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By Stetz A.W.

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Ni + 1, . . = ni + 1 Now replace ni everywhere by ni − 1. . , ni − 1, . . |ai | . . , ni , . . = √ √ ni . . , ni , . . | . . , ni , . . 13) The effect of ai on the state | . . , ni , . . has been to produce a state in which the number of particles in the i’th state has been reduced by one. Eqn. 13) also tells us what the normalization must be. In short ai | . . , ni , . . = √ ni | . . , ni − 1, . . 2. BOSON STATES 65 Of course if ni = 0, the result is identically zero. ai | .

Ni − 1, . . |ai | . . , ni , . . = √ √ ni . . , ni , . . | . . , ni , . . 13) The effect of ai on the state | . . , ni , . . has been to produce a state in which the number of particles in the i’th state has been reduced by one. Eqn. 13) also tells us what the normalization must be. In short ai | . . , ni , . . = √ ni | . . , ni − 1, . . 2. BOSON STATES 65 Of course if ni = 0, the result is identically zero. ai | . . , ni = 0, . . = 0 The commutation relations are important. Clearly all the ai ’s commute among one another, since it makes no difference in what order the different states are depopulated, and by the same argument, the a†i ’s commute as well.

66) must be analogous to Z[0]. 69) is easily established using the definition (55). D(x − y) is, after all, a Green’s function, and Green’s functions are by definition, inverses of differential operators. That’s what they do for a living. 64), as you will see. 70) The denominator is just Z0 [0], ie. 62) with J = λ = 0. 4. 70) in the case of two fields. 62) with λ = 0. 70) reveals that what we have just calculated is ϕ(x1 )ϕ(x2 ) . Let’s pause for a moment to see how all the pieces fit together.

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