Translation invariant models in QFT without ultraviolet cutoffs
aa r X i v : . [ m a t h - ph ] J un Translation invariant models in QFTwithout ultraviolet cutoffs
Fumio Hiroshima ∗ November 17, 2018
Abstract
The translation invariant model in quantum field theory is considered byfunctional integrations. Ultraviolet renormalization of the translation invari-ant Nelson model with a fixed total momentum is proven by functional inte-grations. As a corollary it can be shown that the Nelson Hamiltonian withzero total momentum has a ground state for arbitrary values of coupling con-stants in two dimension. Furthermore the ultraviolet renormalization of thepolaron model is also studied.
In this paper we consider an ultraviolet (UV) renormalization of the Nelson model H ( P ) with a fixed total momentum P ∈ R by functional integrations.The Nelson model describes an interaction system between a scalar bose fieldand particles governed by a Schr¨odinger operator. The interaction is linear in a fieldoperator and the model is one of a prototype of interaction models in quantum fieldtheory. The Nelson Hamiltonian can be realized as a self-adjoint operator H on aHilbert space and the spectrum of H has been studied so far from several point ofview. See Appendix B for the Nelson model. In the case where external potentialis dropped in H , the Hamiltonian turns to be translation invariant, and it can berealized as the family of self-adjoint operators H ( P ) indexed by the so-called totalmomentum P ∈ R . The spectrum of H ( P ) is studied for every P ∈ R , and thedifference of spectral property of H ( P ) from every P is interesting. ∗ Faculty of Mathematics, Kyushu University, Fukuoka, Japan H ( P ), we prepare tools used in this paper. Theboson Fock space F over L ( R ) is defined by F = ∞ M n =0 [ ⊗ ns L ( R )] . (1.1)Here ⊗ ns L ( R ) describes n fold symmetric tensor product of L ( R ) with ⊗ s L ( R ) = C . Let a ∗ ( f ) and a ( f ), f ∈ L ( R ), be the creation operator and the annihilationoperator, respectively, in F , which satisfy canonical commutation relations:[ a ( f ) , a ∗ ( g )] = ( ¯ f , g ) , [ a ( f ) , a ( g )] = 0 = [ a ∗ ( f ) , a ∗ ( g )] . Note that Here ( f, g ) denotes the scalar product on L ( R ) and it is linear in g andanti-linear in f . We also note that f a ∗ ( f ) and f a ( f ) are linear. Denote thedispersion relation by ω ( k ) = | k | . Then the free field Hamiltonian H f of F is thendefined by the second quantization of ω , i.e., H f = d Γ( ω ), and the field momentumoperator P f by P f µ = d Γ( k µ ) and we set P f = ( P f 1 , P f 2 , P f 3 ). They satisfy e − itH f a ∗ ( f ) e − itH f = a ∗ ( e − itω f ) , e − itH f a ( f ) e − itH f = a ( e itω f ) . (1.2)The field operator φ = φ ( ˆ ̺ ) on which UV cutoff ˆ ̺ is imposed is defined by φ = 1 √ (cid:16) a ∗ ( ˆ ̺/ √ ω ) + a ( e ˆ ̺/ √ ω ) (cid:17) . (1.3)Here ˆ ̺ denotes the Fourier transform of a cutoff function ̺ satisfying ˆ ̺/ √ ω ∈ L ( R ),and e ˆ ̺ ( k ) = ˆ ̺ ( − k ). Definition 1.1 (Translation invariant Nelson Hamiltonian) H ( P ) is a linear oper-ator on F and is defined by H ( P ) = 12 ( P − P f ) + H f + gφ, P ∈ R . (1.4)Before going to discussion on H ( P ) we have to mention the self-adjointness of H ( P ).We decompose H ( P ) as H + H I ( P ) to show the self-adjointness, where H = 12 P f 2 + H f ,H I ( P ) = 12 | P | − P · P f + gφ. Under the assumptionsˆ ̺/ √ ω ∈ L ( R ) , ˆ ̺/ω ∈ L ( R ) , ˆ ̺ ( k ) = ˆ ̺ ( − k ) (1.5)2e see that k φF k ≤ (1 / √ k ˆ ̺/ω kk H / F k + k ˆ ̺/ √ ω kk F k ) follows for F ∈ D ( H f )and φ is symmetric. Then the interaction H I ( P ) is well defined, symmetric and it isinfinitesimally H -bounded, i.e., for arbitrary ε >
0, there exists a b ε > k H I Φ k ≤ ε k H Φ k + b ε k Φ k for all Φ ∈ D ( H ). Thus by the Kato-Rellich theorem H ( P ) is self-adjoint on D ( H )for every P ∈ R . Throughout this paper we assume condition (1.5).The purpose of this paper is to show UV renormalization (=the point chargelimit) of H ( P ). It is remarked that the point charge limit, ˆ ̺ → H ( P ) can beactually achieved in a similar manner to [Nel64a] by functional analysis. While thepurpose of this paper is to prove the point charge limit by functional integrations.Machinery used in this paper is similar to [GHL13], where it plays an importantrole that e − tH is positivity improving. Semi-group e − tH ( P ) is, however, not positivityimproving for P = 0. Despite this fact we can achieve UV renormalization by usinga diamagnetic inequality derived from functional integration.This paper is organized as follows. In Section 2 we show UV renormalization.Section 3 is devoted to showing the existence of a renormalized ground state. InSection 4 we consider the polaron model. In Appendix we briefly introduce euclideanquantum field theory and the Nelson model. Let λ > ̺ ε ( k ) = e − ε | k | / | k |≥ λ , ε > H ε ( P ) = 12 ( P − P f ) + H f + gφ ε , ε > , (2.2)where φ ε is defined by φ with ˆ ̺ replaced by ˆ ̺ ε . Here ε > H ε ( P ) as ε ↓
0. Precisely we can showthe existence of a self-adjoint operator H ren ( P ) such that e − T ( H ε ( P ) − E ε ) → e − T H ren ( P ) (2.3)by functional integrations, where E ε = − g Z | k | >λ e − ε | k | ω ( k ) β ( k ) dk (2.4)3enotes the renormalization term and the propagator β is given by β ( k ) = 1 ω ( k ) + | k | / . (2.5)Notice that E ε → −∞ as ε ↓
0. Our main theorem shows (2.3) for all P ∈ R . Theorem 2.1 (UV renormalization)
Let P ∈ R . Then there exists a self-adjointoperator H ren ( P ) such that s − lim ε ↓ e − T ( H ε ( P ) − E ε ) = e − T H ren ( P ) , T ≥ . (2.6)We carry out the proof by functional integration and obtain E ε as the diagonal termof a pair interaction potential on the paths of a Brownian motion. A Feynman-Kac type formula of (
F, e − T H ε ( P ) G ) is constructed for F, G ∈ F and P ∈ R . Denote H − k ( R n ) = { f ∈ S ′ R ( R n ) | ˆ f ∈ L ( R n ) , ω − k/ ˆ f ∈ L ( R n ) } (2.7)endowed with the norm k f k H − k ( R n ) = Z R n | ˆ f ( x ) | | x | − k dx . Recall that a Euclideanfield is a family of Gaussian random variables { φ E ( F ) , F ∈ H − ( R ) } on a probabil-ity space ( Q E , Σ E , µ E ), such that the map F φ E ( F ) is linear, and their mean andcovariance are given by E µ E [ φ E ( F )] = 0 and E µ E [ φ E ( F ) φ E ( G )] = 12 ( F, G ) H − ( R ) . See Appendix A for the detail. Let ( B t ) t ∈ R be the 3-dimensional Brownian motionon the hole real line on the Wiener space. Let E [ · · · ] be the expectation with respectto the Wiener measure starting from zero. Lemma 2.2 (Feynman-Kac type formula)
Let
F, G ∈ F . Then it follows that ( F, e − T H ε ( P ) G ) = E h(cid:16) J − T e i ( P − ˆ P f ) B − T F, e − φ E ( R T − T δ s ⊗ ˜ ̺ ε ( ·− B s ) ds ) J T e i ( P − ˆ P f ) B T G (cid:17) E i , (2.8) where ˆ P f = d Γ( − i ∇ k ) and ˜ ̺ ε ( x ) = (cid:16) e − ε |·| / ⊥ λ / √ ω (cid:17) ∨ ( x ) , and δ s ( x ) = δ ( x − s ) isthe one-dimensional Dirac delta distribution with mass on s . Proof.
See [Hir07] and Section A. (cid:3) orollary 2.3 (Positivity improving) Let P = 0 . Then e − T H ε (0) is positivity im-proving. Proof.
By Lemma 2.2 we have(
F, e − T H ε (0) G ) = E h(cid:16) J − T e − i ˆ P f · B − T F, e − φ E ( R T − T δ s ⊗ ˜ ̺ ε ( ·− B s ) ds ) J T e − i ˆ P f · B T G (cid:17) E i . (2.9)Since J t and e i ˆ P f · B T are positivity preserving, and J ∗ t J s = e −| t − s | H f is positivityimproving, we have ( F, e − T H ε (0) G ) ≥ F ≥ G ≥
0. We can also deducethat (
F, e − T H ε (0) G ) = 0 in the same way as [Hir07]. Then ( F, e − T H ε (0) G ) > (cid:3) In order to prove Theorem 2.1 we need two ingredients:(1) convergence (2.6) on the Fock vacuum,(2) uniform lower bound of H ε ( P ) with respect to ε .Let 1l = { , , , · · · , } ∈ F be the Fock vacuum. In particular, for F = 1l = G , wecan see the corollary below. Corollary 2.4 (Vacuum expectation)
It follows that (1l , e − T H ε ( P ) E (cid:20) e iP · ( B T − B − T ) e g S ε (cid:21) , (2.10) where S ε = Z T − T ds Z T − T dtW ε ( B t − B s , t − s ) (2.11) is the pair interaction given by the pair potential W ε : R × R → R : W ε ( x, t ) = Z | k |≥ λ ω ( k ) e − ε | k | e − ik · x e − ω ( k ) | t | dk. (2.12) Proof.
This follows directly from Lemma 2.2. (cid:3)
It can be seen that the pair potential W ε ( B t − B s , t − s ) is singular at the diagonalpart t = s . We shall remove the diagonal part by using the Itˆo formula. We introducethe function ̺ ε ( x, t ) = Z | k |≥ λ e − ε | k | e − ik · x − ω ( k ) | t | ω ( k ) β ( k ) dk, ε ≥ , (2.13)5here β ( k ) is given by (2.5), and it is shown by the Itˆo formula that Z Ss W ε ( B t − B s , t − s ) dt = ̺ ε (0 , − ̺ ε ( B S − B s , S − s ) + Z Ss ∇ ̺ ε ( B t − B s , t − s ) · dB. (2.14)Here ̺ ε (0 ,
0) can be regarded as the diagonal part of W ε and turns to be a renor-malization term, since ̺ ε (0 , → −∞ as ε →
0. Let S ren ε = S ε − T ̺ ε (0 , , ε > , which is represented as S ren ε = S ODε + 2 Z T − T ds Z [ s + τ ] s ∇ ̺ ε ( B t − B s , t − s ) ds ! · dB t − Z T − T ̺ ε ( B [ s + τ ] − B s , [ s + τ ] − s ) ds. (2.15)Here 0 < τ < T is an arbitrary number, S ODε denotes the off-diagonal part which isgiven by S ODε = 2 Z T − T ds Z T [ s + τ ] W ε ( B t − B s , t − s ) dt and [ t ] = − T ∨ t ∧ T , and the integrand is given by ∇ ̺ ε ( X, t ) = Z | k |≥ λ − ike − ikX e −| t | ω ( k ) e − ε | k | ω ( k ) β ( k ) dk. Proposition 2.5 (1) It holds that lim ε ↓ E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) e g S ren ε − e g S ren0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = 0 . (2.16) (2) There exists a constant c > such that for all ε ≥ , E (cid:20) e g S ren ε (cid:21) ≤ e T c (2.17)Here S ren0 = 2 Z T − T ds (cid:18)Z t − T ∇ ̺ ( B t − B s , t − s ) ds (cid:19) · dB t − Z T − T ̺ ( B T − B s , T − s ) ds. Proof.
This can be proven by a minor modification of [GHL13, Section 2]. (cid:3)
From this proposition we can derive the lemma below immediately.
Lemma 2.6
It follows that lim ε ↓ (1l , e − T ( H ε ( P )+ g ̺ ε (0 , E (cid:20) e iP · ( B T − B − T ) e g S ren0 (cid:21) . (2.18)6 .4 Existence of ground states for P = 0 We shall show a uniform lower bound of H ε ( P ) + g ̺ ε (0 ,
0) with respect to ε ≥ E ε ( P ) = inf σ ( H ε ( P )). Corollary 2.7 (Diamagnetic inequality)
Let ε > . Then E ε (0) ≤ E ε ( P ) followsfor every P ∈ R . Proof.
By functional integral representation (2.8) it follows that | ( F, e − T H ε ( P ) G ) | ≤ ( | F | , e − T H ε (0) | G | ) . This yields the inequality E ε (0) ≤ E ε ( P ). (cid:3) Intuitively ϕ T g = e − T H ε (0) / k e − T H ε (0) k is a sequence converging to a ground state. Let γ ( T ) = (1l , ϕ T g ) , i.e., γ ( T ) = (1l , e − T H ε (0) (1l , e − T H ε (0) . (2.19)The useful lemma concerning the existence and absence of ground states is thelemma below. Lemma 2.8
There exists a ground state of H ε (0) if and only if lim T →∞ γ ( T ) > . Proof.
This proof is taken from [LMS02]. Suppose that inf σ ( H ε (0)) = 0 andset lim T →∞ γ ( T ) = a . Suppose that a = 0 and the ground state ϕ g exists. Thenlim T →∞ e − T H ε (0) = 1l { } ( H ε (0)). Since ϕ g > e − T H ε (0) is positivityimproving, it follows that a = (1l , ϕ g ) >
0. It contradicts a >
0. Thus the groundstate does not exist. Next suppose a >
0. Then p γ ( T ) ≥ ε for sufficiently large T .Let dE be the spectral measure of H ε (0). Thus we have p γ ( T ) = R ∞ e − T u dE ( R ∞ e − T u dE ) / ≤ R δ e − T u dE + R ∞ δ e − T u dE ( R δ e − T u dE ) / . Then we can derive that p γ ( T ) ≤ ( R δ e − T u dE ) / E ([0 , δ ]) / + e − T δ ( R δ e − T u dE ) / = E ([0 , δ ]) / + 1( R δ e − T ( u − δ ) dE ) / . Take T → ∞ on both sides above, we have √ ε ≤ E ([0 , δ ]) / . Thus taking δ ↓
0, wehave √ ε ≤ E ( { } ) / . Thus the ground state exists. (cid:3) Using the lemma above we can show the existence of the ground state of H ε (0).7 emma 2.9 For all ε > , H ε (0) has the ground state and it is unique. Proof.
The uniqueness follows from the fact that e − tH ε (0) is positivity improving.It remains to show the existence of ground state, which is proven by using Lemma2.8. By the Feynman-Kac type formula we have γ ( T ) = (cid:16) E [ e g R T dt R T dsW ε ] (cid:17) E [ e g R T − T dt R T − T dsW ε ] . By the reflection symmetry of the Brownian motion we see that γ ( T ) = E [ e g R T dt R T dsW ε ] E [ e g R − T dt R − T dsW ε ] E [ e g R T − T dt R T − T dsW ε ]and also the Markov property yields that γ ( T ) = E [ e g R T dt R T dsW ε + g R − T dt R − T dsW ε ] E [ e g R T − T dt R T − T dsW ε ] . Then we obtain that γ ( T ) = E [ e g R T − T dt R T − T dsW ε − g R − T dt R T W ε ] E [ e g R T − T dt R T − T dsW ε ] . Notice that Z − T dt Z T dsW ε ≤ Z R | k |≥ λ e − ε | k | ω ( k ) (cid:0) − e − ω ( k ) T (cid:1) dk ≤ Z R | k |≥ λ e − ε | k | ω ( k ) dk. Hence we conclude that γ ( T ) ≥ exp − g Z R | k |≥ λ e − ε | k | ω ( k ) dk ! > T >
0. Then the lemma follows. (cid:3)
In this section we show a uniform lower bound of the bottom of the spectrum of H ε ( P ) + g ̺ ε (0 ,
0) with respect to ε >
0. Thanks to the diamagnetic inequality, theestimate of the uniform lower bound for any P can be reduced to that of P = 0.We note that the diamagnetic inequality E (0) ≤ E ( P ) can be derived through afunctional integration in Corollary 2.7. 8 emma 2.10 There exists C ∈ R such that H ε ( P ) − g ̺ ε (0 , > − C , uniformlyin ε > . Proof.
Let ϕ g be the ground state of H ε (0). Since e − tH ε (0) is positivity improving,we see that (1l , ϕ g ) = 0 and then E ε (0) − g ̺ ε (0 ,
0) = − lim T →∞ T log(1l , e − T ( H ε (0) − g ̺ ε (0 , > − C by Proposition 2.5, where C is independent of ε >
0. By the diamagnetic inequality E ε (0) ≤ E ε ( P ) we then derive that E ε ( P ) − g ̺ ε (0 , ≥ − C. Then the lemma follows. (cid:3)
Now we extend the result from Fock vacuum 1l to more general vectors of theform F ( φ ( f ) , . . . , φ ( f n )), with F ∈ S ( R n ), where φ ( f ) stands for a scalar fieldgiven by φ ( f ) = 1 √ a ∗ ( ˆ f / √ ω ) + a ( e ˆ f / √ ω )) . Consider the subspace D = (cid:8) F ( φ ( f ) , . . . , φ ( f n )) | F ∈ S ( R n ) , f j ∈ H − / ( R ) , j = 1 , ..., n, n ≥ (cid:9) , which is dense in F . Lemma 2.11 (1) Let ρ j ∈ H − / ( R ) for j = 1 , , and α, β ∈ C . Then lim ε ↓ ( e αφ ( ρ ) , e − T ( H ε ( P )+ g ̺ ε (0 , e βφ ( ρ ) ) = E (cid:20) e iP · ( B T − B − T ) e g S ren0 + ξ (cid:21) , (2.21) where ξ = ξ ( g ) = ¯ α k ρ / √ ω k + β k ρ / √ ω k + 2 ¯ αβ ( ρ / √ ω, e − T ω ρ / √ ω )+ 2 ¯ αg Z T − T ds Z R dk ˆ ρ ( k ) p ω ( k ) 1l | k |≥ λ e −| s − T | ω ( k ) e − ikB s + 2 βg Z T − T ds Z R dk ˆ ρ ( k ) p ω ( k ) 1l | k |≥ λ e −| s + T | ω ( k ) e − ikB s . (2) Let Φ = F ( φ ( u ) , . . . , φ ( u n )) and Ψ = G ( φ ( v ) , . . . , φ ( v m )) ∈ D . Then lim ε ↓ (Φ , e − T ( H ε ( P )+ g ̺ ε (0 , Ψ)= (2 π ) − ( n + m ) / Z R n + m dK dK ˆ F ( K ) ˆ G ( K ) E (cid:20) e iP · ( B T − B − T ) e g S ren0 + ξ ( K ,K ) (cid:21) , (2.22)9 here ξ ( K , K ) = −k K · u/ √ ω k − k K · v/ √ ω k − K · u/ √ ω, e − T ω K · v/ √ ω ) − ig Z T − T ds Z R dk K · ˆ u ( k ) p ω ( k ) 1l | k |≥ λ e −| s − T | ω ( k ) e − ikB s + 2 ig Z T − T ds Z R dk K · ˆ v ( k ) p ω ( k ) 1l | k |≥ λ e −| s + T | ω ( k ) e − ikB s and u = ( u , ..., u n ) , v = ( v , ..., v m ) . Proof. (1) follows from Lemma 2.2. (2) follows fromΦ = F ( φ ( u ) , . . . , φ ( u n )) = (2 π ) − n/ Z R n ˆ F ( k , · · · , k n ) e i P nj =1 k j φ ( u j ) dk · · · dk n and Lemma 2.6. (cid:3) Now we can complete the proof of the main theorem.
Proof of Theorem 2.1.
Let
F, G ∈ H and C ε ( F, G ) = (
F, e − t ( H ε ( P )+ g ̺ ε (0 , G ). ByLemma 2.2 we obtain that C ε ( F, G ) is convergent as ε ↓
0, for every
F, G ∈ D .Since D is dense in H , by the uniform bound k e − t ( H ε ( P )+ g ̺ ε (0 , k < e tC obtainedby Lemma 2.10 we can see that { C ε ( F, G ) } ε converges also for all F, G ∈ H by asimple approximation. Let C ( F, G ) = lim ε ↓ C ε ( F, G ). Hence | C ( F, G ) | ≤ e tC k F kk G k , and there exists a bounded operator T t such that C ( F, G ) = (
F, T t G ) , F, G ∈ H by the Riesz theorem. Thus s − lim ε ↓ e − t ( H ε ( P )+ g ̺ ε (0 , = T t follows. Furthermore,we also see thats − lim ε ↓ e − t ( H ε ( P )+ g ̺ ε (0 , e − s ( H ε ( P )+ g ̺ ε (0 , = s − lim ε ↓ e − ( t + s )( H ε ( P )+ g ̺ ε (0 , = T t + s . Since the left-hand side above is T t T s , the semigroup property of T t follows. Since e − t ( H ε ( P )+ g ̺ ε (0 , is a symmetric semigroup, T t is also symmetric. Moreover by thefunctional integral representation (2.22) the functional ( F, T t G ) is continuous at t = 0 for every F, G ∈ D . Since D is dense in H and k T t k is uniformly bounded,it also follows that T t is strongly continuous at t = 0. Then T t , t ≥
0, is stronglycontinuous one-parameter symmetric semigroup. Thus the semigroup version ofStone’s theorem [LHB11, Proposition 3.26] implies that there exists a self-adjointoperator H ren ( P ), bounded from below, such that T t = e − tH ren ( P ) , t ≥ . Hence the proof is completed by setting E ε = − g ̺ ε (0 , (cid:3) Let E ren ( P ) = inf σ ( H ren ( P )). 10 orollary 2.12 (Diamagnetic inequality) It holds that E ren (0) ≤ E ren ( P ) . Proof.
From inequality | ( F, e − T ( H ε ( P ) − E ε ) G ) | ≤ ( | F | , e − T ( H ε (0) − E ε ) | G | ) it followsthat | ( F, e − T H ren ( P ) G ) | ≤ ( | F | , e − T H ren (0) | G | ). Then the corollary follows. (cid:3) d = 2 Let us suppose d = 2 . In the case of d = 2 we can procedure the renormalization similar to the case of d = 3. The renormalization is however not needed in the case of d = 2, since ̺ ε (0 ,
0) converges to the finite number ̺ (0 ,
0) as ε →
0. One important conclusionof Theorem 2.1 is the existence of a ground state of H ren (0) for d = 2. Lemma 3.1
It follows that γ ( T ) = (1l , e − T H ren (0) (1l , e − T H ren (0) > exp (cid:18) − g Z R | k |≥ λ ω ( k ) dk (cid:19) > Proof.
By (2.20) we have γ ( T ) = (1l , e − T H ε (0) (1l , e − T H ε (0) ≥ exp − g Z R | k |≥ λ e − ε | k | ω ( k ) dk ! > . Take the limit of T → ∞ on both sides we can derive (3.1). (cid:3) Theorem 3.2 (Existence of the ground state)
For arbitrary values of g , H ren (0) has a ground state ϕ ren such that (1l , ϕ ren ) = 0 . Proof.
By Lemma 3.1 we havelim T →∞ (1l , e − T H ren (0) (1l , e − T H ren (0) > exp (cid:18) − g Z R | k |≥ λ ω ( k ) dk (cid:19) > . (3.2)On the other hand we see thatlim T →∞ (1l , e − T H ren (0) (1l , e − T H ren (0) k P g k , where P g denotes the projection to the subspace Ker( H ren − inf σ ( H ren )). By (3.2)we derive that k P g k >
0, which implies H ren has a ground state ϕ ren such that(1l , ϕ ren ) = 0. (cid:3) Polaron model
We introduce the polaron model in this section. The polaron model is similar to H ε ( P ), and the UV renormalization can be seen in a similar manner to the Nelsonmodel. The polaron Hamiltonian is defined by H pol ( P ) = 12 ( P − P f ) + N + g Φ , P ∈ R , (4.1)where N denotes the number operator andΦ = 1 √ (cid:16) a ∗ ( ˆ ̺/ω ) + a ( ˜ˆ ̺/ω ) (cid:17) . Note that the test function is ˆ ̺/ω which is different from the test function ˆ ̺/ √ ω of the Nelson Hamiltonian. We discuss UV renormalization of the polaron model.The discussion is however easier than that of the Nelson model. Let ˆ ̺ ( k ) = e − ε | κ | / ,and H pol ( P ) with ˆ ̺ ( k ) = e − ε | κ | / is denoted by H pol ε ( P ). The vacuum expectationof e − T H pol ε ( P ) is given by(1l , e − T H pol ε ( P ) E (cid:20) e iP · B T e g S pol ε (cid:21) , (4.2)where S pol ε = Z T ds Z T dtW pol ε ( B t − B s , t − s ) (4.3)is the pair interaction for the polaron model and the pair potential is given by W pol ε ( x, t ) = Z | k |≥ λ ω ( k ) e − ε | k | e − ik · x e −| t | dk. (4.4)We can see that W pol ε ( x, t ) = 2 π | x | Z ∞ λ | x | e − εu sin uu due −| t | . Let W pol0 ( x, t ) = Z | k |≥ λ ω ( k ) e − ik · x e −| t | dk and we see that W pol ε ( x, t ) → W pol0 ( x, t ) for each ( x, t ) as ε ↓
0. Then it holds thatlim ε ↓ E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) e g S pol ε − e g S pol0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = 0 . (4.5)From this we can prove the lemma below immediately. Note that any renormaliza-tion is not needed. 12 emma 4.1 It follows that lim ε ↓ (1l , e − T H pol ε ( P ) E (cid:20) e iP · B T e g S pol0 (cid:21) . (4.6)Hence the theorem below is proven in the same way as the Nelson model. Theorem 4.2 (UV renormalization)
Let P ∈ R . Then there exists a self-adjointoperator H pol0 ( P ) such that s − lim ε ↓ e − T H pol ε ( P ) = e − T H pol0 ( P ) , T ≥ . (4.7) Corollary 4.3 (Removal of infrared cutoff)
It follows that lim λ → (1l , e − T H pol0 ( P ) E (cid:20) e iP · B T e g π R T dt R T ds e −| t − s || Bt − Bs | (cid:21) . (4.8) Proof.
It can be seen that W pol0 ( x, t ) = Z | k |≥ λ ω ( k ) e − ik · x e −| t | dk ≤ π + δ | x | e −| t | with some constant δ , and lim λ → W pol0 ( x, t ) = π | x | e −| t | for each x . It can be also checked that E (cid:20) e g π R T dt R T ds e −| t − s || Bt − Bs | (cid:21) is finite in the lemmabelow. Then the Lebesgue dominated convergence theorem yields the corollary. (cid:3) Lemma 4.4 E (cid:20) e g π R T dt R T ds e −| t − s || Bt − Bs | (cid:21) is finite. Proof.
We separate [0 , T ] × [0 , T ] into two regions as Z T dt Z T ds = Z T dt Z Tt ds + Z T dt Z t ds. By the Schwarz inequality we have E (cid:20) e g π R T dt R T ds e −| t − s || Bt − Bs | (cid:21) ≤ (cid:18) E (cid:20) e g π R T dt R Tt ds e −| t − s || Bt − Bs | (cid:21)(cid:19) / (cid:18) E (cid:20) e g π R T dt R t ds e −| t − s || Bt − Bs | (cid:21)(cid:19) / = (cid:18) E (cid:20) e g π R T dt R Tt ds e −| t − s || Bt − Bs | (cid:21)(cid:19) / (cid:18) E (cid:20) e g π R T ds R Ts dt e −| t − s || Bt − Bs | (cid:21)(cid:19) / . (4.9)13e estimate both sides of (4.9). By Jensen’s inequality we have E (cid:20) e g π R T dt R Tt ds e −| t − s || Bt − Bs | (cid:21) ≤ Z T dtT E (cid:20) e g π R Tt dtT e −| t − s || Bt − Bs | (cid:21) . We estimate E (cid:20) e g π R Tt dtT e −| t − s || Bt − Bs | (cid:21) . Let ( F t ) t ≥ be the natural filtration of the Brow-nian motion ( B t ) t ≥ . We can see that E (cid:20) e g π R Tt dsT e −| t − s || Bt − Bs | (cid:21) = E (cid:20) E (cid:20) e g π R Tt dsT e −| t − s || Bt − Bs | | F t (cid:21)(cid:21) = E (cid:20) E B t (cid:20) e g π R Tt dsT e −| t − s || B − Bs − t | (cid:21)(cid:21) = Z R dy (2 πt ) − / e −| y | / (2 t ) E y (cid:20) e g π R Tt dsT e −| t − s || B − Bs − t | (cid:21) = Z R dy (2 πt ) − / e −| y | / (2 t ) E y (cid:20) e g π R Tt dsT e −| t − s || Bs − t − y | (cid:21) . Since the potential V ( x ) = | x | − is a Kato class potential, we havesup y E y (cid:20) e g π R Tt dsT e −| t − s || Bs − t − y | (cid:21) ≤ e a ( T − t ) with some a . Hence E (cid:20) e g π R T dt R Tt ds e −| t − s || Bt − Bs | (cid:21) < ∞ . Similarly it can be shown that E (cid:20) e g π R T ds R Ts dt e −| t − s || Bt − Bs | (cid:21) < ∞ and hence (4.9) is finite. (cid:3) A Schr¨odinger representation and Euclidean field
In this section Hilbert spaces H − / ( R ) and H − ( R ) are given by (2.7). It is wellknown that the boson Fock space F is unitarily equivalent to L ( Q, µ ), where thisspace consists of square integrable functions on a probability space ( Q, Σ , µ ). Con-sider the family of Gaussian random variables { φ ( f ) , f ∈ H − / ( R ) } on ( Q, Σ , µ )such that φ ( f ) is linear in f ∈ H − / ( R ), and their mean and covariance are givenby E µ [ φ ( f )] = 0 and E µ [ φ ( f ) φ ( g )] = 12 ( f, g ) H − / ( R ) . Given this space, the Fock vacuum 1l F is unitary equivalent to 1l L ( Q ) ∈ L ( Q ),and the scalar field φ ( f ) is unitary equivalent to φ ( f ) as operators, i.e., φ ( f ) isregarded as multiplication by φ ( f ). Then the linear hull of the vectors given by theWick products : Q nj =1 φ ( f j ) : is dense in L ( Q ), where recall that Wick product isrecursively defined by: φ ( f ) : = φ ( f ): φ ( f ) n Y j =1 φ ( f j ) : = φ ( f ) : n Y j =1 φ ( f j ) : − n X i =1 ( f, f i ) H − / ( R ) : n Y j = i φ ( f j ) :14his allows to identify F and L ( Q ), which we have done in (2.8), i.e., F ∈ H can beregarded as a function R N ∋ x F ( x ) ∈ L ( Q ) such that R R N k F ( x ) k L ( Q ) dx < ∞ .To construct a Feynman-Kac type formula we use a Euclidean field. Considerthe family of Gaussian random variables { φ E ( F ) , F ∈ H − ( R ) } with mean andcovariance E µ E [ φ E ( F )] = 0 and E µ E [ φ E ( F ) φ E ( G )] = 12 ( F, G ) H − ( R ) on a chosen probability space ( Q E , Σ E , µ E ). Note that for f ∈ H − / ( R ) the rela-tions δ t ⊗ f ∈ H − ( R ) and k δ t ⊗ f k H − ( R ) = k f k H − / ( R ) hold, where δ t ( x ) = δ ( x − t ) is Dirac delta distribution with mass on t . The family of identities used in(2.8) is then given by J t : L ( Q ) → E , t ∈ R , defined by the relationsJ t L ( Q ) = 1l E and J t : m Y j =1 φ ( f j ) : = : m Y j =1 φ E ( δ t ⊗ f j ) : . Under the identification F ∼ = L ( Q ) it follows that(J t F, J s G ) E = ( F, e −| t − s | H f G ) F for F, G ∈ F . B The Nelson model
The Nelson Hamiltonian H is a self-adjoint operator acting in the Hilbert space L ( R ) ⊗ F ∼ = Z ⊕ R F dx, which is given by H = ( −
12 ∆ + V ) ⊗
1l + 1l ⊗ H f + g Z ⊕ R φ ( x ) dx, (B.1)where the interaction is defined by φ ( x ) = 1 √ (cid:16) a ∗ ( ˆ ̺/ √ ωe i ( · ,x ) ) + a ( e ˆ ̺/ √ ωe − i ( · ,x ) ) (cid:17) .H is self-adjoint on D ( H p ) ∩ D ( H f ). A point charge limit of H , ˆ ̺ ( k ) → H has a ground state.15e see the relationship between H and H ( P ). The total momentum P tot ,µ isdefined by P tot ,µ = − i ∇ µ ⊗
1l + 1l ⊗ P f µ , µ = 1 , ,
3. Let V = 0 in H . Then H becomes a translation invariant operator, which implies that[ H, P tot ,µ ] = 0 , µ = 1 , , . Thus H can be decomposed with respect to the spectrum of total momentum P tot ,µ and it is known that H ∼ = Z ⊕ R H ( P ) dP. (B.2) Acknowledgments:
The author acknowledges support of Challenging ExploratoryResearch 15K13445 from JSPS, and thanks for the kind hospitality of 51 WinterSchool of Theoretical Physics Ladek Zdroj, Poland, 9 - 14 February 2015. Moreoverhe also thanks Tadahiro Miyao who gives an idea to solve Lemma 2.10 which is akey ingredient in this paper.
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