[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"lesson:en:73-qft-functional-methods":3},"\u003Cp>The canonical formulation describes fields as operators evolving in time. The functional integral takes another view: it sums over every possible field configuration. This language makes symmetries, Feynman diagrams, and effective field theory especially transparent.\u003C\u002Fp>\n\n\u003Cdiv class=\"key-concept\">\n  \u003Ch4>An integral over functions\u003C\u002Fh4>\n  \u003Cp>An ordinary integral sums over numbers. A functional integral sums over entire functions $\\phi(x)$. The symbol $\\mathcal{D}\\phi$ formally denotes a measure on this configuration space. In practice, the problem is first defined on a lattice or with another regulator, and then one studies the continuum limit.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\n\u003Ch2>Amplitude and Action\u003C\u002Fh2>\n\u003Cp>For a scalar field, the transition amplitude takes the form:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$Z=\\int\\mathcal{D}\\phi\\,e^{iS[\\phi]\u002F\\hbar}$$\u003C\u002Fdiv>\n\u003Cp>Every field history contributes a phase fixed by its action. When $S\\gg\\hbar$, neighboring configurations cancel except near stationary points with $\\delta S=0$. Classical field equations emerge by stationary phase.\u003C\u002Fp>\n\n\u003Ch2>The Generating Functional\u003C\u002Fh2>\n\u003Cp>Introduce an external source $J(x)$:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$Z[J]=\\int\\mathcal{D}\\phi\\,\\exp\\left[\\frac{i}{\\hbar}\\left(S[\\phi]+\\int d^4x\\,J(x)\\phi(x)\\right)\\right]$$\u003C\u002Fdiv>\n\u003Cp>Functional derivatives with respect to $J$ insert fields into the integral:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$\\langle0|T\\phi(x_1)\\cdots\\phi(x_n)|0\\rangle=\n\\left.\\frac{1}{Z[0]}\\left(\\frac{\\hbar}{i}\\right)^n\n\\frac{\\delta^n Z[J]}{\\delta J(x_1)\\cdots\\delta J(x_n)}\\right|_{J=0}$$\u003C\u002Fdiv>\n\u003Cp>These correlation functions are the objects from which the LSZ formula extracts scattering amplitudes.\u003C\u002Fp>\n\n\u003Ch2>Free Theory and the Propagator\u003C\u002Fh2>\n\u003Cp>For a quadratic action, the integral is Gaussian and can be computed exactly. The Feynman propagator $\\Delta_F$ appears in the result:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$Z_0[J]=Z_0[0]\\exp\\left[-\\frac{i}{2\\hbar}\\int d^4x\\,d^4y\\,J(x)\\Delta_F(x-y)J(y)\\right]$$\u003C\u002Fdiv>\n\u003Cp>The propagator is the inverse of the quadratic differential operator with the appropriate causal prescription. It is not the path of a hidden particle; it is a correlation function.\u003C\u002Fp>\n\n\u003Ch2>Interactions and Diagrams\u003C\u002Fh2>\n\u003Cp>For an interaction $\\lambda\\phi^4\u002F4!$, expand the exponential in powers of $\\lambda$. Wick's theorem organizes all contractions. Each contraction gives a propagator, each interaction term gives a vertex, and the combinatorics supplies symmetry factors. Feynman diagrams are therefore a graphical ledger of perturbation theory.\u003C\u002Fp>\n\n\u003Cdiv class=\"key-concept\">\n  \u003Ch4>Connected, irreducible, and effective\u003C\u002Fh4>\n  \u003Cp>$W[J]=-i\\hbar\\ln Z[J]$ generates connected diagrams. Its Legendre transform, the effective action $\\Gamma[\\phi_c]$, generates one-particle-irreducible vertices. The equation $\\delta\\Gamma\u002F\\delta\\phi_c=0$ contains the equations of motion corrected by quantum effects.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\n\u003Ch2>Wick Rotation\u003C\u002Fh2>\n\u003Cp>The oscillating real-time integral is often difficult to define. Setting $t=-i\\tau$ formally gives Euclidean time:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$Z_E=\\int\\mathcal{D}\\phi\\,e^{-S_E[\\phi]\u002F\\hbar}$$\u003C\u002Fdiv>\n\u003Cp>The weight then resembles a Boltzmann distribution. This connection enables lattice gauge-theory simulations. Returning to real time requires precise analytic conditions, and sign problems can make some theories numerically very difficult.\u003C\u002Fp>\n\n\u003Ch2>Ward Identities\u003C\u002Fh2>\n\u003Cp>A change of variables in the functional integral leaves $Z$ invariant if the measure and action respect the symmetry. This produces identities among correlation functions. For gauge symmetry, Ward-Takahashi identities enforce current conservation and constrain renormalization. If the measure is not invariant, a quantum anomaly can appear.\u003C\u002Fp>\n\n\u003Ch2>Exercises\u003C\u002Fh2>\n\u003Col>\n  \u003Cli>Explain why the logarithm of $Z[J]$ selects connected diagrams.\u003C\u002Fli>\n  \u003Cli>What physical distinction separates a propagator from a classical trajectory?\u003C\u002Fli>\n  \u003Cli>Why is a Euclidean weight better suited to Monte Carlo methods?\u003C\u002Fli>\n\u003C\u002Fol>\n\n\u003Ch2>Key Takeaways\u003C\u002Fh2>\n\u003Cul>\n  \u003Cli>The functional integral sums over field configurations weighted by $e^{iS\u002F\\hbar}$.\u003C\u002Fli>\n  \u003Cli>The generating functional produces correlation functions through functional differentiation.\u003C\u002Fli>\n  \u003Cli>Feynman diagrams organize a perturbative expansion; they are not literal histories.\u003C\u002Fli>\n  \u003Cli>The effective action collects quantum corrections to classical equations.\u003C\u002Fli>\n  \u003Cli>Wick rotation connects quantum field theory and statistical physics under controlled analytic assumptions.\u003C\u002Fli>\n\u003C\u002Ful>\n",1786528186726]