Explicit expressions for creation/annihilation operator of the free scalar field

In summary, the conversation discusses the calculation of the commutation relation between creation and annihilation operators for fields and their conjugate momenta. The steps involved in the calculation are described and the final result is determined to be zero due to the presence of a delta function. This explanation addresses a potential confusion in the last step of the calculation.
  • #1
masudr
933
0
I've been trying to work my way through some of my lecture notes, and have hit this snag. (n.b. I use [itex]k_0 \equiv +\sqrt{\vec{k}^2 + m^2}[/itex])

We have
[tex]a(q) = \int d^3 x e^{iqx} \{ q_0 \phi(x) + i \pi(x) \} [/tex]
[tex]a^{\dagger}(q) = \int d^3 x e^{-iqx} \{ q_0 \phi(x) - i \pi(x) \}[/tex]

To calculate the commutation relation between these operators, we simply multiply them out as required, and substitute the canonical commutation relation between fields and their conjugate momenta.

I work through the relatively tedious steps and get

[tex][a(q),a(p)] = \int d^3 x d^3 y e^{i(qx-py)} \delta^3(\vec{x} - \vec{y}) (q_0 - p_0)[/tex]
[tex]= \int d^3 x e^{i(q-p)x} (q_0 - p_0)[/tex]
[tex]= \int d^3 x e^{i(q_0-p_0)x^0} e^{i(\vec{q}-\vec{p})\cdot\vec{x}} (q_0 - p_0)[/tex]

In my notes, the next step is to replace [itex]\vec{q}[/itex] with [itex]\vec{p}[/itex] and so get 0. However, if we integrate over x, surely we are left with a loose delta function outside an integral, which would mean that [itex][a(q),a(p)] = 0 \Leftarrow q=p[/itex] which I know is wrong.

Can anyone explain that last step? Any textbooks I've seen assume this is trivial and just go on to state the commutation relation between the creation/annihilation operators rather than calculating it.
 
Last edited:
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  • #2
Never mind. After doing the integral, I got

[tex]
(q_0 - p_0) \delta^3(\vec{q}-\vec{p}) e^{i(q_0-p_0)t}
[/tex]

which conspires to be zero when the delta function is non-zero because of the first term in brackets, and is zero everywhere else because of the delta function.
 
  • #3
Correct!
 
1.

What is an explicit expression for the creation/annihilation operator of the free scalar field?

An explicit expression for the creation/annihilation operator of the free scalar field is a mathematical representation of the operator that creates or annihilates particles in a free scalar field theory. It is commonly written in terms of the field operators and their derivatives.

2.

How is an explicit expression for the creation/annihilation operator derived?

An explicit expression for the creation/annihilation operator is derived using the canonical quantization procedure, which involves promoting the classical field to a quantum operator and imposing commutation relations. This leads to the creation/annihilation operators being written in terms of the field operators and their conjugate momenta.

3.

What is the significance of the explicit expression for the creation/annihilation operator in quantum field theory?

The explicit expression for the creation/annihilation operator is significant in quantum field theory because it allows us to calculate physical quantities such as particle number and energy using the quantum field theory formalism. It also plays a key role in the formulation of Feynman diagrams and perturbation theory.

4.

How does the explicit expression for the creation/annihilation operator change in different field theories?

The explicit expression for the creation/annihilation operator can change in different field theories depending on the type of particles being described. For example, in a free scalar field theory, the expression involves only the field operators and their derivatives, while in a fermionic field theory, it also includes spinor operators.

5.

Are there any applications of the explicit expression for the creation/annihilation operator in other areas of physics?

Yes, the explicit expression for the creation/annihilation operator has applications in other areas of physics such as condensed matter physics, where it is used to describe the behavior of quasiparticles in systems such as superconductors and semiconductors. It is also used in quantum information theory to study the creation and manipulation of qubits.

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