QFT in curved spacetime: Hawking radiation, Unruh effect

Reconcile quantum mechanics with special relativity by deriving the Klein-Gordon equation from relativistic energy and momentum, and introduce quantum field theory with quantized fields and the d'Alembert operator.

Derive the Klein-Gordon equation from classical field theory by varying the action with respect to the scalar field phi, using a Lagrangian density that depends on phi and its derivatives.

Quantize the classical field by postulating field–momentum commutators with a Dirac delta, derive the Klein-Gordon Hamiltonian, and expand the real scalar field into plane waves using a(k) and a†(k).

Quantize the Klein-Gordon field by expressing phi and its canonical momentum pi in terms of a(k) and a†(k), derive their commutator, and obtain the Hamiltonian in terms of these operators.

Derive and analyze the commutators of a(k) and a dagger(k′) for the Klein-Gordon field, and express the Hamiltonian in terms of a(k) and a dagger(k) to obtain the spectrum.

From a single field Lagrangian density, derive the energy-momentum tensor as a conserved object, with t00 as the Hamiltonian density and t0i as the momentum components.

Reframe the Klein-Gordon Hamiltonian, isolate the finite part h_n, and show that annihilation and creation operators lower or raise energy by omega, defining the spectrum.

Identify the ground state (vacuum) as the state annihilated by A(k), then build one- and two-quanta states with a†(k) giving energies ω(k) and ω1+ω2 with momenta k and k1+k2.

Demonstrate that the invariant volume element D3P over omega is preserved in QFT by rewriting delta(pμ pμ − m^2) in terms of p0 with roots ±omega and using delta function identity.

Introduce light cone coordinates in a moving frame, relate them to proper time tau, and set up the groundwork for the Unruh effect and Hawking radiation using u and v.

Derive Bogoliubov transformations between field modes in Minkowski and accelerating frames, defining alpha and beta coefficients and linking annihilation operators to reveal the Unruh effect.

Derives the Bogoliubov transformations and the alpha_omega and beta_omega coefficients, showing how their magnitudes satisfy |alpha_omega|^2 = e^{2 pi omega / a} |beta_omega|^2, linking to the Unruh effect.

Learn how Bogoliubov transformations and the normalization condition relate alpha and beta, using commutation relations to approach the Unruh effect and Hawking radiation.

Show how Bogoliubov transformations connect alpha and beta modes to the Minkowski vacuum, yielding a Bose-Einstein–like spectrum for accelerated observers—the Unruh effect.

Estimate the lifetime of black holes using mass cubed scaling from Hawking radiation, illustrating orders of magnitude—from sun-mass black holes (~10^74 s) to tiny ones near Planck time (~10^-44 s).

Derive gravity as an emergent, entropic force from the holographic principle and the Unruh effect, using boundary bits on a sphere and equipartition to recover Newton's law.

Explore how a non-rotating black hole's horizon behaves thermally, linking Hawking temperature and area via loop quantum gravity, area spectra, and the Immirzi parameter.

Derive the path integral for a single particle, connecting the Schrödinger evolution to a sum over all q(t) paths weighted by exp(iS[q]), with kernel, completeness relation, and the action S[q].

Understand how the path integral kernel sums over trajectories between initial and final field configurations, weighted by e^(i S/ħ). Relate this to the Lagrangian, fields, and z under Wick rotation.

Derive the path integral from a classical field theory to connect classical mechanics with quantum field theory, using Schwinger's principle and the exponential of i times the action over ħ.

Derive the Schrodinger equation from the path integral, using Feynman's method in one dimension with a one-particle Lagrangian and a propagator framework.

Explore the double slit experiment through path integrals, derive interference patterns from Gaussian slit distributions with a free-particle action, and visualize results in MATLAB.

The lecture links path-integral analysis of the double-slit to standard textbook results, showing how small slit width and geometry yield interference maxima via momentum and wavelength.

Show how a Feynman path integral linked to the double-slit experiment is solved by completing the square and evaluating a Gaussian integral under convergence conditions.

Examine why quantum gravity is hard by expanding the Einstein-Hilbert action around Minkowski space, showing how higher-order terms in the metric perturbation h grow with energy and threaten consistency.

The Einstein field equations cannot be quantized due to nonlinearity and background-independent spacetime; explore perturbative quantization, semiclassical gravity, loop quantum gravity, string theory, path integral, and Wheeler-DeWitt quantum gravity.

Examine the Casimir effect arising from boundary conditions on the electromagnetic field between conducting plates. Consider speculative extensions to gravity and curved space time, including near black holes.

Derive covariant action for a scalar field coupled to gravity, perform a Wick rotation to the Euclidean framework, and define the Euclidean action using the covariant d'Alembert operator and potential.

Reformulates the eigenvalue problem of the operator f in a generalized Hilbert space to compute the functional determinant and the effective action, via a Hermitian operator O related to F.

derive the finite part of functional determinants with zeta function regularization, by expressing det o as exp(-d/ds zeta_o(s) at s=0) and using analytic continuation of zeta_o(s).

Explore how the heat kernel and its trace relate to the zeta function to extract quantum corrections to general relativity, including Ricci scalar and Ricci tensor–based terms, and higher orders.

Compute the heat kernel for the operator describing a scalar field’s interaction with gravity, using perturbation theory around flat space to derive the curved space effective action.

Compute the heat kernel trace from k0 and k1 matrix elements. Use Fourier transform of the delta function in 2ω dimensions and Gaussian integration to yield the heat kernel.

Compute the diagonal heat kernel in curved space using first-order metric perturbations and the Ricci scalar, revealing gravitational corrections and why general relativity is not renormalizable.

Derive the Lorentz Lie algebra from the representation theory of the Lorentz group, using infinitesimal transformations and antisymmetric m_mu_nu, obtaining the commutation relations that relate angular momentum and boosts.

Explore how gamma matrices form the Clifford algebra and define spinor transformations under Lorentz symmetry, showing that a 360-degree rotation yields minus the identity for spinors.

we prove that the vector transformation generators satisfy the lorentz lie algebra, with sigma_mu nu written as minus i and the commutator reproducing the algebra in index notation.

Derives that the spinor transformation generators obey the Lorentz Lie algebra by relating sigma mu nu to gamma matrices and their commutators.

Explore how to compute the exponential of a matrix via diagonalization: A = lambda d lambda inverse, e^A = lambda e^d lambda inverse, with eigenvectors and eigenvalues in 2x2 example.