For those finite-matrix representations of the Lorentz group of rotations/boosts with spin $(A,B)\oplus(C,D)$ that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincare group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin $(A,B)\oplus(C,D).$

The Lorentz Group with Dual-Translations and the Conformal Group

In our previous work [J. Math. Phys. 49, 033513 (2008)] two problems remain to be resolved. One is that we lack a minimal group to replace GL(4,C), the other is that the Equation of Motion (EoM) for fermion has no mass term. After careful investigation we find these two problems are linked by conformal group, a subgroup of GL(4,C) group. The Weyl group, a subgroup of conformal group, can bring about the running of mass, charge etc. while making it responsible for the transformation of interaction vertex. However, once concerning the generation of the mass term in EoM, we have to resort to the whole conformal group, in which the generators $K_\mu $ play a crucial role in making vacuum vary from space-like (or light-cone-like)to time-like. Physically the starting points are our previous conclusion, $\vec E^2-\vec B^2\neq 0$ for massive bosons, and the two-photon process yielding $e^+ e^-$ pair. Finally we get to the conclusion that the mass term of strong interaction is linearly relevant to (chromo-)magnetic flux as well as angular momentum.

Structure Group and Fermion-Mass-Term in General Nonlocality

In the light of their relationships with renormalization, in this paper we associate the scaling transformation with nonlocal interactions. On one hand, the association leads us to interpret the nonlocality with locally symmetric method. On the other hand, we find that the nonlocal interaction between hadrons could be test ground for scaling transformation if ascribing the running effects in renormalization to scaling transformation. The nonlocal interaction Lagrangian turns out to vary under scaling transformation, analogous to running cases in renormalization. And the total Lagrangian becomes scale invariant only under some extreme conditions. The conservation law of this extreme Lagrangian is discussed and a contribution named scalum appears to the spin angular momentum. Finally a mechanism is designed to test the scaling effect on nonlocal interaction.

/pdf/scaling-transformation-for-nonlocal-interactions-2tvr5ktkus.pdf

Scaling Transformation for Nonlocal Interactions

Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method (RGM). We start with the renormalization group equation (RGE), in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invariant under certain transformations (generators) of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

New method of applying conformal group to quantum fields

In our previous work (Wang J. Math. Phys. 49, 033513 (2008)) two problems remain to be resolved. One is that we lack a minimal group to replace GL(4,C), the other is that the Equation of Motion (EoM) for fermion has no mass term. After careful investigation we find these two problems are linked by conformal group, a subgroup of GL(4,C). The Weyl group, a subgroup of conformal group, can bring about the running of mass, charge etc. while making it responsible for the transformation of interaction vertex. However, once concerning the generation of the mass term in EoM, we have to resort to the whole conformal group, in which the generators K
 μ
 play a crucial role in making vacuum vary from space-like (or light-cone-like)to time-like. Physically the starting points are our previous conclusion, $\vec E^{2}-\vec B^{2}\neq 0$
 for massive bosons, and the two-photon process yielding e
 +
 e
 − pair. Finally we get to the conclusion that the mass term of strong interaction is linearly relevant to (chromo-)magnetic flux as well as angular momentum.

Pure spinor formalism and non-integrable exponential factors are used for constructing the conformal-invariant wave equation and Lagrangian density for massive fermion. It is proved that canonical Dirac Lagrangian for massive fermion is invariant under induced projective conformal transformations.

Pure Spinor Formalism for Conformal Fermion and Conserved Currents

pdf/pure-spinor-formalism-for-conformal-fermion-and-conserved-27i7xstcc5.pdf

Pure spinor formalism for conformal fermion and conserved currents

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Pure Spinor Formalism for Conformal Fermion and Conserved Currents

Pure spinor formalism for conformal fermion and conserved currents