Stochastic UV renormalization

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1 Stochastic UV renormalization Fumio Hiroshima Faculty of Mathematics, Kyushu University joint work with MGubinelli and J Lőrinczi 2014/10/15 Tohoku University F Hiroshima UV renormalization 1

2 Nelson model 1 Nelson model 2 History of UV renormalization 3 Stochastic UV renormalization 4 Translation invariant model 5 Weak coupling limit 6 Concluding remarks F Hiroshima UV renormalization 2

3 Nelson model Gaussian random variables Gaussian prob sp (Q,Σ, µ), where Q = S R(R 3 ) Canonical pair ϕ, f = ϕ( f ), ϕ Q, f S R (R 3 ) Gaussian random variables E µ [ϕ( f )] = 0 E µ [ϕ( f )ϕ(g)] = 1 2 ( f,g) L 2 (R 3 ) ϕ( f ) can be extended from f S R (R 3 ) to f L 2 (R 3 ) f ϕ( f ) is linear over C (ϕ( f ), f L 2 (R 3 )) Gaussian rv F Hiroshima UV renormalization 2

4 Nelson model Free field Hamiltonian Boson Fock space L 2 (Q, µ) = F ( : m j ϕ( f j ) :,: n j ϕ(g j ) : ) F = 0 if n m = F = n=0 F (n), where F (n) = span{: n j ϕ(g j ) :} is n-particle subspace A : L 2 (R 3 ) L 2 (R 3 ) Fanctor = Γ(A) : F F by Γ(A) : j A 1 = Γ(A) 1 ϕ( f j ) :=: ϕ(a f j ) : j Ex ω = = U t = Γ(e itω ) is a one-para unitary group H f such that Γ(e itω ) = e ith f for all t R H f free field Hamiltonian, formally "H f = ω(k)a (k)a(k)dk" F Hiroshima UV renormalization 3

5 Nelson model Nelson model Total Hilbert space H = L 2 (R 3 ) F UV cutoff function h ε with UV cutoff parameter ε > 0 ĥ ε (k) = e ε k 2 /2 k 1l k >λ L 2 (R 3 ) ε > 0 Nelson Hamiltonian with UV parameter ε > 0: H ε = H p 1l + 1l H f + gϕ(h ε ( x)) H p = x /2 +V (x) is Schrödinger op on L 2 (R 3 ) F Hiroshima UV renormalization 4

6 History of UV renormalization Nelson renormalization UV renormalization H ε H 0? ε 0 Removal of UV cutoff: limĥ ε (k) = 1 1l k >λ L 2 (R 3 ) ε 0 k Thm E Nelson, JMathPhys5 (1964) Let E ε = g2 2 k >λ e ε k 2 k where E ε as ε 0 Then β(k)dk, β(k) = lim(h ε E ε z) 1 = UG 1 (H 0 z) 1 U G ε 0 1 k + k 2 /2, where U G is a unitary operator called Gross transform F Hiroshima UV renormalization 5

7 History of UV renormalization E Nelson tried to have an alternative proof in [ Proc Conference on Analysis in Function Space, W T Martin and I Segal (eds), p 87, MIT Press, 1964] But he failed! Thm Gubinelli+FH+Lőrinczi (GHL) JFA 2014 lime T (H ε E ε ) = e T H 0 ε 0 F Hiroshima UV renormalization 6

8 Stochastic UV renormalization STEP (1) path int rep and diagonal part Fock vacuum 1l F, & f,h L 2 (R 3 ) is fixed Set V = 0 for simplicity By (Lőrinczi-FH-Betz), [ ] ( f 1l,e 2T H ε h 1l) H = dxe x f (B T )h(b T )e g2 2 S ε Pair interaction S ε = T T T ds dtw ε (B t B, t s) T Pair potential W ε (X,t) = 1 e ε k 2 1l 2 k k >λ e ix x e k t dk The diagonal part of S ε is singular = E ε F Hiroshima UV renormalization 7

9 Stochastic UV renormalization T Off diagonal part of S diagonal part of S T T Off diagonal part of S T Figure: S ε F Hiroshima UV renormalization 8

10 Let φ ε (x,t) = 1 2 Ito formula: Stochastic UV renormalization φ ε (B T B s,t s) φ ε (0,0) = = T s T s S ε = 2 φ ε (B t B s,t s)db t + φ ε (B t B s,t s)db t T T T ds s e ε k 2 e ik x k t 1l k >λ β(k)dk k T s T W ε (B t B s,t s)dt S ε = Y ε + Z ε + 4T φ ε (0,0) and Y ε = 2 T Z ε = 2 T T S ren ε = S ε 4T φ ε (0,0) ds T T s s ( 1 2 x + t )φ ε(b t B s,t s)dt W ε (B t B s,t s)dt φ ε (B t B s,t s) db t, φ ε (B T B s,t s)ds F Hiroshima UV renormalization 9

11 Stochastic UV renormalization STEP (2) vacuum expectation Let a(ε) = ( f 1l,e 2T (H ε+g 2 φ ε (0,0)) h 1l) = Thm (Vacuum exp) GHL, JFA 14 (0) a(ε) < ( ε > 0) (1) S0 ren st lima(ε) = E x ε 0 R 3 (2) C st a(ε) f h e CT ε 0 R 3 E x [ ] f (B T )h(b T )e g2 2 Sren 0 dx [ ] f (B T )h(b T )e g2 2 Sren ε dx F Hiroshima UV renormalization 10

12 Stochastic UV renormalization STEP(3) extension to a dense subspace Dense subspace D = span{ f F(ϕ( f 1 ),,ϕ( f n )); f L 2 (R 3 ),F S (R n ),n 1} H For F,G D we can easily show the existence of the limit: lim ε 0 (F,e 2T (H ε+g 2 φ ε (0,0)) G) Furthermore we can see the path int rep of above limit F Hiroshima UV renormalization 11

13 Stochastic UV renormalization STEP(4) The most difficult step lim ε 0 (F,e 2T (H ε+g 2 φ ε (0,0)) G) for F,G H? It is enough to show the existence of a lower bound C: H ε + g 2 φ ε (0,0) > C, ε > 0 Note that φ ε (0,0) + but infσ(h ε ) Nelson showed it by functional analysis How can we prove by path int? F Hiroshima UV renormalization 12

14 Stochastic UV renormalization Outline of the idea Adding a dummy potential δ x 2 : H ε (δ) = H ε + δ x 2 + g 2 φ ε (0,0) H ε (0) = H ε + g 2 φ ε (0,0) Let φ T g = e T H ε(δ) f 1l/ e T H ε(δ) f 1l gs We can show that by path int rep lim T ( f 1l,φ T g ) > 0 H ε (δ) has a ground state φ g (δ) > 0 ae [H Spohn, 1999] F Hiroshima UV renormalization 13

15 Stochastic UV renormalization From (1) (φ g (δ), f 1l) 0 for 0 f L 2 (R 3 ), (2) ( f 1l,e 2T H ε(δ) h 1l) f h e CT, it follows that infσ(h ε (δ)) (1) = 1 2T lim log( f T 1l,e 2T Hε(δ) f 1l) (2) C, C is independent of δ > 0 F Hiroshima UV renormalization 14

16 Stochastic UV renormalization (F,e 2T H ε(δ) F) e 2TC F 2, F H (F,e 2T H ε(0) F) e 2TC F 2 by Lebesgue domconv Thm (uniform lower bound) GHH JFA14 ịnfσ(h ε + g 2 φ ε (0,0)) C F Hiroshima UV renormalization 15

17 STEP(5) Stochastic UV renormalization Hence lim ε 0 (F,e 2T (H ε+g 2 φ ε (0,0)) G) for F,G H By Riesz rep theorem, T t st lim ε 0 (F,e t(h ε+g 2 φ ε (0,0)) G) = (F,T t G) T t ic C 0 semigroup, and T t = e t H 0 Then we obtain that lim ε 0 e t(h ε+g 2 φ ε (0,0)) = e th 0 F Hiroshima UV renormalization 16

18 Translation invariant model Translation invariant model (Fiber Hamiltonian) V = 0 = H = R 3 H(P)dP Nelson model with total momentum P acting on F e itp f = Γ(e it( i ) ) H(P) = 1 2 (P P f ) 2 + ϕ(φ) + H f, P R 3 H ε (P) = 1 2 (P P f ) 2 + ϕ(φ ε ) + H f UV renormalization for each P? F Hiroshima UV renormalization 17

19 Translation invariant model In the same was as H ε we can show that lim ε 0 (1l,e T (H ε(p) E ε ) 1l) But P < p = ground state of H ε (P) Thm (UV ren for fiber Hamiltonian) H 0 (P) such that lim ε 0 e T H ε(p) = e T H 0(P) for P R 3 Proof By path int rep of e T H ε(p) we can see that (1) infσ(h ε (0)) infσ(h ε (P)) diamagnetic inequality, (2) H ε (0) has a ground state φ g (0) and φ g (0) > 0, (3) (1l,φ g (0)) 0 Hence the uniform lower bound, H ε (0) E ε > C, exists H ε (P) E ε > C follows! F Hiroshima UV renormalization 18

20 Weak coupling limit Weak coupling limit Many body Nelson model where h p = H ε = h p 1l + 1l H f + N j=1 ( 1 2 j) +V (x 1,,x N ) H ε (κ) = h p 1l + κ 2 1l H f + κ N j=1 N ϕ(h ε ( x j )), j=1 ϕ(h ε ( x j )), κ > 0 E ε (κ) = g2 N R3 e ε k 2 κ 2 2(2π) 3 k κ 2 k + k 2 /2 1l k >λ dk F Hiroshima UV renormalization 19

21 Weak coupling limit Effective potentials Thm (Effective potential) where lim lim ( f 1l,e t(h ε(κ) E ε (κ)) h 1l)(HJMP98) ε 0 κ = lim κ lim ε 0 ( f 1l,e t(h ε(κ) E ε (κ)) h 1l)(GHLJFA14) =( f,e th eff h), h eff = 1 2 N j +V (x 1,,x N ) g2 j=1 4π i< j 1 x i x j F Hiroshima UV renormalization 20

22 Summary Concluding remarks E Nelson: E ε is derived from a commutator, and the Gross trans U G is needed GHL: E ε is a diagonal part of pair int and the Gross trans is not needed Applications 1 model on mfd: BM diffusion proc 2 Non-local Nelson model: BM Lévy proc + m 2 1l + 1l H f + ϕ(h ε ( x)), 3 Translation invariant case F Hiroshima UV renormalization 21

23 Referemces Concluding remarks Bach-Fröhlich-Sigal (Adv Math 97,CMP98) GS for g 1 Gérard (AHP 00), Spohn(LMP 00) GS g Fefferman (Adv Math 00) Stability of matter FH (JMP99, CMP 01,JFA05) uniqueness and sa g Betz-FH-Lőrinczi-Minlos-Spohn (RMP01) Gibbs meas Griesmer-Lieb-Loss (Inv Math 01) GS g FH-Spohn(AHP 01,JMP05) enhanced binding and eff mass Hirokawa-FH-Spohn(Adv Math 05) GS with UV ren FH-Lőrinczi (JFA 08) model with spin Gérard-FH-Panati-Suzuki (CMP12,JFA12,LMP12) model on mfd FH (Adv Math 14) Non-local model F Hiroshima UV renormalization 22