I QED replacing photon field with current in 3-point function

handrea2009
Messages
2
Reaction score
3
TL;DR Summary
QED: replacing photon field with current in the definition of the renormalized 3-point function vertex
I am self-studying QFT in the Schwartz book "Quantum Field Theory and the Standard Model", currently I am struggling to understand the all-orders proof that ##Z_1=Z_2## using Ward-Takahashi identity (page 352).

He states that ## -ie_R\Gamma^\mu ##, which is the sum of the 1PI contributions to matrix elements for the 3-point function ##\langle \psi(x_1)A_\nu(x)\bar\psi(x_2)\rangle## with externa legs amputated , can be formally defined as follow:

$$-ie_R\Gamma^\mu(p,q_1,q_2)(2\pi)^4\delta^4(p+q_1-q_2) \\
\equiv -ie_R\int d^4x d^4x_1 d^4x_2 e^{ipx}e^{iq_1x_1}e^{-iq_2x_2}\\(iG)^{-1}(\not q_1) \langle j^\mu(x) \psi(x_1) \bar\psi(x_2)\rangle (iG)^{-1}(\not p+\not q_1) \tag {19.78}
$$

I don't understand how he gets to replace in ##\langle \psi(x_1)A_\nu(x)\bar\psi(x_2)\rangle## the ##A_v## photon field with the current ##j^\mu = \bar\psi\gamma^\mu \psi##.

The only clue I could find is on page 281 where we have the following formula which comes from Schwinger-Dyson equation:

$$
\square_{\alpha\beta}^k \square_{\mu\nu} \langle A_\nu(x)...A_\beta(x_k)...\rangle = \langle j_\mu(x)...j_\alpha(x_k)...\rangle
$$

so basically you can remove ##A_\mu## field and insert current ##j_\mu##.

However, even using that result it seems to me that in the formula (19.78) in the right-hand-side we have a wrong extra ##ie_R## factor, since we already have one which comes from exploding the expectation value ##\langle ... \rangle ## ( see (7.77) ):

$$
\langle j^\mu(x) \psi(x_1) \bar\psi(x_2)\rangle \equiv \langle \Omega|T\{j^\mu(x)\psi(x_1) \bar\psi(x_2)\}|\Omega\rangle = \langle 0|T\{j^\mu_0(x)\psi_0(x_1) \bar\psi_0(x_2)e^{-ie_R\int \bar\psi_0\gamma^\mu\psi_0 A_\mu}\}|0\rangle_{no bubbles}
$$

where ## -ie_R\bar\psi\gamma^\mu\psi A_\mu ## is the interaction term in the QED Lagrangian density
 
Last edited:
Physics news on Phys.org
I believe in the book the formula on page 281:

$$
\square_{\alpha\beta}^k \square_{\mu\nu} \langle A_\nu(x)...A_\beta(x_k)...\rangle = \langle j_\mu(x)...j_\alpha(x_k)...\rangle \tag{14.152}
$$

is missing an ##e## factor on the RHS, this seems to be confirmed by the Schwinger-Dyson equation ##(14.117)## which is used to get the ##(14.152)##:

$$
\square^x_{\mu\nu}\langle A^\nu(x)A^\alpha(y)\bar\psi(z_1)\psi(z_2)\rangle = \\
e \langle j_\mu(x)A^\alpha(y)\bar\psi(z_1)\psi(z_2)\rangle -i\delta^4(x-y)\delta^\alpha_\mu\langle\bar\psi(z_1)\psi(z_2)\rangle \tag{14.117}
$$

In that way we have:

$$
\square^{\mu\nu}_x \langle A_\nu(x)\psi(x_1)\bar\psi(x_2)\rangle = e_R\langle j^\mu(x)\psi(x_1)\bar\psi(x_2)\rangle
$$

If I substitute that in the ##(19.78)## and as a check I do the calculation at leading order, I correctly get ##\Gamma^\mu = \gamma^\mu##
 
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Back
Top