Rigorous Functional Integration with Applications to Nelson s and the Pauli-Fierz Model

Transcription

1 Rigorous Functional Integration with Applications to Nelson s and the Pauli-Fierz Model József Lőrinczi Zentrum Mathematik, Technische Universität München and School of Mathematics, Loughborough University

2 Nelson Model 1 Definition (Nelson s Hamiltonian) H N = H p + H f + H i on L 2 (R d, dx) F with H p = ( (1/2) + V(x)) 1 H f = 1 R d ω(k)a (k)a(k)dk H i = ( ϱ(k)e R ik x a(k) + h.c. ), ϱ δ UV cutoff d ω(k) self-adjoint on D(H p ) D(H f ) if ϱ ω 1/2, ϱ ω 1 L 2 unique, strictly positive ground state Ψ L 2 F if ϱ ω 3/2 L 2 IR cutoff [Spohn 1998]

3 Pauli-Fierz Model with spin 2 Definition (Pauli-Fierz model with spin) H PF = 1 2 (σ ( i 1 ea))2 + V H f = 1 2 ( i 1 ea)2 + V H f e 2 σ B with A µ (x) := 1 2 j=±1 j=±1 e µ (k, j)e ν (k, j) = δ µν k µk ν k 2 ( ) ϱ(k) e µ (k, j) e ik x a(k, j) + h.c. dk ω(k) ground state: Bach-Fröhlich-Segal 1999, Griesemer-Lieb-Loss 2001

4 Aims 3 derive and prove qualitative behaviour of system, i.e., for A of interest (Ψ, AΨ) L 2 F how serious are IR & UV cutoffs, i.e., does there exist a ground state in L 2 (R d, dx) F if cutoff conditions are lifted how robust are qualitative properties, i.e., by removing cutoff do ground state expectations change Method: use tools of stochastic analysis via Feynman-Kac formula - rough paths analysis - Lévy processes - cluster expansion

5 Feynman-Kac formula for Nelson s model 4 use jointly ground state transform and Wiener-Itô transform e thp = P(φ) 1 -process R t X t R d path measure dn 0 t (X) = e R t 0 V(Xs)ds dw t (X) e th f = Ornstein-Uhlenbeck process R t ξ t S (R d ) path measure G, E G [ξ t (f)] = 0 E G [ξ t (f)ξ s (g)] = f(k)ĝ(k)(2ω(k)) 1 e ω(k) t s dk R d H i (ξ t ϱ)(x) Theorem (Feynman-Kac-Nelson) (F, e t H N G) L 2 = F(X 0, ξ 0 )G(X t, ξ t ) e R t 0 (ξt ϱ)(xs)ds d(nt 0 G) }{{} P t = path measure int. syst.

6 Structure of path measure 5 mixture of Gaussian and Gibbsian ( ) P T ( ) = P T ( X)dN T (X) on C [ T, T], R d S (R d ) with dn T = 1 e R T R T T T W(Xt Xs,t s)dtds dnt 0 Z T Z T = e R T T (ξt ϱ)(xt)dt d(nt 0 G) = e R T R T T T W(Xt Xs,t s)dtds dnt 0 W(x, t) = 1 4 Rd ϱ(k) 2 ω(k) cos(k x)e ω(k) t dk Theorem N = lim T N T = P = lim T P T

7 Gibbs measure on path space 6 Theorem if e = ϱ(x)dx small and V(x) x 2a, a > 1, then (1) N = lim Tn N Tn (2) N uniquely supported on C(R, R d ) (3) N -a.s. X t C(log( t + 1)) 1/(a+1) + Q(X) (4) C, γ > 0 s.t. F, G bounded cov N (F(X s ), G(X t )) C sup F sup G s t γ + 1 (5) T > 0, N T N 0 and dn T /dn 0 dn/dn 0 L-Minlos 2001, Betz-L 2003

8 The two pictures 7 Fock space quantization H 0 := L 2 (R d S (R d ), dp 0 ) L 2 (R d, dx) F H N H N Φ Ψ Euclidean quantization H := L 2 (R d S (R d ), dp) E P [F(X 0, ξ 0 )G(X t, ξ t )] = ( F, T t G) H T t = e theuc Euclidean Hamiltonian, s.a., semibd 1 is g.s. of H euc ; unique since T t positivity improving

9 Ground state expectations: general theorem 8 Theorem P P 0 = Φ 2 = dp dp 0 Theorem ] B s.a. in L 2 (G), E N [ B : e ξ(f ± X ) : 2 L 2 (G) <, g L (R d ) [ (Φ, g B Φ) H 0 = E N M B g(x 0 ) e 2 R 0 R ] W(X 0 s X t,s t)dsdt with M B = ( ) : e ξ(q X ) :, B : e ξ(q+ X ) : L 2 (G) q ± X = R ± ϱ(k) cos(k X t )e ω(k) t dt Betz, Hiroshima, Minlos, L, Spohn 2002

10 Boson number distribution 9 with π n : F F (n), p n := (Φ, 1 π n Φ) H 0 0 I := ϱ 2 ω 3 dk, w := W(X s X t, s t)dsdt R 0 Corollary (1) [ p n = (1/n!) E N ( 2w) n e 2w] (2) p n (I n /n!) e I (3) (Φ, e αn Φ) H 0 <, α > 0 for massive bosons p n (D n /n!) e I, D < I

11 Average field strength and fluctuations 10 with χ := ψ 2 Φ(x, ξ) 2 dg(ξ), V ω := 1/ω 2 Corollary (1) (Φ, ξ(k)φ) H 0 = ϱ(k) (2π) d/2 ω(k) 2 (2) (Φ, ξ(x)φ) L 2 (P 0 ) = (χ V ω ϱ)(x) 1 x, as x (3) field fluctuations increase on coupling particle to field

12 Particle localization 11 Lemma (Diamagnetic Inequality) with f, g H 0 ( (f, e t H N g) H 0 e ti f L 2 (G), e t H p g L 2 (G) Theorem with V(x) x 2a, C 1, C 2 > 0 χ(x) C 1 e C 2 x a+1 if total charge small, then C 3, C 4 > 0 χ(x) C 3 e C4 x a+1 ) L 2 (N 0 )

13 IR divergence: general 12 Definition (IR divergence) H N is IR divergent if has no ground state in H 0 Theorem (Characterization of IR divergence) suppose N exists; then: H N is IR divergent P P 0 L-Minlos-Spohn 2002a

14 Cases 13 Theorem (3D IR divergence) no IR assumption; if d = 3, then (1) particle charge small = H N IR divergent (2) ϱ 0, lim x V(x) = = H N IR divergent Theorem (Higher-dimensional IR regularity) if d 4, then H N has unique g.s. Φ H 0 and Φ 2 = dp/dp 0 Corollary H N IR divergent H N has no ground state in L 2 (R d ) F in 3D H 0 H resp. H N H euc in > 3D H 0 H resp. H N H euc

15 3D IR-renormalization 14 Theorem with h s.t. ĥ R, even, bd and ĥ(0) = 1 HN ren ϱ(k) ( ) = dk e ik x ĥ(k) a(k) + h.c. R 3 2ω(k) R3 ) dk (e ϱ(k) 2 ω(k) 2 ĥ(k) ik x ĥ(k) 1 + H p H f is unitary equivalent with H euc and has a unique strictly positive ground state L-Minlos-Spohn 2002b

16 UV divergence 15 point charge limit ϱ(x) δ(x) scaling charge distribution ϱ Λ (k) = ϱ(k/λ) UV limit Λ ground state energy (perturbatively) E Λ = e 2 ϱλ (k) 2 2ω(k) 1 ω(k) + k 2 /2 dk + O(e4 ) log Λ

17 UV operator renormalization Gross transform T Λ = e ϱλ (k) 1 ( ) 2ω(k) ω(k) + k 2 e ik x a(k) + h.c. dk /2 Gross transformed Nelson Hamiltonian e T H N e T = terms dependent on a vector potential + E Λ Theorem (UV renormalized Hamiltonian) H UV N := lim Λ (etλ H N e TΛ E Λ ) exists, and for small e is self-adjoint and bd below Nelson 1964

18 Regularized functional interaction interaction (ill defined) W(x, t) = R 3 1/(2 k ) cos(k x)e k t dk Definition (regularized interaction) W ε (x, t) = R 2 k e ε k cos(k x)e k t dk, ε > 0 3 Lemma ε > 0, [ T, T] R T t WT(X) ε = 2 dt T T +4Tϕ ε (0, 0) 2 ϕ ε (X t X s, t s)dx s T T ϕ ε (X t X T, t + T)dt with ϕ ε (x, t) = R 3 (2 k ( k + k 2 /2)) 1 e ε k 2 cos(k x)e k t dk

19 Renormalized functional interaction Definition (renormalized regular pair potential) W ε T(X) = W ε T(X) 4Tϕ ε (0, 0) Lemma (1) W T ε(x) converges to a random variable W T 0 (X) as ε 0 (2) with ψ(x, t) = R 3 ( k ( k + k 2 /2) 2 ) 1 cos(k x)(e k t 1) T W T(X) 0 = T T T T T T dx s 2 ψ(x t X s, t s)dx t s ψ(x T X s, T s)dx s 2ψ(X T X T, 2T) ψ(x t X T, t + T)dX t

20 Renormalized Gibbs measure Lemma e small = W 0 I (X) exponentially integrable wrt Brownian bridge Theorem (renormalized Gibbs measure) T > 0, small e > 0, x, y R 3 dµ T (X x, y) := w lim ε 0 dµ ε T(X x, y) Gubinelli-L 2007a&b e W ε T (X) R T := w lim ε 0 ZT ε dw (x, y) 0 = e ew T (X) R T dw Z T 0 (x, y) T V(Xs)ds T V(Xs)ds x,y T (X). x,y T (X)

21 FK-Formula for PF model with spin Theorem (F, e thpf G) = E x,σ lim ε 0 et σ=±1 [ dx e R ] t 0 V(Bs)ds J 0 F(B 0, σ 0 )e Ut(ε) J t G(B t, σ t )dp 0 Q with U t (ε) = ie + 3 t A µ (j s λ( B s ))db µ s t µ=1 0 0 t+ 0 H on (B s, σ s, s)ds log ( H off (B s, σ s, s) εψ ε (H off (B s, σ s, s))) dn s σ t = σ( 1) Nt, σ = ±1 H on (x, σ, s) = e 2 σb 3(j s λ( x)) H off (x, σ, s) = e 2 (B 1(j s λ( x)) iσb 2 (j s λ( x))).

22 Consequences Corollary (Energy comparison inequality) max E(0, B1 2 + B2 2, 0, B 3) E(0, E(0, B3 2 + B2 1, 0, B 2) B2 2 + B2 3, 0, B 1) E(A, B 1, B 2, B 3 ). Corollary (Boson sector decay) small enough α > 0 = (Ψ, e αn Ψ) < Hiroshima-L 2007a&b