Quantum field theory and the Ricci flow Daniel Friedan Department of Physics & Astronomy Rutgers the State University of New Jersey, USA Natural Science Institute, University of Iceland Mathematics Colloquium Reykjavík, October 20, 2008 1 / 22
Aside Two topics I m interested in at the moment: The bundle G 2 S 6 = G 2/SU(3) and the Yang-Mills flow. Using 1+1d quantum field theories for quantum computation. 2 / 22
Quantum mechanics of a single degree of freedom Start with one degree of freedom: q R. The Hilbert space of states is H = L 2(R), consisting of wave functions ψ(q). The time evolution is generated by a hamiltonian operator H: q(t) = e ith q e ith H = H, H 0 The Heisenberg algebra:»i q, q = i The free particle hamiltonian: 2 «H = 1 2 q 2 3 / 22
The harmonic oscillator The harmonic oscillator hamiltonian: H = 1 2 2 q 2 «+ 1 2 E 2 q 2 E > 0 destruction/creation operators: a = 1 2 q + Eq «a = 1 2 q + Eq «[a, a ] = E H = a a + 1 2 E [H, a ] = Ea [H, a] = Ea eigenstates: Hψ n = (ne + 1 2 E)ψn aψ0 = 0, ψn = (a ) n ψ 0, n = 1, 2, 4 / 22
N degrees of freedom in a line q(0ɛ) 0ɛ q(1ɛ) 1ɛ q(2ɛ) 2ɛ q(3ɛ) 3ɛ q(x) x q(l ɛ) q(l) L ɛ L = Nɛ boundary condition: q(0) = q(l) (i.e., x ɛ N ) Hilbert space: H = x L 2(R) = L 2(R N ) states: ψ(q(1ɛ), q(2ɛ),, q(nɛ)) hamiltonian: H = 1 X 1 2 ɛ 2 q(x) + 1 «[q(x) q(x 2 1)]2 2 x (local!!!) N translation symmetry: x x + ɛm (mod ɛn), q(x) q(x + ɛm) 5 / 22
Equivalent to N harmonic oscillators Fourier transform in N : q k = 1 X e ikx q(x) p k = 1 X e ikx i N N q(x) x x k = 2πm ɛn = 2πm L 1 ɛ N [p k, q k ] = iδ k,k H = X k 1 2 p k p k + 1 2 E k 2 q k q k E k = 1 (2 2 cos ɛk)1/2 ɛ a k = 1 2 ( ip k + E k q k ) [a k, a k ] = 0 [a k, a k ] = E k δ k,k H = X k a k a k + 1 «2 E k [H, a k ] = E k a k [H, a k ] = E k a k The ground state: a k ψ 0 = 0 Hψ 0 = X k 1 2 E k ψ 0 Note that E 0 = 0. The zero-mode q 0 is a free particle. 6 / 22
The continuum limit Send N, ɛ 0 holding L = Nɛ fixed. Now x parametrizes the circle of length L. The oscillators are indexed by k = 2πm ɛn = 2πm L m j 12 N,, 12 ff N The oscillator energies become E k = 1 ɛ (2 2 cos ɛk)1/2 k for k 1 ɛ The hamiltonian has a limit, once the ground state energy is subtracted, H X k 1 2 E k H ren = X a k a k k 2π L In taking the limit, we have discarded the states with energy eigenvalues of order 1/ɛ. 7 / 22
Quantum fields Define δ δq(x) = 1 ɛ q(x) xq = q(x) q(x ɛ) ɛ In the continuum limit, H = ɛ X x " 1 2» δ δq(x), q(x ) «# 2 δ + 1 δq(x) 2 ( xq)2 xq q x δ(x x ) q(x) and δ δq(x) become operator-valued distributions on the circle, and H X k 1 2 E k H ren = L 0 dx " 1 2 : δ δq(x) «2 : + 1 «2 q :# 2 : x 8 / 22
Discussion The continuum limit is physically motivated. We quite often investigate physical systems in which many degrees of freedom are distributed locally in space, separated by distances of order ɛ, much smaller than our experimental apparatus can probe. With our limited resources, we cannot produce or detect the high energy states that we discarded in taking the continuum limit. Everything our instruments can detect will be described to good accuracy by a continuum limit. In the continuum limit, what we observe is described by an algebra of operator-valued distributions O α(x) smeared by smooth functions. A quantum mechanics with such an algebra of observables (and satisfing a few additional conditions) is a quantum field theory. Can we say anything about what quantum field theories are possible? The only substantial progress on this question has been for qfts in 1+1 dimensions (1 space dimension and 1 time). 9 / 22
Discussion (cont.) So far, we have discussed the most trivial example, the free massless scalar field in 1+1 dimensions. It is exactly solvable, H being quadratic in Heisenberg operators, so we can take the continuum limit explicitly. We have effective techniques perturbation theory for studying hamiltonians that are close to free (i.e., close to quadratic). First, a trivial generalization: q i (x) i = 1,, n H = 1 X nx 1 2 ɛ 2 q i (x) + 1 h i «2 q i (x) q i (x 1) 2 2 x i=1 = 1 X 1 ɛ 2 δij q i (x) q j (x) + 1 «2 δ ij xq i xq j x H is the sum of n free massless scalar hamiltonians. The Hilbert space is the tensor product of n free massless scalar Hilbert spaces. 10 / 22
The path integral The free particle, H = 1 2 2 q 2, as path integral q T e TH q 0 = Dq e S(q) S(q) = over paths q(τ) from q(0) = q 0 to q(t ) = q T. T 0 dτ 1 ( τ q)2 2 This is just the path integral for the heat kernel. Analytically continue in T to get the real time evolution operator e ith. Derivation: q T e TH q 0 = dq T δ dq 2δ dq δ q T e δh q T δ q 3δ e δh q 2δ q 2δ e δh q δ q δ e δh q 0» q τ+δ e δh q τ = exp δ 2 qτ+δ q 2 τ δ 11 / 22
The path integral (2) Path integral for the harmonic oscillator, H = 1 2 2 q 2 + 1 2 E 2 q 2, q T e TH q 0 = Dq e S(q) S(q) = T 0 dτ 1 2 ( τ q)2 + 1 «2 E 2 q(τ) 2 For the free massless scalar field, q T e TH q 0 = Dq e S(q) S(q) = T 0 dτ ɛ X x 1 2 ( τ q)2 + 1 «2 ( xq)2 Now the integral is over paths τ q(x, τ) for each x, i.e., an integral over all maps (x, τ) q(x, τ). Write this in the continuum S(q) = (leaving the cutoff ɛ implicit). T dτ L 0 0 dx 1 2 τ q τ q + 1 «2 xq xq 12 / 22
The path integral (3) or, writing x µ = (x, τ), S(q) = d 2 x 1 2 γµν µq νq where γ µν is the 2d metric: (ds) 2 = γ µνdx µ dx ν = (dx) 2 + (dτ) 2 A property that was hidden in the hamiltonian formalism becomes manifest: 2d euclidean symmetry 1 + 1d Poincaré symmetry after analytic continuation to real time more generally, 2d covariance: γ µν(x, τ) relativistic quantum field theory 13 / 22
Perturb n free massless scalars For n free massless scalar fields q i (x, τ), S(q) = d 2 x 1 2 δ ij γ µν µq i νq j We want to investigate the possible continuum limits of small, non-quadratic, perturbations S S + S. The perturbations should be local in the q i (x, τ) and covariant in 2d. S = d 2 x q k1 ( µ1 q i 1 ) Further limit perturbations by requiring them to be dimensionless (to be justified later). Dimensional analysis: 1 ɛ x µ is a number, so dim(x µ ) = 1 ɛ µ is a number, so dim( µ) = +1 S(q) is a number, so dim(s) = 0 therefore dim(q i ) = 0 So S can contain any number of q i, and exactly two derivatives µ. 14 / 22
The general dimensionless perturbation S(q) = d 2 x 1 2 δ ij + g ij,k q k + 1 «2! g ij,k 1 k 2 q k 1 q k 2 + γ µν µq i νq j The numbers g ij,k1 k 2 are called coupling constants. Define the q-dependent matrix g ij (q) = δ ij + g ij,k q k + 1 2! g ij,k 1 k 2 q k 1 q k 2 + so S(q) = d 2 x 1 2 g ij(q) γ µν µq i νq j This is covariant in R n. The matrix g ij (q) is a Riemannian metric on R n (ds) 2 = g ij (q)dq i dq j 15 / 22
The general nonlinear model Now it is more or less obvious that we should regard R n as a coordinate patch on a general n-dimensional manifold M. The path integral is an integral over maps (x, τ) q(x, τ) M q T e TH q 0 = Dq e S(q) S(q) = d 2 x 1 2 g ij(q) γ µν µq i νq j H = Dq = Y (x,τ) M dx 1 2 g ij (q) q i (x) dvol(q(x, τ)) q j (x) + 1 «2 g ij xq i xq j H = x L 2(M) Again, everything is understood to be cutoff at distance ɛ. The question is: can we take the continuum limit ɛ 0? 16 / 22
The general nonlinear model (2) Introduce a parameter α 0, Dq e S(q) S(q) = d 2 x 1 1 2 α g ij(q) γ µν µq i νq j The path integral is dominated by the constant maps q(x, τ) = q 0 M. Write q(x, τ) = q 0 + δq(x, τ) Dq e S(q) = dq 0 Dδq e S(q 0+δq) M S(q 0 + δq) = d 2 x 1 1 2 α g ij(q 0 + δq) γ µν µδq i νδq j Scale δq i (α ) 1/2 δq i to see that this is a free theory plus small perturbations. We have effective techniques (Feynman diagrams) for systematically evaluating the path integral as a formal power series in α. [Simple example: M = S 1 (or M = T n ). Each fluctuation integral is free, only the zero-mode integral is nontrivial. This example is an excellent exercise.] 17 / 22
The renormalization group flow Imagine increasing ɛ slightly, ɛ ɛ + ɛ. The claim is that we can make a compensating change in the action, S S + S, so that everything we can observe remains the same. Start with our original path integral, with cutoff ɛ and action S(φ). Imagine integrating out a small fraction 2 ɛ/ɛ of the integration variables q i (x, τ). We are left with an integral over the remaining variables q i (x, τ), but with a slightly modified action S + S. Now our integration variables are slighlty less dense on the x, τ surface. Essentially, the cutoff is now ɛ + ɛ. But nothing observable in the path integral has changed. In principle, S + S contains all possible local interactions. Dimensional analysis tells us that terms with more than two derivatives µ are suppressed by positive powers of ɛ, so we are justified in ignoring them. In the Feynman diagram expansion the formal power series in α this dimensional analysis is rigorous. So S + S = S(q) = d 2 x 1» 1 2 α g ij(q) + 1 α g ij(q) γ µν µq i νq j 1 «ɛ α g ij(q) = β ij (q) ɛ 18 / 22
The renormalization group flow (2) We have constructed a flow ɛ 1 ɛ α g ij(q) = β ij (q) such that, if we increase ɛ and move our coupling constants along the flow, nothing observable changes. There exists a continuum limit if the flow can be run backwards, all the way to ɛ = 0. The only circumstance where we know this can be done is when the flow goes backwards to a fixed point β ij = 0 ( ultraviolet safety ). (We can always define the continuum limit order by order in α.) β ij can be calculated as a formal power series in α, to all orders. It is a polynomial in the derivatives of g ij (q), covariant on M, so each term in the formal power series must be a polynomial in the curvature tensor and its covariant derivatives ɛ 1 ɛ α g ij = β ij = R ij + 1 2 α R iklm R kjm j + O(α 2 ) A slightly subtlety: the fixed point equation is actually β ij = i v j + j v i since a change of metric that is just a reparametrization of the q i is just a change of variables in the functional integral. 19 / 22
The Ricci flow The Ricci flow d dt g ij = R ij was used by Perelman (following on Hamilton and Thurston) to prove the Poincaré conjecture in n = 3 dimensions. The renormalization group flow for the general nonlinear model ɛ 1 ɛ α g ij = R ij + 1 2 α R iklm R kjm j + O(α 2 ) looks closely related. But note that it does not go to the Ricci flow when α 0. We would need to define Then the rg flow becomes t = 1 α ln ɛ d dt g ijg ij = R ij + 1 2 α R iklm R kjm j + O(α 2 ) which does become the Ricci flow in the limit α 0. 20 / 22
Elaborations more 2d symmetry (2d supersymmetry) 1d boundaries and boundary couplings (boundary conditions) 21 / 22
An analytic version of Riemannian manifold? geometric objects = analytic objects Riemannian manifolds = 1+1d quantum field theories Can any mathematical use be made of this? 22 / 22