Wigner's inert infinite spin representations and possible relations with ...

arXiv:1601.02477v3 [physics.gen-ph] 30 Mar 2016

Wigner’s inert infinite spin representations and possible relations with the enigmatic dark matter dedicated to the memory of Robert Schrader

Bert Schroer present address: CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil permanent address: Institut f¨ ur Theoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin, Germany September 2015 Abstract Positive energy ray representations of the Poincar´e group are naturally subdivided into three classes according to their mass and spin content: m>0, m=0 finite helicity and m=0 infinite helicity. For a long time the localization properties of the massless infinite spin class remained unknown before it became clear that such matter does not permit compact spactime localization and its generating covariant fields are localized on semi-infinite spacelike strings. We study the problem of possible couplings of string-local higher spin fields to normal (standard model) matter within an extended new renormalization setting. Whereas the problem remains unresolved for finite higher spin matter, it is shown that, with the possible exception of gravitational reactiveness, infinite spin matter is inert. This makes it an interesting candidate for further studies in connection with the still enigmatic astrophysical dark matter.

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Wigner’s infinite spin representation and stringlocalization

Wigner’s famous 1939 theory of unitary representations of the Poincar´e group P was the first systematic and successful attempt to classify relativistic particles according to the intrinsic principles of relativistic quantum theory [1]. As we know nowadays, his massive and massless spin/helicity class of positive energy ray representations of P does not only cover all known particles, but their ”covariantization” [2] leads also to a complete description of all covariant point-local free fields. For each representation there exists a field of lowest short distance

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dimension; all other ”elementary” fields are obtained by applying derivatives, and by forming Wick-ordered products of elementary fields one arrives at the ”local equivalence class” of composite fields. The only presently known way to describe interactions in four-dimensional Minkowski space is to start from a scalar interaction density in terms of Wickproducts of free fields with the lowest short distance dimension and use it as the starting point of the cutoff- and regularization-free causal perturbation theory [3]. These free fields do not have to be Euler-Lagrange fields; perturbative QFT can be fully accounted for in terms of interaction densities defined in terms of free fields obtained from Wigner’s representation theory without referring to any classical parallelism. All positive energy representations are ”induced” from irreducible representations of the ”little group”. This subgroup of the Lorentz group is the stability group of a conveniently chosen reference momentum on the forward mass shell H+ , respectively the forward surface of the light coneV+ . For m > 0 this is a rotation subgroup of the Lorentz group and for m = 0 the noncompact Euclidean subgroup E(2). Whereas the massive representation class (m > 0, s = n2 ), of particles with mass m and spin s covers all known massive particles (the first Wigner class), the massless representations split into two quite different classes. For the finite helicity representations the E(2) subgroup of Lorentz-”translations” are trivially represented (”degenerate” representations), so that only the abelian rotation subgroup U (1) ⊂ E(2) remains; this accounts for the semi-integer helicity ± |h| , |h| = n2 representations (the second Wigner class). The third Wigner class consists of faithful unitary representations of E(2). Being a noncompact group, they are necessarily infinite dimensional and their irreducible components are characterized in terms of a continuous Pauli-Lubanski invariant κ. Since this invariant for massive representations is related to the spin as κ2 = m2 s(s + 1) one may at first think that the properties of this infinite spin matter can be studied by considering it as a limit m → 0, s → ∞ with κ fixed. However it turns out that the (m, s) spinorial fields do not possess such an infinite spin limit. Our main result concerning the impossibility of quantum field theoretical interactions between WS with normal matter depends among other things on the absence of such an approximation; for this reason we will refer to these representations briefly as the ”Wigner stuff” (WS). This terminology is also intended to highlight some of the mystery which surrounded this class for the more than 6 decades after its discovery and which also the present paper does not fully remove. For a long time the WS representation class did not reveal its quantum field theoretic localization properties. The standard group theoretical covariantization method to construct intertwiners [2], which convert Wigner’s unitary representations into point-like covariant wave functions and their associated quantum fields, did not work for the WS representations. Hence it is not surprising that attempts in [4] (and more recently in [5]), which aim at the construction of wave equations and Lagrangians, fell short of solving the issue of localization. In fact an important theorem [6] dating back to the 70s showed that it is not possible to associate pointlike Wightman fields with these representations. 2

Using the concept of modular localization, Brunetti, Guido and Longo showed that WS representations permit to construct subspaces which are ”modular localized” in arbitrary narrow spacelike cones [7] whose core is a semi-infinite string. In subsequent work [8] [9] such generating string-local covariant fields were explicitly constructed in terms of modular localization concepts. In the same paper attempts were undertaken to show that such string-local fields cannot have point-local composites. These considerations were strengthened in [10]. A rigorous proof which excludes the possibility of finding compact localized subalgebras (generating point-local fields) was finally presented in an seminal paper by Longo, Morinelli and Rehren [11]. Being a positive energy representation, WS shares its stability property and its ability to couple to gravity with the other two positive energy classes; hence it cannot be dismissed from the outset as being unphysical. In view of the unsolved problems concerning the relation of the increasing amount of unexplained astrophysical data concerning the still mysterious dark matter, it is tempting to explore the possibility of a connection between this (according to some physicist) ”biggest enigma of the 21st century” and the theoretical puzzle of the WS class. This re-opens a problem which Weinberg temporarily closed in the first volume of his textbook by stating that ”nature does not use the WS representations” [2]. The aim of the present paper is to convert the question of whether nature uses WS matter into a theoretical problem of understanding its largely unexplored properties. Such a theoretical question is meaningful because QFT (in contrast to quantum mechanics) is fundamental in the sense that the wealth of its physical consequences can be traced back to different manifestations of its underlying causal localization principle. In the context of quantum theory this principle is much more powerful than its classical counterpart. The concept of modular localization permits to address structural problems of QFT in a completely intrinsic way which avoids the use of ”field-coordinatizations”. An illustration of the power of this relatively new concept is the proof of existence of a certain class of two-dimensional models starting from the observations that certain algebraic structures in integrable d=1+1 models can be used to construct modular localized wedge algebras [12]. In the work of Lechner and others this led to existence proofs for integrable models with nontrivial short distance behavior together with a wealth of new concepts (see the recent review [13] and literature cited therein). Even in renormalized perturbation theory modular localization has become useful in attempts to replace local gauge theory in Krein space by string-local fields in Hilbert space [16]. In [7] it was essential to extract localization properties directly in the form of modular localized subspaces since Weinberg’s group theoretic method of constructing covariant local field within the standard intertwiner formalism does not work for WS. In an unpublished previous note [17] I tried to address the problem of a possible connection between WS and dark matter. But the recent gain of knowledge from modular localization regarding attempts to unite WS with normal matter 3

under the conceptual roof of AQFT in [11], as well as new insights coming from perturbative studies of couplings involving string-local fields [16] [20], led to a revision of my previous ideas. In [11] it was shown that the attempt to unite normal matter together with WS in a nontrivial way1 under the conceptual roof of AQFT leads to an unexpected suspicious looking loss of the so-called ”Reeh-Schlieder property” for compact localized observable algebra. The R-S property states that the set of state vectors obtained by the application of operators from a compact localized subalgebra of local observables to the vacuum is ”total” in the vacuum Hilbert space. The possibility to manipulate large distance properties of states in the vacuum sector by applying operators localized in a compact spacetime region O to the vacuum is considered to be a universal manifestation of vacuum polarization. It plays an important role in the DHR superselection theory [14] and its breakdown in the presence of WS asks for further clarification. It turns out that, different from point-local interactions where the power-counting requirement dint sd ≤ 4 for renormalizability is the only requirement for the perturbative existence of a model, string-local interactions must fulfill an additional quite restrictive condition which prevents total delocalization in higher orders. The main result of the present paper is that this additional condition cannot be fulfilled in couplings of WS to normal matter. The reason is that in contrast to massless finite helicity matter which can be obtained as a massless limit from string-local massive matter, the class of WS remains completely isolated; in particular the fields are not simply the m → 0, s → ∞ limit of string-local covariant spin s fields with fixed Pauli-Lubanski invariant κ2 = m2 s(s + 1). This leaves only the possibility that, apart from interactions with gravity as a consequence of the positive energy property, WS cannot interact with normal matter. In mathematical terminology: normal matter tensor-factorizes with WS and the Reeh-Schlieder property is then that of a compact localized observable subalgebra in the tensor factor of normal matter. A world in which the WS matter only reacts with gravity may be hard to accept from a philosophical viewpoint. But after we got used to chargeless leptons which only couple to the rest of the world via weak and gravitational interactions, the step to envisage a form of only gravitationally interacting kind of matter is not as weird as it looks at first sight; in particular when astrophysical observations reveal that on the one hand the presence of a new form of matter is indispensable for attaining a gravitational balance in agreement with galactic observations, but at the same time require a high degree of inertness (”darkness”) with respect to ordinary matter. Apart from the fact that any positive energy matter couples to gravity, Wigner’s (m, s) classification contains no information about possible interactions (strong, electromagnetic, weak). As will be argued in the following, the noncompact localization properties permit however to support the idea that in1 Excluding the trivial possibility of a tensor product of WS with the world of ordinary matter (total inertness except with respect to classical gravity).

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teractions of WS with normal matter are not possible; apart from its reactiveness to classical gravitation, WS is inert. In contrast to the local observables of noncompact string-like massless tensor potentials interacting among themselves or with s < 1 matter, covariant stringlocal fields contain no local observables at all and hence the presence of stringlocal states created by WS fields acting on the vacuum cannot be registered in particle counters. The possible galactic presence of WS is restricted to pure gravitational manifestations. The paper is organized as follows. The next section presents a ”crash course” on Wigner’s theory of positive energy representations of the Poincar´e group including the explicit construction of string-local WS free fields and their two-point functions. The third section highlights an important restriction on renormalizable couplings involving string-local fields which is necessary to avoid higher order total delocalization. In section 4 it is explained why this perturbative restriction is violated for WS which is the cause of its inertness. Section 5 addresses the problem of the role of string-localization in the construction of the correct energy-momentum tensor for higher spin quantum (i.e. acting in Hilbert space) fields which is the prerequisite for the Einstein-Hilbert gravitational coupling. The concluding remarks point at problems arising from the identification of WS with dark matter.

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Matter as we (think we) know it and Wigner’s infinite spin ”stuff”

The possible physical manifestations of WS matter can only be understood in comparison to normal matter. Hence before addressing its peculiarities it is necessary to recall the localization properties of free massive and finite helicity zero mass fields. It is well known that all point-local massive free fields can be described in terms of matrix-valued functions u(p) which intertwine between the creation/annihilation operators of Wigner particles [2]. Their associated covariant fields are of the form ˙

ψ A,B (x) =

1 3/2

(2π)

Z

˙

˙

(eipx uA,B (p) · a∗ (p) + e−ipx v A,B (p) · b(p))

d3 p 2p0

(1)

The intertwiners u(p) and their charge-conjugate counterpart v(p) are rectangular (2A + 1)(2B + 1) ⊗ (2s + 1) matrices which intertwine between the unitary (2s + 1)-component Wigner representation and the covariant (2A + 1)(2B + 1) dimensional spinorial representation labeled by the semi-integer A, B˙ which characterize the finite dimensional representations of the covering of the Lorentz 5

group SL(2, C). the a# (p), b# (p) refer to the Wigner particle and antiparticle creation/annihilation operators and the dot denotes the scalar product in the 2s + 1 dimensional spin space. For a given physical spin s there are infinitely many spinorial representationindices of the homogeneous Lorentz group; their range is restricted by [2] ˙ m>0 (2) A − B˙ ≤ s ≤ A + B,

For explanatory simplicity we restrict our subsequent presentation to integer spin s; for half-integer spin there are similar results. All fields associated with integer spin s representation can be written in ˙ of degree terms of derivatives acting on symmetric tensor potentials (A = B) s with lowest short distance dimension dssd = s + 1. For s = 1 one obtains the divergenceless (∂ · AP = 0) Proca vector potential AP µ with dsd = 2, whereas for s = 2 the result is a divergence- and trace-less symmetric tensor gµν with dsd = 3. Free fields can also be characterized in terms of their two-point functions whose Fourier transformation are tensors in momenta instead of intertwiners. For s = 1 on obtains Z

P  d3 p 1 pµ pν −ip(x−x′ ) P P ′ P e M (p) A (x)A (x ) = , Mµν (p) = −gµν + 2 (3) µν 3 0 2p m (2π) and for higher spin the M ′ s are symmetric tensors formed from products of gµν and of p′ s (P stands interchangeably for ”Proca” or ”point-like”) For m = 0 and finite integer helicity h the two dimensional ± |h| helicity representation replaces the 2s + 1 component spin. Despite this difference, the covariant fields turn out to be of the same form (1), except that (2) is now replaced by the more restrictive relation (4) A − B˙ = |h| , m = 0

which excludes all the previous tensor potentials but permits their field strengths (which are tensors of degree |h| and dsd = |h| + 1 with mixed symmetry properties). This is well-know in case of |h| = 1 where there exist no massless point-local vector potential A = 1/2 = B˙ who’s curl is associated to the electromagnetic field strength. The absence of point-local tensor potentials in (4) results from a clash between point-local spin s tensor potentials and Hilbert space positivity. Gauge theory substitutes the non-existent point-local Hilbert space vector potential by one in an indefinite Krein space and the prescriptions by which one extracts a physical subtheory from a Krein space lead to gauge theory. This is also a clash between the classical Lagrange formalism (positivity has no place in classical physics) and the most basic Hilbert space positivity on which quantum theory’s probability hinges. It shows the limitation of that parallelism to classical theory

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better known as ”quantization”2. The problem can be resolved in two ways; either one sacrifices positivity or one gives the Hilbert space a chance to determine the tightest localization which is consistent with positivity, which turns out to be localization on semiinfinite spacelike strings x + R+ e, e2 = −1. Beware that there is no relation between string-local fields and string theory. Whereas the change from pointlocal to string-local fields for s ≥ 1 is required in order to uphold Hilbert space positivity, ST has no conceptual compass, it is the result of a playful spirit to extend the game of QFT. The first solution leads to a physically restricted theory in Krein space in which all gauge dependent fields are physically void. The advantage is only computational since point-local fields are computationally easier (however this does not apply to the explicit extraction of the physical data with the help of the BRST ghost formalism which remains involved). The perturbation theory of string-local fields turns out to be more demanding, but as a reward one obtains a full QFT in which all fields are physical (though, as for point-local s < 1 interactions in Hilbert space, not all operators represent local observables). String-local tensor potentials also exist for massive fields. The string-local counterpart of the point-local massive two-point functions (3) turn out to be s (p; e.e′ ) = −gµν − Mµν

pν e′µ pµ pν e · e ′ pµ e ν + + (p · e − iε)(p · e′ + iε) p · e − iε p · e′ + iε

(5)

Its massless limit is of the same form, except that the momentum p is on the + boundary of the positive lightlike surface H0+ of the forward light cone Hm ; This has to be taken into account in the Fourier transformation to x-space. The more complicated form as compared to the simpler −gµν in the Feynman gauge setting is the prize to pay for improving the high energy behavior while preserving positivity and securing the existence of a massless limit. The only way I know which secures positivity is the intertwiner representation of covariant fields as linear combinations of Wigner creation/annihilation operators. Lagrangian quantization account for positivity only for s < 1 interactions; in all higher spin cases it leads to indefinite metric which only permits a partial return to positivity in case of a gauge formalism in Krein spaces. I am not aware that physical (Hilbert space) s > 1 energy-momentum tensors have been constructed in the existing literature. The best description of the interacting massless theory is to first calculate the renormalized massive correlation functions and then take their massless limit. This has the advantage of performing perturbation theory in the simple Wigner Fock particle space and leaving the reconstruction of the massless limit (in which this physical description of the Hilbert space in terms of particle states is lost) to the application of Wightman’s reconstruction theorem to the limiting correlation functions. 2 Not to be confused with ”second quantization” which is an unfortunate terminology for a functorial relation between Wigner’s representation theory of particles and the associated quantum free fields acting in a Wigner-Fock Hilbert space.

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Massive theories are simpler from a conceptual viewpoint because the presence of a mass gap permit to use the tools of scattering theory and the identification of the Hilbert space with a Wigner Fock space. Whereas it is plausible that the asymptotic short distance behavior of the Hilbert space setting is correctly accounted for in terms of the asymptotic freedom properties of gauge theories, the problems related to long-distance properties as confinement remain outside the physical range of the gauge setting. There is a very efficient way to derive the relation between the point-local Proca potential and its string-local counterpart. Integrating the latter along the space-like direction e, one obtains a string-local scalar field φ(x, e) Z ∞ Z 1 d3 p µ P ipx φ(x, e) := dλe Aµ (x + λe) = (6) (e u(p, e) · a(p) + h.c.) 2p0 (2π)3/2 0 1 e · e′ 1 u(p, e) := u(p) · e , M φ,φ = 2 − ip · e m (p · e − iε) (p · e + iε) where the inner product in the first line refers to the 3-dim. spin space and the denominator is simply the Fourier transform of the Heavyside function. The φ two-point function can either be computed from carrying out the line integrals on the Proca two-point function or by using the intertwiner. The string-local vector is defined as Z ∞ P Aµ (x, e) := eν Fµν (x + λe)ds, Fµν := ∂µ AP (7) ν − ∂ν Aµ 0

which leads to the two-point function (5). The three fields turn out to be linearly related Aµ (x, e) = Apµ (x) + ∂µ φ(x, e)

(8)

which either can be derived from the previous definition or by defining the three fields in terms of their intertwiners in which case one obtains the relation as a linear relation between intertwiners. By changing the λ-measure dλ → κ(λ)dλ one can improve the short distance behavior and get arbitrarily close to dsd = 0, but this will be of no avail3 for enlarging the scope of renormalizability since this would destroy the linear relation (8) which is essential for improving the renormalizability properties of interactions involving higher spin s ≥ 1 fields while keeping the Hilbert space positivity. The relation (8) looks like a gauge transformation; indeed the extension of this relation to interactions with matter fields suggests a formal connection between point-local ψ(x) and is string-local counterpart which has the expected exponential form ψ(x, e) = ψ(x) exp igφ. But in contrast to gauge theory these relations intertwine between string-local fields and their more singular pointlocal siblings within the same string-local relative localization class; in fact this 3 In particular the lowering of d sd is of no help for s < 1 fields. There is no Elko trick which improves the short distance properties of s = 1/2 fields. The proposal in ([15], formula ()) shows a total misunderstanding of what QFT is about.

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formula (after making it precise in terms of normal products) may be seen as the definition of an e-independent dsd = ∞ singular pointlike counterpart of a polynomially bounded string-local field. This construction can be extended to all integer spin fields. The divergencefree Proca potential is replaced by divergence- and trace-free symmetric tensor potentials Aµ1 ,...µs of tensor rank s. Iterated integration along a space-like direction e leads to s string-local φ tensor fields of lower rank Z φµ1 ..µk (x, e) = dλ1 ..dλs−k eν1 ..eνs−k Aν1 ,...νs−k ,µ1 ,...µk (x + λ1 e + ..λs−k e) (9) Again one can construct the e- dependent intertwiners from these relations. The extension of (8) spin s relates the point-local tensor-potential AP to its stringlocal counterpart and the symmetrized contributions from the derivatives of the string-local tensor escorts φ...(x, e) of A....(x, e) (9) Aµ1 , ..µs (x, e) = AP µ1 , ..µs (x) + sym

s X

∂µ1 ,.. ∂µk φµk +1,..µs

(10)

k=1 P (x, e) + sym ∂µ φν + ∂µ ∂ν φ gµν (x, e) = gµν

(11)

where the second line is the special case of the connection between the traceand divergence-less s = 2 point-local symmetric tensor and its string-local counterpart including the two string-local s < 2 escorts φµ and φ. The appearance of these lower spin φ-escorts is characteristic for the change of massive point-local fields into their string-local siblings acting in a Hilbert space. They are important new fields which depend on the same degrees of freedom (the Wigner creation/annihilation operators) as the other two operators. For s = 1 the scalar escort field φ may be seen as the QFT analog of the bosonic Cooper pairs which are the result of a reorganization of the condensed matter degrees of freedom in the superconducting phase 4 . Without the formation of Cooper pairs from existing condensed matter degrees of freedom it is not possible to convert the long-range classical vector potentials into its short range counterparts within the superconductor (as anticipated by London). The relation between long range massless and short range massive potentials requires the presence of the φ; in fact it is not possible to formulate massive QED as a renormalizable theory in Hilbert space without the presence of these scalar escorts, but there is no need of additional H fields. The reason why additional degrees of freedom in the form of H-fields are indispensable in the case of self-interacting massive vectormesons is quite deep but bears no relation to physical spontaneous symmetry breaking. The lower spin escort fields in (10) have no massless limit, but together with the Proca tensor potentials their presence is necessary for the construction of 4 This remark is intended to counteract the claim that one needs addional H-fields in order to ”fatten vector potentials by swallowing Goldstone particles”. The physical reason why H ′ s are needed in the presence of self-interacting massive vector mesons is quite different (see below).

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the massive string-local potential; all these fields are relatively local and act in the same Wigner-Fock Hilbert space. Only the correlation functions of the degree s string-local tensor potential possess a massless limit. The string-local A... in (10) is related to the point-local field strength Fµ1 ..µs ,ν1 ..νs = as {∂µ1 ..∂µs Aν1 ,..νs } µ,ν

(12)

where the as imposes antisymmetry between the µ − ν pairs. As in the previous case of the vector potential (7), the string-local tensor potentials with fulfill (10) can be obtained in terms of iterated integrations along e starting from the field strength. The field strength tensor is the lowest rank point-local tensor field5 . With appropriate changes these results have analogs for semi-integral spin. Before passing to the string-local fields of the WS class it may be helpful to collect those properties which turn out to be important for a comparison with higher spin string-local fields. • Whereas pointlike massive tensor potentials have short distance dimension dsd = s + 1, their string-local counterparts have dsd = 1 independent of spin. Hence there are always first order string-local interaction densities within the power-counting limit dint sd ≤ 4, but whether they can be used in a consistent perturbative renormalization setting is another story. • String-local tensor potentials are smooth m → 0 limits of their massive counterpart. They inherit the dsd = 1 from their massive counterpart. The lowest point-local fields in the same representation class are field strengths (tensors of rank 2s with mixed symmetry properties). K • The point-local dK sd = 1 zero mass vector-potentials Aµ of local gauge theory act in an indefinite metric Krein space. The physical prize for resolving the clash between point-like localization and Hilbert space positivity is the sacrification of the Hilbert space. The physical range of the resulting gauge theory is restricted to local observables which act in a smaller Hilbert space which does not include fields. The latter contain no information about the physical relation between Hilbert space positivity and Einstein-causal physical localization. What makes gauge theory useful is the fact that the perturbative unitary on-shell S-operator is gauge invariant. The absence of local observables excludes the use of gauge theory in WS models.

Whereas the two-point functions of point-local massive free fields are polynomial in p, their string-local counterparts have a rational p-dependence (5) (6). The family of all WS interwiners for a given Pauli-Lubanski invariant κ has been computed in [8], their two point-functions are transcendental functions of p, e which are boundary values of from Im(e) ∈ V + . 5 For s = 2 this tensor has the same mixed symmetry property as the linearized Riemann tensor whereas the symmetric second rank tensor deserves to be denoted as gµν .

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A particular simple WS intertwiner with optimal small and large momentum space behavior (corresponding to the minimal choice for string-local massless s ≥ 1) has been given in terms of an exponential function in [10] u(p, e)(k) = expi

~k(~e −

p− ~) e− p

−κ

p·e

(13)

here k is a two-component vector of length κ; the Hilbert space on which Wigner’s little group E(2) acts consists of square integrable functions L2 (k, dµ(k) = δ(k 2 − κ2 )dk) on a circle of radius κ. The vector arrow on e and p refer to the projection into the 1-2 plane, and the and e− , p− refer to the difference between the third and zeroth component. The most general solution of the intertwiner relation differs from this special one by a function F (p·e) which is the boundary value of a function which is analytic in the upper half-plane [8]. The two-point function is clearly a J0 Bessel function. The calculation in [10] was done in a special system. Writing its argument in a covariant form one obtains 6 M W S (p, e) ∼ J0 (κ |w(p, e)|)exp − iκ( with w2 (p, e) = −(

1 1 − ) ′ p · e − iε p · e + iε

(14)

e′ e − ′ )2 e · p − iε e · p + iε

The exponential factor compensates the singularity of J0 at e · p = 0. Note that the Pauli-Lubanski invariant κ has the dimension of a mass so that the argument of the two-point function of the string-local field has the correct engineering dimension den = 1 of a quantum field. The main purpose of this calculation is to convince the reader that there are explicitly known transcendental WS intertwiner and two-point functions whose associated propagators have a well behaved ultraviolet and infrared behavior. As already mentioned, the physical reason why these fields are nevertheless excluded from appearing in interaction densities is that higher orders lead to a complete delocalization; this will be explained in the next section.

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The problem of maintaining higher order stringlocalization

It is well known that the only restriction for point-local interaction densities is the power-counting inequality dint sd ≤ 4. Since the minimal short distance dimension of point-local spin s fields7 is s + 1, there are no point-local renormalizable interactions involving s ≥ 1 fields. String-local free fields on the other hand have an s-independent short distance behavior with dsd = 1, so that one always 6I

am indepted to Henning Rehren for showing me the covariantization of K¨ ohler’s result. (without the added specification ”gauge”) are always acting in Hilbert space.

7 Fields

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can find polynomials of maximal degree 4 which represent interaction densities within the power-counting limitation. Point- and string-local fields represent two different descriptions of the same spin s quantum matter, just like two different coordinatizations in geometry. For s ≥ 1 the use of the point-local coordinatization of interaction densities becomes too singular; the breakdown of the power-counting bound dint sd ≤ 4 leads to singular interacting fields (unbounded increase of dsd with perturbative order). The bad aspect of such a singular (non Wightman) behavior is not primarily the polynomial unboundedness in momentum space, but rather the fact that the perturbative counterterm formalism leads to an ever increasing number of undetermined counterterm parameters which destroys the predictive power. There are two explanations for the cause of this situation, either the interaction, density is incompatible with the principles of QFT or the model is consistent but the point-local coordinatization is too singular for the application of the rules of renormalized perturbation theory. In the latter case the string-local field coordinatization may lead to reduction of the short distance singularity within dint sd ≤ 4 and in this way save the model. If this fails one falls back to square one, this time without remedy. There are good reasons to believe that perturbative renormalization in the new string-local setting accounts correctly for the existence or failure of a QFT with a prescribed field content. This structural correctness does not imply that the perturbative series converges and in many models this has been established; but I am not aware of arguments which exclude asymptotically convergent for weak coupling g → 0. There are also convincing arguments that there is nothing to be gained by weakening localization beyond semi-infinite strings e.g. to go to operators localized on spacelike hypersurfaces. These arguments are based on a theorem which states that in the presence of a mass gap the content of a QFT model can always be described in terms of semi-infinite spacelike strings8 . Fields which even in their string-local coordinatization cannot be used for defining interaction densities of renormalizable model will be called inert. The main claim of the present work is that WS fields fall into this category whereas for higher finite spin fields s ≥ 2 the question whether they are ”reactive” or remain inert remains unsettled (see next section). Formally renormalizable string-local interaction densities within the powercounting bound come with a physical hitch. Unless they fulfill an additional requirement it is not possible to maintain the string-localization in higher orders. In that case the result will be a complete delocalization and hence the principles of QFT exclude such interactions. In the next section it will be shown that the presence of WS fields in a interaction density leads to such a situation. On the other hand free WS fields fulfill all general localization- and stabilityrequirements (energy-positivity) of QFT and consequently cannot be excluded as unphysical [11]. This justifies the terminology ”inert matter” used in the next 8 The theorem has been proven in the setting of algebraic QFT in which ne works with arbitrary tight spacelike cones (whose core is a spacelike string) [18].

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section. The remainder of this section will address the problem of upholding string-localization in higher orders which will be taken as the defining property of ”reactive (or dynamic) quantum matter”. As a simple nontrivial illustration we start with a string-local interaction density L of massive QED L = Aµ (x, e)j µ (x)

(15)

Here Aµ (x, e) is a massive string-local vector potential (5) and j µ is the conserved current of a massive complex scalar field. These fields act in a WignerFock Hilbert space of the corresponding Wigner particles, and since dsd (Aµ ) = 1 and hence the short distance dimension dint sd (L) = 4 (15) stays within the power-counting bound dint sd = 4 of renormalizability. The string-local L is related to its dsd = 5 point-local counterpart LP as LP = AP · j = L − ∂ µ Vµ , Vµ (x, e) := ∂µ φj µ Z Z (1) (1) LP d4 x = Ld4 x, i.e. S (1) = SP = SS

(16) (17)

where the second line follows since in the presence of a mass gap the divergence of Vµ does not contribute to the adiabatic limit which represents the first order S-matrix. In other words one splits the dsd = 5 point-local density into its string-local dsd = 4 counterpart and a dsd = 5 divergence term which can be disposed of in the adiabatic (on-shell) S-matrix limit. In this way one solves two problems in one stroke, on the one hand one expresses the (first order) S-matrix in terms of a dsd = 4 interaction density, and at the same time the e-dependence disappears in the first order on-shell S-matrix. The linear relation between L and its point-local counterpart LP is a consequence of the linear relation between the massive point-local Proca potential and its string-local dsd = 1 counterpart (8). The lowering to dsd < 1 by using instead of dλ another measure µ(λ)dλ would be possible, but this would destroy the linear relation (16) and as a result also the relation (17) which is the basis of the e-independence of S 9 . For the following it is convenient to formulate the e-independence in terms of a differential calculus on the d = 1 + 2 dimensional directional de Sitter space. The differential form of the relation (8) reads de Aµ = ∂µ u, u = de φ de (L − ∂ µ Vµ ) = de L − ∂ µ Qµ = 0, Qµ := de Vµ

(18) (19)

Hence A, φ, L, V are dsd = 1 zero-forms whereas u, Q are exact dsd = 1 oneforms; together with the exact two-form uˆ (6) they exhaust the linear with AP µ linear related relatively local dsd = 1 forms. 9 Recent suggestions (”Elko*) that the mere lowering of d sd = 3/2 to 1 for s = 1/2 (probably to lower the power-counting bound of the 4-Fermi interaction) reveal a total misunderstanding of QFT.

13

We will refer to the relation expressing the e-independence in terms of a closed zero form as the ”L, V (or L, Q) relation” (19). It is a necessary condition for the e-independence of S and for the dependence of interacting field correlation only on those e′ s of the fields which correlate and not on the e′ s of inner propagators in Feynman diagrams which contribute to these correlations. As the independence of S from gauge-fixing parameters in gauge theory, this e-independence in the Hilbert space setting results from cancellations between different contributions in the same order; but different from unphysical gauge dependent correlation functions, correlations of charge-carrying stringlocal fields are expectation values of physical fields in an extended Wightman setting (endpoint x- and directional e-smearing). As not every use of point-local fields in a Krein space can be called a gauge theory, not every formalism in Hilbert space using string-local fields qualifies as a QFT. As one needs gauge symmetry and gauge invariance for point-like s = 1 fields in Krein space, one cannot obtain physically relevant descriptions of interacting s ≥ 1 fields in Hilbert space without the delocalization preventing L, V condition (not a symmetry condition!). But whereas classical gauge theory is a natural and useful extension of Maxwell’s theory, it clashes with the most important quantum theoretical positivity requirement and leads to the appearance of ghosts (an appropriate name for objects without a quantum status). In order to secure the e independence in higher orders we must extend the relation (19) to higher order time-ordered products (second order for simplicity): ′

(de + de′ )T LL′ − ∂ µ T Qµ L′ − ∂ µ T LQ′µ = 0

(20)

If it where not for the distributional singularities of T -products at coalescent points, this would follow from (19). For the second order S-matrix we only need the one particle contraction component (”tree approximation”). For massive spinor QED the relation is fulfilled in term of the standard free field propagator. The more singular scalar QED contains dsd = 2 derivatives ∂ϕ which according to the minimal scaling rules of the divergence and regularization free Epstein-Glaser renormalization theory lead to a delta counterterm hT ∂µ ϕ∗ ∂ν′ ϕ′ i = ∂µ ∂ν′ hT ϕ∗ ϕ′ i + cgµν δ(x − x′ )

(21)

where the imposition of the relation (20) fixes the parameter c with the expected result of an induced second order term g 2 gδ(x − x′ )Aµ Aµ ϕ∗ ϕ. Note that none of the arguments of classical gauge theory (as the replacement ∂ → D = ∂ + igA) has been used; the result is solely a consequence of the causal localization principles and Hilbert space positivity. There are some interesting foundational aspects of this otherwise trivial calculation. The independent fluctuation in e and e′ do not allow to set e = e′ in off-shell correlations (5) ; the different iε prescriptions for e and e′ in the off-shell propagator prevent this. The on-shell e-independence corresponds to the second order gauge invariance of the scattering amplitude; individual contributions are generally e-dependent and upon setting e = e′ lead to infinite fluctuations. 14

The ”magic” of L, Vµ pairs with (18) is that on the one hand they they permit to use the lower short distance dimension of string-local fields (and in this way lower the power-counting bound of renormalizability); on the other hand they also guaranty the e-independence of the S-matrix since the derivative contributions in (20) disappear in the adiabatic S-matrix limit Z (de + de′ )S (2) = 0, S (2) ∼ T LL′ (22) The extension of the Bogoliubov S-matrix formalism to quantum fields leads to correlation functions of interacting string-local fields. As the S-matrix is independent of the e′ s of the inner propagators after summing over sufficiently many contributions in a fixed perturbative order, the correlation functions of interacting fields only depend on the e′ s of those fields. A new phenomenon as compared with gauge theory is that the higher order interactions spread the e-dependence also to those fields which entered the first order interaction density as point-local fields10 . In fact the interacting matter fields in the new string-local setting of renormalization theory are string-local in a stronger sense than the vector potentials which remain linearly related with their point-local field strengths. In the limit of massless string-local vectormesons the correlation functions define a new theory; the particle setting in a Wigner-Fock Hilbert space disappears and the strings of the charge-carrying fields become ”stiff” and cause a spontaneously breaking of Lorentz invariance in charged sectors [19]. The L, Vµ (or L, Qµ ) pair property (18) is a necessary condition for maintaining string-localization; it permits to sail between Scilla of nonrenormalizability and the Charybdis of total delocalization. Heuristically speaking it provides a compensatory mechanism between contributions to the same order which prevents the total delocalization resulting from the integration of inner strings x + R+ e over over x. The main point of the present work is the argument that in the presence of WS fields prevent the fulfillment of the L, Vµ condition so that WS matter can only exist in the interaction-free form. There is another important physical aspect of the L, Vµ pair property. The escort field φ plays an essential physical role in it; although φ does not add new degrees of freedom a Hilbert space setting of perturbation theory would not be possible without its presence Heuristically speaking the transition from long range massless string-local vector potentials to their short range massive counterpart is not possible without the appearance of the φ escort. This is somewhat reminiscent of the presence of the bosonic Cooper pairs in the BCS description of superconductivity; without their presence it is not possible to have short ranged vector potentials. As the φ in massive QED they are not the result of additional degrees of freedom; they arise from rearrangements of existing condensed matter degrees of freedom in the low temperature phase. 10 The perturbation theory of interacting string-local fields is still in its beginnings. First results will be the subject of forthcoming work by Jens Mund..

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The QFT analog of the BCS or the Anderson screening mechanism is the screened ”Maxwell charge” [16] i.e. Z jµ := ∂ ν Fµν , Qscr = j0 (x)d3 x = 0 (23) Z ∂ µ jµ = 0, QSSB = j0 (x)d3 x = ∞, long dist.divergence (24) The screening property only depends on the massive field strength and not on the kind of matter to which it couples (which may be complex or Hermitian matter). This includes non-interacting massive vector mesons for which jµ ∼ AP µ . Spontaneous symmetry breaking on the other hand reveals itself in form of a conserved current whose charge diverge instead of being zero. Renormalizable models are uniquely specified by their field content, and the A-H content is different from that of Aµ interacting with a complex field. The shifts in the field space of models with Mexican hat potentials do not constitute the definition of SSB in QFT. They are only a (quasiclassically suggested) computational trick which under certain conditions leads to SSB. The defining property of SSB is the divergence of global charges of conserved currentsRi.e. models in which the Noether theorem cannot be inverted (∂j = 0 but Q = j0 = ∞). This is the case for Goldstone type models of scalar selfinteractions, but the argument breaks down as soon as couplings to massive vector mesons (more general couplings to s ≥ 1 fields) enter. In that case no Mexican hat potential needs to be added (no ”Higgs mechanism”), rather the A-H interaction induces a fourth degree H self-interaction, and the the charge of the resulting identically conserved Maxwell current of a massive field strength is screened which is the extreme opposite of SSB. There remains the historically interesting question why the ”fattening (of the massless vector meson) by swallowing the Goldstone boson” argument which imposed a SSB interpretation onto the screening property became that popular and was never corrected11 . A metaphor may be useful as a temporary placeholder for an incompletely understood situation, but it is almost uncorrectable and may impedes progress after it solidified during more than 4 decades. But what this has to do with the WS problem? The answer is that the correct understanding of higher spin s ≥ 1 interactions matters, and the nonabelian Higgs model is the lowest spin interaction for which a new mechanism for s ≥ 1 which has no s < 1 counterpart becomes relevant. Let us first recall some facts about the abelian Higgs model [16] [20]. The application of the L, V requirement to the coupling of a massive vector meson to a Hermitian H field proceeds as follows. The point-local interaction with the lowest short distance dimension dP,int = 5 is LP = mAP · AP H. sd Converting it into a string-local L, Vµ pair, one obtains (easy to check by the 11 The fact that a Nobel prized discovery was made at the same time in at least 3 independent papers using almost identical calculations and wordings is extraordinary.

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use of the free Klein-Gordon equation for H and relation (8)):     ← → → m2 1 ← L = m A · (AH + φ ∂ H) − H φ2 H , Vµ = m Aµ φH + φ2 ∂ µ H 2 2 (25) L − ∂V = LP = mAP · AP H, de (L − ∂V ) = 0 In this case the on-shell e-independence requirement (20) in second and third order tree approximation leads to a much richer collection of induced terms than in the case of scalar massive QED [16] (for a gauge-theoretic derivation see [21] section 4.1). Whereas in the QED case this requirement induces only the A · Aϕ∗ ϕ term, the induction in case of an interaction with a Hermitian field leads besides the expected A · AH 2 , A · Aφ2 terms (which as in scalar QED can be absorbed into a changed time-ordered product) also to second order induced H 4 , φ4 , H 2 φ2 terms from A-A contractions as well as to an additional first order H 3 term [16] [20]. The coupling strengths of these second order12 induced terms are fixed in terms of the 3 parameters of the elementary model-defining Aµ , H fields, namely the coupling strength and (ratios of) the two masses m, mH . The result is the same as that of the formal calculation based on the SSB Higgs mechanism, except that now the Mexican hat potential is induced from causal string-localization and not postulated [16] . There is no question as to which derivation is supported by the physical principles of QFT. The string-local formulation is required by maintaining positivity for s ≥ 1 interactions. The ultimate test is of course to ask the constructed model itself in which case it tells us that its would-be symmetry charge is not divergent but zero13 . QFT is a foundational quantum theory in which all properties of a model are intrinsic i.e. do not depend on the way by what prescriptions it has been obtained. Different from quantum mechanics, perturbative renormalizable interactions of QFT are uniquely fixed in terms the field content and possible imposed inner symmetries which maintain relations between coupling constants in higher orders. It suffices to start with a particular coupling of these which contains the field content (in the present case the A·AH coupling); the renormalization counterterm formalism and (for s ≥ 1) the L, Q induction requirement completes the interaction which represents the unique interaction with the given field content. The idea of selecting and maintaining a preferred coupling (say only A · AH without the H-self-interactions) is not compatible with the s ≥ 1 induction and also clashes with the counterterm formalism of renormalization theory. Any prescription which leads to a Aµ , H field content ends up to describe that unique 12 In higher (4th ) order one also expects the appearance of new counterterm parameters as known from point-local interactions. 13 The Mexican hat argument leads only to SSB if the conserved charge diverges (the definition of SSB) and this is only the case if no s ≥ 1 fields are present.

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theory but the question what are its intrinsic properties bears generally no relation to the way jn which one has obtained that theory. A Maxwell current of a massive field strength (the only current of the abelien Higgs model) is always screened and never SSB. As mentioned before the above calculations have a counterpart in gauge theory where formally the string-independence of the S-matrix corresponds to its gauge invariance in terms of the BRST s operation i.e. sS = 0 (the ”CGI operator formulation” in [21]). In the standard functional formulation of BRST it is hard to distinguish on- from off-shell and there is no protection against misreading the second order on-shell BRST requirement as an off-shell SSB prescription. The relation with the WS problem becomes clear after understanding the situation of massive self-interacting vector mesons. There is no problem with setting up a first order L, Q pair and calculating second order corrections. But now the induced terms contain an uncompensated nonrenormalizable dsd = 5 term [16]14 . The only mechanism which could maintain renormalizability in this situation is that for which supersymmetry became popular, namely short distance compensations between different spin components. In the present context this amounts to an extension of the field content by a coupling of the massive vector potentials to an additional field in the hope that there will appear a compensating singular second order term of the same form. The new field should have a lower spin (in order not to worsen the short distance situation) and the same Hermiticity property as Aµ i.e. it should be a H-field. The compensation works in this case and converts the extended model into a renormalizable string-local QFT [20] (or [21] in gauge theoretical setting in Krein space). It attributes a fundamental role to the H coupling which is consistent with the principles of QFT15 . We are now prepared to discuss the problem of consistent couplings of higher spin fields and in particular of WS fields.

4

Reactive and inert fields for s≥ 1

Perturbative renormalization theory is at best asymptotically convergent in the sense weak coupling limits. Its impressive quantitative success in the description of electro-weak interactions and its consistency in accounting for hadronic collisions adds to its credibility. The new string-local Hilbert space setting for interactions involving s ≥ 1 fields strengthens the confidence that perturbative QFT is a consistent way of finding out whether interactions with a concrete prescribed field content are possible. The conceptual reasons for this optimism is based on two observations 14 The full second order L, Q calculations have not yet been published but the formalism is similar to that in the CGI setting of gauge theory [21] [22]. 15 Note that d sd = 5 contributions which have to be compensated do not occur in SSB models.

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• Any QFT with local observables and a mass gap is generated by stringlocal fields of which point-local fields constitute a special case. In other words to generate a QFT with s ≥ 1 one does not need field operators with weaker than string-like localization (e.g. field localized on spacelike hypersurfaces). This is the consequence of a theorem in algebraic QFT [18] which supports the naturalness of string-localization; but it does not answer the question which models need this extension to string-localization. • Whereas massive point-local bosonic or fermionic spin s fields have minimal short distance dimensions dsd = s + 1, their string-local siblings have an s-independent dsd = 1 which permits to define string-local interaction densities L within the power counting bound dsd ≤ 4 for each s. But this fulfillment of the first order pcb is not sufficient. In contrast to point-local s < 1 interactions string-local L will generically delocalize in higher orders. The heuristic picture is that the dependence of inner propagators on the individual fluctuating e′ s will not compensate by adding up all contributions in a fixed order and hence neither the requirement of the e-independence of S nor that of vacuum expectation values of fields from inner e′ s can be satisfied. Such an L cannot be an interaction density of a QFT. The necessary condition to prevent a total delocalization turns out to be the condtion that L belongs to an L, V pair such that L − ∂V is a closed zero form in the d = 1 + 2 de Sitter space of spacelike directions. • Violations of the power-counting bound in higher order should cancel after extending the model by couplings to lower spin fields (eneralization of the phenomenon of Aµ selfinteractions) Previously such compensations between different spins have been welcomed as a fringe-benefit of supersymmetry, but in the situation of self-interacting Aµ they are the raison d’ˆetre for the H-particle. The systematic construction of higher spin interactions which fulfill these requirements is nontrivial and will not be attempted here. This suggest the following definition Definition 1 A field of spin s ≥ 1 is called ”inert” if (a) it is not possible to use it in first order string-local L, Vµ pair with dsd (L) ≤ 4 or (b) if the pair condition can be satisfied but dint sd > 4 terms in higher orders cannot be compensated by extension of the field content. Such fields exist only in the form of free fields; hence ”inert” is synonymous with ”non-reactive”. In the previous section it was shown that the Aµ in massive QED is reactive and above it was argued that massive A′ν s in self-interactions can be incorporated into an H-extended reactive model. An example for which the failure of the L, Q pair condition with a renormalizable L is the tri- or quadri-linear self-coupling of a string-local s = 2 massive tensorpotential gµν (x, e). But the problem of whether this condition can be fulfilled gµν couplings to lower spin fields can only be answered by additional calculations. In fact not having done such calculatations we do not know whether there are s > 1 analogs of the self-interacting s = 1 A, H model. 19

The question whether for s ≥ 2 there exist reactive fields requires detailed calculations and presently remains open, but what one can exclude is the reactiveness of W S fields. To show that (apart from gravitational manifestations of positive energy representations) they are inert, one must show that zero mass infinite spin fields form an isolated class which, unlike massless finite spin fields, cannot be approximated by massive representations. Infinite spin fields as (14) are string-local scalar fields which, unlike finite spin fields, cannot be approximated by a massive spin s tensor fields in the zero mass limit at fixed Pauli-Lubanski invariant κ2 = s(s + 1)m2 . Therefore they remain outside string-local formalism which extends the point-local s < 1 Hilbert space renormalization theory to positivity preserving s ≥ 1 interactions; there is simply no L involving a WS field whose one-form de L has the form of a divergence. The present formalism breaks down in the massless limit: the Wigner-Fock Hilbert space whose existence is guarantied by the validity of scattering theory in the presence of a mass gap disappears and the V and Q operators, which played an important role in securing the e-independence of the S-matrix, diverge as 1/m (as compared to the logarithmic infrared divergence of the S-matrix). The only object which remains infrared finite (at least in the absence of confinement) are the positivity respecting correlation functions of string-local fields. As shown by Wightman, the Hilbert space and the operators in it can be reconstructed from the correlation functions [23]. The absence of zero mass point-local potentials and the definition of their well-defined string-local counterpart in terms of the massless limit of the massive string-local localization is the only way to get to the massless string-local fields even in the case of free fields. At this point gauge theory fails completely; the point-local vector potentials and the point-local charge-carrying matter fields of gauge theory are pure physical fictions; physical electrons cannot be described by point-local fields [14]. Whereas gauge theory may correctly account for the short distance asymptotic freedom behavior, it becomes totally misleading at long distances (the confinement problem); the correct analogy of long-ranged (Coulomb) potentials are string-local matter fields resulting from interactions with string-local massless vector potentials. Whether a gauge theoretic description is possible for s > 1 tensor potentials remains open, but for a field without local observables as WS this is certainly impossible.

5

The problem of the s ≥ 2 E-M tensor, coupling to gravity

cl The classical energy-momentum tensor Tµν is a trace- and divergence-less quadratic expression in terms of classical fields which can conveniently be obtained within the Lagrangian formalism. For low spin s ≤ 1 fields the Lagrangian quantization leads to the same free fields and energy momentum tensor as that obtained in Wigner’s representation theoretic quantum setting.

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However for s ≥ 2 the quantum free fields start to differ from their classic counterparts; point-local tensor potentials and their fermionic counterparts are not solutions of Euler-Lagrange equation, and Euler-Lagrange fields are incompatible with positivity; their quantization requires to use indefinite metric Krein spaces. A point-local E-M tensor of integer spin s is expected to be of the form P λ1 ..λn P + gµν T P Tµν ≃ as ∂µ AP λ1 ..λn ∂ν A

(26)

Here AP is the point-local symmetric trace- and divergence-less free tensor field which appeared in (10 section 2), and as denotes antisymmetrization in µ and ν with the λ-indices of the symmetric tensors on which the ∂ ′ s act. All products are Wick-ordered and T P is chosen such that Tµν is trace-and divergence-free. For s = 1 the E-M tensor is the well-known expression 1 P ≃ Fµκ Fνκ − δµν Fκλ F κλ Tµν 4

(27)

P The problem with (26) for s ≥ 2 is that dsd (Tµν ) = 2+2s > 4 whereas a E-M tensor whose conserved charges can be related to the Poincar´e generators should have dsd = den = 4. If we could use the string-local tensor Tµν (x.e) this property is automatically fulfilled and the problem changes to that of e-independence of the charges. The relation (10) between the tensors AP ...(x) and their string-local siblings A...(x, e) involve the symmetrized derivatives of the string-local escort fields. The contributions in P Tµν = Tµν + ∂ κ Rµν,κ (x, e) (28)

where R involves at least one escort field, have the form of a divergence. The idea is that these e-dependent divergence terms do not contribute to the limit which defines the global charges, so that e-dependent conserved current of the Poincar´e charges obtained from the E-M tensor have the the desired property dsd (T ) = den (T ) = 4. The problem of whether conserved string-local currents associated to s > 1 fields can lead to e-independent charges is of great importance since without a positive answer the QFT of the textbooks concerning conservation laws would not permit an extension to higher spin nor could one use expectation values of E-M tensors of higher spin fields on the right hand side of the Einstein Hilbert equation. In order to relate the inert WS matter to the ”darkness” of dark matter one would have to show that WS permits a formulation in infinite extended models of curved spacetime. The problems of formulating point-local interacting QFT in curved spacetime has been solved in terms of concepts from algebraic QFT [24]. To extend these constructions to string-local fields which do not permit a Lagrangian formulation seems possible since already the cited pointlocal perturbative formalism does not use Lagrangian quantization but rather is based on interaction densities of free fields and their time ordered products.

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Calculations of backreaction of scalar point-local quantum matter on curved spacetime in Hadamar states have been carried out in [25]. It should be possible to extend such calculations to WS fields in curved spacetime background. To estimate the effect of WS on the galactic gravitational balance such calculation would be necessary. The problem whether the inertness of WS elevates it to a contender for dark matter will not be addressed in the present paper. Another problem which will remain open is the question of inertness of massive or massless finite higher spin matter.

6

Concluding remarks

The existence of forms of inert matter enriches the discussion about dark matter in an interesting way. The problem with proposing candidates as WIMPS, cold dark matter, axions,..is that there is no natural explanation how this galaxy penetrating stuff shields itself on the one hand against direct terrestrial or astrophysical observations while on the other hand it cintributes an enormous amount to the gravitational balance. The existence of stuff as WS which is inherentlly inert and, as a particular kind of positive energy matter, only reveals its presence through gravitational manifestations may be hard to digest from a philosophical point of view. We have gotten used to matter which only interacts weakly, but matter which exclusively couples to classical gravity appears only acceptable if its existence is in a deep way related to the unknown quantum aspects of gravity. As the present paper shows inertness may not be limited to infinite spin but could also occur for finite higher spin. This can only be settled by looking at the problem of implementation of the L, V renormalization requirement and verify that a compensation with lower spin interactions is not possible. There remain valid astrophysical objections and questions as: how could noncompact localized inert matter have gotten into our universe and what was its role in the formation of ordinary matter after the big bang? Here high temperature phase transitions may have played a role in converting a high temperature compact localized and more reactive phase of infinite spin matter into its present noncompact inert vacuum form. Unlike other dark matter candidates, WS has not been invented to explain dark matter16 . As a rather big Wigner representation class being characterized in terms of the continuous invariant κ, it made its debut already in Wigner’s 1939 paper which was written in the same decade in which Zwicky discovered dark matter from his analysis of galactic observations. What was missing for nearly seven decades was an understanding of its unexpected causal localization properties on which the present ideas in this paper are based. Any particle counter observation of a new form of matter for which there are good reasons to interpret it as a manifestation of the ubiquitous galactic 16 As indicated in the title, the paper is primarily meant as a presentation of new ideas in QFT; the only connection of WS with dark matter is its apparent astrophsical (and terrestrial) inertness.

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dark matter will eliminatete WS from the list of possible future dark matter candidates. But this would not diminish WS important role as a catalyzer of new ideas concerning the interplay between Hilbert space positivity, localization and short distance behavior for interactions involving higher spin s ≥ 1 fields. Acknowledgement: I thank Jakob Yngvason and Jens Mund for their continued interest in the issue of string-local fields associated to the third Wigner class whose construction was the result of a past joint collaboration [8]. I am indebted to Christian K¨ohler for sending me a copy of his thesis before publication. Last not least I thank Jos´e Gracia-Bondia for a critical reading of the manuscript.

References [1] E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40, (1939) [2] S. Weinberg, The Quantum Theory of Fields I, Cambridge University Press 1991 [3] H. Epstein and V. Glaser, Ann. Inst. Poincar´e A19, (1973) 211 [4] L. F. Abbott, Massless particles with continuous spin indices, Physical Review D, 13(8), 2291 (1976). [5] P. Schuster and N. Toro. A gauge field theory of continuous-spin particles, JHEP 20, (2013) 10 [6] J. Yngvason, Zero-mass infinite spin representations of the Poincar´e group and quantum field theory, Commun. Math. Phys. 18 (1970), 195 [7] R. Brunetti, D. Guido and R. Longo, Modular localization and Wigner particles, Rev. Math. Phys. 14, (2002) 759 [8] J. Mund, B. Schroer and J. Yngvason, Commun. Math. Phys. 268, (2006) 621 [9] J. Mund, String–localized quantum fields, modular localization, and gauge theories, New Trends in Mathematical Physics (V. Sidoravicius, ed.), Selected contributions of the XVth Int. Congress on Math. Physics, Springer, Dordrecht, 2009, pp. 495 [10] Ch. K¨ ohler, On Localization Properties of Quantum Fields with zero mass and infinite Spin, University of Vienna thesis 2015 [11] R. Longo, V. Morinelli and K-H. Rehren, Where Infinite Spin Particles Are Localizable, arXiv:1505.01759 [12] B. Schroer, Modular localization and the d=1+1 formfactor program, Ann. of Phys. 275 (1999) 190 23

[13] G. Lechner, Algebraic constructive quantum field theory: integrable models and deformation techniques, arXiv:1503.0322 [14] R. Haag, Local Quantum Physics, Springer Verlag Heidelberg 1996 [15] D. V. Ahluwalia, A Lorentz covariant local theory of fermions with mass dimension one, arXiv:1601.03188 [16] B. Schroer, Peculiarities of massive vectormesons, Eur. Phys. J. C (2015) 75:365 [17] B. Schroer, Is inert matter from indecomposable positive energy “infinite spin” representations the much sought-after dark matter?, arXiv:0802.2098v3, unpublished [18] D. Buchholz and K. Fredenhagen, Locality and the structure of particle states, Commun. Math. Phys. 84, (1982) 1 [19] J. Fr¨ohlich, G. Morchio and F. Strocchi, Phys. Lett. 89B, [20] B. Schroer, Beyond gauge theory: Hilbert space positivity and its connection with causal localization in the presence of vector mesons, to appear [21] G. Scharf, Quantum Gauge Theory, A True Ghost Story, John Wiley & Sons, Inc. New York 2001 [22] M. Duetsch, J. M. Gracia-Bondia, F. Scheck and J. Varilly, Quantum gauge models without classical Higgs mechanism, Eur. Phys. J. C 80, (2012) 599 [23] R. S. Streater and A. S. Wightman, PCT Spin and Statistics and all that, New York : Benjamin 1964 [24] R. Brunetti and K. Fredenhagen, Quantum Field Theory on Curved Backgrounds, arXiv:0901:2063 [25] D. Dappiagi, K. Fredenhagen and N. Pinamonti, Stable cosmological models driven by a free quantum scalar field, Phys. Rev. D 77, (2008) 194915

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