Lab Notes for a Scientific Revolution (Physics)

April 13, 2008

Lab Note 5: The Central Role in Physics, of the Dirac Anticommutator g^uv=(1/2){gamma^u,gamma^v}

Filed under: General Relativity,Gravitation,Physics,Science — Jay R. Yablon @ 11:38 pm

   I would like to take a break from my current work on Kaluza-Klein, and focus on the central importance to physics of the Dirac anticommutator relationship \eta ^{\mu \nu } \equiv {\tfrac{1}{2}} \left(\gamma ^{\mu } \gamma ^{\nu } +\gamma ^{\nu } \gamma ^{\mu } \right)\equiv {\tfrac{1}{2}} \left\{\gamma ^{\mu } ,\gamma ^{\nu } \right\}, when generalized to a non-zero gravitational field in the form g^{\mu \nu } \equiv{\tfrac{1}{2}} \left\{\Gamma ^{\mu } \Gamma ^{\nu } +\Gamma ^{\nu } \Gamma ^{\mu } \right\}\equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}.  In particular, when g^{\mu \nu } \ne \eta ^{\mu \nu } but rather include a gravitational field g^{\mu \nu } (x)=\eta ^{\mu \nu } +\kappa h^{\mu \nu } (x), then also the \Gamma ^{\mu } \ne \gamma ^{\mu } , but rather include a “square root” gravitational field h^{\mu } (x) which may be defined as \Gamma ^{\mu } (x)\equiv \gamma ^{\mu } +\kappa h^{\mu } (x).  Combining all the foregoing, this means that \kappa h^{\mu \nu } \equiv {\tfrac{1}{2}} \kappa \left[h^{\mu } \gamma ^{\nu } +\gamma ^{\mu } h^{\nu } +h^{\nu } \gamma ^{\mu } +\gamma ^{\nu } h^{\mu } \right]+{\tfrac{1}{2}} \kappa ^{2} \left[h^{\mu } h^{\nu } +h^{\nu } h^{\mu } \right].

   We also note that in perturbation theory, non-divergent perturbative effects are, in the end, captured in a correction to the vertex factor given by \overline{{\rm u}}(p)\gamma ^{\mu } {\rm u(p)}\to \overline{{\rm u}}(p)\left(\gamma ^{\mu } +\Lambda ^{\mu } \right){\rm u(p)} operating on a Dirac spinor {\rm u(p)}.  That is, the bare vertex \gamma ^{\mu } becomes the dressed vertex \gamma ^{\mu } +\Lambda ^{\mu } .  By then associating the perturbative \Lambda ^{\mu } with \kappa h^{\mu } just specified,  we raise the possibility that gravitational and perturbative descriptions of nature may in some way be interchangeable.  More to the point: when we consider perturbative effects in particle physics, we may well be considering gravitational effects without knowing that this is what we are doing.  The inestimable benefit of gravitational theory over  perturbation theory is that it is non-linear and exact.  The inestimable benefit of perturbation theory over gravitational theory is that we know something about how to achieve its renormalization.  Perhaps by developing this link further via the vitally-central physical relationship g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, we can infuse the exact, non-linear character of gravitational theory into perturbation theory, and the renormalizability of perturbation theory into gravitational theory.  Recognizing that “Lab Notes” is in the nature of a scientific diary, this, in any event, is the starting point for this lab note.

   Now, there are two main directions in which to exploit the connection g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, and both need to be considered.  First, we may start with the metric tensor g_{\mu \nu } from a known, exact solution to Einstein’s equation, calculate its associated \Gamma ^{\mu } , and then employ \Gamma ^{\mu } in the Dirac equation, in the form 0=\left(\Gamma ^{\mu } \left(i\partial _{\mu } +eA_{\mu } \right)-m\right)\psi .  Using the Schwarzschild solution as the basis, I have done this in detail, in a paper linked at Magnetic Moment Anomalies of the Charged Leptons.  If you would like an “Executive Summary” of this paper, you may obtain this at What the Magnetic Moment Anomaly May Tell Us About Planck-Scale Physics.  What is especially noteworthy, is that the magnetic moment anomaly can perhaps be understood as a symptom of gravitational effects near the Planck scale.

   I actually wrote the above detailed paper in September, 2006, but never posted it anywhere, because as soon as it was written, I went off into writing the related ArXiV paper at titled Ward-Takahashi Identities, Magnetic Anomalies, and the Anticommutation Properties of the Fermion-Boson Vertex.  This paper illustrates the second direction in which to exploit the connection g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}.  Here, we start with a known \Gamma ^{\mu } , then use g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}  to obtain the associated g^{\mu \nu } , and then use this as a metric tensor in the usual way.  In this paper, in particular, we start with the perturbative vertex factor \Gamma ^{\mu } \equiv \gamma ^{\mu } +\Lambda ^{\mu } =F_{1} \gamma ^{\mu } +{\tfrac{1}{2}} F_{2} i\sigma ^{\mu \nu } (p'-p)_{\nu } from equation (11.3.29) of Weinberg’s definitive treatise The Quantum Theory of Fields, then we calculate the anticommutators g^{\mu \nu } \equiv {\tfrac{1}{2}} \left\{\Gamma ^{\mu } ,\Gamma ^{\nu } \right\}, and then we use the g^{\mu \nu } as a metric tensor.  What is fascinating about this approach, is that the \Gamma ^{\mu } (p_{\mu } ) are specified in momentum space, rather than spacetime.  This means that g^{\mu \nu } =g^{\mu \nu } (p_{\mu } ) deduced therefrom define a non-Euclidean momentum space, rather than a non-Euclidean spacetime.  This may open up a whole new branch of physics dealing with — I’ll say it again — Non-Euclidean Momentum Space.  As we know from Heisenberg, spacetime is conjugate to momentum space, \left[x^{\mu } ,p^{\nu } \right]=i\hbar \delta ^{i\mu \nu } .  The central result of this paper, arrived at via the Ward-Takahashi Identities which are central to renormalization, is that interaction vertexes are a measure of curvature in momentum space, and the strength of the interaction at a vertex is proportional to the momentum space curvature, see Figures 1 and 2.  This may place particle physics onto a firm geometric footing, but rooted in the geometry of momentum space.

   What I have not yet gotten to, is the question of how to use \left[x^{\mu } ,p^{\nu } \right]=i\hbar \delta ^{i\mu \nu } , as between a non-Eclidean spacetime and a non-Euclidean momentum space,  because in the real world, we have both.  That is a project for the remaining free time in my day, between 3AM and 5AM.  😉

   As always, these are lab notes, representing “work in progress.”  I welcome comments and contributions, as always.



1 Comment »

  1. Helo,

    I do not think that the initiative which you have is unique it would be good to connect you on the site of the forum of physics and to consult the work of one of the comedians on a work similar to yours.
    I also invite you to consult the site of A C E on a new physics the NAP, and to read the article on the gravitation:
    ” NAP applied to gravitation and the implications for Einstein’s theory of special and general relativity.”
    The reading of this article shows the possibility of following another way of calculation of the gravitational phénoméne and electromagnetique.

    Comment by A C ELBEZE — December 22, 2008 @ 1:14 pm | Reply

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