M ay 2 01 8 Quantum-first gravity

This paper elaborates on an intrinsically quantum approach to gravity, which begins with a general framework for quantum mechanics and then seeks to identify additional mathematical structure on Hilbert space that is responsible for gravity and other phenomena. A key principle in this approach is that of correspondence: this structure should reproduce spacetime, general relativity, and quantum field theory in a limit of weak gravitational fields. A central question is that of “Einstein separability,” and asks how to define mutually independent subsystems, e.g. through localization. Familiar definitions involving tensor products or operator subalgebras do not clearly accomplish this in gravity, as is seen in the correspondence limit. Instead, gravitational behavior, particularly gauge invariance, suggests a network of Hilbert subspaces related via inclusion maps, contrasting with other approaches based on tensor-factorized Hilbert spaces. Any such localization structure is also expected to place strong constraints on evolution, which are also supplemented by the constraint of unitarity. ∗ Email address: giddings@ucsb.edu 1. The quantum mechanics first approach A profound challenge for modern fundamental physics is to find a theory of gravity consistent with quantum reality. Various attempts to quantize gravity have met with serious difficulties. But, an alternate approach is to begin with the goal in mind and consider what structure an intrinsically quantum-mechanical theory should have in order to describe gravity. Indeed, the rigidity of quantum mechanics suggests that the need for a theory to fit within quantum mechanics could be tightly constraining, and thus provide a new direction which is supplemented by other important clues that we already have about the quantum nature of gravity. Such a “quantum-first” approach to gravity has been advocated in [1-3], and also recently by Carroll and collaborators[4-6], and will be elaborated further here. In contrast, previous approaches have started with a classical theory, such as general relativity (GR), and attempted to apply a set of rules to quantize the theory. These approaches have encountered vexing problems: at first that of non-renormalizability, but perhaps more profoundly, also the problem of respecting the basic quantum principle of unitarity, when one accounts for black hole formation and decay. Similar comments can be made regarding attempts to quantize strings; this seems to avoid non-renormalizability but still has not resolved the “unitarity crisis” associated with black holes. The troubles with attempts to quantize gravity suggest taking a different tack – that of “geometrizing” quantum mechanics. While it is certainly conceivable that we would have to alter quantum mechanics to describe gravity, a worthy and well-motivated goal is thus to investigate whether gravity can be described in an inherently quantum-mechanical framework. (If that is not possible, this is also critical information.) A starting point here is a suitably general framework for quantum mechanics. For example, ref. [9-12] proposed a “generalized quantum mechanics,” but this is still too closely tied to the notion of quantizing a classical theory and is not sufficiently general. This motivated a proposal[1] for the essential postulates of quantum mechanics – those of “universal quantum mechanics” (UQM). In brief, as outlined in [1], these are the existence of a linear space of states with an inner product (“Hilbert space”), and the existence of linear hermitian operators that are interpreted as providing quantum observables; the postulates include unitarity, e.g. the S-matrix, in appropriate 1 For an earlier but somewhat different approach to describing quantum structure for gravity, see [7,8].