Abstract
Ultrahyperfunctions (UHF) are the generalization and extension to the complex plane of Schwartz’ tempered distributions. This effort is an application to Einstein’s gravity (EG) of the mathematical theory of convolution of Ultrahyperfunctions developed by Bollini et al. [1] [2] [3] [4]. A simplified version of these results was given in [5] and, based on them; a Quantum Field Theory (QFT) of EG [6] was obtained. Any kind of infinities is avoided by recourse to UHF. We will quantize EG by appealing to the most general quantization approach, the Schwinger-Feynman variational principle, which is more appropriate and rigorous that the popular functional integral method (FIM). FIM is not applicable here because our Lagrangian contains derivative couplings. We follow works by Suraj N. Gupta and Richard P. Feynman so as to undertake the construction of an EG-QFT. We explicitly use the Einstein Lagrangian as elaborated by Gupta [7], but choose a new constraint for the ensuing theory. In this way, we avoid the problem of lack of unitarity for the S matrix that afflicts the procedures of Gupta and Feynman. Simultaneously, we significantly simplify the handling of constraints, which eliminates the need to appeal to ghosts for guarantying unitarity of the theory. Our approach is obviously non-renormalizable. However, this inconvenience can be overcome by appealing to the mathematical theory developed by Bollini et al. [1] [2] [3] [4] [5]. Such developments were founded in the works of Alexander Grothendieck [8] and in the theory of Ultradistributions of Jose Sebastiao e Silva [9] (also known as Ultrahyperfunctions). Based on these works, an edifice has been constructed along two decades that are able to quantize non-renormalizable Field Theories (FT). Here we specialize this mathematical theory to discuss EG-QFT. Because we are using a Gupta-Feynman inspired EG Lagrangian, we are able to evade the intricacies of Yang-Mills theories.
Highlights
Quantifying Einstein gravity (EG) remains an open question, a kind of supreme desideratum for quantum field theory (QFT)
The failure of some attempts in this direction is due to the fact that 1) they appeal to Rigged Hilber Space (RHS) with undefined metric, 2) problems of non-unitarity, and 3) non-renormalizablity issues
We deviate from his work by using a different Einstein’s gravity (EG)-constraint, facing a problem similar to that posed by Quantum Electrodynamics (QED)
Summary
Quantifying Einstein gravity (EG) remains an open question, a kind of supreme desideratum for quantum field theory (QFT). We face a theory similar to QED, but endow with unitarity at all finite orders in power expansions in G (gravitation constant) of the EG Lagrangian This was attempted without success first by Gupta and by Feynman, in his Acta Physica Polonica work [10]. Quantifying a non-renormalizable field theory is tantamount to suitably defining the product of two distributions (a product in a ring with zero-divisors in configuration space), an old problem in functional theory tackled successfully in [1] [2] [3] [4] [5]. The independent term in the Laurent expansion yield the convolution value This translates to configuration space the product-operation in a ring with divisors of zero.
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