Computation of propagation amplitudes for KG field

In summary, the conversation is about a specific equation for the spacetime Klein-Gordon field from Peskin and Schroeder's book. The discussion then moves on to the computation of the propagation amplitude for a particle to go from one spacetime point to another, and the question of why certain terms in the equation vanish. The expert summarizer explains that the terms vanish due to the operators acting on the vacuum state, and the only non-vanishing term is expressed as a delta function.
  • #1
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Note: I'm posting this in the Quantum Physics forum since it doesn't really apply to HEP or particle physics (just scalar QFT). Hopefully this is the right forum.

In Peskin and Schroeder, one reaches the following equation for the spacetime Klein-Gordon field:
$$\phi(x,t)=\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}\Big(a(p)e^{-ip\cdot x}+a^\dagger(p)e^{ip\cdot x}\Big)$$
Then they say that the propagation amplitude for a particle to go from a spacetime point x to a spacetime point y is [itex]\langle 0\vert\phi(x)\phi(y)\vert 0\rangle[/itex] where the ket |0> is the vacuum. I understand this up to here. But then they start computing it as follows (equation 2.50):
$$\langle 0\vert\phi(x)\phi(y)\vert 0\rangle = \int\frac{d^3p\,d^3q}{(2\pi)^6}\frac{1}{2\sqrt{E_{p}E_{p'}}}\Big(a(p)e^{-ipx}+a^\dagger(p)e^{ipx}\Big)\Big(a(q)e^{-iqy}+a^\dagger(q)e^{iqy}\Big)$$

Clearly there will now be four terms when you expand the parentheses, and the book claims that all of these vanish except for the term with [itex]\langle 0\vert a(p)a^\dagger(q)\vert 0\rangle[/itex]. Two questions:

a) Wouldn't this term vanish also since a(p) kills the vacuum bra, producing a zero?

b) Why doesn't the term with [itex]a^\dagger(p)a^\dagger(q)[/itex] stay? In that term there are no annihilation operators to kill the vacuum, so surely the term should vanish.
 
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  • #2
Theage said:
a) Wouldn't this term vanish also since a(p) kills the vacuum bra, producing a zero?

No. ##a_p## annihilates ##|0\rangle## which clearly means ##a^{\dagger}_p## annihilates ##\langle 0|##, not ##a_p##.

Theage said:
b) Why doesn't the term with [itex]a^\dagger(p)a^\dagger(q)[/itex] stay? In that term there are no annihilation operators to kill the vacuum, so surely the term should vanish.

See above.
 
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  • #3
You forgot to "sandwich" the right-hand side of your equation between ##\langle 0|## and ##|0 \rangle##. Then it becomes immediately clear that all expressions with a annihilation operator acting to the right and a creation operator acting to the left (it becomes an annihilation operator when letting it act on the left argument of a scalar product) to the vacuum state, gives 0. The only non-vanishing term is thus indeed
$$\langle 0|a(p) a^{\dagger}(q) 0 \rangle =\langle a^{\dagger}(p) 0|a^{\dagger}(q)0 \rangle=(2 \pi)^3 \delta^{(3)}(\vec{q}-\vec{p}).$$
 
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  • #4
Again, it's all due to the damn bra-ket notation which is confusing. The equation should read:

## \langle 0, a (p) a^{\dagger} (q) 0\rangle = \langle a^{\dagger} (p) 0, a^{\dagger}(q)0\rangle ##
 

1. What is the KG field and why is it important in computation of propagation amplitudes?

The Klein-Gordon (KG) field is a quantum field theory that describes the behavior of spinless particles, such as the Higgs boson. It is important in computation of propagation amplitudes because it provides a mathematical framework for understanding how these particles interact and propagate through space and time.

2. How are propagation amplitudes computed for the KG field?

Propagation amplitudes for the KG field are computed using a mathematical technique called Feynman diagrams. These diagrams represent the different possible interactions and exchanges of particles that contribute to the overall propagation amplitude.

3. What are the challenges in computing propagation amplitudes for the KG field?

One of the main challenges in computing propagation amplitudes for the KG field is dealing with infinities that arise in the calculations. This is known as the problem of renormalization and requires the use of various techniques to cancel out these infinities and obtain meaningful results.

4. How do computation of propagation amplitudes for the KG field contribute to our understanding of the universe?

By computing propagation amplitudes for the KG field, scientists are able to make predictions about the behavior of particles and their interactions. This helps us better understand the fundamental forces and building blocks of the universe, and can also lead to the development of new technologies and applications.

5. Are there any real-world applications of computation of propagation amplitudes for the KG field?

Yes, computation of propagation amplitudes for the KG field has numerous applications in fields such as particle physics, cosmology, and quantum computing. For example, it is used in the calculation of cross-sections for particle collisions at accelerators, and in the study of the early universe and its evolution.

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