only a non-zero vaccum expectation value... I am confused about this different or maybe same point of views.  In this post:  , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seem that the validity of last part, the appearance of a mass term for $A$, is guaranteed if we have a non-zero equilibrium value $\phi_0$ to expand around. I do not see that the requirement that the phase of the field $\phi$ need to be fixed at some particular value to generate mass term. Thus it seems to me it is not true that SSB is really indispensable for Higgs mechanism.  To put it simply:  The spontanously breaking of what is attributed to Higgs mechanism?   local gauge symmetry global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics. In higgs mechanism, the really broken symmetry is a global one. Mathematically, it is similar in looking as fixing a gauge, but one should not think it as a spontanously breakdown of local gauge symmetry. other.   Is SSB really indispensable for Higgs mechanism?    yes, Higgs mechanism is relied on the SSB of some symmetry (above question), the other approches of description eventually has spontanously broke some symmetry. No, the SSB is just one way to describe Higgs mechanism (or even not a complete way), what is really need is the non-zero vaccum expectation value, for example in the linked post the requirement for the mass term to occur is to have some non-zero expectation value of $\phi$ to expand around, we do not need the phase of the field to be fixed, thus the symmetry is not broken. Other.   some reference materials:    States that SSB of local gauge symmetry is impossible.  in the abstract states that:       gauge symmetries merely reflect a redundancy in the state description   and therefore the spontaneous breaking can not be an essential   ingredient. Indeed, as already shown by Higgs and Kibble, the   mechanism can be explained in terms of gauge invariant variables,   without invoking spontaneous symmetry breaking     In the introduction it says:      In particular, we emphasize that global U(1) phase rotation symmetry,   and not gauge symmetry, is spontaneously violated, and show that the   BCS wave function is, contrary to claims in the literature, fully   gauge invariant
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Coupling constants run with the energy scale $\mu$. But what is exactly this energy scale. My question is, if I have a physical process, how do I compute $\mu$?
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As of this writing it has been made clear to me that classical physics' Luminiferous aether was a terriblly poor discriptor of space. With the advent of Special Relativity and General Relativity, that aether ceased to be applicable. Today I began reading on the Higgs boson, and the Higgs field. How are those theories different from or similar to the seemingly irrational Luminiferous aether?  P.S. I am a first year student at Bakersfield Community College. If this question seems naive, you now have an explanation.  (EDIT) To me the similarity rests upon their fundamental influences regarding static positions in space. I currently only accept models in which static spacial positions are impossible. Contrastingly relative positions seem quite reasonable. In my humble opinion my naivatisms inhibit me from absorbing and understanding the Higgs boson. If they actually are similar, public protest would probably be more apparant. However, I have put forth my best effort to pose an direct question here.  P.S. The questions Where does space go when it falls? and What happens to time lost when matter accelerates? come much earlier in my studies, and they seem relevant.
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Earlier questions under a similar title referred to the short range of the strong force. My question is completely different. I'd like to know why gluons are considered massless in the Standard Model (regardless of the range of the strong force). For example, if mass is a result of the Higgs interactions, then why do gluons not interact with the Higgs boson? In addition to the theoretical reasons, is there any experimental evidence of gluons being actually massless? (I do realize that such experiments are not easy, because of the confinement.)  In the past, neutrinos were also considered massless, but not anymore. Is there a similar possibility for gluons to have perhaps a small invariant mass or is there an overriding reason for them to be definitely massless?  EDIT: The answer I've been looking for is deep in the comments below and not immediately apparent. To make it clear, I am repeating it here: To interact with a boson, a particle (the Higgs) must have a charge mediated by this boson. The Higgs has a weak charge and therefore interacts with the W and Z bosons thus giving them mass. The Higgs does not have the electric or color charge and therefore does not interact with the photon or gluons thus leaving them massless.
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Some scenarios describing the fate of the matter vs  tug of war on the universe involve the acceleration of the universe increasing to the point that it ends up ripping apart even atoms. This is called the .  This makes no sense to me. It looks like all of these   (GR) models of the universe assume it has a uniform isotropic distribution of matter and energy. This works great at long scales, but it is also clearly wrong even at the length scale of the separation of galaxies.  The density of atomic nuclei remains the same even though the universe is expanding even as we speak. I don't think the "scale" in the  can be interpreted so literally. Or maybe said better, it's got to break down when the cosmological horizon distance gets to the scale where the isotropic assumption breaks down, right?  How can scientists be claiming that runaway expansion will rip apart atoms?
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Emphasis is on the irreducible. I get what's special about them. But is there some principle that I'm missing, that says it can only be irreducible representations? Or is it just 'more beautiful' and usually the first thing people tried?   Whenever I'm reading about some GUT ($SU(5)$, $SO(10)$, you name it) people usually consider some irreducible rep as a candidate field. Also, the SM Lagrangian is constructed in this way. (Here, experimental evidence of course suggests it.)
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I assumed that the 1-dimensional Lie algebra was simple since I cannot think of any proper non-trivial ideal it could have (you either have no elements, or once you have one element you span the space). However every classification of simple Lie algebras I have seen does not include a 1-dimensional Lie algebras. Is there a reason for this or is the 1-dimensional Lie algebra somehow simple?  Thank you!
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I have read before in one of Seiberg's articles something like, that gauge symmetry is not a symmetry but a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations.  Regarding this I have a few questions:   Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc? Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)? Are there analogs or other examples to this idea, of introducing fake degrees of freedom to facilitate the calculations or to build interactions, in classical physics? Is it like introducing the fictitious force if one insists on using Newton's 2nd law in a noninertial frame of reference?
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What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?    If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$.  I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem.  Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too.   Do we know the limit for making up these objects? What is the fundamental justification for order two?