Consequences of Godel Theorems on Third Quantized Theories Like String Field Theory and Group Field Theory

TL;DR

This paper explores how G"odel's theorems impact the consistency and completeness of third quantized theories like string and group field theory, framing them as formal axiomatic systems.

Contribution

It formulates axioms for third quantized theories and analyzes the implications of G"odel's theorems on their logical consistency and completeness.

Findings

Third quantized theories can be structured as formal axiomatic systems.

G"odel's theorems imply inherent limitations in the consistency and completeness of these theories.

The analysis highlights fundamental logical constraints in quantum gravity models.

Abstract

The observation that spacetime and quantum fields on it have to be dynamically produced in any theory of quantum gravity implies that quantum gravity should be defined on the configuration space of fields rather than spacetime. Such a theory is described on the configuration space of fields rather than spacetime, which is a third quantized theory. So, both string theory and group field theory are third-quantized theories. Thus, using axioms of string field theory, we motivate similar axioms for group field theory. Then, using the structure of these axioms for string field theory and group field theory, we identify general features of axioms for any such third quantized theory of quantum gravity. Thus, we show that such third-quantized theories of quantum gravity can be formulated as formal axiomatic systems. We then analyze the consequences of G\"odel theorems on such third quantized theories. We thus address problems of consistency and completeness of any third quantized theories of quantum gravity.