Quantum Gravity: from Gravitational Waves to Gravitons

Explore how a small perturbation h mu nu around the Minkowski metric reshapes g mu nu, enabling the determinant-based rewrite of the action and the emergence of the graviton.

Demonstrate that the determinant of a matrix equals the exponential of the trace of its natural logarithm, using diagonalization and the series definitions of exp and log.

Rewrite the square root of minus the determinant of the metric as an exponential of the trace log of the perturbation, enabling a perturbative expansion.

Rewrite the natural log of a metric perturbation as a series, deriving tr ln(delta + h) = h − h^2/2 + O(h^3) and translating to index notation.

The lecture expands sqrt(-g) in h_mu nu, with h as its trace, yielding sqrt(-g) approximately 1 + 1/2 h - 1/4 h_mu nu h^{mu nu} + 1/8 h^2 + O(h^3).

Derive the variation of a matrix determinant as the trace of its inverse times the perturbation, proving the Jacobi identity with general relativity's metric determinant.

Demonstrate the Neumann series for a matrix inverse, with b inverse equal to the sum of (I − b)^k. Assume convergence as n goes to infinity.

Explore expressing the inverse metric g^{mu nu} as a perturbative expansion about the Lorentz metric, yielding eta^{mu nu} minus h^{mu nu} plus second-order corrections h^{mu}_{alpha} h^{alpha nu}.

Express the connection to first order in h, using g_mu nu = eta_mu nu + h_mu nu and g^{mu nu} = eta^{mu nu} - h^{mu nu}, rewriting Einstein's equations toward graviton description.

Rewrite the derivative of the connection as one half of d_mu d_nu h, using the symmetry of h and the Minkowski metric to raise indices, up to second order.

Derive the first-order Ricci tensor R_mu nu^1 from a perturbed metric, using h_mu nu and the d'Alembert operator, and note higher-order o(h^2) corrections.

Compute the second-order part of the Ricci tensor by inserting Christoffel symbols from the perturbation h, contract with eta, and relate to the second-order Einstein tensor, laying groundwork for gravitons.

Explore how perturbing the metric rewrites the general relativity action into a second-order lagrangian for a massless spin-2 field, revealing the graviton dynamics.

Demonstrate how the h_mu nu transformation leaves the Lagrangian invariant, detailing cancellations of cross terms and total derivatives that preserve the action.

Derive a gauge condition from the Lagrangian symmetry, obtaining a transversality-like constraint on h mu nu, and show how the field and wave equations illuminate gravitational waves and gravitons.

Vary the lagrangian with respect to h mu nu and chi nu to derive the perturbed-metric field equations, including symmetry considerations and total-derivative terms.

Apply the derivative and the d'Alembert operator to the perturbed metric, cancel terms, and obtain equation two.

Vary lagrangian with respect to k_nu to derive the vector field's equation of motion, yielding chi_nu equals H_mu nu and confirming it as equation three tied to a prior constraint.

Substitute chi nu in terms of h mu nu to derive the gravitational wave equation using the d'Alembert operator on h mu nu and its source t mu nu.

Rewrite the gravitational wave equation by multiplying by a new h_mu_nu to define T_mu_nu, showing equivalence to d'Alembert h and guiding the solution approach.

Solve the wave equation by replacing the source with a four-dimensional Dirac delta, derive the delta function in momentum space, and set up convolution with a general source.

Enforce causality by inserting an epsilon in the propagator denominator, analyze p0 poles, and close the contour accordingly; relate momentum-space delta plus to position-space propagators from a Dirac delta source.

Derive the full solution to Einstein's field equations by convolving the propagator with the four-dimensional delta source to obtain h_mu_nu, and discuss causality and the path integral.

It introduces the path integral formulation of quantum field theory for gravity, replacing the classical field h_mu_nu with its vacuum expectation and computing vacuum-to-vacuum transition amplitudes using a source T_mu_nu.

Derive the vacuum-to-vacuum transition amplitude for a weak gravitational field by expanding the metric into a small perturbation h_mu nu and employing quantum vacuum expectations.

Rewrite the vacuum to vacuum transition amplitude in momentum space by substituting delta mu nu sigma rho and applying the Fourier transform, connecting the result to polarization tensors and gravitons.

introduces the completeness formula in Minkowski space with four vectors p, p bar, and polarization vectors, derives a frame where p plus minus simplify, and discusses implications for vacuum amplitude.

Apply completeness formula to rewrite Minkowski metric in momentum and polarization tensors. Relate the log of the vacuum-to-vacuum amplitude to momentum-space expressions via the d'Alembert operator, noting gauge symmetry arbitrariness.

Rewrite the vacuum-to-vacuum transition amplitude in a compact form by defining symmetric tensors and evaluating sums over lambda and lambda prime, using Kronecker delta relations.

Explore how quantum field theory describes gravitons, defining graviton polarization tensors, two polarization states, and their role in the vacuum-to-vacuum transition amplitude using the completeness relation and momentum integration.

Recovering the classical result, this lecture derives the zero-plus zero-minus inner product and shows the vacuum evolution equals exp(i times the integral of H dt), yielding gravitational potential energy.

Relate gravitons to Newtonian gravity via the gravitational potential and eight pi g; define the average number of gravitons of arbitrary momenta and polarizations and its Poisson emission statistics.

Explore polarization of gravitational waves propagating along the z axis, using perpendicular vectors and polarization tensors, then interpret dilation along x, contraction along y, and shear under a 45-degree rotation.

Visualizes the two polarization states of gravitational waves and how the tensor epsilon_mu_nu deforms a ring of test particles, causing dilation or contraction at 45-degree orientations.

Derive the path integral for a single particle from the Schrödinger equation, showing time evolution as a sum over all paths weighted by e^{i S}, with no single trajectory privileged.

Explore the path integral formulation summing over infinite trajectories via the kernel, action, and Lagrangian, with hbar and Wick rotation to Euclidean space.

Explore the Fourier transform of the Heaviside step function by decomposing the integral into cosine and sine parts, producing delta and signum results within distribution theory.

Derive the Poisson distribution from a time interval by modeling events with rate lambda and small time chunks, using binomial to Poisson limits via Stirling’s approximation.