Causal Localizations of the Massive Scalar Boson

The positive operator valued localizations (POL) of a massive scalar boson are constructed and a characterization and structural analyses of their kernels are obtained. In the focus of this article are the causal features of the POL. There is the well-known causal time evolution (CT). Recently a POL by Terno and Moretti, which is a kinematical deformation of the Newton-Wigner localization (NWL) and belongs to the here fully analyzed class of finite POL, is shown by V.Moretti to comply with CT. A further POL with CT treated here, which is in the same class, is the only one being the trace of a projection valued localization (like NWL) with CT. - Causality imposes a condition CC, which implies CT but is more restrictive than CT. Extending Moretti's method it is shown rigorously that the POL of the class introduced by Petzold et al. satisfy CC. Their kernels are called causal kernels, of which a rather detailed description is achieved. One the way there the case of one spatial dimension is solved completely. This case is instructive. In particular it directed Petzold et al. and subsequently Henning, Wolf to find their basic one-parameter family $K_r$ of causal kernels. The causal kernels are, up to a fixed energy factor, normalized positive definite Lorentz invariant kernels. A full characterization of the latter is attained due to their close relation to the zonal spherical functions on the Lorentz group. Finally these considerations discharge into the main result that $K_{3/2}$ is the absolute maximum, viz. $|K|\le K_{3/2}$ for all causal kernels $K$.


Introduction
As well-known relativistic massive particle position would be described in a fully satisfactory manner by the Newton-Wigner localization (NWL), since above all NWL provides an abundance of boundedly localized states, if there were not the requirement of causality.The so-called Einstein causality in the most direct interpretation requires from localization that the probability of localization in a region of influence cannot be less than that in the region of actual localization.NWL violates blatantly this principle as it localizes frame-dependently.There is no state of the particle for which the particle is boundedly localized by NWL in different spacelike hyperplanes.
So one is induced to determine the probability of localization in a region no longer by the expectation value of an orthogonal projection attributed to the region as occurs for NWL, but rather of a positive operator with spectrum in the unit interval thus giving rise to a positive operator valued localization (POL).By now PO-localization is a settled concept.For a brief discussion and several references see for instance the passage in [4, sec.III.F].
In sec. 4 we construct the POL for a massive scalar boson.The result is the explicit formula in the theorem (2), which furnishes myriads of different POL.In order to recognize their features it is very effective to describe a POL in a concise manner by its kernel.A useful characterization and structural analyses of the POL kernels are given by the theorem (11).
To every POL a Euclidean covariant position operator is attributed, which is the first moment of the POL.The physical relevance of the POL is underlined by the fact that in case of a real smooth POL kernel the position operator coincides with the Newton-Wigner position operator.The proof is due to V. Moretti.Recently in [17] a POL T T M by Terno and Moretti is studied in detail showing furthermore that T T M is a kinematical deformation of NWL with almost localized states for every region with nonempty interior and which obeys causal time evolution (see CT in sec.10).From general criteria for POL kernels we obtain rather short proofs of these properties, see sec.10.1, (48).What is more, (47) provides a simple explicit formula for a sequence of localized states at any point of the space.Actually (47) is a byproduct of the proof of the easy criterion (46) in sec.17 on POL kernels for the existence of point localized sequences of states.
In sec.10 we treat also the POL with trace CT T tct , which has the noted properties of T T M and which is the unique POL being the trace of a projection valued localization with CT.T tct is also a kinematical deformation of NWL (15), all of which are given by the formula (9.3).
However one cannot content oneself with POL obeying CT as the latter satisfies only partially the requirements of causality.If T is a POL, ∆ a spacelike region, and σ any spacelike hyperplane, then as expounded in sec.11 causality imposes the condition T (∆) ≤ T (∆ σ ) (CC) were ∆ σ denotes the region of influence of ∆ in σ, i.e., the set of all points in σ, which can be reached from some point in ∆ by a signal not moving faster than light.CT concerns just the case that ∆ and σ are parallel.POL which satisfy CC are called causal.
The causality principle formulated above let one think of the probability of localization as a conserved quantity reigned by an associated density current.Petzold and his group [8] were apparently the first to study the conserved covariant four-vector currents with a positive definite zeroth component.The outcome on POL kernels is reported in sec.11.1.See (55) for a clear-cut result in this regard.
Petzold et al. [9] argue that their POL obey CT.Probably they were the first to introduce and to treat the concept of CT.However they do not take under consideration the full causality requirement CC.Actually, extending Moretti's method [17] we succeed in (56) to prove rigorously that their POL are causal.Moreover, we mention particularly a communication by R.F.Werner, which suggests that most probably these are the only causal POL.Therefore we refer quite simply to their kernels (11.1) as the causal kernels.
A small digression (sec.12) concerns the case of one spatial dimension which we completely solve by the formula in (22).This case is rather instructive since it directed Petzold et al. [8] and subsequently Henning, Wolf [11] to find the particular causal kernels K r (11.3), which turn out to be fundamental.Multiplying a causal kernel by an element of G yields again a causal kernel.Here G denotes the set of all continuous positive definite Lorentz invariant normalized kernels (28).What is more, any causal kernel is the product of some element of G and a fixed energy factor.Therefore it is important to know G.In (32) a complete description of G is attained employing means of group representation theory.
Each G ∈ G determines by the Reproducing Kernel Hilbert Space (RKHS) construction a continuous unitary representation of SL(2, C) in separable Hilbert space (sec.14.1).By this construction G is closely related to the matrix element of a SU (2)-invariant vector.Since SL(2, C) is locally compact tame with countable basis, SL(2, C) admits an integral representation of the latter by the zonal spherical functions on SL(2, C) (14.7).
The question is to find out which G ∈ G determine a causal kernel.By (42) these are only G's, which are derived from the principal series and are not extreme.We mention the inversion formula in (35) concerning these elements.
At this point we are ready for the main result (43).Put m > 0 mass, k, p ∈ R 3 momentum, ǫ(k), ǫ(p) energy.The causal kernel is maximal.Actually |K(k, p)| < K 3/2 (k, p) holds for all k = p and all causal kernels K = K 3/2 (44).The relation |K| ≤ K 3/2 ≤ 1, where 1 is the kernel of the NWL distinguished by its abundance of boundedly localized states, suggests that the POL with kernel K 3/2 is the causal POL with the best localization features.Further studies should work on this distinguished POL, which satisfies several prerequisites.Above all it should be clarified whether there are point localized sequences of states for this POL.
Presumable, like (24) in the case of one spatial dimension, not all G ∈ G with G ≤ G 3/2 (the second factor of K 3/2 ) determine a causal kernel.We rather guess that the causal kernels are exactly where it suffices to consider r < 2, since K r /K 3/2 ∈ G for r ≥ 2 by (26).
Finally here is a brief guide to the article.
The sections 2-7 are concerned with the POL and their kernels.Their content is summarized in (2), (6.1), and (11).
Section 9 deals with the class of finite POL.The concluding result is (9.3).In section 10 causal time evolution CT is introduced and two finite POL with CT are studied.
Section 11 deduces the causality condition CC for POL and introduces the class of causal kernels (11.1).There in 11.2 is also an overview of the known results on causal kernels.Section 12 treats the case of one spatial dimension, which is completely solved by the formula in (22).
Section 17 introduces the concept of a point localized sequence of states.The criterion (46) and its corollary (47) regard all POL kernels.
The discussion in the final section 18 incites to find answers to two outstanding questions of great relevance.
The appendices C and D provide space to some cumbersome technical details, whereas the appendices A and B house all which regards the CT criterion (52) and the result on CC (56), respectively.

Some facts on representations of the Euclidean and Poincaré group
We recall some facts on the reps1 of the universal covering groups Ẽ = ISU (2) and P = ISL(2, C) of the Euclidean group and the Poincaré group, respectively.P acts on R 4 as g Here Λ : SL(2, C) → O(1, 3) 0 is the universal covering homomorphism onto the proper orthochronous Lorentz group.Identifying Λ(SU (2)) ≡ SO(3), SU (2) acts on R 3 .Representing Minkowski space by R 4 the Minkowski product is given by

Irreps of Ẽ
Up to unitary equivalence the irreducible mutually inequivalent reps of Ẽ are U 0,j on C 2j+1 for j ∈ N 0 /2 and U ρ,s on L 2 (S 2 ρ ) for ρ > 0, s ∈ Z/2 with the sphere S 2 ρ := {p ∈ R 3 : |p| = ρ} endowed with the rotational invariant measure normalized to 1. Explicitly one has • U 0,j (b, B) := D (j) (B) and |p| for all p ∈ R 3 \ {0}, whence the Wigner rotation on the right hand side leaves e 3 invariant and hence is diagonal. 2

Particular reps of Ẽ
We will be concerned with the reps U (s) , s ∈ Z/2, of Ẽ on L 2 (R 3 ) given by for p = 0. Note that κ(0, B) is not defined.The obvious unitary equivalence yields the decomposition of U (s) into irreps.It shows in particular that U (s) and U (s ′ ) for s = s ′ are disjoint, i.e., that any two subreps of the latter are inequivalent.

Induced reps of Ẽ from SU(2)
A projection valued measure (PM) on R 3 , which is Euclidean covariant, is called a Wightman localization (WL).By Mackey's imprimitivity theorem every WL is Hilbert space isomorphic to a rep of Ẽ induced from the subgroup SU (2) together with the related system of imprimitivity.
Ẽ of Ẽ induced from the irrep D (j) , j ∈ N 0 /2, of SU (2) and the system of imprimitivity E read The PM operator E can (∆) multiplies by the indicator function 1 ∆ .F denotes the unitary Fourier integral transformation with kernel (2π) −1/2 e − i xy in L 2 .The WL (D are irreducible, mutually inequivalent, and complete up to unitary equivalence.The decomposition into subreps is obtained by the unitary transformation 2 The standard helicity cross section reads which is completed to a WL by the PM X (j) F E can F −1 X (j)−1 .
One concludes that every sum ⊕ ι U (sι) is the subrep of a rep with an Euclidean covariant PM forming an WL.Hence generally this holds true for every rep, which is unitarily equivalent to such a sum.

Decomposition of
By the foregoing considerations it is easy to verify

How to get PO-Localizations
A positive operator valued measure T on the Borel sets B(R 3 ) (POM), which is Euclidean covariant, is called a PO-localization (POL).Recall that a POM E constituted by orthogonal projections is denoted by PM, and a Euclidean covariant PM is called a Wightman localization (WL).Given a rep W of P and a POM T such that (W | Ẽ , T ) is a POL, then by [5, (9) Theorem] there is a unique Poincaré covariant extension (W, T ).This means that there is just one map on S, the set of all Lebesgue measurable subsets of spacelike hyperplanes of Minkowski space, still denoted by T , which satisfies ).The elements of S are called regions.Henceforth let W be a rep of P with positive mass spectrum and finite spinor dimension.This means equivalently that W is a finite orthogonal sum of reps from sec.2.4.
Regarding parts of (1) see e.g.[25, sec.II] for the general theory.Let (U, E) be a WL acting on the Hilbert space K. Let H be a U -invariant subspace of K and j : H → K the identical injection.Then the POL j * (U, E)j is called the trace (or compression) of (U, E) on H.
(1) Theorem.Let W be a rep of P on H.
(3) Definition.The finite dimensional spinor spaces, considered as finite dimensional D-invariant subspaces of S, are for J ∈ N 0 , ν j ∈ N 0 with dim(S f in ) = J j=0 ν j (2j + 1).A POL T is called finite if it is the trace of an WL with finite dimensional spinor space.The spinor choice e is called finite if the range of e lies in a finite dimensional spinor space S f in .
Obviously a POL T is finite if and only if T = T e for some finite spinor choice e.We will be concerned with finite POL in sec.9 providing an equivalent characterization of the latter as kinematic deformations of the NWL.This expression is due to Moretti [17] thus calling a POL proposed by Terno [23].We give a rigorous definition of this term (14).Besides the POL of Terno-Moretti in sec.10 a further physically relevant finite POL is treated.
Let C be a unitary operator on L 2 (R 3 , S), (Cϕ)(p) := C 0 ϕ(p), where C 0 is unitary on S such that C 0 = ⊕ j C 0,j with C 0,j acting on the multiplicity space of D (j) Ẽ .Then C and XF E can F −1 X −1 commute and Cj e = j e ′ for e ′ := C 0 e.Thus T e = T e ′ .

WL and NWL
NWL is for Newton-Wigner localization.-Let (W m,0,η , E) be a Poincaré covariant WL.Recall W m,0,η | Ẽ = D (0) Ẽ .According to the imprimitivity theorem reported in sec.2.3, (W m,0,η | Ẽ , E) = S(D (0) Ẽ , F E can F −1 )S −1 for some unitary S.This implies that S and D Hence referring to (4.1), S 0 ≡ C and E = j * e F E can F −1 j e for e = s The NWL E N W is given by e = 1.About the uniqueness of 6 Kernels of the POL Let T be any P OL of a massive scalar boson.Then there is a spinor choice e with T = T e .From (4.1) it follows for φ ∈ L for every region ∆ ⊂ R Obviously t e is a measurable positive definite kernel5 on R 3 \ {0} with t e (p, p) = 1 for all p.It is also rotation invariant, i.e., t e (Rk, Rp) = t e (k, p) for all rotations R, as we will see explicitly below.Clearly T and its kernel t determine each other.
(4) Corollary.Let (b (l) ) l be an orthonormal basis of S. Then holds for all k, p = 0 and . If e is a finite spinor choice then (1) holds for an orthonormal basis (b (l) ) l of S f in , and the sum is finite.
(5) Definition.Let T be a POL and t its kernel.t is called finite if holds for all k, p = 0 with finitely many measurable f l : R 3 → C.
For an explicit expression of t e (6.2) choose an orthonormal basis (b (j,n) ) j,n of S 0 according to the orthogonal sum S 0 = ⊕ j ∞C so that e(ρ) = ∞ j=0 ∞ n=1 e j,n (ρ) b (j,n) with e j,n (ρ) := b (j,n) , e(ρ) holds for every spinor choice e.If e is a finite spinor choice then, adapting the enumeration, e j,n = 0 if j > J or n > ν j for some j.
(6) Corollary.Every set of measurable functions 0 < ρ → e j,n (ρ) ∈ C with j n |e j,n (ρ)| 2 = 1 determines a kernel of a POL for the massive scalar boson represented by W m,0,η and vice versa by for p = 0, k = 0, where 4) Here Szegö's notation for the Legendre polynomials ).The first three Legendre polynomials read , whence again by Cauchy-Schwarz inequality In view of the comment on (2) we note (8) Corollary.Let e and e ′ be two spinor choices.Suppose t e = t e ′ .Then for every j ∈ N 0 and all (σ, ρ) 3), (6.4).Hence by assumption f = f ′ .This implies λ j = λ ′ j being the coefficients of the expansion in orthonormal Legendre polynomials in L 2 (−1, 1).

Characterization of the POL kernels
For a map consider the properties (i) to be measurable, (ii) to be normalized, i.e., K(p, p) = 1 for all p, (iii) to be a positive definite kernel, (iv) to be a positive definite separable kernel, i.e. the RKHS associated to K (see e.g.[7]) is separable, and (v) to be rotation invariant, i.e., K(Rk, Rp) = Kk, p) for all k, p and rotations R. 6 Recall that (iii) implies whence, if (ii) holds, |K| ≤ 1.
(9) Proposition.Let K satisfy (i)-(iv).Then there is a measurable feature map J on R 3 \ {0} in a separable Hilbert space H J for K, i.e.K(k, p) = J k , J p , such that Proof.Let H J denote the RKHS associated to the kernel K and put Obviously every POL kernel t e satisfies (i)-(v).In particular see (6.2).The converse holds true, too.
. This is easy using the formula on φ, T (∆)φ in the proof of ( 9).
Let K be rotation invariant.Introduce the map |k||p| for all k, p = 0.
(11) Theorem.The kernels of the POL of the massive scalar boson are exactly the measurable normalized rotational invariant positive definite separable kernels K on R 3 \ {0}.They are given by for all k, p = 0 with (k j ) j=0,1,2,... any sequence of measurable positive definite kernels k j on ]0, ∞[ (not excluding k j = 0 for some j) satisfying j k j (ρ, ρ) = 1 for all ρ.K and (k j ) j determine each other uniquely.The sum converges everywhere.Recall (7.4).One has Proof.In view of ( 10) and ( 6) it remains to add the following consideration regarding the POL kernels.Choose a measurable feature map Φ j for k j (see e.g.[7]) and let e j,n (ρ) be the coefficients of Φ j (ρ) with respect to some ONB.Then σ) e j,n (ρ) for all σ, ρ, j.Further by the assumption on (k j ) one has j,n |e j,n (ρ)| 2 = 1 for all ρ.Let t e denote the kernel corresponding to (e j,n ) according to (6).Note k(σ, ρ, •) = j k j (σ, ρ)P j in L 2 (−1, 1), where the sum converges everywhere due to (6.3).So K is the kernel of T e and hence equals t e .

Change to shell rep
In place of the momentum reps W m,j,η of P from sec.2.4 frequently it is convenient, because of the simpler transformation formulae, to use the unitarily equivalent shell reps on L 2 (O m,η , C 2j+1 ) of functions on the mass shell O m,η := {p ∈ R 4 : p 0 = ηǫ(p)} equipped with the Lorentz invariant measure Note that for this X m,η is unique up to a constant factor of modulus 1. Hence in the case of the massive scalar Boson the Poincaré covariant POM (W shell,m,0,η , T shell e ) are given by (cf.(4.1)) η , e spinor choice (8.1)Then for Φ, Φ ′ square integrable and In particular with respect to the shell representation the kernel of the NWL E N W reads ǫ(k)ǫ(p).
9 How to get a POL from Newton-Wigner Localization Moretti [17] proves a formula for the POL proposed by Terno [23] (sec.10.1) calling it a kinematic deformation of the NWL.Generalizing we like to use Moretti's expression for a certain kind of POL (14).It turns out that just the finite POL (3) are of this kind (15).
For the momentum operator P = (P 1 , P 2 , P 3 ) and a bounded measurable function f : R 3 → C let f (P ) be the related operator defined by the functional calculus.On momentum state space L 2 (R 3 , C), f (P ) is the multiplication operator by f given by (f the result follows from (6.1) and ( 7).
(14) Definition.A POL T is called a kinematic deformation of the NWL if holds for some finitely many measurable bounded f l : R 3 → C.
(15) Theorem.Let T be a POL and t its kernel.Then the statements (a), (b), and (c) are equivalent.
(c) T is a kinematic deformation of the NWL (14).
Proof.Assume (a).Then by (4) t(k, p) = l f l (k) f l (p) for all p, k = 0 with some finitely many measurable bounded f l : R 3 → C thus showing (b).(b) implies (c) since (9.1) follows by (13).Now assume (c).Then (9.1) and ( 13) imply t(k, p) = L l=1 f l (k) f l (p) for all k, p = 0, where t is the kernel of T .-We enter into the proof of ( 9) for for every linear combination of the J p .Therefore D ′ (B)J p := J B•p determines a unitary operator D ′ (B) and Hence T is the POL (7.3) with kernel t.Since H J is finite dimensional, (a) follows.
In conclusion we specify how to get the kinematical deformations T of the NWL.( 16) is an application of ( 15) and (11).

POL with causal time evolution
Let (U, T ) be a POL and (V, U ) a rep of the little kinematical group R ⊗ Ẽ.Then (V, U, T ) is said to be a POL with causal time evolution if holds for all regions ∆ ⊂ R 3 and all times t ∈ R Here ∆ t := {y ∈ R 3 : ∃ x ∈ ∆ with |y − x| ≤ |t|} is the region of influence of ∆.CT simply means that after time t, respectively before time t, the probability of localization in ∆ t is not less than originally in ∆.In case of a POL for a massive scalar boson V (t) is given by W m,0,η (t).
Probably the first to introduce and to treat this concept of CT is the Petzold group [9, sec.3].There CT follows from the existence of a nonspacelike conserved current associated to the probability density of localization.This relation is adapted by Moretti [17] in an advanced manner thus showing rigorously CT for the POL introduced by Terno [23].In case of WL, CT is examined in [21] and thoroughly studied in [2] and [4], [5].
The POL with CT for a massive scalar boson known so far in the literature are the (not finite) POL with conserved covariant kernels by Petzold et al. [8] and Henning, Wolf [11], which we study in the following sections, and the POL by Terno-Moretti.We add here the finite POL with trace CT.
Henceforth we will consider the particle case η = + only, as the antiparticle case η = − is quite analogous.Let H denote the energy operator, which in momentum rep equals the multiplication operator by ǫ(p) = m 2 + |p| 2 1/2 .

POL by Terno-Moretti
In Moretti [17] it is shown that T T M (10.2) is a POL with CT.Below we give a simplified proof of this based on the general criteria ( 11), (16), and (52).
is the kernel of the finite POL with CT Proof.Obviously t T M is a finite continuous rotational invariant positive definite normalized kernel on R 3 \ {0} and hence a finite POL kernel (11).Recall (16).It remains to show CT.For this we apply (52).Indeed, induced by [17] we show that t T M is conserved timelike definite due to j = (j 1 , j 2 , j 3 ) given by Show condition (51)(ii): Since j is real symmetric, it suffices to consider real coefficients c a .Put C := a c a , D i := a c a p a.i /ǫ(p a ), and , whence the claim.
is the kernel of the finite POL with CT (10.6) Actually, (W m,0,+ | R⊗ Ẽ , T tct ) is the only one which is the trace of a relativistic quantum system of finite spinor dimension with CT by a WL.
The proof is postponed to app.D.

Two spinor choices for T tct
(19) Proposition.The spinor choice e for which T tct = T e according to the construction in sec.4 is The proof is postponed to app.D.
On the other hand, following (6) one obtains immediately Hence a dilation of (W m,0,+ | Ẽ , T tct ) corresponding to (10.8) is the WL (W ′ | Ẽ , E ′ ) for This is rather interesting, since here (W ′ , E ′ ) is the NWL and therefore the time evolution of (W ′ , E ′ ) is not causal [5, (2) Theorem].

Causal POL
CT for a POL is a requirement of the fact that there is no propagation faster than light.Clearly limited propagation affects all kinematical transformations.For instance, according to NWL, if a massive particle localized in a bounded region is subjected to a boost of however small rapidity then there is a non-vanishing probability to observe the particle arbitrarily far away.This frame-dependence of NWL (already described by Weidlich, Mitra 1964 [24] and still currently discussed, see e.g.[5,3]) is an acausal behavior.In fact, causality requires from localization that the probability of localization in a region of influence cannot be less than that in the region of actual localization.More precisely, let ∆ be a region, i.e., a measurable subset of a spacelike hyperplane of Minkowski space, and let σ be any spacelike hyperplane, then is the region of influence of ∆ in σ.It is the set of all points x in σ, which can be reached from some point z in ∆ by a signal not moving faster than light.Therefore (W, T ) (sec. 3) is called a causal POL if holds for all ∆ ∈ S, σ spacelike hyperplane.The causality condition CC is thoroughly studied in [5].There causal POL for the Dirac electron and the Weyl fermions are revealed.Apparently the causality condition has not been analyzed elsewhere.Recently, along a tentative "new and different operational interpretation of the notion of spatial position" in [17] a causality condition corresponding to CC is shown to hold.Let us verify explicitly that CT is a special case CC.Indeed, for ∆ ⊂ {0} × R 3 , t ∈ R, and σ := {−t} × R 3 check ∆ σ = (−t, 0, I 2 ) • ∆ t , whence CT by covariance of T .

Causal kernels
As argued above CC, which implies CT, is an indispensable property for a localization.For this one is highly interested in the kernels of causal POL.We study the promising class K of conserved covariant POL kernels (B.1) introduced by Petzold et al. [8].Indeed, (56) shows that a POL with kernel K ∈ K is causal.Moreover, most probably K comprises even all kernels of causal POL.Therefore we refer quite simply to their kernels (11.1) as the causal kernels.
One deals with the kernels on R 3 of the kind

Known results
The problem is to find the functions g so that K is positive definite.Let us report the results, which in the literature are achieved so far including the personal communications of R.F. Werner with proofs.Put (a) The known solutions for K in (11.1) are deduced in [8] and [11].They read (e) In [9] it is argued that a POL with kernel K ∈ K obeys CT.
12 Causal kernels for one spatial dimension As recommended by [8, footnote to (3.2)] and since there is still some interest in this topic [13], we look first at the instructive case of one spatial dimension.Here, indeed, g 1/2 arises naturally and turns out to furnish the maximal solution K 1 1/2 (21)(c).A complete description of the causal kernels is given in (22).Accordingly, the product of K 1  1/2 with a normalized positive definite kernel is a causal kernel and every causal kernel is of this kind.-Call The question is about the functions g of causal kernels on R.
(iv) K 1 (sinh x, sinh y) = h(x, y) g cosh(x−y) Proof.Verify the claim by elementary computations.Positive definiteness of h is obvious.
(a) First we note that a real kernel G(x, y) on R is positive definite if and only if the kernel G 1 (x, y) := h 1 (x)h 1 (y)G(x, y) for h 1 (x) := cosh x/2 √ cosh x is positive definite.Indeed, as to the less trivial implication let G 1 be positive definite.Let c 1 , . . ., c n ∈ R and x 1 , . . ., Hence G is positive definite.

It remains to check
x.This equation has the only solution x = 0 if 0 < ς ≤ 2, whence f (0) = 1 is the maximum.(If ς > 2, there are exactly three solutions, whence f (0) = 1 is a local minimum.) There is a consequence of (22) for K in (11.1).Choose any p 0 ∈ R 3 with |p 0 | = 1.One observes that K(ap 0 , bp 0 ) equals K 1 (a, b) for a, b ∈ R. Therefore K 1 is positive definite if K is positive definite.By (22) this implies the following necessary condition for K being causal.

Definiteness of gr and K r
We return to causal kernels on R 3 .Recall g r (t) = 2 r (1 + t) −r , r > 0 and let K r denote the map K in (11.1) for g = g r .Recall that K r is positive definite for r ≥ 3/2 (11.3).
The literature is not clear what about the values 0 < r < 3/2.By [8, Eq. (3.10)]K 1 is not positive definite.In (27) we show that K r is not positive definite for 0 < r < 3/2.
Similarly there is the question about the definiteness of gr .Clearly it is positive definite for r ≥ 3/2.Actually it holds true exactly for r ≥ 1/2 as we are going to show simply adapting the proof of [11,Theorem].
Exploiting a further formula [11, (8)] ) we complete the result by Proof.Since gr is not positive definite for 0 < r < 1/2, so is K r .Hence we may assume 1/2 ≤ r < 3/2.We use again the expansion (7.5) and show that the 0th coefficient of K r is not positive definite.By (13.3) the latter equals k 0 (ρ i , ρ j ) < 0, thus ending the proof.
We like to point out that an easy proof of the fact that gr , 0 < r < 1  2 and K r , 0 < r < 3  2 are not positive definite is given in (39).

Lorentz invariant kernels
The kernels g on R 3 given by g(k, p) := g ǫ(k)ǫ(p) − kp for some g : (a) G ∈ G is automatically continuous.Indeed, continuity of G follows from (14.1) and ( 12).
(b) G ∈ G is automatically real-valued since it is symmetric and positive definite.
The crucial facts, why we are interested in G, are (e) and (f).Up to now the only elements of G we know are the maximal element 1 and gr , r ≥ 1/2 and pointwise limits of convex combinations of these.

Positive definite Lorentz invariant kernels, reps of SL(2, C) with SU(2)-invariant vectors
Let G be a positive definite kernel on R 3 .We consider the RKHS H G associated to the kernel G (see e.g.[7]).
Lorentz invariance of G gives rise to a rep of SL(2, C).For A ∈ SL(2, C) and p ∈ R 3 let A • p denote the spatial vector part of A • p for p := ǫ(p), p .Moreover, let A κ denote the boost e κσ3/2 in direction (0, 0, 1) with rapidity κ ∈ R. One has . By linear extension U becomes a unitary homomorphism on V := span{G p : p ∈ R 3 }.Since the latter is dense in H G , U extends further to a unitary homomorphism on This is quite obvious.Indeed, recall that V is dense in H G .Then even span{G p : [15] for more information) it suffices to show the weak continuity of U .First note that obviously Similarly one replaces g by g 0 ∈ H G showing the assertion.
One reminds that the matrix element in (14.2) is invariant under unitary equivalence transformations and that U G (B)G 0 = G 0 holds for all B ∈ SU (2), i.e., G 0 is SU (2)invariant.
Proof.Note g(1) = 1 and that g is continuous.We show at once that g is real.The Then there is for κ = cosh −1 t, whence the claim by (14.1).
In (30) the carrier space H U of U is not assumed to be separable.One concludes The matrix elements g in (14.3) are closely related to the zonal (or elementary) spherical functions on SL(2, C), i.e., the SU (2)-bi-invariant functions φ on SL(2, C) associated to irreps U on SL(2, C) given by φ(A) := Γ 0 , U (A)Γ 0 , where up to normalization Γ 0 is the unique SU (2)-invariant vector (see below).Recall that for every , whence e κ , e −κ are the eigenvalues of A * A, and since τ A κ τ −1 = A −κ (see the proof of (30)).

G is the set of convex combinations of its irreducible elements
The theory of reps of SL(2, C) is treated exhaustively in [18], to which we will refer in the sequel.The irreps of SL(2, C) of the principal series are characterized by a pair (m, λ) ∈ Z × R of invariant numbers.For m = 0 they do not contain SU (2)-invariant vectors = 0, and for (0, λ) there is up to a phase just one SU (2)-invariant normalized vector.(0, λ) and (0, −λ) determine equivalent reps.The matrix element (14.3) reads See [18, III §11.5 formula after Eq. ( 10)] for Γ 0 , U (A 2κ )Γ 0 with ρ = 2λ.For λ = 0 it is not positive and hence gp λ is not positive.The irreps of SL(2, C) of the supplementary series are characterized by one invariant number λ ∈]0, 1[.They all contain just one one-dimensional SU (2)-invariant subspace.The matrix element (14.3) reads ] for ρ = 2λ.It is positive.Hence, by (31), gs λ is positive for 0 ≤ λ ≤ 1.The cases λ = 0, 1 hold by continuity.Actually for the case λ = 0, and the trivial rep yields the case λ = 1.Note the ordering We call g p λ and g s λ irreducible and gp λ , gs λ the irreducible elements of G.
(32) Theorem.Every G ∈ G is given by G = g for with κ = cosh −1 t, where µ p and µ s are Borel measures such that Proof.We apply (31).So let U be a rep of SL(2, C) in a separable Hilbert space having normalized SU (2)-invariant vectors.Being semisimple connected, SL(2, C) is a locally compact tame group with countable basis.Therefore by [14, Second Part § 8.4 Theorem 3] it allows the direct integral (continuous sum) decomposition into its primary components such that up to unitary equivalence.Here µ is a probability measure on the set Λ of equivalence classes of irreps of SL(2, C), W λ is a primary component written in the form U λ ⊗ I λ .U λ belongs to the class λ ∈ Λ, and I λ is the trivial rep of dimension n(λ) ∈ N∪{0, ∞}.
(39) Effectiveness of NC.The necessary condition for positive definiteness of rotational invariant normalized kernels introduced above is shown to be not satisfied by gr for 0 < r < 1 2 and by K r for 0 < r < 3 2 thus furnishing straightforward alternative proofs to (26) and ( 27), by which these kernels are not positive definite.Moreover, what is important, the necessary condition holds by equality for the kernels g1/2 and K 3/2 .
Proof.The claim is easily checked.

Localization in bounded regions
As known a POL T of the massive scalar boson with CT, i.e., for which time evolution is causal, does not localize the boson in any bounded region ∆ ⊂ R 3 , which means that T (∆) has no eigenstate with eigenvalue 1.One has the general result (45) Theorem [5, (8) Theorem].Let T be a POL with CT.Let the relativistic relation H ≥ |P | hold.Suppose that there is a state localized in the region ∆.Then ∆ is essentially dense, i.e., ∆ \ N = R 3 for every Lebesgue null set N .
However, the property T (∆) = 1 may be regarded as physically equivalent to the presence of the eigenvalue 1.Indeed, as norm 1 means that the system can be localized within that region ∆ by a suitable preparation, not strictly but as accurately as desired.So for obvious physical reasons one is interested in POL with ||T (B)|| = 1 for every however small open ball B = ∅.We call them separated.In this case for every b ∈ R 3 there is a sequence (φ n ) of states satisfying The main result of this section is the following criterion on the separateness of a POL.
(46) Theorem.Let K be the kernel on R 3 \ {0} of a POL T for a massive scalar boson.Suppose that (i) lim λ→∞ K(k, λp) exists for every k and almost all p (ii) lim λ,λ ′ →∞ K(λp, λ ′ p) = 1 for almost all p (iii) there is k 0 such that, for any λ n → ∞, K(k 0 , λ n p) does not vanish for almost all p.
Then T is separated.
Proof.We use the representation (7.3) of T .Let H J be the RKHS associated to the kernel K (see e.g.[7]) and put J p := K(•, p).The orthogonal projection P on L 2 (R 3 , H J ), (P ϕ)(p) := J p , ϕ(p) J p maps onto the subspace j L 2 (R 3 ) .There is the representation D of the dilation group acting on L λ .The result holds by [4, Theorem 7] if the strong limit Q := lim λ→∞ P λ exists with Q = 0.
The integrand 1 vanishes for almost every p when λ, λ ′ → ∞ due to (i), (ii) and since |K| ≤ 1.So the integral vanishes by dominated convergence.We conclude P λ ϕ − P λ ′ ϕ 2 → 0 for λ, λ ′ → ∞.Actually this result follows for all ϕ ∈ L 2 (R 3 , H J ) since the set of all ϕ of the above kind is total in L 2 (R 3 , H J ) as {J k : k ∈ R 3 \ {0}} is total in H J .So P λ converges strongly to some projection Q.
Hence a POL T of the massive scalar boson with causal kernel (11.1) is a candidate for a non-separated POL, i.e., T (B) < 1 might hold for some small ball B = ∅.

Discussion
A POL can be regarded as a physically accessible data set, namely the set of the probabilities of localization of the particle in every state in every region.Therefore among the infinitely many POL for a massive scalar boson (2) only a particular one describes truly the position of the boson.The question is how to identify this POL.Many, as e.g.NWL, are ruled out by the requirement of causality CC (sec.11).The remaining causal POL are still infinitely many.We studied thoroughly the causal POL related to a conserved density current analyzing the set K of causal kernels (11.1).
In search of the best POL we recall the observation that KG ∈ K if K ∈ K and G ∈ G (28).Probably multiplying K by G deteriorates the localization features of the corresponding POL.More generally, if K, K ′ ∈ K with K ′ ≤ K, then K is better than K ′ ?Take as evidence the fact K ′ ≤ K ≤ 1 with 1 the kernel of NWL distinguished by its abundance of boundedly localized states.If it were true, the POL with kernel K 3/2 , being the greatest element (43), would be the only valid POL.
For the localization features of the POL T 3/2 with kernel K 3/2 it is decisive to know whether ||T 3/2 (B)|| = 1 for all balls B = ∅.Assume this.Then T 3/2 is separated (sec.17) and for every point b ∈ R 3 there is a sequence of states localized at b (17.1), which is very satisfactory from a physical point of view.
However, also the case ||T 3/2 (B * )|| = δ < 1 for some ball B * = ∅ were rather interesting.Assume this.By translation covariance B * may be centered at 0. The probability for the boson of being localized in B * is at most δ in every state.Note (e) Finally let ∆ be any measurable set.Since µ is tight, there are compact K (i) ⊂ ∆ satisfying µ(∆ \ C) = 0 for C := i K (i) .By (d), (1) holds for C (n) := n i=1 K (i) .Now proceed as in (c).Hence (1) holds for ∆.
The applications of (50) in ( 52) and (56) regard the case T ′ = T as they concern a POL T of the massive scalar boson, which due to the Poincaré covariance (3.1) is defined on all spacelike hyperplanes.Moreover the particular case in (50) of parallel hyperplanes ε and σ = {t} × R 3 for t ∈ R and for V = R 3 is used for the CT criterion below.
(51) Definition.Let t be the kernel of POL T of a massive scalar boson.Suppose that t is continuous.t is said to be (i) conserved, (ii) timelike definite, if there are continuous Hermitian kernels j i : R 3 \ {0} × R 3 \ {0} → C, i = 1, 2, 3 satisfying respectively (52) Theorem.Let T be a POL of a massive scalar boson.Suppose that its kernel t is continuous and conserved timelike definite.Then time evolution is causal.
Proof.We treat the particle case η = + (the antiparticle case is analogous) and use the shell rep in sec.for x ∈ R 4 , and Φ square integrable and √ ǫ Φ integrable.Recall that j i is bounded.

. 1 )
for every open ball B around b.This means that by a suitable preparation the system can be localized around b as good as desired, thus distinguishing b from any other point.According to [4, sec.G], any (φ n ) satisfying (17.1) is called a sequence of states localized at b.