A pr 2 00 8 SOLUTION OF A LINEARIZED MODEL OF HEISENBERG ’ S FUNDAMENTAL EQUATION

Abstract. We propose to look at (a simplified version of) Heisenberg’s fundamental field equation (see [2]) as a relativistic quantum field theory with a fundamental length, as introduced in [1] and give a solution in terms of Wick power series of free fields which converge in the sense of ultrahyperfunctions but not in the sense of distributions. The solution of this model has been prepared in [5] by calculating all n-point functions using path integral quantization. The functional representation derived in this part is essential for the verification of our condition of extended causality. The verification of the remaining defining conditions of a relativistic quantum field theory is much simpler through the use of Wick power series. Accordingly in this second part we use Wick power series techniques to define our basic fields and derive their properties.

1. Introduction 1.1. Motivation and outline of paper. Heisenberg's fundamental field equation (see [2]) γ µ ∂ ∂x µ ψ(x) ± l 2 γ µ γ 5 : ψ(x)ψ(x)γ µ γ 5 ψ(x) : = 0 (1.1) contains a parameter l of the dimension of length and accordingly one might speculate that this parameter can play the rôle of the fundamental length of a quantum field theory with a fundamental length as introduced in [1]. Unfortunately, nobody knows to solve this equation. However there is a simplification of Heisenberg's equation which is solvable in the sense of classical field theory, namely the system of equations for a Klein-Gordon field φ and a spinor field ψ. It is this system of coupled equations which we discuss in the framework of [1]. In [5], this system with the Lagrangian density is quantized by the method of path integral, that is, the n-point Schwinger functions are calculated by the Euclideanized lattice approximation (with infinitesimal spacing, in the framework of nonstandard analysis) of following path integral: where ψ 1 = ψ, ψ 2 =ψ. After renormalization we obtain the continuous limit of the lattice Schwinger functions. Then the Wightman functions are obtained by Wick rotation of the Schwinger functions. If these Wightman functions satisfy the axioms of the relativistic quantum field theory, then, by the reconstruction theorem, we can construct the operator valued generalized functions φ(x) and ψ(x). We understand that these fields φ(x) and ψ(x) are the solutions of the system defined by the Lagrangian density (1.3) according to standard interpretation of renormalization procedure.
In this paper, we try to construct the quantum fields φ(x) and ψ(x) which satisfy the system of differential equations (1.2), then show that these fields satisfy the axioms of the relativistic quantum field theory. In [5], it is shown that the Wightman functions of ψ(x) are not tempered distributions used in the usual Wightman axioms but tempered ultra-hyperfunctions which are used to formulate the quantum field theory with a fundamental length in [1].
In Section 2 we show that the n-point functionals constructed in this way satisfy the spinor version of the functional characterization of our condition of extended causality of [1]. In order to verify the remaining defining conditions of our relativistic field theory with a fundamental length we use Wick power series to define the theory. Accordingly, in this second part we construct an operator valued generalized function ψ(x) satisfying (1.2). The basic idea to solve the system (1.2) is quite natural: Take a Klein-Gordon field of mass m and suppose that we can show the following three statements: C) the free Dirac field ψ 0 (x) is a multiplier for the field ρ and so, define the field ψ(x) = ψ 0 (x)ρ(x), (1.5) and calculate . Thus, if A) -C) hold, the operator-valued ultra-hyperfunction ψ(x) satisfies Equation (1.2).
In [1] statement A) is shown together with the fact that the fields φ(x), ρ(x) and ρ * (x) satisfy the axioms of ultra-hyperfunction quantum field theory (UHFQFT). In Section 3 the convergence of the Wick power series for ρ(x) =: e gφ(x) 2 : is recalled form [1]. In the next section the important differential equation is proven. Then in order to prepare the treatment of Dirac fields, in Section 5 the axioms of UHFQFT with a fundamental length ℓ, for general type of (in particular spinor) fields are presented. In order to show statement C), we study some properties of ρ(x) which follow from the axioms of UHFQFT in Section 5. In Section 7 it is shown that the pointwise product (1.5) of two operator-valued tempered ultrahyperfunctions is well-defined and thus statement C) can be established; and it is shown that φ(x), ψ(x) = ψ 0 (x)ρ(x) and ψ(x) = ρ * (x)ψ 0 (x) =ψ 0 (x)ρ * (x) satisfy all axioms of UHFQFT for general type fields as presented in Section 4, and their Wightman functions are the same ones obtained in [5] using path integral methods.
1.2. Localization properties of tempered ultra-hyperfunctions. As announced, in Section 2 we are going to show that the system of n-point functionals as constructed in the first part satisfy the condition of extended causality. Since this condition is based on the localization properties of tempered ultra-hyperfunctions we explain here briefly the technical realization of these localization properties. To simplify matters we use a simple one-dimensional model first. Denote ⊂ C, and let T (T (−ℓ, ℓ)) be the set of functions f holomorphic in T (−ℓ, ℓ) and rapidly decreasing in any T [−k, k] ⊂ T (−ℓ, ℓ). Then for |a| < ℓ, we get The above equality implies the following two facts.
(A) and (B) say: Elements in T (T (−ℓ, ℓ)) ′ do not allow to distinguish between {0} and {−a}, if |a| < ℓ, but if |a| > ℓ then elements in T (T (−ℓ, ℓ)) ′ can be used to distinguish between the locations {0} and {−a}. Such a length ℓ is considered to be the fundamental length. T (T (−∞, ∞)) ′ is called the space of the tempered ultrahyperfunctions, where T (T (−∞, ∞)) = lim ∞←ℓ T (T (−ℓ, ℓ)) is the space of rapidly decreasing entire functions. T (T (−ℓ, ℓ)) ′ is the space of tempered ultrahyperfunctions whose carrier are contained in T (−ℓ, ℓ). The standard locality condition of quantum field theory in terms of Schwartz distributions is extended using the notion of carrier of analytic functionals (functionals over the test-function space of analytic functions) instead of the notion of support of Schwartz distributions.
For a field φ(x) satisfying the standard Wightman axioms, the twopoint functional (Φ, φ(x)φ(y)Ψ) is a functional over the test-function space S(R 2·4 ), i.e., a tempered distribution. However, for the field ψ(x) satisfying Equation (1.2), (Φ, ψ(x)ψ(y)Ψ) is not a functional over the test-function space S(R 2·4 ) but, as shown in sections 2 and 7 of this paper, a functional over the test-function space T (T (L ℓ ′ )) for any Thus such a functional can distinguish two events occurring at x 1 and x 2 if the distance between x 1 and x 2 is greater than ℓ, and cannot distinguish them if the distance is smaller than ℓ. In that sense, the field ψ(x) does not define a local field but a quasi-local field with a fundamental length ℓ.
It is quite interesting that the parameter l with the dimension of length contained in Equations (1.2) is essentially the fundamental length in the sense of this theory.

Verification of extended causality
In this section we are going to prove that the system of functionals (5.7) of Part I (see [5]), i.e., the functionals on T (T (R 4n )) where A(z) is the n × n symmetric matrix whose entries a j,k are given by a j,k = a k,j = 2h r j h r k l 2 D (−) m (z j − z k ) for r j = ±1, h ±1 = e ±iπ/4 , j < k and a j,j = 1, and where the paths Γ j are R 4 + i(y 0 j , 0, 0, 0) for appropriately chosen constants y 0 j , satisfies the spinor version of condition (R3) of extended locality as presented in [1]. For convenience we recall this condition here: (R3) (Condition of extended causality): For all n = 2, 3, . . . and all j = 1, . . . , n − 1 denote is a complex neighborhood of light cone V . Then, for any ℓ ′ > ℓ, is extended continuously to T (W ℓ ′ j ).
Remark 2.1. In our previous paper [1], we defined a complex neighbourhood V ℓ by But we found that to treat the present model, the neighbourhood (2.2) is convenient, and by this change of the ℓ-neighbourhood of V , our theory [1] is not affected.
The transposition of z j and z j+1 causes the change of a j,j+1 = a j+1,j : We consider the matrix B = (b i,j ) obtained from A by the change of j-th and j + 1-th rows and j-th and j + 1-th columns. Then we have det A = det B. Next we consider the matrix C = (c j,k ) obtained from B by changing only b j,j+1 = b j+1,j = a j,j+1 = a j+1,j , i.e., c j,j+1 = c j+1,j = . . , z n ) is also expressed by the sum of products of the two-point functions of the Dirac field as in the scalar case, and for space-like separated x j , x j+1 (y 0 j − y 0 j+1 = 0) and other y 0 . . , z n ). In order to proceed we need some estimates for D Let ω(|p|) = |p| 2 + m 2 and introduce the auxiliary function d|p|.
Then we have and for Im z ≤ 0 and Im x = 0, the estimate If we put t = ω(|p|) = |p| 2 + m 2 then |p| = √ t 2 − m 2 , and the equation can be continued by: and a = min ± |x 0 − iǫ ± |x||. Then, if a > ℓ m (l), the estimate Proof. We have the following inequalities.
As a solution of the inequality This completes the proof.
3. Convergence of Wick power series for ρ(x) =: e gφ(x) 2 : Our starting point are the well-known results of Jaffe [4] on formal Wick power series of free fields. If we consider the power series of a free field φ then we have the following theorem.
Theorem 3.1 (Theorem A.1 of [4]). In the sense of formal power series the following identity holds Proof. The chain of identities shows that 1≤i<j≤n {g i g j } r ij = A(R), and thus we get Assume that for some σ > 0 lim sup Then Theorem 6.3 of [1] says that the power series (3.1) defines an ultra-hyperfunction quantum field with fundamental length ℓ if φ is a massless free field. Now consider In this case we find for the above limit σ = 2|g|. Suppose that the 0 < t ij 's satisfy 1≤i<j≤n t ij < 1 2|g| .
Then the power series ∞ r ij =0; 1≤i<j≤n This shows the convergence of the vacuum expectation value in the sense of tempered ultra-hyperfunctions, and moreover implies the strong convergence of ). For the definition and basic properties of the testfunction space T (T (R 4 )) of tempered ultrahyperfunctions we refer to [1].
with r j = ±1. Then the vacuum expectation values of these fields are given by

5)
where A is the n × n symmetric matrix whose entries a j,k are given by Note that the result (3.5) is the same as the corresponding result in [5].
Proof. The equation can be considered as an equation for the following two power series of the variable p: and by inserting p = √ 2φ(x) and using Wick products we get, as a formal series : (−(hφ(x)) 2 ) n : /n!.
Let h r j = e ir j π/4 and denote σ (j) (x) =: e it j √ 2hr j φ(x) :. Then Corollary 2.2 says and thus we get

Verification of the equation
We begin by recalling some basic facts about Wick products of free fields which are then used to study Wick polynomials and Wick power series.
This shows that that is, Let D 0 be the set generated by the vectors of the form where ρ (k) (x) is one of φ(x), ρ(x) and ρ * (x), and Φ ∈ D 0 . Then we have seen in the previous section that is strongly convergent, and by (4.2) This shows that We write the last expression as That is, the formal expression (which is difficult to give a direct meaning) Then by (4.3), the above expression equals and this is equal to in the sense of generalized functions. In the above understanding, we have is defined by the Wick power series

Wightman's Axioms for general type fields
In Wightman's scheme, the concept of a relativistic quantum field φ (κ) of type κ plays a fundamental role. Such a field, for example a scalar, tensor or spinor field, has a finite number of Lorentz components φ (κ) j (j = 1, . . . , r κ ).
The field components φ are densely defined linear operators in a complex Hilbert space H. They are not assumed to be bounded.
Here we state Wightman's axioms for the ultra-hyperfunction quantum field theory [1]. For the neutral scalar fields, these axioms are the axioms in [1].
is a tempered ultrahyperfunction. It is supposed that the vacuum vector Φ 0 is contained in D and that D is taken into itself under the action of the operators φ Moreover it is assumed that there exist indicesκ, such that φ W.V. Poincaré-covariance of the fields: According to the type of the field, there is a finite dimensional real or complex matrix represen- i.e., for any f ∈ T (T (R 4 )) and Ψ ∈ D, x − a)). We have V (κ) (−1) = ±1. If V (κ) (−1) = 1, then the field is called a tensor field. If V (κ) (−1) = −1, then the field is called a spinor field. W.VI. Extended causality or extended local commutativity: Any two field components φ (κ) j (x) and φ (κ ′ ) l (y) either commute or anticommute if the distance between x and y is greater than ℓ: a) The functionals and can be extended continuously to T (T (L ℓ )) in some Lorentz frame, for arbitrary elements Φ, Ψ in the common domain D of the field operators b) The carrier of the functional is dense in H.
At the end of this section we complete the proof of the condition of extended causality in the form of axiom WVI by showing that this axiom is equivalent to Condition (R3) for the Wightman functionals which has been verified in Section 2.

Conclusion
After the condition of extended causality had been verified in its functional version (Section 2), this second part of our study of a linearized model of Heisenberg's fundamental equation established first the convergence of the Wick power series ρ(x) =: e il 2 φ(x) 2 := ∞ n=0 (il 2 ) n n! : φ(x) 2n : through Wick power series techniques. It turns out that this power series converges in the sense of tempered ultra-hyperfunctions but not in the sense of (tempered) Schwartz distributions.
Next through the use of further Wick product techniques it is shown that this field ρ satisfies the differential equation (in the sense of operatorvalued tempered ultra-hyperfunctions) where we used the abbreviation ∂ µ = ∂ ∂x µ . Finally, in order to solve the system (1.2) by the ansatz ψ(x) = ψ 0 (x)ρ(x) (8.1) with ψ 0 being a free Dirac field two results have been established, namely a) the concept of a relativistic quantum field with a fundamental length of general type κ (i.e., a scalar, tensor or spinor field) generalizing the case of a scalar field presented in [1] and b) the free Dirac field ψ 0 is a multiplier of the field ρ. Then it follows that the field ψ in (8.1) is a relativistic quantum field with a fundamental length of spinor type which satisfies the system (1.2). The interpretation and the motivation of our use of the concept of a quantum field theory with a fundamental length can also be found in the introduction to part I and in [1].
We find it a very remarkable fact that the length parameter l in the linearized version of Heisenberg's fundamental equation can be interpreted as the fundamental length in the sense of our theory of relativistic quantum field theory with a fundamental length as developed in [1].
As important physical consequences we mention that therefore the solution of the linearized version of Heisenberg's fundamental equation falls in the class of quantum field theories for which the PCT and spinstatistic theorems hold and for which a scattering theory is available.