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Reading Zee's QFT book he defines the interacting partition function as follows

$$Z(J,\lambda)=\int\mathcal{D}\varphi \exp{\left(i\int d^4x \left[\frac{1}{2}((\partial\varphi)^2-m^2\phi^2)-\frac{\lambda}{4!}\varphi^4+J\varphi\right]\right)}.\tag{I.7.11}$$

We can solve the partition function as an expansion in $J$ as follows

$$Z(J,\lambda)= Z(0,0)\sum_{s = 0}^\infty \frac{i^s}{s!} \int d^4x_1\cdots d^4x_s J(x_1) \cdots J(x_s)G^{(s)}(x_1,\cdots,x_s).\tag{I.7.13}$$

Zee states that the power of $J$ would indicate the number of particles in the process. In his previous section I.4, he used delta functions as sources where $$J(x) = \delta^3(x-y_1) + \delta^3(x-y_2).$$ From what I understand, if the field is initially in the 0 state then this source would create a particle at spacetime locations $y_1$ and $y_2$. However, I am not sure how to destroy a particle using the source. If there is a way to write a source as $J(x) = Sources + Sinks$. Would plugging this source into the partition function collapse the partition function into

$$Z(J,\lambda) = Z(0,0)\frac{i^s}{s!}G^{(s)}(x_1,\cdots, x_s)$$

where $s$ is the number of particles in the process?

My intuition tells me that $Z(J,\lambda)$ is the amplitude for a free field to create $a$ particles from the vaccuum and interact such that there are $b$ particles and then these particles then get annihilated into the vacuum where $a + b = s$.

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  1. Yes, the source $$J(x)~=~J_E(x)+J_A(x)$$ is really a sum of emission and absorption sources. Ref. 1 writes:

    We also know that $J(x)$ corresponds to sources and sinks.

    That both could in principle be present is a crucial part of a Lorentz covariant theory. Whether a source is interpreted as an emission or absorption sources, depends on the causal order of the spacetime points involved. E.g. a scattering process in one inertial frame can look like a partial production in another inertial frame, cf. e.g. Ref. [2].

  2. Example. A prototype of a (worldline, emission, absorption) source for a charged point particle would be $$\begin{align} J(x)~=~&q \int_{-\infty}^{\infty} \!\mathrm{d}\tau~\delta^4(x-x(\tau)),\cr J_E(x)~=~&q \int_{\tau_0}^{\infty} \!\mathrm{d}\tau~\delta^4(x-x(\tau)),\cr J_A(x)~=~&q \int_{-\infty}^{\tau_0}\!\mathrm{d}\tau~\delta^4(x-x(\tau)), \end{align} $$ respectively, where the notion of charge $q$ depends on the theory.

  3. Perhaps the issue is easier to parse in a Fourier transformed picture, where external particles carry definite incoming or outgoing 4-momenta. There an $n$-point correlation function are related to $S$-matrix elements involving $n$ external particles, cf. the LSZ reduction formula.

References:

  1. A. Zee, QFT in a nutshell; section 1.4 + eq. (I.7.13).

  2. T. Banks, Modern QFT, 2010; section 1.2.

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  • $\begingroup$ If the mathematical form of the sources is a delta function what do the corresponding sinks look like? $\endgroup$ Feb 25, 2023 at 9:39
  • $\begingroup$ same but with opposite sign $\endgroup$ Feb 25, 2023 at 9:45
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    $\begingroup$ Is the particle represented by J a particle that is represented by the $\varphi$ field or another external field? i.e. is this source creating a particle in the $\varphi$ field along this world line? $\endgroup$ Feb 26, 2023 at 12:11

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