# Maxwell propagator in Kaku's QFT book

• hyungrokkim
In summary, in Michio Kaku's QFT book, he illustrates problems with direct quantization due to gauge invariance by writing the action of the Maxwell theory in terms of the vector potential 'A' and rearranging it to get an expression involving a second-order derivative. He then throws away the 4-divergence and uses partial integration and Stokes' theorem to get a form that allows for easier handling of Gaussian integrals. This technique is not unique to Kaku's book and can also be applied to the scalar field. The occurrence of the inverse of the d'Alembertian may seem unusual, but it is still possible.
hyungrokkim
In Michio Kaku's QFT book, p. 106, he writes:

[To illustrate problems with direct quantization due to gauge invariance]
let us write down the action [of the Maxwell theory] in the following form:
$$\mathcal L=\frac12 A^\mu P_{\mu\nu}\partial^2A^\nu$$
where
$$P_{\mu\nu}=g_{\mu\nu}-\partial_\mu\partial_\nu/(\partial)^2$$
The problem with this operator is that it is not invertible. [...]​

I don't understand his notation. Normally, the same Lagrangian is written
$$\mathcal L=-\frac14F^2$$

When factoring out $$A^\mu$$, how does he get the $$\partial^2$$?

hyungrokkim said:
In Michio Kaku's QFT book, p. 106, he writes:

[To illustrate problems with direct quantization due to gauge invariance]
let us write down the action [of the Maxwell theory] in the following form:
$$\mathcal L=\frac12 A^\mu P_{\mu\nu}\partial^2A^\nu$$
where
$$P_{\mu\nu}=g_{\mu\nu}-\partial_\mu\partial_\nu/(\partial)^2$$
The problem with this operator is that it is not invertible. [...]​

I don't understand his notation. Normally, the same Lagrangian is written
$$\mathcal L=-\frac14F^2$$

When factoring out $$A^\mu$$, how does he get the $$\partial^2$$?

Kaku wrote everything in terms of the vector potential 'A', rearranged the resulting "expression" to get another "expression + 4-divergence" , and threw away the 4-divergence (you can throw away a 4-divergence in a Lagrangian). This is a typical technique, not unique to Kaku's book.

In these kind of expression you often want to get an expression like

$$\phi Y \phi$$

where Y is an expression involving a second order derivative, and with some contraction depending on what phi exactly is. The reason is that these kind of expressions give you Gaussian integrals which you know how to handle. The way to get them is via partial integration and using Stokes;

$$\partial A \partial B = \partial(A \partial B) - A \partial^2 B$$

The first term gives a boundary condition after integration and can be discarted after suitable boundary conditions.

Maybe you should try it for the scalar field; the action of the scalar field can be rewritten as

$$\int d^n x \phi(\partial^2 + m^2)\phi$$

So $Y = \partial^2 + m^2$. You can do the same thing for the vector field A.

I did figure it out, thanks for the replies. What threw me off was the (unusual, but still possible) occurrence of $$(\partial^2)^{-1}$$, i.e., the inverse of the d'Alembertian.

## 1. What is the Maxwell propagator in Kaku's QFT book?

The Maxwell propagator, also known as the Feynman propagator, is a mathematical tool used in quantum field theory to calculate the probability amplitude for a particle to travel from one point to another in spacetime.

## 2. How is the Maxwell propagator derived?

The Maxwell propagator is derived from solving the equations of motion for a quantum field, specifically the Maxwell field, using the Feynman path integral method. This involves summing over all possible paths the particle could take between the two points in spacetime.

## 3. What is the physical significance of the Maxwell propagator?

The Maxwell propagator is a fundamental tool in calculating the probability amplitude for electromagnetic interactions in quantum field theory. It allows us to make predictions about the behavior of particles and their interactions based on fundamental principles of quantum mechanics.

## 4. How does the Maxwell propagator differ from other propagators?

The Maxwell propagator is specific to the electromagnetic field and takes into account the spin of the particles involved, unlike other propagators which may be used for scalar or fermionic fields. It also includes a term for the photon's polarization, which is necessary for calculating probabilities in electromagnetic interactions.

## 5. What are the limitations of using the Maxwell propagator?

The Maxwell propagator is a mathematical tool that can only be applied to systems that follow the principles of quantum field theory. It also assumes that spacetime is flat and does not take into account the effects of gravity. In addition, it does not account for higher-order corrections, which can become significant in certain situations.

• Quantum Physics
Replies
5
Views
658
• Quantum Physics
Replies
3
Views
1K
• Quantum Physics
Replies
24
Views
2K
• Quantum Physics
Replies
3
Views
987
• Calculus and Beyond Homework Help
Replies
3
Views
909
• Quantum Physics
Replies
14
Views
2K
• Quantum Physics
Replies
3
Views
4K
• Quantum Physics
Replies
7
Views
3K
• Quantum Physics
Replies
10
Views
2K
• Quantum Physics
Replies
4
Views
1K