# Creation and destruction of particles in the Path integral

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 $$a + b = s$$.

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.

• If the mathematical form of the sources is a delta function what do the corresponding sinks look like? Commented Feb 25, 2023 at 9:39
• same but with opposite sign Commented Feb 25, 2023 at 9:45
• 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? Commented Feb 26, 2023 at 12:11