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Advances in Theoretical and Mathematical Physics

A New Approach to Quantising Systems: II. Quantising on a Category of Sets

Christopher John Isham

Published Jan 1, 2003• 19 citations
Publisher

Abstract

In [1], a new approach was suggested for finding quantum structures that might, in particular, be used in potential approaches to quantum gravity that involve non-manifold models for space and/or space-time (for example, causal sets).This involved developing a procedure for quantising a system whose configuration space-or historytheory analogue-is the set of objects in a (small) category Q.In the present paper, we show how this theory can be applied to the special case when Q is a category of sets.

DOI
10.4310/atmp.2003.v7.n5.a3

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In [1], a new approach was suggested for finding quantum structures that might, in particular, be used in potential approaches to quantum gravity that involve non-manifold models for space and/or space-time (for example, causal sets).This involved developing a procedure for quantising a system whose configuration space-or historytheory analogue-is the set of objects in a (small) category Q.In the present paper, we show how this theory can be applied to the special case when Q is a category of sets.
The Causal Set hypothesis for Quantum Gravity asserts that the smooth Lorentzian spacetime manifold of General Relativity is only an approximation to a fundamental microscopic discrete structure: a locally-finite partially-ordered set. The points of this set are akin to the point-events of the spacetime manifold, and the order-relations among these points are akin to the causal-relations among these point-events. The local-finiteness condition implies a fundamental discreteness and provides an important [conformal] factor needed for the continuum approximations. In this dissertation, we address two aspects toward the formulation of a quantum dynamics for causal sets. The first aspect concerns the dynamics of a zero-mass classical scalar field on a background causal set, Poisson-sampled from an Alexandrov interval in (1 + 1)-dimensional Minkowski spacetime. Our numerical simulations suggest that actions can be defined for such a scalar field, expressed solely in terms of the causal set and the scalar field, which approximates the corresponding classical action on the continuum Minkowski spacetime. This result may help us describe matter and gauge fields in a local and Lorentz-invariant way on a causal set, a discrete structure now being considered as the arena for spacetime physics. More importantly, it may help us formulate an action for the causal set itself, which would be used in a sum-over-histories approach to describe the quantum dynamics of causal sets. The second aspect concerns Quantum Measure Theory, a new approach to quantum dynamics inspired by the sum-over-histories approach, which is better suited to formulating the dynamics of causal sets and of other alternative structures for spacetime. To better understand the mathematical structure underlying this new approach, we derive some algebraic identities involving the sum-rules of the Quantum Measure and its generalizations. We conclude with a discussion on possible applications and open problems concerning these aspects.
Causal set theory starts out from a central result in general relativity according to which the causal structure determines, up to a so-called ‘conformal factor’, the geometry of a relativistic spacetime. Space and time, it seems, must be part and parcel of the ontology of any physical theory, of any theory with a credible claim to being a physical theory, that is. After all, physics is the science of the fundamental constitution of the material bodies, their motion in space and time, and indeed of space and time themselves. There are two areas in physics in which space or spacetime may emerge from something qualitatively rather different: wave function monism in quantum mechanics and quantum theories of gravity. The former case concerns a particular interpretative issue in non-relativistic quantum mechanics. The severity and the characteristics of the disappearance of spacetime differ from one approach to quantum gravity to another.
We develop a nonlinear quantum theory of Newtonian gravity consistent with an objective interpretation of the wavefunction. Inspired by the ideas of Schrodinger, and Bell, we seek a dimensional reduction procedure to map complex wavefunctions in configuration space onto a family of observable fields in space-time. Consideration of quasi-classical conservation laws selects the reduced one-body quantities as the basis for an explicit quasi-classical coarse-graining. These we interpret as describing the objective reality of the laboratory. Thereafter, we examine what may stand in the role of the usual Copenhagen observer to localise this quantity against macroscopic dispersion. Only a tiny change is needed, via a generically attractive self-potential. A nonlinear treatment of gravitational self-energy is thus advanced. This term sets a scale for all wavepackets. The Newtonian cosmology is thus closed, without need of an external observer. Finally, the concept of quantisation is re-interpreted as a nonlinear eigenvalue problem. To illustrate, we exhibit an elementary family of gravitationally self-bound solitary waves. Contrasting this theory with its canonically quantised analogue, we find that the given interpretation is empirically distinguishable, in principle. This result encourages deeper study of nonlinear field theories as a testable alternative to canonically quantised gravity.
This paper presents a novel framework for examining causality through a fundamentally non-temporal perspective, grounded in mathematical logic and category theory. We demonstrate that standard logical operations, including implication, conjunction, disjunction, and quantification, are inherently non-temporal, suggesting that causality itself might be better understood as a logical rather than temporal relationship. Our framework provides: (1) A rigorous mathematical foundation for non-temporal causality, based on set theory and mathematical logic; (2) A reinterpretation of quantum mechanical phenomena, including wave function collapse and entanglement, as manifestations of non-temporal logical relations; (3) A new perspective on causal set theory that challenges the assumed primacy of temporal ordering in discrete spacetime approaches. The framework naturally addresses several persistent puzzles in physics, including the measurement problem in quantum mechanics, the emergence of time in quantum gravity, and the apparent tension between locality and quantum correlations. We develop formal proofs showing how temporal causation might emerge as a special case of more general non-temporal logical relations, with implications for quantum foundations, relativity theory, and the unification of quantum mechanics with gravity. This approach suggests new directions for understanding the foundations of physics, the nature of time, and the relationship between logic and physical reality.
This paper reviews the histories approach to quantum mechanics. This discussion is then applied to theories of quantum gravity. It is argued that some of the quantum histories must approximate (in a suitable sense) to classical histories, if the correct classical regime is to be recovered. This observation has significance for the formulation of new theories (such as quantum gravity theories) as it puts a constraint on the kinematics, if the quantum/classical correspondence principle is to be preserved. Consequences for quantum gravity, particularly for Lorentz symmetry and the idea of “emergent geometry”, are discussed. 1