Abstract
The asymptotic behavior of the effective mass of the so-called Nelson model in quantum field theory is considered, where is an ultraviolet cutoff parameter of the model. Let be the bare mass of the model. It is shown that for sufficiently small coupling constant of the model, can be expanded as . A physical folklore is that as . It is rigorously shown that , with some constants , , and .
1. Introduction and Main Results
The model considered in this paper is the so-called Nelson model [1], which describes a nonrelativistic nucleon with bare mass interacting with a quantized scalar field with mass . The nucleon is governed by a Schrödinger operator. Let us first define the Nelson Hamiltonian. We use relativistic unit and employ the total momentum representation. Then the Hilbert space of states is the boson Fock space over which is given by where denotes the -fold symmetric tensor product and Then can be written as , where . The Fock vacuum is defined by . Let , be the annihilation operator and , the creation operator on , which are defined by and , where is the symmetrizer, the domain of operator , and the norm on . They satisfy canonical commutation relations as follows: on a suitable dense domain, where and is the inner product on (linear in the second variable). Let be a self-adjoint operator on . Then we define the self-adjoint operator on by , wherewith . Here, for a closable operator denotes the closure of The operator is called the second quantization of . The free energy of the scalar field is given by , where is considered as a multiplication operator on . Similarly the momentum of the scalar field is given by . The coupling of the nucleon and a scalar field is mediated through the Segal field operator defined bywhere is a cutoff function given by . Here is the form factor with infrared cutoff and ultraviolet cutoff , which are defined by The Nelson Hamiltonian with total momentum is given by a self-adjoint operator on as follows:where is a coupling constant. Let be the energy-momentum relation (the infimum of the spectrum ) defined byThen the effective mass is defined by Here denotes the three-dimensional Laplacian in the variable . We are concerned with the asymptotic behavior of as the ultraviolet cutoff goes to infinity. It is however a subtle problem. Removal of the ultraviolet cutoff through mass renormalization means finding sequences and such that , , and converges. Since we can see that is a function of , to achieve this, we want to find constants and such thatIf we succeed in finding constants and such as in (10), scaling the bare mass as where with an arbitrary positive constant , we have The mass renormalization is, however, a subtle problem, and unfortunately, we cannot yet find constants and such as in (10). For that reason we turn to perturbative renormalization, by which we try to guess the proper value of . Main results obtained in this paper are summarized as follows.
Theorem 1. Let . Then is an analytic function of and can be expanded in the following power series for sufficiently small :
Theorem 2. There exists a strictly positive constant such that
Theorem 3. There exist some constants and such that
From Theorems 2 and 3, if , it is suggested that . So, the mass of the Nelson model is renormalizable for sufficiently small .
The effective mass and energy-momentum relation have been studied mainly in nonrelativistic electrodynamics. Spohn [2] investigates the upper and lower bound of the effective mass of the polaron model from a functional integral point of view. Hiroshima and Spohn [3] study a perturbative mass renormalization including fourth order in the coupling constant in the case of a spinless electron. Hiroshima and Ito [4, 5] study it in the case of an electron with spin . Bach et al. [6] show that the energy-momentum relation is equal to the infimum of the essential spectrum of the Hamiltonian for . Fröhlich and Pizzo [7] investigate energy-momentum relation when infrared cutoff goes to 0.
2. Analytic Properties
In order to investigate the effective mass in a perturbation theory we have to check the analytic properties of .
2.1. Analytic Family in the Sense of Kato
Lemma 4. is an analytic family in the sense of Kato.
Proof. We prove is an analytic family of type (A). We see thatwhere and . Hence all we have to do is to prove the following facts. (a).(b)There exist real constants , , and such that for any We prove (a) at first. Since , we have . Additionally, since , we have Furthermore, since is a nonnegative and injective self-adjoint operator on , it follows that . Hence we have . Together with them, (a) is proven. Next we prove (b). Let be an arbitrary vector in Then we have Since , we have . HenceSince ,hold. HenceFrom triangle inequality, we have . In addition,Since and are strongly commutative and nonnegative self-adjoint operators on , holds. Hence . Then we haveFrom (19) and (23), (b) is proven. Hence is an analytic family of type (A). Since every analytic family of type (A) is an analytic family of in the sense of Kato, it is an analytic family in the sense of Kato.
We denote the ground state of by
Lemma 5. is analytic in and if and are sufficiently small. is strongly analytic in and if and are sufficiently small.
Proof. From [8, Theorem XII.9], (1) follows, and from [8, Theorem XII.8], (2) follows.
2.2. Formula
In this section we expand with respect to .
Lemma 6. The ratio can be expressed aswhere
Proof. Since is symmetry, , we haveSince , for any , holds. Taking a derivative with respect to on both sides above, we haveHere denotes the derivative or strong derivative with respect to , and Setting and , we haveThis expression and the definition of the effective mass prove the lemma.
2.3. Perturbative Expansions
We define operators and by and . Then . Moreover, let andSince is symmetry , we haveSince , is not injective. However, we define the operator (for notational simplicity we write for in what follows) on as follows. HereWe define the subspace of as .
Lemma 7. It holds that .
Proof. Let Then . Hence the lemma follows.
Lemma 8. Let . Then , , and the recurrence formulasfollow, withand is given by
Proof. We have by substituting in (30). Since is the ground state energy of , Hence . Since is the ground state of , can be . We can find that for holds in the same way as [3]. From now we set , , , and means (strong)derivative with respect to .holds for . Differentiating (37) with respect to , we have Hence and we haveSubstituting and into (39) and taking into account , we have . Differentiating (37) times with respect to , we also have By the induction on , we have and Substituting and into both sides above, we have From now on, we shall prove where we set by induction for , and Since , , . Moreover, since we have , where . Assume that the assumption of the induction holds when . ThenIt is derived that , , by and (46). By the assumption of the induction, , , holds. When , it holds thatwhere means that is omitted. By the assumption of the induction, the supports of the functions and are or . Furthermore, holds. By the assumption of the induction, the support of the right hand side is or . Hence we have or , . We can prove , , and or , , in a similar way. From the discussion so far, we have Hence we have where and are some constants. Since , , and , . Hence (33) and (34) are proven. By the discussion so far, (35) are also proven. We can derive (36) by (33) and .
3. Main Theorems
For notational simplicity we set and for . Let
Theorem 9. Let . Then is an analytic function of and can be expanded in the following power series for sufficiently small :
Proof. By the power series (29), we haveBy Lemma 8, if and only if both and are even or odd. Then we have From the fact that both and are analytic functions of and Lemma 6, we have the following power series: Since is an analytic function of , we can write We note that Hence if is sufficiently small, then we have the following power series:This proves the theorem.
Theorem 10. There exists strictly positive constant such that .
Proof. From (59), we have Therefore . Since we haveSubstituting into (27) and using (25), we haveIn addition, by setting , we have . Since holds. Hence we haveDifferentiating both sides of (63) with respect to , we have Substituting into both sides, we have Therefore and Since , we havewhere is some constant. By , (62), (64), and (68), we have It is also seen that Thus we have Changing variables into polar coordinate, we have Since , the improper integral converges. It is trivial to see that . Thus the theorem follows.
Lemma 11. It follows that for .
Proof. By (64), we have Assume that holds when . Differentiating both sides of (63) times with respect to and substituting , we have however, when is odd. Since , and Thus andSince , , and , by the assumption of induction, Hence holds when .
Lemma 12. It holds that , , and the recurrence formulas
Proof. The first and second expressions are proven in Theorem 10. From (75), it follows that These prove the lemma.
Lemma 13. It is proven that can be expanded as where are given by
The proof of Lemma 13 is given in the next section. The asymptotic behaviors of terms as is given in the lemma below. Only two terms and logarithmically diverge, and other terms converge as .
Lemma 14. (1)–(3) follow the following:(1)There exist some constants and such that .(2)There exist some constants and such that .(3)For .
The proof of Lemma 14 is technical and also given in the next section.
Lemma 15. It holds that
Proof. From (36), we have and . It implies (80).
Now we are in the position to state the main theorem in this paper.
Theorem 16. There exist some constants and such that
Proof. We have . Since , we haveWe also have . Then and we also haveBy (82) and (83), Theorem 10, and Lemmas 13, 14, and 15 we can conclude the theorem.
4. Proof of Lemmas 13 and 14
In this section we prove Lemmas 13 and 14.
4.1. Proof of Lemma 13
From (59) and , we have . HereUsing recurrence formulas (33), (34), and (76), we have Substituting them into (85), we have We estimate 21 terms above. We can however directly see that as follows: We can compute remaining terms as
Thus the lemma follows.
4.2. Proof of Lemma 14
Proof of and . Changing variables to polar coordinates, we have where We define , , , , and as ThenIn addition, follows. Since and , . HenceLet satisfy Suppose that . Since , holds. Therefore we have . Since , we have and . Thus . So, follows. When , we have Thenwhere From (93), (94), and (97), follows.
The proof of is similar to that of . Then we omit it.
Proof of . We redefine , , , , , and asThen we haveSince , we haveLet be . Since and ,follows. Let be . Since and ,holds. When , since , we have . Then . So,follows. From (102), (103), and (104), we have Using this, we see that . Since and , we haveSince , Then we haveFrom (101), (106), and (108), it follows thatFrom (100) and (109), the lemma follows.
Proof of . We redefine , , , and asWe have Thenholds. Since , we haveWe have in the same way as the proof of . Since and , we haveFrom (112), (113), and (115), the lemma follows.
Proof of . We define as and redefine as Then we have . We divide in the following way.whereSince , is decreasing in .Since , we have . Hence . Therefore we have , and similarly . When , we haveThenWhen , we have . Then it holds that Thus we haveWhen , we haveFrom (119), (121), (122), (124), and (125), it follows that HenceWhen , we haveSince , we have . Then we haveFrom (119), (124), (128), and (129), it follows that HenceThen by (120), (127), and (131), we have Since is decreasing and bounded below, it converges as . This fact proves the lemma.
Proof of . We redefine as Then we have . We define as Step 1. We define asOur first task is to prove that exists for all . Since , is increasing in . Let Thenholds. We haveLet . Since , it holds thatThen from (138) and (140), it follows thatHence from (139) and (141), it follows that . Therefore we haveLet . Since , we haveSince , we haveHence from (143) and (144), it follows that . Therefore we haveFrom (137), (142), and (145), it follows that Since is increasing in and bounded above for all , it converges as goes to infinity.
Step 2. Our second task is to prove that converges when goes to infinity. Let be holds for all , and by Step 1 there exists Sincefrom Cauchy convergence condition, for any , there exists such that if , . Then for and all , Therefore holds. Since family of functions on satisfies uniform Cauchy conditions, it converges uniformly on . Since is a Jordan measurable bounded closed set of , the function is continuous on . Hence is continuous on . Since both and are integrable on Jordan measurable set , by uniform convergence theorem, we have It implies that converges as .
Proof of . We redefine , , and as Then . We have in the same way as (118). Since is decreasing in . Since , we have Similarly, we have Hence Since is decreasing and bounded below, it converges as .
Proof of . We redefine , , , , and as We have .
Step 1. Our first task is to prove that exists for all . Since , is increasing in . We havein the same way as (137). When , it holds thatWhen , it also holds that Then we haveHenceLet . Since , we haveThenThereforeFrom (160), (164), and (167), it follows that . Since is increasing in and bounded above, it converges as goes to infinity.
Step 2. Our second task is to prove converges as goes to infinity. This step is the same as that of .
Proof of . We redefine , , and asThen we have , andin the same way as . We define and asThen we have . We redefine and byThenSince , is decreasing in . We divide in the following way:Let . Then we havein the same way as (124). Let . Then we also haveThereforeHenceLet . Then we haveIn addition, since , we can see that ThereforeThen we haveHenceThen we haveSince is decreasing and bounded below, it converges as . Since , is also decreasing in . Let . Then ThereforeThenHenceSince is decreasing in and bounded below, it converges. Since both and converge, converges.
Proof of . We redefine , , , , , , and aswhereFurthermore, we define asThen we have , andin the same way as the proof of . Since , it holds thatSince and are decreasing in . Let . ThenLet . Then we haveWe haveUsing (197) and (198), we haveFrom (189), (196), and (199), it follows that Hence we have Since is decreasing in and bounded below, it converges. When and , from (189) and (196), it holds that Hence we have Since is decreasing in and bounded below, it converges. We haveFrom (190), (196), and (204), we have Hence Since is decreasing in and bounded below, it converges. Since converge, converges by (195).
5. Concluding Remarks
(1) The Nelson model is defined as the self-adjoint operatoracting in the Hilbert space . Here is an external potential and In the case of , is translation invariant and the relationship between and is given by
Furthermore the ground state energy of coincides with that of .
(2) We show that and . It is also expected that diverges and the signatures are alternatively changed. Hence may converge but it is not trivial to see it directly.
(3) The relativistic Nelson model is defined by replacing with the semirelativistic Schrödinger operator in (207); that is, Then it follows that
where . Then the effective mass of is defined in the same way as that of . We are also interested in seeing the asymptotic behavior of as . However is a nonlocal operator and then estimates are rather complicated.
Another interesting nonlocal model is the so-called semirelativistic Pauli-Fierz model defined by where is a quantized radiation field. See [9] for the detail. Then it follows that where . It is also interesting to investigate the asymptotic behavior of the effective mass of the semirelativistic Pauli-Fierz model.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Susumu Osawa is grateful to Asao Arai for helpful comments and financial support. This work is financially supported by Grant-in-Aid for Science Research(B) 16H03942 and Grant-in-Aid for challenging Exploratory Research 15K13445 from JSPS.