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It is said that interaction with the Higgs field gives rest mass to fundamental particles.

When a particle is moving, is it then correct to say that there is more of the same interaction that gives it rest mass, thus accounting for its full relativistic mass?

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Alan, I prefer the term invariant mass over rest mass and your question provides an example motivating that preference.

The terms in a relativistic Lagrangian for some QFT must be Lorentz scalars; they must be invariant under Lorentz transformations, i.e., they must have the same value in all reference frames.

When the Higgs field has a vacuum expectation value, the Standard Model Lagrangian can be written such that the matter and gauge fields have associated "mass" terms. These terms are, of course, Lorentz invariant.

So, in this way, it is clear that the interaction results in an invariant mass, one that is the same in all reference frames.

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No.

Interaction with Higgs field doesn't depend on motion. That's why Rest Mass is not variable.

Relativistic Mass has nothing to do with Higgs mechanism. It varies because every interaction with mass in relativistic physics are affected by modified space-time scale. For better understanding, energy setup of physics takes care of that. $$E=\sqrt {(mc^2)^2+(pc)^2}$$ It says mass is energy, so there's no reason why it can't be changed.

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Most of this I like, but what do you mean by "so there's no reason why it can't be changed"? –  David Z Jul 8 '12 at 0:32
    
@David The classical mental model says mass should be constant, but energy can vary. The sentence is to break down that. –  Sachin Shekhar Jul 8 '12 at 6:51
    
"break down that" - do you mean you want to say that the fact that mass is invariant but energy is not is incorrect? If that's the case, then that statement is wrong (unless you specify that by "mass" you mean relativistic mass, but then that's just energy under another name). –  David Z Jul 8 '12 at 6:56
    
@David Why do you always stress on that. If I entered relativistic physics, why would I need to say "relativistic mass". All mass based interactions see only one type of mass in relativistic physics. "Relativistic mass" word is used to distinguish with classical mass (now, rest mass of SM). –  Sachin Shekhar Jul 8 '12 at 7:10
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Because it's easy for people to become confused by the difference between relativistic mass and rest mass. It's important to have a convention for the meanings of variables, and the convention that is emerging is that "mass" or $m$ refers to rest mass. If you use the word "mass" or the variable $m$ to mean relativistic mass, you will not be able to effectively communicate with most physicists. –  David Z Jul 8 '12 at 7:28
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