Free complex scalar field - showing operators are creation and annihlation?
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Using the method of canonical quantization we can show that for a free scalar field we have:
ϕ(x)=∫dp~(a(p⃗ )e−ipx+b†(p⃗ )eipx)ϕ(x)=∫dp~(a(p→)e−ipx+b†(p→)eipx)
where a(p⃗ )a(p→) and b†(p⃗ )b†(p→) are two operators. It turns out that a(p⃗ )a(p→) and b†(p⃗ )b†(p→) are creation and annihilation operators, but every argument I have seen for this has been fairly hand-wavy.
So my question is: What is the easiest method to show that a(p⃗ )a(p→) and b†(p⃗ )b†(p→) have to be creation and annihilation operators and can be nothing else?
The definition of creation and annihilation operators is the Fock definition, as given e.g. on .
ϕ(x)=∫dp~(a(p⃗ )e−ipx+b†(p⃗ )eipx)ϕ(x)=∫dp~(a(p→)e−ipx+b†(p→)eipx)
where a(p⃗ )a(p→) and b†(p⃗ )b†(p→) are two operators. It turns out that a(p⃗ )a(p→) and b†(p⃗ )b†(p→) are creation and annihilation operators, but every argument I have seen for this has been fairly hand-wavy.
So my question is: What is the easiest method to show that a(p⃗ )a(p→) and b†(p⃗ )b†(p→) have to be creation and annihilation operators and can be nothing else?
The definition of creation and annihilation operators is the Fock definition, as given e.g. on .
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