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A Note on the Renormalized Square of Free Quantum Fields in Space-Time Dimension D > 4

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A Note on the Renormalized Square of Free Quantum Fields in Space-Time Dimension D > 4 A Note on the Renormalized Square of Free Quantum Fields in Space-Time Dimension d > 4 Sergio Albeverio, Song Liang no. 247 Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer- sität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, September 2005 A note on the renormalized square of free quantum fields in space-time dimension d ¸ 4 Sergio ALBEVERIO ,¤ Song LIANG y Abstract We consider the renormalized or Wick square of the free quantum field, which is well-defined as a Hida distribution. It is known that this is a random variable for space-time dimension d · 3. In this report, by considering the characteristic function, we show that this Wick square is not a random variable in dimension d ¸ 4. Keywords: quantum fields, Wick powers, free field model, Hida distribution, L´evy-Khinchine representation. AMS-classification: 81T08, 46F25 1 Introduction Let m0 > 0 be a fixed number and d ¸ 4. Let (Á; ¹0) be Nelson’s Euclidean d free field on R of mass m0, i.e., ¹0 is the centered Gaussian measure on ¤1). Institute of Applied Mathematics, University of Bonn, Wegelerstr. 6, D53115 Bonn (Germany), and SFB611, 2). BiBoS, CERFIM(Locarno); Acc. Arch. (USI), Mendrisio (Switzerland), 3). Dip. di Matematica, Universit`adi Trento (Italy) y1). Institute of Applied Mathematics, University of Bonn, Wegelerstr. 6, D53115 Bonn (Germany), 4). Graduate School of Information Sciences, Tohoku University (Japan). 5). Financially supported in part by Alexander von Humboldt Foundation (Germany) and Grant-in-Aid for the Encouragement of Young Scientists (No. 15740057), Japan Society for the Promotion of Science. Contacting address: Song Liang, Graduate School of Information Sciences, Tohoku Uni- versity, Aramaki Aza Aoba 09, Aoba-ku, Sendai 980-8579, Japan [email protected] 1 0 d 2 ¡1 d d S (R ) with the covariance (¡∆ + m0) . Also, let f 2 S(R ), S(R ) being Schwartz’ function space of rapidly decreasing smooth test functions. Then R Á(f) ´ Rd Áxf(x)dx (in the distributional sense) is well-defined as a random variable. We also consider the quantity : Á2 :(f), which is well-defined as a Hida distribution for all d (for this concept see, e.g., [12]). Here : ¢ : stands for the Wick power. In Section 2, by using analytic continuation of the relativistic free field, we give the precise definition of : Á2 :(f). : Á2 :(f) is by definition the Wick power of the Euclidean (quantum) free field (over the Euclidean space-time Rd). In Section 3, we recall the definition and the basic properties of Hida distributions, and use them to confirm that : Á2 :(f) is a Hida distribution. This Wick square field : Á2 :(f) has been studied in the d · 3 dimensional case, for example, in [1], [3], [10], [11], [14]. In these references, it is showed in particular that : Á2 :(f) is a random variable for d · 3. It is a natural question to ask whether this quantity : Á2 :(f) is still a random variable for d ¸ 4. One can easily see that it can not be square integrable, moreover, in Sec- tion 4, we shall see that the natural candidate for its characteristic function turns out to be the function identically zero for d ¸ 4, which implies that indeed : Á2 :(f) is not a random variable (see Proposition 4.1 and Theorem 4.2). Remark 1 We want to emphasize that, for d · 3, Á2 :(f) is a random 1 1 2 M(f) variable with a representation of the characteristic function (at 2 )) as e , with M(f) expressed as in (4.3) (this has been discussed in [11]). (4.3) can indeed be proven by a limit procedure, starting from a regularization of : Á2 :(f)). From this, it follows that for d · 3, : Á2 :(f) is an infinitely divisible random variable, and in particular one can construct from it other PN 2 2 Euclidean fields of the form i=1 : Áj :(f), with : Áj :(f) independent fields distributed as : Á2 :(f), and N 2 N; these in turn can be extended to any N 2 R. This motivated our study, in the hope of proving (4.3) also for d ¸ 4 and thus obtaining a rich class of (infinitely divisible) Euclidean random fields for d ¸ 4. However, our rigorous results leave little hope for such a construction. We also want to remark that [3] showed that the relativistic Wick square is not infinitely divisible (in a sense explained in [3]) in any space-time dimension. 2 2 Definition of the relativistic resp. Euclidean Wick square In this section, we first recall the definition of the Wick square of the rela- 2 tivistic free field ÁR. We denote this Wick square by : ÁR :. We then recall 2 the definition : ÁE : of the corresponding Euclidean Wick square, by analytic continuation of the relativistic free field (see also [2], [10], [12], [7], and the references therein). Let us first recall the relativistic free field. We consider the Fock space 1 (n) (n) F = ©n=0F , where F is the tensor product of n one particle spaces (n) (1) (0) F = Sy[F1 ­ ¢ ¢ ¢ ­ F ], F ´ C. Here ”Sy” stands for the symmetrizer. F (1) is the space of complex valued functions on Rd square integrable with + 2 2 d respect to the measure dΩ (p) = ±(p + m0)θ(p0)d p (mod the class of func- tions of 0 norm with respect to this measure), where ± is the Delta-measure 2 2 2 2 at 0, p = (p0; p1; ¢ ¢ ¢ ; pd¡1), p ´ ¡p0 + (p1 + ¢ ¢ ¢ + pd¡1), and θ(p0) = 1fp0>0g. 2 The fields ÁR and : ÁR : are defined according to the following: For any (n) (n) n 2 N, let M be the set of all functions in F such that jΦ(p1; ¢ ¢ ¢ ; pn)j(1+ Pn 0 ® i=1 jpi j ) is bounded for p0 ¸ 0, for each ® 2 N. Let D be the linear span of the sets M(n) (n ¸ 0). For any Φ 2 D, the component of Á (f)Φ ´ R R ÁR(x)f(x)dxΦ (the integral being in the distributional sense, i.e., ÁR is an operator-valued distribution acting on the vector Φ) in F (n) is given by p ³ Xn n 1 e (ÁR(f)Φ) = ¼ p f(¡pk)Ψ(p1; ¢ ¢ ¢ ; pˆk; ¢ ¢ ¢ ; pn) n k=1 p Z ´ + e + n + 1 dΩ (n)f(n)Ψ(n; p1; ¢ ¢ ¢ ; pn) ; 2 (n) and the component of : ÁR :(f)Φ in F is given by 1 ³ ´ (: Á2 :(f)Φ)(n) = T (fe)Φ(n¡2) + T (fe)Φ(n) + T (fe)Φ(n+2) ; R 2 (2;n;0) (2;n;1) (2;n;2) R with fe being the Fourier transform of f (defined by fe(p) = eixpf(x)dx, d (n¡2+2j) (n) f 2 S(R )), and the operators T(2;n;j)(g): M !M , j = 0; 1; 2, given by ¡1=2 (T(2;n;0)(g)Ψ)(p1; ¢ ¢ ¢ ; pn) = (n(n ¡ 1)) 3 Xn (n¡2) £ g(¡pk1 ¡ pk2 )Ψ(p1; ¢ ¢ ¢ ; pˆk1 ; ¢ ¢ ¢ ; pˆk2 ; ¢ ¢ ¢ ; pn); Ψ 2 M ; k16=k2;k1;k2=1 (T(2;n;1)(g)Ψ)(p1; ¢ ¢ ¢ ; pn) = Xn Z + (n) dΩ (n1)g(n1 ¡ pk1 )Ψ(n1; p1; ¢ ¢ ¢ ; pˆk1 ; ¢ ¢ ¢ ; pn); Ψ 2 M ; k1=1 1=2 (T(2;n;2)(g)Ψ)(p1; ¢ ¢ ¢ ; pn) = ((n + 2)(n + 1)) ZZ + + (n) £ dΩ (n1)dΩ (n2)g(n1 + n2)Ψ(n1; n2; p1; ¢ ¢ ¢ ; pn); Ψ 2 M ; where ˆ¢ means that the corresponding component is excluded. (See also [14] and [20]). Under the above definition, it is known that the corresponding Wightman functions 2 2 Wn(f1; ¢ ¢ ¢ ; fn) = (Ω0; : ÁR :(f1) ¢ ¢ ¢ : ÁR :(fn)Ω0); (0) with Ω0 ´ (1; 0; ¢ ¢ ¢ ; 0; ¢ ¢ ¢) 2 F the (relativistic) Fock vacuum and (¢; ¢) the scalar product in F , satisfy the Wightman Axioms [21] (see also [19], [13]). Therefore, the corresponding Schwinger functions (of the Euclidean field) satisfy the Osterwalder-Schrader Axioms (see [16] and e.g., [18]). More 2 2 precisely, by axioms, there exist Wn(x1; ¢ ¢ ¢ ; xn) = (Ω0; : ÁR :(x1) ¢ ¢ ¢ : ÁR : (xn)Ω0) such that (in the distributional sensen) Z Wn(f1; ¢ ¢ ¢ ; fn) = f1(x1) ¢ ¢ ¢ fn(xn)Wn(x1; ¢ ¢ ¢ ; xn)dx1 ¢ ¢ ¢ dxn: Rdn Let 0 0 0 0 Sn((x1; ~x1); ¢ ¢ ¢ ; (xn; ~xn)) ´ Wn((ix1; ~x1); ¢ ¢ ¢ ; (ixn; ~xn)); 0 d¡1 with (xj ; ~xj) 2 R £ R , j = 1; ¢ ¢ ¢ ; n, and define the Schwinger functions corresponding to the Wightman functions by Z Sn(f1; ¢ ¢ ¢ ; fn) ´ f1(x1) ¢ ¢ ¢ fn(xn)Sn(x1; ¢ ¢ ¢ ; xn)dx1 ¢ ¢ ¢ dxn; Rdn d d for any (f1; ¢ ¢ ¢ ; fn) 2 S6=(R ) with S6=(R ) denoting the set of all functions which vanish (together with their derivatives) on each hyperplane yi ¡yj = 0, i; j = 1; ¢ ¢ ¢ ; n. Then Sn satisfies the Osterwalder-Schrader Axioms. There- fore, there exists a Hilbert space HE, a (Euclidean) vacuum Ω0;E and a field 2 2 2 : ÁE :(f) such that Sn(f1; ¢ ¢ ¢ ; fn) = (Ω0;E; : ÁE :(f1) ¢ ¢ ¢ : ÁE :(fn)Ω0;E)HE . 4 One can raise the question whether these Schwinger functions also satisfy Nelson’s Axioms [15], [18] (in particular whether they satisfy Nelson’s pos- itivity condition), so that the corresponding (Euclidean Wick square) field 2 d (: ÁE :(f); f 2 S(R )) is not only an ”abstract operator field family”, but also a family of random variables (defined on a common probability space). We show however in Section 4 that this is not the case, since the only candi- tate for the corresponding characteristic function is trivial (see Proposition 4.1 and Theorem 4.2). 3 Hida distributions 2 For some purposes like describing ”singular interactions”, the space L (d¹0) is too small, and we need to define a bigger space (this corresponds to extend- ing L2(Rd) to S0(Rd), the space of Schwartz tempered distributions, which is given as the dual of the subspace S(Rd), the Schwartz space of rapidly decreasing smooth test functions, in L2(Rd).
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