String field theory and non-abelian tensor gauge fields

1、String Field TheoryNon-Abelian Tensor Gauge Fields and Possible Extension of SM George Savvidy Demokritos National Research Center Athens Phys. Lett. B625 (2005) 341. A21 (2006) 4959 . A21 (2006) 4931Fortschr. Phys. 54 (2006) 472 Prog. Theor. Phys.117 (2007) 729 - Takuya TsukiokaHep-th/0604118 Hep-t

2、h/ 0704.3164 - Jessica BarrettHep-th/ 0706.0762 - Spyros Konitopoulos Patras 2007String Field TheoryExtended Non-Abelian gauge transformationsField strength tensors Extended current algebra as a gauge groupInvariant Lagrangian and interaction verticesPropagating modes Higher-spin extension of the St

3、andard ModelString Field The multiplicity of tensor fields in string theory grows exponentially Lagrangian and field equations for these tensor fields ? Search for the unbroken phase ?Wittens generalization of gauge theoriesField strength tensor isand transforms “homogeneously” under gauge transform

4、ations for any parameter of degree zero.The gauge invariant Lagrangian is topological invariant - the star product is similar to the wedge product !Open string field takes values in non-commutative associative algebra The gauge transformations are defined as:There is no analogue of the usual Yang-Mi

5、lls action, as there is no analogue of raising and lowering indices within the axioms of this algebra. The other possibility is the integral of the Chern-Simons formwhich is invariant under infinitesimal gauge transformations L. P. S. Singh and C. R. Hagen. Lagrangian formulation for arbitrary spin.

6、 I. The boson case. Phys. Rev. D9 (1974) 898L. P. S. Singh and C. R. Hagen. Lagrangian formulation for arbitrary spin. II. The fermion case. Phys. Rev. D9 (1974) 898, 9103. C.Fronsdal. Massless fields with integer spin, . D18 (1978) 36244. J.Fang and C.Fronsdal. Massless fields with half-integral sp

7、in, Phys. Rev. D18 (1978) 3630. Particles, Sourses, and Fields (Addison-Wesley, Reading, MA, 1970)and the corresponding equations describe massless particles of helicity The Lagrangian and equations are invariant with respect to the gauge transformation:Free field Lagrangian Free field theories exhi

8、bit reach symmetries. Which one of them can be elevated to the level of symmetries of interacting field theory?In our approach the gauge fields are defined as rank-(s+1) tensorsand are totally symmetric with respect to the indicesA priory the tensor fields have no symmetries with respect to the inde

9、x the Yang-Mills field with 4 space-time componentsthe non-symmetric tensor gauge field with 4x4=16 space-time componentsthe non-symmetric tensor gauge field with 4x10=40 space-time componentsThe extended non-Abelian gauge transformation of the tensor gauge fields weshall define by the following equ

10、ations:The infinitesimal gauge parameters are totally symmetric rank-s tensorsAll tensor gauge bosons carry the same charges as ,there are no traceless conditions on the gauge fields. In general case we shall get and is again an extended gauge transformation with gauge parameters Gauge AlgebraExtend

11、ed gauge algebra Difference with K-K spectrumThe field strength tensors we shall define as:The inhomogeneous extended gauge transformation induces the homogeneous gauge transformation of the corresponding field strength tensorsYang-Mills Fields First rank gauge fieldsIt is invariant with respect to

12、the non-Abelian gauge transformationThe homogeneous transformation of the field strength iswhereThe invariance of the Lagrangian Its variation is The first three terms of the Lagrangian are:The Lagrangian for the rank-s gauge fields is (s=0,1,2,) and the coefficient is The gauge variation of the Lag

13、rangian is zero: The Lagrangian is a linear sum of all invariant forms It is important that: Every term in the sum is fully gauge invariant Coupling constants g_s remain undefined Lagrangian does not contain higher derivatives of tensor gauge fields All interactions take place through the three- and

14、 four-particle exchanges with dimensionless coupling constant g The Lagrangian contains all higher rank tensor gauge fields and should not be truncated It is invariant with respect to gauge transformationEquation of motion isThe Free Field EquationsFor symmetric tensor fields the equation reduces to

15、 Einstein equationfor antisymmetric tensor fields it reduces to the Kalb-Ramond equation In momentum representation the equation has the form:where 16x16 matrix has the formThe rank of this matrix depends on momentumWithin the 16 fields of non-symmetric tensor gauge field of the rank-2 only three po

16、sitive norm polarizations are propagating and the rest of them are pure gauge fields.On the non-interacting level, when we consider only the kinetic term of the full Lagrangian, these polarizations are similar to the polarizations of the graviton and of the Abelian anti-symmetric B field.But the int

17、eraction of these gauge bosons carrying non-commutative internal charges is uniquely defined by the full Lagrangian and cannot bedirectly identified with the interactions of gravitons or B field.Interaction VerticesThe VVV vertexThe VTT vertexInteraction VerticesThe VVVV and VVTT verticesHigher-Spin Extension of the Standard Model Standard Mod