The discussion centers on the bosonic field operator defined as $$W=e^{\imath f \phi(x)}$$, where ##f## is a parameter. When this operator acts on the vacuum state ##|0\rangle##, it generates a "coherent" state characterized by an uncertain number of particles, diverging from the original Fock basis. The Taylor series expansion of the exponential does not converge, necessitating a regularization approach, such as introducing a physical cutoff, as referenced in Itzykson and Zuber's work.
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You get a "coherent" state in which the number of particles is uncertain. To see that, expand the exponential in the Taylor series and split the field into creation and destruction operators.QFT1995 said:If i had a bosonic field ##\phi(x)## and I took the exponential in the following way to get the operator $$W=e^{\imath f \phi(x)}$$ where ##f## is a parameter what effect would this have when acting on the vacuum ##|0\rangle##? Is it analogous to the space translation operator? Will it transform the vacuum into some state?