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As I have understood it, the Standard Model includes particles that carry the different forces, e.g. the electromagnetic (EM) force, the gravitational (G) force. When talking about EM fields such as visible light or microwaves, the associated particle is said to be the photon. But what about a static EM field without any electricity, like a common household magnet? How does that magnet communicate its force? Via photons?

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Hello and welcome to the cite.
Yes -- the carrier for any electromagnetic interaction is a photon. Including ordinary magnets.

You might ask: "Why then can't I see those photons?" Well, even from classical point view, the photons can have different wavelenghts. And with the magnet you deal with really large wavelenghts, while one cannot "see" radiowaves. While also I think that one cannot even use the concept of "interacion carrier" without working in a context of quantum field theory anyway.

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so photons of really large wavelengths means photons of really low energies? Is the magnet constantly loosing energy by emitting these photons or does it also absorb photons of the same wavelength? Are the photons emitted isotropically or do they follow the field lines in some way? Should I include these questions in the question rather than in a comment? –  Dirty Harry Jan 19 '11 at 11:27
    
edit your question at your leasure, be sure to follow the faq –  Argus Jul 10 '12 at 0:12

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