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I understand that defining the Vacuum is important in Field Theory, why? Is this because it is the 'ground' state, before particles are added, so defines the 'background'?

I assume its not important in Quantum Mechanics, is that right?

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Try playing with the integers without the number 0. –  Chris Gerig Jun 13 '12 at 6:37
    
Didn't the Greeks get pretty far without it? –  Mozibur Ullah Jun 13 '12 at 6:38

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In QFT, particles are generally understood as excitations of the vacuum. Thus the vacuum is as indispensible in quantum field theory when studying particles as the ground state of a system is in quantum mechanics, when one studies its excitations.

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Not an expert in QM, but I do know that there are virtual particles which have a well-established and studied effect on 'real' particles. The vacuum is a never-still zoo of virtual particles that have a vast number of interactions with each other and the 'real' matter. This means there is never actually, truly, nothing there.

Virtual particles are in fact (within my very limited understanding) one of the main mechanisms by which certain forces arise between nucleons. Read up on it - it's incredible yet true and fascinating.

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