Advanced Quantum Field Theory: intuition with path integrals

The lecture shows how adding a source to the quadratic zero-dimensional action yields a shifted field and a Gaussian integral, producing Z(J) = Z0 exp( (1/(2 hbar)) J^T M^{-1} J ).

Expand path integrals around the action's global minimum, use the gaussian integral and Hessian determinants to derive the free theory partition function Z0, and set the stage for Feynman diagrams.

We analyze a zero-dimensional quantum field theory with a mass term and phi^4 interaction, computing Z via a perturbative hbar expansion and noting its asymptotic series and Feynman diagram connections.

Explore how feynman diagrams arise from combinatorics in a four-valent phi4 theory, linking the partition function ratio z over z0 to labeled vacuum graphs, edges, and propagators.

Define the Wilsonian effective action w as minus h bar log Z, and show it is given asymptotically by a sum over connected graphs with weight factors and symmetry factors.

Generalize feynman rules to two fields with a four-valent phi squared chi squared vertex. Propagators are h-bar over M squared, and the vertex factor is minus lambda over h-bar.

Explore the two-field Wilsonian effective action with Phi and Chi, featuring quadratic terms and a Phi^2 Chi^2 interaction, and apply Feynman rules to compute correlation functions via connected graphs.

Directly compute the effective action from the path integral by integrating out chi, matching prior Feynman-rule results for two fields, using Gaussian integrals and a log expansion.

Directly calculate the phi squared expectation value from the path integral, derive the propagator, and introduce renormalization via the action and effective action, without using Feynman rules.

Learn how integrating out the y field in a one-dimensional quantum field theory yields an effective S_f(x) with a Gaussian determinant of -d^2/dt^2 + M^2 + (lambda/2) x^2.

See how path integrals reproduce the position–momentum commutation relation for a free particle by discretizing paths, considering t−, t, t+, and taking limits.

derives how correlation functions correspond to time-ordered products in a one-dimensional quantum field theory, using path integrals and time evolution of operators.

Analyze the renormalization group by separating a field theory into low energy and high energy modes at scale lambda zero, yielding a scale dependent effective action and partition function.

Derive the Callan-Symanzik equation from integrating out high-energy modes, reveal how couplings run with scale via beta functions, and define anomalous dimension gamma_phi and wave function renormalization.

Derive how endpoint correlation functions transform under coordinate scaling and energy-scale changes, revealing the scaling and anomalous dimensions of the field.

Explore scale invariant theories via two-point correlation functions and zero beta functions, revealing a Delta phi driven power-law where gamma_lambda two scales as lambda^{-2} (x-y)^{-2 Delta phi}.

Explore how a massive scalar (Klein–Gordon) field in three dimensions yields a Yukawa-type potential, contrasting classical and path integral quantum treatments through Green's functions and correlation functions.

Derive the Yukawa potential in three dimensions from the Klein–Gordon field via Fourier transforms, yielding a short-range potential of minus e^{-m r}/(4 π r).

Explore how coupling constants renormalize under the local potential approximation, showing the running of couplings via integrating out high-energy modes and deriving a beta-function‑like equation.

Derive quantum corrections to the propagator from the path integral by manipulating the action in position and Fourier domains, including interaction terms and discretization steps.

Explore how discretized Fourier modes and phase cancellations in the path integral yield quantum corrections to the propagator; derive perturbative Feynman rules from a discretized action.

Derive the discrete propagator from the path integral using the discrete Fourier transform in the frequency domain, and expand to first order in lambda to reveal quantum corrections.

Describe the first-order quantum correction to the propagator from the path integral, computing b/z0 via Gaussian integrals, and relate it to a loop diagram with a q-integral.

Examine how delta z and delta m squared renormalize the scalar propagator at one loop, enforcing the pole at m0 and deriving delta m squared.

Derive one-loop renormalization of quartic coupling, including delta lambda and the effective lambda f, and show lambda f equals lambda in scheme as lambda0 to infinity and lambda1 to zero.

Derive the Schrodinger equation from the path integral by following Feynman’s method for a one-dimensional particle in a potential, using the Lagrangian, small time steps, and Gaussian integrals.

Explore the double slit experiment via path integrals, deriving the propagation amplitude with gaussian slit profiles and implementing a Matlab simulation to show interference patterns depending on slit separation delta.

We connect path integral calculations of the double-slit experiment to familiar physics-book results, showing how small slit width and geometry produce interference patterns and the classical limit.

Explore how complex contour integration and the residue theorem evaluate complex Gaussian integrals of e^{i a x^2} for a>0, connecting to real Gaussian integrals.

Derive the Lagrange duplication formula using beta and gamma functions, with integral representations and variable changes, to show gamma(z) gamma(z+1/2) = 2^{1-2z} sqrt(pi) gamma(2z).

Derive the Green function from the differential operator d^2/dt^2 - M^2 by Fourier transforming the equation and using the residue theorem, yielding G(t,t') = - e^{-M|t-t'|}/(2M).

Derive the volume and surface of an n-dimensional sphere using radial and angular integrals, relate omega_n to gamma functions, and express V_n(R) as sqrt(pi) R^n / gamma(n/2+1).

Learn the quantum harmonic oscillator via the path integral, using the non-relativistic Lagrangian, classical trajectory plus endpoint-fixed perturbations, and the propagator to reach the energy spectrum.

Divergent series appear in physics and quantum field theory; truncating a path integral expansion yields meaningful results, illustrating asymptotic series and how divergences inform physics.