Sketch of Path Integral Formalism

Before we step into the path integral formalism, we give an overall sketch as well as some rules of our construction here. Beware, that the rules are not necessarily accepted or respected by all the academics.

Sketch of Theoretical Construction

The construction will start from analyzing the general path integral of scalar field from some stable state to itself (usually it is vacuum state \(|\Omega\rangle\))

\[\langle \Omega|\Omega\rangle = N\int\mathcal{D}\varphi\exp\{-\frac{1}{2}\mathrm{i}\int\varphi[\partial^2-m^2]\varphi\} = 1\]

where \(N\) is a normalization factor. The last equality comes from the property stable, which means it will not change under the situation of free field, which will determine the normalization factor. Now, we construct a quantity called generating functional by adding an arbitrary potential \(J\)

Stage 1

\[\begin{split}W_0[J] = \langle\Omega|\Omega\rangle |_J &= N\int\mathcal{D}\varphi\exp\{-\mathrm{i}\int\frac{1}{2}\varphi[\partial^2-m^2]\varphi - J\varphi\}\\ &= \exp\{-\frac{1}{2}\mathrm{i}\int J\Delta_FJ\}\end{split}\]

The above generating functional includes merely the scalar field without any other actual interaction terms (the \(J\) term is an auxiliary term with no physical significance). If the Lagrangian includes the interaction terms \(\mathcal{L}_{\text{int}}\), the generating functional will be

Stage 2

\[\begin{split}W[J] &= \exp\{-\mathrm{i}\int\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\} W_0[J] \\ &= \exp\{-\mathrm{i}\int\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\}\exp\{-\frac{1}{2}\mathrm{i}\int J\Delta_FJ\}\end{split}\]

As is indicated, this expression deals with some stable state. Now we hope to generalized it into other ordinary states, like from state \(\alpha\) to state \(\beta\). This forms the well-renowned S-matrix (scattering matrix)

Stage 3

\[S = \langle \beta|\alpha\rangle = S[J]|_{J=0} = \left.\exp\{\mathrm{i}\int\varphi_{\text{as}}(\partial^2 - m^2)\frac{\delta}{\delta J}\}W[J]\right|_{J=0}\]

where \(\varphi_{\text{as}}\) is the asymptotic field corresponding to the initial and final state. The last equality is the famous LSZ reduction formula.

The theoretical construction will end at S-matrix. For a connection with the experiment, scattering cross-section can be derived through the S-matrix, which is directly measurable in the scattering experiment.

Some Rules

It has been stated before that we will not introduce any contents from canonical quantization into our construction of the path integral formalism. Therefore, actually we are not able to have concepts like state which is widely used in quantum theory. In this case, we have to use some configuration of fields as state (like asymptotic field). But this would imply that there must be some subtle identical relationship between field and state, which is somehow rejected by canonical quantum field theory.

As a matter of fact, the problem caused by the lack of field operator and successive state only arises when constructing S-matrix, which means that stage 1 and stage 2 will not be affected anyway. In our philosophy, the path integral formalism, as a parallel companion of canonical quantization, should be free from the concepts in canonical form. Therefore, we will try our best not to invoke concepts like field operator and the corresponding state, or at least push them as later as possible.