INTEGRAL KERNELS OF THE RENORMALIZED NELSON HAMILTONIAN (Mathematical aspects of quantum fields and related topics)

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(1)69. INTEGRAL kernels KERNELS of OF the THE renormalized RENORMALIZED Integral NELSON Hamiltonian HAMILTONIAN Nelson Hiroshima*∗, Fumio Hiroshima Faculty of Mathematics, Kyushu University University††. Abstract In this article we consider the ground state of the renormalized Nelson HamilHamil‐ tonian in quantum field theory by using the integral kernel of the semigroup gengen‐ erated by the Hamiltonian. By introducing an infrared cutoff, the existence of the ground state is shown and the expectation values of observables with respect to the ground state are given in terms of a probability measure.. 1. Introduction. This joint work This is is aa joint work with with Oliver Oliver Matte Matte [19]. [19]. Since Since the the end end of of the the last last century century several several interaction models between quantum mechanical matters and quantum fields fields have been investigated; investigated; the the Pauli-Fierz Pauli‐Fierz model model [25] [25] in in non-relativistic non‐relativistic quantum quantum electrodynamics electrodynamics and spin-boson spin‐boson model have been typical examples. There are a lot of contributions to studying ground states of models. Here the ground state describes an eigenvector associated with the bottom of the spectrum of a self-adjoint self‐adjoint operator. The Hamiltonian of the interaction system can be realised as a self-adjoint self‐adjoint operator and we are interested in investigating the ground state of the Hamiltonian, e.g., the existence of the ground state and its properties. In this article we discuss the ground state of a renormalized Hamiltonian introduced by by Edward Edward Nelson Nelson in in 1964 1964 [23, [23, 24] 24] to to consider consider the the removal removal of of ultraviolet ultraviolet cutoffs. cutoffs. This This ∗ *e e-mail: ‐mail: †\dagger_{744}. hiroshima@math.kyushu-u.ac.jp hiroshima@math.kyushu‐u.ac.jp 744 Motooka, Nishi-ku, Nishi‐ku, Fukuoka, Japan, 819-0395 819‐0395. 1.

(2) 70 model is nowadays the so-called so‐called renormalized Nelson Hamiltonian. The renormalized Nelson Hamiltonian describes a linear interaction between non-relativistic non‐relativistic spinless nunu‐ cleons and spinless scalar mesons, where the non-relativistic nucleons are governed by non‐relativistic 2 dN a Schr¨ odinger operator acting in L (R Schrödinger L^{2}(\mat hbb{R}^{dN}),) , where NN denotes the number of nucleons and dd the spatial dimension. The physical reasonable choice is dd=3 = 3.. In this article we assume that NN=1 = 1 and dd=3 = 3 for simplicity. In mathematics field field operator 1 ¯ φ(f ) = √ (a† (fˆ) + a(fˆ)) 2. \phi(f)=\frac{1}{\sqrt{2} (a^{\dag er}(\hat{f})+a(\overline{\hat{f} ). (1.1) (1.1). can be defined L2 (R3 ),, but in physics φ(x) = δ(· − x).. defined for ff\in ∈ defined by φ(f L^{2}(\mathbb{R}^{3}) \phi(x) is defined \phi(f)) with ff=\delta(\cdot-x) It is not straightforward however to define φ(x). It is common to define φ(x) define \phi(x) . define \phi(x) as the limit of φ(f → ∞,, where ff_{n}arrow\delta(\cdot-x) → ∞ in some sense. ff_{n}n is called n ) as nnarrow\infty n → δ(· − x) as nnarrow\infty \phi(f_{n}). cutoff function or ultraviolet cutoff cutoff function in this article. The Nelson Hamiltonian is cutoff. defined defined by introducing cutoff cutoff functions and it can be realised as a self-adjoint self‐adjoint operator acting in the Hilbert space given by 2 3 H \mathscr{H= }=L^{2L }(\math(R bb{R}^{3})\)otim⊗ es \matF hscr{F,}, 2 3 \mathscr{F} denotes the boson Fock space over L where F defined by L^{2}(\mat(R hbb{R}^{3)}) defined ∞ 2 3n F \mathscr= {F}=\opl⊕us_{n=0 n=0}^{\in[L fty}[L_{sym sym}^{2}(R (\mathbb{R}^)], {3n})], 2 3n 2 2 0 where L ‐functions with L L_{ssym ym}^{2}(\(R mathbb{R}^{3n})) denotes the set of symmetric LL^{2} -functions L_{ssym ym}^{2}(\(R mathbb{R})^{0})== \mathbb{C. C} . We (n) 2 3n set F \mathscr{F}^{(= n)}=L_{Lsym}sym ^{2}(\mat(R hbb{R}^{3n})) and (n) F F | \∃m = 0 for \forall≥ n\geqm}. m }. \math0scr{F}= _{0}= {Φ { \Phi\in∈\mathscr{F}| exists m such such that that Φ\Phi^{(n)}=0 for ∀n. \mathscr{F} . Subtracting a renormalization term F \mathscr{F0}_{0} is called the finite finite particle subspace of F. from the Nelson Hamiltonian, we can define H_{\infty}. define the renormalized Nelson Hamiltonian H ∞. A crucial point is that H H_{\infty} defined by the limit of the Nelson Hamiltonian and ∞ is defined consequently it is given as a semi-bounded semi‐bounded quadratic form. Then it is impossible to \mathscr{H} .. Recently see M\emptyset 1ler and H_{\infty} ∞ as see an an explicit explicit form form of of H as an an operator operator in in H Recently J. J. Møller and O. O. Matte Matte [22] [22] 2.

(3) 71 71 −T H∞ however succeeded in constructing a Feynman-Kac explicitly. Feynman‐Kac type formula of ee^{-TH_{\infty}} More precisely it is shown that F, e−T H∞ G H = dxEx [(F (B0 ), KT G(BT ))F ] .. (F, e^{-TH_{\infty} G)_{\mathscr{H} = \int_{\mathbb{R}^{3} dxE^{x}[(F(B_{0}), K_{T}G(B_{T}) _{\mathscr{F} ]. R3. Here (·, \mathscr{F,}, (B 3‐dimensional Brownian motion t )t≥0 (\cdot, \cdo·) t)_{\matF hscr{F} is the inner product on F (B_{t})_{t\geq 0} denotes 3-dimensional and K K_{T} T is an integral kernel which is of the form T. ˜. (U ) −T Hf a(U ) e− 0 V (Bs )ds eTaH_{f}}e^{a(\overl e ine{U})}\dagger. e . K K_{T}=e^{T =j_{0}^{T}V(B_{s})ds}e^{a(U)}e^{†. In In this this article article we we shall shall show show (1) (1) and and (2) (2) below: below: (1) H_{\infty} ∞ has (1) If If an an infrared infrared cut cut off off is is introduced, introduced, then then H has the the ground ground state state ϕ and it it is is \varphig_{g} and unique up to multiple constants. φ) is (2) g = ϕg (x, \phi) (2) The The ground ground state state ϕ\varphi_{g}=\varphi_{g}(x, is localized localized in in the the sense sense of of Gaussian Gaussian domination domination with respect to field field operators φ, \phi , and super-exponential super‐exponential decay of the number of bosons: ∞ 2 e2βn

(4) ϕ(n) ∀β > 0. g

(5) L2 (R3x ×R3n ) < ∞,. \sum_{n=0}^{\infty}e^{2\betan}\Vert\varphi_{g}^{(n)}\Vert_{L^{2}(\mathb {R} _{x}^{3}\cros \mathb {R}_{k}^{3n}) ^{2}<\infty,\foral\beta>0. k. n=0. Above Above (1) (1) and and (2) (2) can can be be proven proven by by using using Feynman-Kac Feynman‐Kac type type formula formula mentioned mentioned above. above.. 2 2.1. Renormalized Nelson Hamiltonian Definition of the Nelson Hamiltonian with cutoffs Definition. In this section we define define the renormalized Nelson Hamiltonian as a self-adjoint self‐adjoint operator \Lambda \mathcal{H} . The Nelson Hamiltonian with ultraviolet cutoff acting in H. Λ is defined cutoff defined by H 1l ⊗ Hf H_{f}+H_{I}. + HI . Λ = Hp ⊗ 1l +11+]1\otimes H_{\Lambda}=H_{p}\otimes We explain H H_{f}f and H H_{I}I below. First p, H H_{p}, 1 Hp = − Δ + V 2. H_{p}=- \frac{1}{2}A+V 3.

(6) 72 2 3 denotes a Schr¨ odinger operator acting in L Schrödinger L^{2}(\mat(R hbb{R}^{3). }) . Here we assume that the mass of the particle is one and we shall give an assumption on VV below. Operator H f = dΓ(ω) H_{f}=d\Gamma(\omega) \omega : \mathscr{F} defined denotes the free field field Hamiltonian acting in F defined by the second quantization of ω: n ∞ ω(kj ) , dΓ(ω) = 0 ⊕ ⊕n=1. d \Gamma(\omega)=0\oplus\{\oplus_{n=1}^{\infty}[\sum_{j=1}^{n}\omega(k_{j})]\}, j=1. where where ω(k) \omega(k) describes describes the the dispersion dispersion relation relation (the (the multiplication multiplication operator operator by by ω(k)). \omega(k) ). Note that H H_{f}f acts as. n ∞ ω(kj ) Φ(n) (k1 , · · · , kn ) Hf Φ = ⊕n=1. H_{f} \Phi=\oplus_{n=1}^{\infty}(\sum_{j=1}^{n}\omega(k_{j}) \Phi^{(n)}(k_{1}, \cdot \cdot \cdot k_{n}) j=1. for Φ · · ,, k_{n}) kn ).. Finally H H_{I}I is the interaction term defined defined by 1 , ·\cdots \Phi== \oplus_{⊕n=0}∞ ^{\inftyΦ }\Phi(n) ^{(n)} (k (k_{1}, n=0. ˆκ,Λ (k) ˆκ,Λ (k) 1 ikx ϕ † −ikx ϕ

(7) dk. HI = √ + a(k)e

(8) a (k)e 2 ω(k) ω(k). H_{I}=\frac{1}{\sqrt{2}\int(a^{\dag er}(k)e^{-ikx}\frac{\hat{\varphi} _{\kap a,\Lambda}(k)}{\sqrt{\omega(k)}+a(k)e^{ikx}\frac{\hat{\varphi}_{\kap a, \Lambda}(k)}{\sqrt{\omega(k)} dk.. † (k) are the formal kernel of the annihilation and creation operators, Here a(k) a(k) and aa\dagger(k) respectively, satisfying canonical commutation relations:. [a(k), a† (k ′ )] = δ(k − k ′ ). [a(k), a^{\dagger}(k')]=\delta(k-k') .. Using this notation we have a(f ) =\intff(k)a(k)dk (k)a(k)dk and aa†\dagger(f)=\int (f ) = ff(k)a\dagger(k)dk (k)a† (k)dk.. Both a(f)= † a(f (f ) are closed operators and they satisfy canonical commutation realtions: a(f)) and aa\dagger(f) [a(f ), a† (g)] = (f¯, g). [a(f), a^{\dagger}(g)]=(\overline{f}, g) .. ϕ cutoff function given by \ˆhat{\vaK,Λ rphi}_{K,\Lambda} is the cutoff ⎧ ⎪ ⎨0, |k| < κ, ϕˆκ,Λ (k) = 1, κ ≤ |k| ≤ Λ, ⎪ ⎩ 0, |k| > Λ. \hat{\varphi}_{\kap ,\Lambda}(k)=\{begin{ar y}{l 0, |k<\ap , 1 \kap \leq|k\leqLambda, 0 |k>\Lambda \end{ar y}. \Lambda such that \kappa and ultraviolet cutoff with infrared cutoff cutoff parameter κ cutoff parameter Λ 00\leq\kappa<\Lambda. ≤ κ < Λ.. 4.

(9) 73 In In order order to to give give an an assumption assumption on on VV we we define define the the Kato-class Kato‐class of of potentials potentials [4, [4, 21]. 21]. d \mathbb{R} d V: \ mathbb{R}^{d}arrow Potential V : R → R is called d-dimensional ‐dimensional Kato-class Kato‐class potential whenever lim sup |g(x − y)V (y)|dy = 0. r ar ow 01\dot{ \imath} m\sup_{x\in \mathb {R}^{d} \int_{|x-y|\leq r}|g(x-y) V(y)|dy=0. r→0 x∈Rd. |x−y|≤r. holds, where function gg depends on the dimension and is given by ⎧ ⎪ d = 1, ⎨|x|, g(x) = − log |x|, d = 2, ⎪ ⎩ 2−d |x| , d ≥ 3.. g(x)=\{ begin{ar y}{l |x, d=1, -\log|x, d=2, |x^{2-d}, d\geq3. \end{ar y}. We introduce assumptions used through this article unless otherwise stated. Assumption Assumption 2.1 2.1 (Dispersion (Dispersion relation relation and and potential) potential) We We assume assume (1) (1) and and (2): (2): (1) = |k|. (1) ω(k) \omega(k)=|k|. (2) (2) VV is is 3-dimensional 3‐dimensional Kato-class Kato‐class potential. potential. It can be seen that H + 1l ⊗ H_{f} Hf . By the H_{I}I is infinitesimally infinitesimally small with respect to H p⊗ H_{p}\oti mes 1]l1+I\otimes Kato-Rellich theorem [20] H is self-adjoint on D(H ⊗ 1 l + 1 l ⊗ H ) and bounded p f and bounded from Kato‐Rellich theorem [20] H_{\Lambda}Λ is self‐adjoint on D(H_{p}\otimes from 11+]1\otimes H_{f}) below. We mention it as proposition below. \Lambda<\infty Proposition 2.2 Let κ\kappa\geq ≥ 00 and Λ < ∞.. Then H H_{\Lambda} self‐adjoint and bounded from Λ is self-adjoint + 1l ⊗ H below on D(H p ⊗ 1l11+]1\otimes f ).. D(H_{p}\otimes H_{f}). 2.2. Definition of of the renormalized Nelson Hamiltonian Definition. According According to to [23], [23], we we introduce introduce the the renormalization renormalization term term defined defined by by 2 |ϕˆκ,Λ (k)| 1 β(k)dk, EΛ = − 2 R3 ω(k). E_{\Lambda}=-\frac{1}{2}\int_{\mathb {R}^{3} \frac{|\hat{\varphi}_{\kap a, \Lambda}(k)|^{2} {\omega(k)}\beta(k)dk,. where β(k) \beta(k) describes a propagator given by −1 1 2 . β(k) = ω(k) + |k| 2. \beta(k)=(\omega(k)+\frac{1}{2}|k|^{2})^{-1}. We notice that lim EΛ = −∞.. \lim_{\Lambdaarrow\infty}E_{\Lambda}=-\infty.. Λ→∞. 5.

(10) 74 Proposition ≥ 0. 0 . Then Proposition 2.3 2.3 (Nelson (Nelson [23]) [23]) Let Let κ\kappa\geq Then there there exists exists aa self-adjoint self‐adjoint operator operator H ≥0 T\geq H_{\infty} ∞ bounded below such that for any T −T (HΛ −EΛ ). −T H∞. \lim kap a r ow\inefty 1\dot{ \imath} me^{-T(H_{\Lambda}-E_={\Lambdea})}=e^{- TH_{\infty} .. κ→∞. Nelson Nelson [23] [23] proved proved the the convergence convergence in in Proposition Proposition 2.3 2.3 in in the the strong strong sense. sense. It It is is however however shown that this convergence is in the uniform sense in e.g. [22]. shown that this convergence is in the uniform sense in e.g. [22]. −T H∞ can be represented in terms of path measures. Let (B We shall see that ee^{-TH_{\infty}} t )t≥0 (B_{t})_{t\geq 0} x \mathcal{Wx }^{x} debe 3-dimensional 3‐dimensional Brownian motion on a probability space (Ω, de‐ (\Omega, \matF, hcal{F}, \matW hcal{W}^{), x}) , where W x \Omega such that W notes a probability measure on Ω (B0 = x) = 1.. Let \mathcal{W}^{x}(B_{0}=x)=1 T −|T −s|ω(k) T −sω(k) e e

(11)

(12) 1l|k|≥κ e−ikBs ds, U˜ = 1l|k|≥κ eikBs ds. U= ω(k) ω(k) 0 0. U= \int_{0}^{T}\frac{e^{-s\omega(k)} {\sqrt{\omega(k)} ]1_{|k\geq\kap a}e^{- ikB_{s} ds, \tilde{U}=\int_{0}^{T}\frac{e^{-|T-s|\omega(k)} {\sqrt{\omega(k)} ] 1_{|k\geq\kap a}e^{ikB_{s} ds.. 3 ∋ k = Both integrals are finite ≥ 00 and R finite for arbitrary κ\kappa\geq \mathbb{R}^{3}\ni k\neq0. 0 . Furthermore since. 2 2 ˜ E[ |U | dk]U|^{2}dk]<\infty < ∞ and E[ U, U \~{˜U}\∈ in L^{2}L(\mat2h(R bb{R}^3{3)}) almost E[\int_{\mathbb{R}^{3}}| E[\int_R{\mat3 h|bb{UR}^|{3} dk] |\tilde{U}|^< {2}dk]<\i∞, nfty , we can see that U, R3 † (U ) and a(U surely. Hence both aa\dagger(U) well‐defined closed operators almost surely. a(U)) are well-defined Here we review the exponentials of annihilation operators and creation operators. B ] for See L2 (R3 ) and See [17, [17, Appendix Appendix B] for the the detail. detail. Let Let ff\in ∈ L^{2}(\mathbb{R}^{3}) and we we define define the the exponential exponential of of. creation operators FF_{f}f by. and the domain is given by . ∞ 1 † n a (f ) n! n=0. F_{f}= \sum_{n=0}^{\infty}\frac{1}{n!}a^{\dag er}(f)^{n}. Ff =. ∞ 1 † n Φ ∈ ∩∞

(13) a† (f )n Φ

(14) < ∞ . n=1 D(a (f ) ) | n! n=0. D(F_{f})= \{\Phi\in\bigcap_{n=1}^{\infty}D(a^{\dag er}(f)^{n})|\sum_{n=0} ^{\infty}\frac{1}{n!}\Vert a^{\dag er}(f)^{n}\Phi\Vert<\infty\}.. D(Ff ) =. \Phi\in∈\mathscr{F}^{(m)} Let Φ F (m) . Thus we have. ∞ . √. m + n − 1··· n!. √. m. \Vert F_{f}\Phi\Vert\leq\Vert\Phi\Vert+\sum_{n=1}^{\infty}\frac{\sqrt{m+n-1} \cdots\sqrt{m} {n!}\Vert f\Vert^{n}\Vert\Phi\Vert<\infty..

(15) Ff Φ

(16) ≤

(17) Φ

(18) +. n=1.

(19) f

(20) n

(21) Φ

(22) < ∞.. F define the exponential of annihilation operators by 0 ⊂ D(F f ) follows. We also define \mathscr{F}_{0}\subset D(F_{f}) ∞ 1 a(f )n n! n=0. G_{f}= \sum_{n=0}^{\infty}\frac{1}{n!}a(f)^{n}. Gf =. 6.

(23) 75 with the domain . ∞ 1 n

(24) a(f ) Φ

(25) < ∞ . | n! n=0. D(G_{f})= \{\Phi\in\bigcap_{n=1}^{\infty}D(a(f)^{n})|\sum_{n=0}^{\infty} \frac{1}{n!}\Vert a(f)^{n}\Phi\Vert<\infty\}.. D(Gf ) =. Φ∈. †. n ∩∞ n=1 D(a(f ) ) ¯. a (f ) We simply write FF_{f}=e^{a(f)}\dagger and G = ea( f = e G_{ff}=e^{a(\overl ine{ff})}) whenever confusion may † a† (f ) ∗ a(f¯) can see that (e (e^{a(f)}\dagger)^){*}\supset⊃e^{a(\eoverline{f})} and this implies that ee^{aa(f)}\(fdagger) is closable. † ) a(f ) dagger is denoted by the same symbol. Similarly the closure of ee^{a(f)} is ee^{aa(f)}\(f same symbol.. arise. Then we The closure of denoted by the. Proposition L2 (R3 ) and f, gg\in∈ Proposition 2.4 2.4 (Algebraic (Algebraic relations) relations) Let Let f, L^{2}(\mathbb{R}^{3}) and PP be be aa polynomial. polynomial. † (f +g) a† (g) a† (f ) \Omega\in \mathscr{F} Suppose that Ω ∈ F be the Fock vacuum. Then (1) e e Ω = ea+g)}\Omega\dagger, Ω, Suppose that be the Fock vacuum. Then (1) e^{a}e^{a(f)}\Omega\dagger(g)\dagger=e^{a(f † (f ) † (f ) a† (f ) a† (f ) a(g) a (¯ g ,f ) a (2) (a(g))e Ω = Pin((¯ , f ))e Ω and Ω = eine{g},f)}e^{a(fe)}\Omega\dagger. Ω. (2) PP(a(g))e^{a(f)}\Omega\dagger=P((\overl e{g},gf))e^{a(f)}\Omega\dagger and (3) (3) ee^{a(g)}e^{a(fe)}\Omega\dagger=e^{(\overl \square B ]. Proof ✷ Proof See See [17, [17, Appendix Appendix B]. † (f ) † (f ) † (f ) a a a We note that ee^{a(f)}\Omega\dagger Ω is an eigenvector for a(f Ω = i(¯ Ω. a(f)) such that a(f a(f)e^{a(f)e )}\Omega\dagger=(\overl ne{gg},, f)fe^{a)e (f)}\Omega\dagger. † (f ) a \mathbb{C}. ee^{a(f)}\Omega\dagger We conclude that the spectrum of a(f Ω is called a coherent vector. a(f)) is C.. √ a† (f ) −tHf Proposition > 0 and D(1/ ω).. Then e Proposition 2.5 2.5 (Boundedness) (Boundedness) Let Let tt>0 and ff\in∈D(1/\sqrt{\omega}) Then both both ee^{a(f)}e^{-tH_{f}}\dagger −tH a(f ) f e^{-tH_{f}}e^{a(f)} e are bounded operators. and e B ]. Proof Proof See See [17, [17, Appendix Appendix B]. T † (U ) − T H ˜) a ˜ =\frac{T}{2}H_{f f f a(inU \~{A}=e^{}}e^{a(\overl e− 2 H e e{U})}. . Let AA=e^{a(U)}e^{-\frac{T}{2}H_{f}}\dagger =e e 2 and A. \square ✷. A and A Ã˜ are bounded. Lemma 2.6 Let κ\kappa\geq ≥ 0. 0 . Then A. 2 Proof Since E[ |U |2 /ωdk] E[\int_{\mathbb{R}^{3}}| U|^{2}/\omega<dk]<\i∞ nfty and E[ E[\int_{\Rmat3hbb{|UR˜}^{3|} |\t/ωdk] ilde{U}|^{2}/\omega<dk]<∞, \infty , the lemma follows from R3 \square Proposition 2.5. ✷ G\in \mathcal{H} Theorem ≥ 0. ∈ H.. Then 0 . Let F, G Theorem 2.7 2.7 (Matte (Matte and and Møller[22]) M\emptyset 1ler[22] ) Let Let κ\kappa\geq Let F, Then it it follows follows that T 1 ˜ F, e−T H∞ G H = ) dxEx e− 0 V (Bs )ds e 2 Sren F (B0 ), AAG(B ,, (2.1) T (2.1). (F, e^{-TH_{\infty} G)_{\mathscr{H} = \int_{\mathbb{R}^{3} dxE^{x}[e^{-\int_{0} ^{T}V(B_{s})ds}e^{\frac{1}{2}S_{ren} (F(B_{0}), A\~{A} G(B_{T}) _{\mathscr{F} ] F. R3. 7.

(26) 76 where phase factor SS_{ren} ren is given by T t ∇ϕ0 (Bs − Bt , s − t)ds dBt − 2 Sren = 2. T. S_{ren}=2 \int_{0}^{T}(\int_{0}^{t}\nabla\varphi_{0}(B_{s}-B_{t}, s-t)ds)dB_{t} -2\int_{0}^{T}\varphi_{0}(B_{S}-B_{T}, s-T)ds 0. and. 0. ϕ0 (Bs − BT , s − T )ds. e−ikX e−|t|ω(k) β(k)1l|k|≥κ dk. 2ω(k). \varphi_{0}(X, t)=\int_{\mathb {R}^{3} \frac{e^{-ikX}e^{-|t\omega(k)} {2\omega(k)}\beta(k)I_{|k\geq\kap a}dk.. ϕ0 (X, t) = −T H∞ e^{-TH_{\infty}}. 0. In particular e. . R3. for TT>0 > 0 is positivity improving, and if H H_{\infty} ∞ has a ground state, then it is unique up to multiple constants. We give several remarks on Theorem 2.7. 2 3 L^{2}2 -functions \mathbb{3R}^{3.}. \mathscr{F} -valued \mathcal{H} with (1) (1) In In (2.1) (2.1) we we identify identify H with L L^{2}(\mat(R hbb{R}^{3;};\matFhscr{F): }) : the the set set of of F ‐valued L ‐fUnctions on on R x. I.e., FF\in \mathcal{H} ∈ H implies that FF(x)\in (x)\mathscr{F} ∈ F for each x.. (2) H_{\Lambda} H_{I}I with gH_{I}I .. Then Λ by (2) We We revive revive aa coupling coupling constant constant gg in in H by replacing replacing H with gH Then renorrenor‐ 2 malization term E E_{\Lambda} coefficient of gg^{2} in the expansion of the Λ is identical with the coefficient = 0 on gg^{2}2 , i.e., ground state energy of H H_{\Lambda} Λ with VV=0 E(g) = EΛ g 2 + an g 2n ,. E(g)=E_{\Lambda}g^{2}+ \sum_{2\leq n}a_{n}g^{2n}, 2≤n. . an g 2n < ∞.. See See [16]. [16].. \Lambda r ow\infty 1\dot{ \imath} m\sum_{2\leq n}a_{n}g^{2n}<\infty. and lim and we we can can see see that that Λ→∞. 2≤n. G\in D with (3) orinczi [13] ∈D F, G (3) Gubinelli,Hiroshima Gubinelli,Hiroshima and and L¨ Lörinczi [13] derived derived (2.1) (2.1) for for F, with some some dense dense D. domain D.. 3 3.1. Ground states Existence of the ground state. The difficulty difficulty in establishing the existence of the ground state in quantum field field theory comes from the fact that the bottom of the spectrum lies in the essential spectrum, not below it, as is the case for usual Schr¨ odinger operator. Let us consider Schr¨ odinger Schrödinger Schrödinger 8.

(27) 77 ∞ operator −Δ/2+V ∈ nfL (R3 ) and |V (x)| → satisfies VV\in L^{\i -\triangle/2+V with external potential VV which satisfies ty}(\mathbb{R}^{3}) |V(x)|arrow 0 V 0 as |x| → ∞.. This assumption yields that V is relatively compact with respect to |x|arrow\infty −Δ/2; V (−Δ/2 + m)−1 is compact for any mm>0 > 0 and the essential spectrum of -\triangle/2;V(-\triangle/2+m)^{-1} −Δ/2 + V is [0, ∞).. Let ee be the bottom of the spectrum of −Δ/2 + V and ee_{0}0 that of -\triangle/2+V [0, \infty) -\triangle/2+V. −Δ/2. + V . Then ee_{0}=0. 0 = 0. -\triangle/2. ee_{0}0 is equal to the bottom of the essential spectrum of −Δ/2 -\triangle/2+V If ee<e_{0} < e0 , then ee is discrete and we can conclude that −Δ/2 + V has the ground state. -\triangle/2+V + V , we denote the bottom of the spectrum of H Consider H H_{\Lambda} H_{\Lambda} Λ . Similar to −Δ/2 Λ and -\triangle/2+V E and E H H_{\Lambda} E_{0}0 , respectively. In the case of the Nelson Λ with no external potential by E Hamiltonian, despite inequality EE<E_{0}, < E0 , EE lies in the bottom of the essential spectrum,. and it is unclear that H H_{\Lambda} Λ admits a ground state. \Lambda<\infty In the case of Λ < ∞ it is shown that the ground state of H H_{\Lambda} Λ exists and it is unique up up to to multiple multiple constants. constants. This This is is due due to to e.g., e.g., [2, [2, 5, 5, 6, 6, 27]. 27]. In In [14] [14] the the existence existence of of a ground state of the renormalized Nelson Hamiltonian without ultraviolet cutoff cutoff is shown but only for sufficiently sufficiently small coupling constants. In this section using FeynmanFeynman‐ Kac type formula mentioned above we can also show \kappa>0. that the ground state of H > 0. H_{\infty} ∞ exists for arbitrary values of coupling constants for κ Note that in our setting the coupling constant is absorbed in coupling function ϕ ˆ \hat{\varpκ,Λ hi}_{\kap a,\Lambd.a}.. Theorem > 0.. Then Theorem 3.1 3.1 (Hiroshima (Hiroshima and and Matte Matte [19]) [19]) Suppose Suppose that that κ\kappa>0 Then the the ground ground exists and it is unique. state of H H_{\infty} ∞ Proof The uniqueness is due to Theorem 2.7. We shall show the existence. Outline of detail. of aa proof proof is is as as follows. follows. See See [19] [19] for for the the

(28) detail. 2 + ν 2 with some artificial constant ν\nu>0. > 0. Step 1: = |k| 1: We assume that ω(k) artificial \omega(k)=\sqrt{| k|^{2}+\nu^{2}} 3 G\subset⊂ \mathbb{RR}^{3} be a bounded and open subset. Let Let G τ\tau_{G}(x)=\inf\{t>0|B_{t}+x\not\in G (x) = inf{t > 0|Bt + x ∈ G} G\} G . In particular when x be the exit time from G. ∈ G, Define the quadratic x\not\in G, τ\tau_{G}(x)=0 G (x) = 0.. Define \mathscr{H}\cross \mathscr{H}arrow \mathbb{C} form Q Q_{t}t : H × H → C by. :. . t 1 ˜ \Phi ), A AΦ(B ) . dxEx 1lτG (x)≥t e 2 Sren e− 0 V (Bs )ds Ψ(B 0 t \Psi(B_{0}). \Psi\cros \Phi\mapsto\int_{\mathb {R}^{3} dxE^{x}[]1_{\tau_{G}(x)\geq t} e^{\frac{1}{2}S_{ren} e^{-\int_{0}^{t}V(B_{s})ds} (. Q Q_{t}t : Ψ × Φ →. R3. 9. , AÃ (Bt)) ].

(29) 78 Thus it can be seen that there exists a self-adjoint H_{G} self‐adjoint operator H G bounded from below −tHG such that (Ψ, Φ) = Qt (Ψ,\Phi) Φ).. The self-adjoint H_{G} self‐adjoint operator H G can be regarded as a (\Psi,ee^{-tH_{G}}\Phi)=Q_{t}(\Psi, 2 ∼ (Q) with some (G) ⊗ F . Under the identifications self-adjoint operator on L = L2L^{2}(Q) self‐adjoint identifications F L^{2}(G)\otimes \mathscr{F} \mathscr{F}\cong probability space (Q, B,{B}, µ), (Q, \mathcal \mu) , and 2 LL^{2}(G)\otimes × Q), (G) ⊗\mathscr{F}\cong F ∼ = L2 (G L^{2}(G\cross Q) ,. −tHG is hypercontractive for t t>0 > 0.. Hence H it can be seen that ee^{-tH_{G}} H_{G} G must have the 2 ground state in L (G × Q) by [12, 26], since λ × µ is a finite measure × QQ and G\cross ground state in L^{2}(G\cross Q) by [12, 26], since \lambda\cross\mu is a finite measure on on G and −tHG G. > 0,, is hypercontractive. Here λ\lambda denotes the Lebesgue measure on G. ee^{-tH_{G}},, tt>0. Step 2: Let ϕ > 0 and we extend ϕ H_{G} \varphiG _{G} be the unique ground state of H \mu>0 \varphiG _{G} to G with µ 2 3 the vector on L (R × Q)Q) by zero-extension, zero‐extension, i.e., L^{2}(\mathbb{R}^{3}\cross ϕG (x, φ) (x, φ) ∈ G × Q ϕ˜G (x, φ) = 0 (x, φ) ∈ G × Q.. \tilde{\varphi}_{G}(x,\phi)=\{ begin{ar ay}{l} \varphi_{G}(x,\phi) (x,\phi)\inG\cros Q 0 (x,\phi)\not\inG\cros Q. \end{ar ay}. 2 3 (R Let ϕ G_{n}\nuparrow↑\mathR bb{R}^3{3} . It can be seen that {ϕ n = ϕGn and G n } is a Cauchy sequence in L \varphi_{n}=\varphi_{G_{n}} \{g_{n}\} L^{2}(\mat hbb{R}^×Q) {3}\cross Q) \Lambda<\infty and lim ϕn exists for each Λ < ∞.. The limit is denoted by ϕ and it is the ground \varphi_{\Λ Lambda} \lim_{nar ow\Lambda}\varphi_{n} n→Λ > 0. state of H H_{\Lambda} \mu>0. Λ with µ Step 3: It is established that if H > 0 has the ground state, then H H_{\Lambda} H_{\Lambda} \mu>0 Λ with µ Λ with. µ\mu=0 = 0 also > 0.. This also has has the the ground ground state, state, since since κ\kappa>0 This trick trick is is used used in in e.g. e.g. in in [5]. [5]. We We denote the ground state of H = 0 by the same symbol ϕ H_{\Lambda} \mu=0 \varphi_{Λ \Lambda}.. Λ with µ \Lambdaarrow\infty Step 4: Suppose that Λ → ∞.. Hence it can be also shown that {ϕ \{\varphi_{\ΛLambda}}\} is a compact −tH∞ 2 3 −tH set in L as Λ → ∞, (R × Q)Q) by using the uniform convergence of ee^{-tH_{\Lambda}Λ} to ee^{-tH_{\infty}} \Lambdaarrow\infty, L^{2}(\mathbb{R}^{3}\cross the the pull-through pull‐through formula, formula, spatial spatial exponential exponential decay decay [2, [2, 11] 11] and and the the Kolmogorov-RieszKolmogorov‐Riesz‐. Fr´ echet type Fréchet type theorem theorem [18]. [18]. This This implies implies that that {ϕ \{\varphi_{\ΛLambda}}\} includes includes aa strongly strongly convergent convergent ′ = ϕg is the ground state of H ϕ subsequence ϕ H_{\infty}. Λ ∞. \varphi_{Λ \Lambda'′} and we can conclude that lim \lim′_{→∞ \Lambda r ow\infty}\varphi_{\Lambda'}=\varphi_{g} Λ \squar e ✷. 3.2. Ground state expectations. Problems we are interested in are the expectation values of observables with respect to O be an observable realised as a self-adjoint the ground state of H H_{\infty} self‐adjoint operator in ∞ . Let O 2. +βN +βφ(f ) \mathscr{H}. We want to estimate (ϕ H (\varphig_,{g}Oϕ , O\varphig_{g). }) . Typical examples of OO are ee^{+\beta N} and ee^{+\beta\phi(f)^{2}} .. 10.

(30) 79 3.2.1. Super-exponential Super‐exponential decay of the number of bosons. + 1l ⊗ H_{f} Hf . The We consider the Nelson Hamiltonian without the interaction: H p ⊗ H_{p}\oti mes 1]l1+1\otimes ground state of it is ff\otim⊗ Ω,, where ff is the ground state of H es\Omega H_{p}p . Then the number of N denotes the number operator bosons of ff\oti⊗ Ω is zero. I.e., (1 l ⊗ NN)f\otimes\Omega=0 )f ⊗ Ω = 0,, where N mes\Omega (]1\otimes. defined = dΓ(1l).. We want to estimate the number of bosons of the ground defined by N N=d\Gamma(11) state ϕ H_{\infty} define the number operator with momenta grater than one by by^{1}1 ∞ . We define \varphig_{g} of H. † N a† (k)a(k)dk.. Then NN=N_{+}+N_{-} = N+ + N− .. We set (k)a(k)dk and N − = + = N_{+}= \int_{||k|≥1 k|\geq a1}a\dagger(k)a(k)dk N_{-}= \int_{||k|<1 k|<1}a\dagger(k)a(k)dk N.. 11\otimes l ⊗ NN by N βN+ ) for any β\beta\geq ≥ 0.0. Lemma 3.2 ϕ\varphi g ∈ _{g}\iD(e n D(e^{\beta N_{+}}). ∞ Proof Let EE be the bottom of the spectrum of H etEtH_{\i e−tH ϕ_{g}g for H_{\infty} ∞ . We have ϕ g _= \varphi {g}=e^{tE}e^{nfty}}\varphi βN tE βN −tH∞ any tt\geq ≥ 0. 0 . Then

(31) e Feynman‐Kac formula we can \Vert e^{\betaϕN}\varg p

(32) hi_{g}= \Vert=e^{etE}\Vert

(33) e e^{\beta N}e^{-teH_{\infty} \varphiϕ_{g}g\Ver

(34) t and by Feynman-Kac see that 1 t ˜ F, eβN e−tH∞ G = . dxEx e 2 Sren e− 0 V (Bs )ds F (B0 ), eβN AAG(B t). (F, e^{\beta N}e^{-tH_{\infty} G)= \int_{\mathbb{R}^{3} dxE^{x}[e^{\frac{1}{2} S_{ren}}e^{-\int_{0}^{t}V(B_{s})ds}(F(B_{0}), e^{\beta N}A\~{A} G(B_{t}) ] R3. It can be also seen that. β 1l. ∗ |k|≥1 U ) βN ee^{βN \beta+N_{A+} AA\~{Ã}== e^{a^{*}e(e^{a\bet(ea j1_{lk1\geq 1}}U)}e^{\ebeta N_{++} e^{e- −tH tH}e^{a(\eovera(linUe{˜U)}),},. βN+N_{+}}e^{-tH} and ee^{\beta e−tH is bounded for tt>\beta > β,, since l (−t|k|+β) βN+ −tH |k|<1 ee^{\beta e = Γ(e−t|k|+β1 ) = Γ(e1l|k|≥1 )Γ(e−t|k|1k|l]1_{| N_{+}}e^{-tH}=\Gamma(e^{-t| k|+\beta]1_{||k|≥1 k|\geq 1}})=\Gamma(e^{] 1_{| k|\geq 1(-t|k|+\beta)}})\Gamma(e^{-t| k|<1}})) + −tH∞ e and 1]l|k|≥1 (−t|k| + β) < 0 for any |k| ≥ 1. eβN G)|

(35)

(36) G

(37) |k|\geq 1 . Then |(F, |(F, e^{\beta N_{+}}e^{-tH_{\infty}}G)| \leq ≤ C\VertC

(38) F F\Vert\Vert G\Vert with 1_{|k|\geq 1}(-t|k|+\beta)<0 −tH βN ∞ + some constant CC depending only tt and β. e is bounded and the lemma \beta . Thus ee^{\beta N_{+}}e^{-tH_{\infty}}. follows.. \square ✷. βN ≥ 00 Lemma 3.3 ϕ\varphi_{g}\in g ∈ D(e D(e^{\beta−N-})) for any β\beta\geq. 1. N+ = dΓ(1l|k|≥1 − = dΓ(1l|k|<1 1N_{+}=d\Gamma(11_{| k|\geq 1})) and N N_{-}=d\Gamma(I_{| k|<1}))... 11.

(39) 80 Proof To show the lemma we use the Gibbs measure associated with the ground \mathbb{R} . Let state ϕ 3‐dimensional Brownian motion on the whole real line R. (B_{t})_{tt\in)\matt∈R hbb{R} be 3-dimensional \varphig_{g} .. Let (B ≤ t\} t}.. We set F sigma‐field generated by {B t = σ(Br ; −t ≤ r ≤ r ; −t ≤ r \mathcal{F}_{t}=\sigma(B_{r};-t\leq r\leqt)t) be the sigma-field \{B_{r};-t\leq r\leq G\mathcal= Define the probability measure µ \mu_{tt(·) }(\cdot) by {G}=\sigσ(∪ ma(\bigcup_{t\t≥0 geq 0}\matF hcal{Ft}_{). t}) . Define 1 dxEx [1lA Lt ] , A ∈ G, µt (A) = Z t R3. \mu_{t}(A)=\frac{1}{Z_{t} \int_{\mathb {R}^{3} dxE^{x}[1 _{A}L_{t}], A\in \mathcal{G},. where Z Z_{t}t denotes the normalising constant and 1. t. ¯. − −t V (Bs )ds S 2 ren e L (Bt )e −tr)f L_{tt}=f=(B_{-tf})f((B B_{t})e^{\f ac{1}{2}\overl ine{s}_{ren}}e^{\int_{-t}^{t} V(B_{s})ds}. with T. . t. . T. \overline{S}_{ren}=2\int_{-T}^{T}(\int_{-T}^{t}\nabla\varphi_{0}(B_{s}-B_{t}, s-t)ds)dB_{t}-2\int_ϕ{-T}^(B{T}\var−phiB_{0}(B,_{ss}-−B_{TT}, s)ds. -T)ds.. S¯ren = 2. . −T. −T. ∇ϕ0 (Bs − Bt , s − t)ds dBt − 2. . 0. s. T. −T. By a direct computation we then have 0 (e−tH∞ f ⊗ 1l, e−βN− e−tH∞ f ⊗ 1l) ds 0t drW −(1−e−β ) −t = lim E e , µ t t→∞ t→∞

(40) e−tH∞ f ⊗ 1l

(41) 2. ( \varphi_{g}, e^{-BN-}\varphi_{g})=\lim_{tar ow\infty}\frac{(e^{-tH_{\infty} f \otimes I,e^{-\beta N-}e^{-tH_{\infty} f\otimes I)}{|e^{-tH}\infty f\otimes] 1|^{2} =\lim_{tar ow\infty}E_{\mu_{t} [e^{-(1-e^{-\beta})\int_{-t}^{0} ds\int_{0}^{t}drW}],. (ϕg , e−BN− ϕg ) = lim where. 1 −|r−s|ω(k) −ik(Br −Bs ) e e dk. ω(k). W= \int_{\kappa\leq|k|\leq 1}\frac{1}{\omega(k)}e^{-|r-s|\omega(k)}e^{-ik(B_{r} -B_{s})}dk.. W = We see that. . κ≤|k|≤1. . |W| \leq\int_{\kappa\leq|k|\lω(k) eq 1}\omega(kdk)^{-3}d<k<\i∞nfty. |W | ≤. −3. κ≤|k|≤1. W is unifprmly bounded with respect to Brownian motion and which implies that W tt\geq ≥ 0. 0 . By the existence of the positive ground state, and. e−tH∞ f ⊗ 1l = ϕg , t→∞

(42) e−tH∞ f ⊗ 1l

(43). tar ow\infty1\dot{\imath}m\frac{e^{-tH_{\infty}f\otimesI}{\Verte^{-tH} \inftyf\otimes]1\Vert}=\varphi_{g}, lim. there exists a probability measure µ \mu_{\i∞ nfty} on (Ω, (\Omega, \matG) hcal{G}) such that 0 ∞ −β (ϕg , e−βN− ϕg ) = Eµ∞ e−(1−e ) −∞ ds 0 W dr .. (\varphi_{g}, e^{-\beta N-}\varphi_{g})=E_{\mu_{\infty} [e^{-(1-e^{-\beta}) \int_{\infty}^{0}ds\int_{0}^{\infty}Wdr}] 12.

(44) 81 81 The \mu_{∞ \infty} is The proof proof of of the the existence existence of of the the measure measure µ is due due to to [15]. [15]. By By the the analytic analytic continuation continuation in β\beta we can extend above identity to whole β\beta\i∈ n \mathC. bb{C} . Thus it follows that 0 ∞ β (ϕg , eβN− ϕg ) = Eµ∞ e−(1−e ) −∞ ds 0 W dr for β\beta\i∈ n \matC. hbb{C}.. (\varphi_{g}, e^{\beta N_{-} \varphi_{g})=E_{\mu_{\infty} [e^{-(1-e^{\beta}) \int_{\infty}^{0}ds\int_{0}^{\infty}Wdr}]. +βN In particular

(45) e > 0,, and the lemma is proven. g

(46) _{g}< \Vert e^{+\beta−N-ϕ }\varphi \Vert<∞ \infty for any β\beta>0. \square ✷. Theorem > 0.. Then Theorem 3.4 3.4 (Hiroshima (Hiroshima and and Matte Matte [19]) [19]) Let Let κ\kappa>0 Then βN ϕ\varphi_{g}\in ), N}) , \forall\beta>0. ∀β > 0. g ∈ D(e D(e^{\beta. Proof By Lemmas 3.2 and 3.3 we have βN βN βN βN βN βN (ϕ

(47) _< (ϕ_g{g},,e^{e\betβN g )_{g= (\varphig_e{g}e^{\betaϕN}\gvarphi) _= {g})=(\varphi a N_{++} ee^{\beta N_{−-} ϕ}\varphi })=(e^{\(e beta N_{+} \+varphiϕ_{gg}, ,e^{e\beta N-}\−varphiϕ_{gg})=)\Vert= e^{

(48) e \beta N_{+} \+varphiϕ_{gg}\

(49)

(50) e Vert\Vert e^{\bet−a N_{ϕ-} \gvarphi {g}\Vert∞. <\infty. \square Hence the theorem is proven. ✷ From Theorem 3.4 we can say that the number of bosons of the ground state of H H_{\infty} ∞ is a few.. 3.2.2. Gaussian dominations. In a similar manner to the proof of the super-exponential super‐exponential decay of ϕ \varphig_{g} we can also show a Gaussian domination of the ground state ϕ by the path measure µ \mu_{\infty}. ∞. \varphig_{g} 2 /2 1 d d 1 † −|x| 1/4 = e x|^{2}/2}/\pi^{1/4} /π . The harmonic Let aa= \f= \varphi(x)=e^{-| rac{1}√{\sqrt2 (x {2} (x+\f+rac{ddx}{dx})) and aa \dagger==\fra√c{12}{\sqr(xt{2} (− x-\frac{dxd}{dx}).) . Set ϕ(x) 2 † ∞ = a a.. The spectrum of hh is given by {n} oscillator in L defined by hh=aa\dagger L^{2}(\mat(R) hbb{R}) is defined \{n\}_{n=0}n=0 ^{\infty} and 2 n † −1/2 −|x| /2 hϕ ( a )ϕ.. Precisely ϕ\varphi_{n}(x)=h_{n}(x)e^{-| h\varphi_{n}=n\varphi_{n} n = nϕn for ϕ n = (n!) n (x) = hn (x)e x|^{2}/2} with some \varphi_{n}=(n!)^{-1/2}(\prod^{n}a\dagger)\varphi n ‐degree polynomial hn (x).. In particular we have n-degree h_{n}(x) 2. lim

(51) e(β/2)|x| ϕn

(52) L2 (R) → ∞... 1\dot{ \imath} m\beta\upar ow 1\Vert e^{(\beta/2)|x^{2} \varphi_{n} \Vert_{L^{2}(\mathb {R}) ar ow\infty. β↑1. (3.1) (3.1). Now we consider the Nelson Hamiltonian without the interaction: H 1+ ]l⊗H 1\otimes H_{f}f . The p ⊗1 H_{p}\ot imes ]l+1 ground state of it is ϕ\varphi_{0}=f\otimes\Omega 0 = f ⊗ Ω,, where ff is the normalised ground state of H H_{p}p . The free field H_{f}f can be regarded as an infinite field Hamiltonian H infinite freedom version of harmonic. 13.

(53) 82 h . We oscillator oscillator h. We have have aa counterpart counterpart of of (3.1). (3.1). Let Let φ(g) \phi(g) be be given given by by (1.1). (1.1). We We have have √ 2 −1/2 (β/2)φ(g) (Ω, Ω) = \(1 − β

(54) ˆ g / ω

(55) ) 1/2} . In particular (\Omega,ee^{(\beta/2)\phi(g)}\Omega)=(1beta\Vert\hat{g}/\sqrt{\omega} \Vert^{2})^{2.

(56) e(β/2)φ(g) ϕ0

(57) = ∞.. \lim_{\beta\upar ow\Vert\hat{g}/\sqrt{\omega}\Vert^{-2} \Vert e^{(\beta/2)\phi (g)^{2} \varphi_{0}\Vert=\infty.. lim √ β↑ ˆ g / ω −2. The renormalized Nelson Hamiltonian H H_{\infty} ∞ has a similar properties. We only mention the statement. √ Theorem > 0.. Suppose Theorem 3.5 3.5 (Hiroshima (Hiroshima and and Matte Matte [19]) [19]) Let Let κ\kappa>0 Suppose that that gˆ\hat/{g}/\sqrtω {\omega}∈ \in √ 2 2 3 2 1 3 l|k|≥κ g ˆ /ω ∈ L (R ) and β < 1/

(58) ˆ g / L ω

(59) . Let φ(g) be given by (1.1). Then L^{2}(\(R mathbb{R}^), {3}), 111_{ and \bet a <1/\Vert \ hat { g } / \sqrt { \ omega} \ Vert ^ { 2 } . Let \phi(g) be given by (1.1). Then |k|\geq\kappa}\hat{g}/w^{2}\in L^{1}(\mathbb{R}^{3}) 2. (β/2)φ(g) ϕ\varphi g ∈ _{g}\D(e in D(e^{(\bet a/2)\phi(g)^{2})). and βK(g)2 1 2 √ 2

(60) e(β/2)φ(g) ϕg

(61) 2 =

(62) √ 2 Eµ∞ e 1−βˆg/ ω ,, 1 − β

(63) ˆ g / ω

(64). \Verte^{(\beta/2)\phi(g)^{2}\varphi_{g}\Vert^{2}=\frac{1}{\sqrt{1-\beta\Vert \hat{g}/\sqrt{\omega}\Vert^{2} E_{\mu_{\infty}[e^{\frac{\betaK(g)^{2} {\imath}-\beta\Vertg/\sqrt{\omega}\Vert^{2} ]. (3.2) (3.2). where K(g) defined by K(g) denotes the random variable defined ∞ 1 e−|r|ω(k) gˆ(k)e−ikBr K(g) = dk dr . 2 −∞ ω(k) κ≤|k| In particular. K(g)= \frac{1}{2}\int_{-\infty}^{\infty}dr\int_{\kap a\leq|k}dk\frac{e^{- |r\omega(k)}\hat{g}(k)e^{-ikB_{r} {\omega(k)}. (β/2)φ(g)2. \lim_{\beta\upar ow\Vert\hat{g}/\sq

(65) ert{\omega}\Vert^{-2} \Vert e^{(\beta/2)\ϕphi (g)^

(66) {2} \var=phi_{g}\V∞. ert=\infty.. lim √ β↑ ˆ g / ω −2. g. Proof In a similar manner to the proof of Theorem 3.4 we have 2 2. iβφ(g) −k β I /2 −iβI (ϕ (\varpghi,_{ge}, e^{i\beta\phi(gϕ)}\vargp)hi_{= g})=E_{E\mu_{µ\∞infty[e } [e^{-k^{2}\beta^{2} 1I_{1}/e2}e^{-i\beta I2_{2],} ],. where . dk. R3. ∞. 2. |k|≥κ. R3. Using the identity. |ˆ g (k)|2 , 2ω(k). I_{1}= \int_{\mathb {R}^{3} dk\frac{|\hat{g}(k)|^{2} {2\omega(k)}, gˆ(k) I = dk1l 2ω(k) I_{2}= \int_{\mathb {R}^{3} dkI_{|k\geq\kap a}\frac{\overline{\hat{g}(k)} {2\omega(k)dse }\int_{-\infty}^{\infty}dse^{-\omeega(k)|s}e^{-ikB_{s}... I1 =. −ω(k)|s| −ikBs. −∞. . e^{-\phi(g)^{2}/2}=(2 \pi)^{-1/2}\int_{\mathbb{eR} e^{-ik\phie(g)}e^{-k^{2}/dk,2}dk,. e−φ(g). 2 /2. = (2π)−1/2. −ikφ(g) −k2 /2. R. and taking analytic continuation of β\beta to some region in the complex plan, we have the \square theorem. ✷ 14.

(67) 83. 4 4.1. Concluding remarks Comparison with H H_{\Lambda} Λ. In this article we show that the renormalized Nelson Hamiltonian has the unique ground state and the number of bosons in the ground state is super-exponential super‐exponential decay which can be proven by using the Gibbs measure derived from Feynman-Kac Feynman‐Kac type formula. In = 0, H H_{\infty} ∞ has In [19] [19] it it is is also also shown shown that that for for κ\kappa=0, has no no ground ground state, state, but but Gross-transformed Gross‐transformed. G ≥ 0. renormalized Nelson Hamiltonian H 0 . We note that H_{\inf∞ ty}^{G} has the ground state for all κ\kappa\geq G H > 0.. We can also see localization H_{\infty} ∞ and H H_{\inf∞ ty}^{G} are unitary equivalent if and only if κ\kappa>0. γ|x| such that

(68) e > 0.. These results are counterparts of the results \Vert e^{\gamma|ϕx|}g\var

(69) phi_< {g}\Vert∞ <\infty for some γ\gamma>0 for H_{\Lambda} Λ established for H established in in [1, [1, 2, 2, 3, 3, 5, 5, 11, 11, 14, 14, 15]. 15].. 4.2. The Nelson model on a Lorenzian manifold. \mathbb{R3}^{3} and In In [10] [10] the the Nelson Nelson model model is is defined defined on on aa static static Lorenzian Lorenzian manifold manifold instead instead of of R and ultraviolet cutoff is removed. It is also interesting to studying ground states of the cutoff defined on a static Lorenzian manifold. The Nelson renormalized Nelson Hamiltonian defined cutoff defined defined on a static Lorenzian manifold has the Hamiltonian with ultraviolet cutoff ground ground state state according according to to local local properties properties (curvature) (curvature) of of the the manifold. manifold. See See [7, [7, 8, 8, 9]. 9]. We conjecture that the renormalized Nelson Hamiltonian defined defined on a static Lorenzian manifold also has the ground state in the same condition on local properties of manifold as those of the Nelson Hamiltonian with ultraviolet cutoff. cutoff.. Acknowledgements: This work is financially financially supported by JSPS KAKENHI 16H03942 , CREST JPMJCR14D6 JPMJCR14D6 and JSPS open partnership joint research with Den16H03942, Den‐ mark 1007867. We also thank Tomoko Eto for typing the manuscript.. References [1] [1] A. Arai. Ground state of the massless Nelson model without infrared cutoff in a non-Fock non‐Fock representation. Rev. Math. Phys., 13:1075–1094, 13:1075−1094, 2001.. 15.

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