# The full path integral of a quantum field theory

Suppose if one is able to do a full path integral of a QFT with an action say $$S[\phi]$$ i.e. $$Z = \int [\mathcal{D}\phi] e^{iS[\phi]}.$$ What can I use $$Z$$ for? Can I use the $$Z$$ like the partition function and compute Helmoltz free energy and entropy like in statistical physics? Apart from using it to normalize the correlators to cancel the vacuum diagrams in QFT what other interesting facts about a QFT can be extracted from $$Z$$?

Edit: If it is not obvious from them question, I also mean apart from computing the correlation functions, what else is $$Z$$ good for?

The short answer is everything.

You should think of $$Z$$ as exactly analogous to the partition function in statistical mechanics. We can introduce an external field $$J(x)$$ to the partition function

$$Z(J)=\int [\mathcal{D}\phi]\exp\Big(iS(\phi)+i\int J\phi\Big)$$ And we can apply functional derivatives $$\delta/\delta J(x)$$ onto $$Z(J)$$ to calculate vacuum expectation values. In particular $$\langle\phi(x_1)\cdots\phi(x_n)\rangle=\frac{(-i)^n}{Z(J)}\frac{\delta}{\delta J(x_1)}\cdots\frac{\delta}{\delta J(x_n)}Z(J)\bigg|_{J=0}$$

A quantum field theory is essentially defined by the set of all correlation functions, so we really do arrive at everything just from $$Z$$. For instance, you can get scattering amplitudes from correlation functions using the LSZ Reduction Formula.

In addition to fewfew4's correct answer, here are some further uses of the path integral, focusing on applications in high energy physics:

• By a Legendre transformation, the log of the partition function $$W[J]=-i \ln Z[J]$$ can be converted into the 1-loop effective action $$\Gamma[\phi]$$ (where $$\phi$$ refers to the fields). This object encodes loop corrections to all the vertices and propagators of the theory. The equations of motion derived from $$\Gamma[\phi]$$ (as opposed to the ones from the classical action $$S[\phi]$$) are satisfied by the vev $$\langle \phi \rangle$$, including all quantum corrections. This idea can be used to compute the effective potential for a scalar field including quantum corrections; this is the Coleman-Weinberg potential.
• The path integral enables a very natural way to describe effective field theory. The effective action or Wilson action $$S_\Lambda[\phi]$$ is the classical action obtained by integrating over all field modes above a scale $$\Lambda$$. In other words, if we write $$\phi = \phi_{\rm l.e.} + \phi_{\rm h.e.}$$, where $$\phi_{\rm l.e.}$$ refers to the modes of the field below $$\Lambda$$ and $$\phi_{\rm h.e.}$$ above, then we can write $$$$e^{i S_{\Lambda}[\phi_{\rm l.e.}]} = \int_{k > \Lambda} \rm D \phi_{\rm h.e.} e^{i S[\phi_{\rm l.e.} + \phi_{\rm h.e.}]}$$$$ Then the correlation functions can be obtained from $$S_{\Lambda}$$ by integrating modes only up to energies $$\Lambda$$ $$$$Z = \int_{\rm k < \Lambda} D \phi e^{i S_\Lambda[\phi]}$$$$ This approach enables us to explain renormalization theory and provides a modern picture for understanding how field theories emerge at low energies. The introduction by Burgess provides an excellent starting point to learn about effective field theories.
• The path integral approach provides an elegant and (relatively) intuitive way to understand how to correctly quantize non-Abelian gauge theories. The Fadeev-Popov ghosts arise as way to represent the Jacobian determinant from changing variables, and the gauge-fixing term comes from an average over different gauge choices. You can read more on Scholarpedia.
• Anomalies (symmetries of the classical theory that fail to be symmetries of the quantized theory) can be understood as the failure of the measure of the path integral to be invariant under a symmetry, that the classical action is invariant under.
• Non-trivial saddle points of the path integral give rise to instantons, classical solutions to the equations of motion of the field theory that contribute to non-perturbative effects (effects that can't be computed in any order of perturbation theory), for example instantons can be used to model the non-trivial QCD vacuum state. They also give rise to the strong-CP problem in QCD, which motivated the proposal of axions, which are now considered a natural dark matter candidate.
• Functional methods also enable elegant derivations of the Ward-Takahashi identities (essentially the integral of a total derivative is zero, which implies relationships among correlation functions following from a symmetry) and the Schwinger-Dyson equations, non-perturbative relations among correlation functions in a quantum field theory.

TL;DR: The path integral approach provides an elegant way to derive important results that are very difficult to obtain in other formalisms. However, the reverse is also true; unitarity of the S-matrix is very difficult to understand from the path integral point of view, and the way unitarity is always checked is by showing the path integral is equivalent to the canonical operator picture and checking unitarity in the canonical language.

From what I can tell, the OP asked "why is $$Z$$ useful" and then fewfew4 answered by explaining why a much more general object is useful. The OP then tacitly accepted the practice of referring to the zero-point function and the generator of all $$n$$-point functions interchangeably by replying that he or she was interested in things "apart from computing the correlation functions".

So there are really two questions here.

1. What is $$Z[J]$$ good for besides computing correlation functions?
2. What is $$Z[0]$$ good for in theories where computing $$Z[J]$$ is too hard (as it almost always is)?

The second question is more interesting. In addition to the source free parts of Andrew's answer, one can consider free energies which become finite when you put the theory on a compact manifold. These are believed to behave monotonically (F-theorem) along renormalization group flows and thereby generalize Zamolodchikov's central function to higher dimensions. In certain cases, this is also related to the monotonicity of relative entropy in quantum information. A similarly defined object in supersymmetric theories, called the Witten index, is essential for determining when the supersymmetry can be broken.

• Can you provide some references ? I would be most grateful. Commented Aug 16, 2021 at 8:39
• Yes, but I'm not familiar enough with this to know the best references. A review of SUSY breaking which works its way up to the Witten index is people.sissa.it/~bertmat/lect7.pdf while the breakthrough in computing path integrals exactly for some interacting QFTs was in 0712.2824. For the zero point function on a sphere, 1405.7271 discusses some of the ambiguities and gives a physical interpretation for the scheme independent part. Commented Aug 16, 2021 at 15:57