Towards a quantum field theory of primitive string fields
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ELEMENTARY PARTICLES AND FIELDS Theory
Towards a Quantum Field Theory of Primitive String Fields* ¨ ** W. Ruhl Department of Physics, Technical University of Kaiserslautern, Germany Received May 5, 2011
Abstract—We denote generating functions of massless even higher-spin fields “primitive string fields” (PSF’s). In an introduction we present the necessary definitions and derive propagators and currents of these PDF’s on flat space. Their off-shell cubic interaction can be derived after all off-shell cubic interactions of triplets of higher-spin fields have become known. Then we discuss four-point functions of any quartet of PSF’s. In subsequent sections we exploit the fact that higher-spin field theories in AdSd+1 are determined by AdS/CFT correspondence from universality classes of critical systems in d-dimensional flat spaces. The O(N ) invariant sectors of the O(N ) vector models for 1 ≤ N ≤ ∞ play for us the role of “standard models”, for varying N , they contain, e.g., the Ising model for N = 1 and the spherical model for N = ∞. A formula for the masses squared that break gauge symmetry for these O(N ) classes is presented for d = 3. For the PSF on AdS space it is shown that it can be derived by lifting the PSF on flat space by a simple kernel which contains the sum over all spins. Finally we use an algorithm to derive all symmetric tensor higher-spin fields. They arise from monomials of scalar fields by derivation and selection of conformal (quasiprimary) fields. Typically one monomial produces a multiplet of spin s conformal higher-spin fields for all s ≥ 4, they are distinguished by their anomalous dimensions (in CF T3 ) or by their mass (in AdS4 ). We sum over these multiplets and the spins to obtain “string type fields”, one for each such monomial. DOI: 10.1134/S106377881210016X
Da = (∂a · ∇) − 1/2(a · ∇)a (de Donder operator).
1. FROM HIGHER-SPIN GAUGE FIELDS TO PRIMITIVE STRING FIELDS We define higher-spin gauge fields h(s) (z; a)
(1)
in D-dimensional flat space z ∈ RD , a ∈ TD (z) satisfying the constraints (valid also off shell) (a∂a )h(s) (z; a) = s · h(s) (z; a),
(2)
2a h(s) (z; a) = 0
(3)
A very practical constraint is to impose de Donder gauge Da h(s) (z; a) = 0
(4)
Derivatives on z are denoted ∇ and on a by ∂a . The trace of the gauge function (s−1) vanishes a (s−1) = 0.
By Fourier transformation we obtain the operators
Fa h(s) (z; a) = 0, Fa = − (a · ∇)Da ,
= ∇ · ∇,
Dˆa = (p∂a ) − 1/2(ap)a ,
(10)
ˆ a. Fˆa = p2 − (ap)D
(11)
Therefore the propagator in momentum space and de Donder gauge is G(s) (p; a, b) =
(5)
A free massless higher-spin field satisfies Fronsdal’s equation
(9)
off shell. It corresponds to Feynman gauge in QED.
whose infinitesimal gauge transformations are δ h(s) (z; a) = s · (a∇)(s−1) (z; a).
(8)
1 (s) A (a, b), p2
(12)
where A(s) (a, b) is double traceless 2a A(s) (a, b) = 2b A(s) (a, b) = 0.
(6)
(13)
This implies (7)
∗
The text was submitted by the author in English. ** E-mail: [email protected]
(s)
A(s) (a, b) = A1 χλs (a, b) (s)
+ A2 a2 b2 χλs−2 (
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