Consider a real scalar field operator $\varphi$. It can be written in terms of creation and anihilation operators as $$\varphi(\textbf{x})=\int \tilde{dk}[ a(k)e^{i\textbf{kx}}+a(k)^{\dagger}e^{-i\textbf{kx}}]$$ where $\tilde{dk}$ is a Lorenz-invariant measure. If $\varphi$ is interpreted as creating a particle at $\textbf{x}$ when acting on the vacuum, what is its action on a generic state? It seems to be creating a superposition of a state with one added quantum of energy through the creation operator, and a state with one less quanta of energy through the annihilation operator.
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As the formula clearly shows, $\phi(x)$ cannot be interpreted as a pure creation operator of any type. It is a combination of creation and annihilation operators. Creation operators are those called $a(k)^\dagger$ and annihilation operators are called $a(k)$. So yes, if $\phi(x)$ acts on a generic state with a well-defined number of particles $N$, it produces a linear superposition of states that have $N+1$ and $N-1$ particles, respectively. When it acts on the vacuum, for example, however, the annihilation operator piece drops out and it creates a 1-particle state. It's somewhat hard to understand what you mean by "interpretation". The only right interpretation is the right calculation. It is an operator that gives something if it acts on a state, and all these answers may be calculated. They shouldn't be interpreted, they should be calculated. |
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