MATH 273 / STATS 395 Topics in Mathematical Physics Spring 2017 Lecture 99: May 17 Lecturer: Sourav Chatterjee Scribe: Sicong (Scott) Zhang

Physics Motivation

Quantum mechanics for single particle in 1D let q be location, t be time.

2 2 A state is ψ(q, t) ∈ L (R) with ||ψ||L2 = 1. |ψ(q, t)| is the probability density at q at time t. Schrodinger’s equation: ∂ψ 1 ∂2ψ i = − + V (q)ψ(q, t) = Hψ ∂t 2 ∂q2

1 ∂2 where V (q) is the potential function, and H = − 2 ∂q2 + V is the energy (Hamiltonion) function. Solution to Schrodinger’s equation: ψ(·, t) = e−itH ψ(·, 0) where e−itH is unitary because H is symmetric (Hermetian?) Feynman’s path integral solution: “integrate over function space”. Z 0 0 0 ψ(q, t) = Kt(q, q )ψ(q , 0)dq where the Schrodinger kernel

Z " Z T Z T !# 1 2 Kt(q0, q1) = exp i q˙(t) dt − V (q(t))dt Dq 2 0 0 where Dq is the “Lebesgue measure” on paths (might not exist rigorously.) The integral is over all paths such that q(0) = q0, q(T ) = q1. Mark Kac developed the real version of path integral solution, for the heat equation

∂ψ 1 ∂2ψ = − V ψ = −Hψ ∂t 2 ∂q2

Solutions are ψ(·, t) = e−tH ψ(·, 0)

Kac proved rigorously that indeed Z 0 0 0 ψ(q, t) = Kt(q, q )ψ(q , 0)dq

99-1 99-2 Lecture 99: May 17

where Z Z T Z T ! 1 2 Kt(q0, q1) = exp − q˙(t) dt − V (q(t))dt Dq. 2 0 0

1 R T ˙ 2 Although Dq is still undefined, exp(− 2 0 (q) dt)Dq is well-defined: it is the probability law of Brownian motion. Take any path q(t), 0 ≤ t ≤ τ, imagine q has analytic continuation to the imaginary axis. Wick rotation trick: q1(t) := q(it). Z T 1 Z T 1 i [ q˙(t)2 − V (q(t))]dt = [ q˙(it)2 − V (q(it))]dt 0 2 0 2

Finitely many particle in 1D

Let ψ(q1, q2, t) be the quantum state of 2 particles in 1D. Schrodinger equation becomes

∂ψ 1 2 1 2 i = − ∂ ψ − ∂ ψ + V (q1, q2)ψ(q1, q2, t) ∂t 2 q1 2 q2

2 2 2 For example, V (q1, q2) can be q1 + q2 + (q1 − q2) . Feynman representation of the Schrodinger kernel is similarly

Z " Z T Z T Z T !# 1 2 1 2 exp i q˙1(t) dt + q˙2(t) dt − V (q(t))dt Dq 2 0 2 0 0

Consider n particles in 1D, each by itself will behave like a simple harmonic oscilator, but there is some weak P 2 P 2 interaction between any pair, i.e. V (q1, q2, ··· , qn) = qi +  (qi − qi+1) . infinitely many particles

QFT deals with infinitely many particles, one at each x ∈ R. (We are considering 1 space dimension and 1 time dimension.) Then q1(t), q2(t), ··· becomes a function q(x, t). R 2 R 2 R R 2 q = qt : R → R, V (q) = q(x) dx + (∂xq) dx or V (q) = U(q(x))dx + (∂xq) dx It is impossible to write down the Schrodinger PDE. However, Feynman integral can still be written down: " !# Z 1 Z Z T Z Z T 1 Z Z T exp i (∂ q)2dtdx − U(q(x, t))dtdx − (∂ q)2dtdx Dq 2 t 2 x R 0 R 0 R 0 integral is over the space of surfaces. (NOTE: where did the last 1/2 come from?) Wick rotation, t → it gives ! Z 1 Z Z T Z Z T 1 Z Z T exp − (∂ q)2dtdx − U(q(x, t))dtdx − (∂ q)2dtdx Dq 2 t 2 x R 0 R 0 R 0 which is exactly the 2D Gaussian free field. Lecture 99: May 17 99-3

Constructible QFT

Arthur Wightman gave a list of criteria that allow you to do the analytic continuation rigorously, and built an actual QFT. Simplification was given by Osterwalder and Schrader. Yang-Mill theory is the limit of lattice gauge theory. But QFT is hard to construct. Many of them are Gaussian, i.e. “trivial”. It has connection with random surface theory, and Liouville quantum gravity.