Abstract
This chapter applies the foundational material from previous chapters to construct a flexible new version of discrete quantum causal theory, which may be applied either in the background dependent context of particles and fields existing on a fixed directed set, or in the background independent context of quantum spacetime structure. Section 6.1 reviews basic quantum theoretic notions, such as the superposition principle and the path summation approach to quantum theory, and discusses some modern contributions to background independent quantum theory by Isham and Sorkin. Section 6.2 introduces the theory of path summation over a multidirected set, which is analogous to path integral methods in conventional quantum field theory. Section 6.3 describes a striking structural feature of discrete causal theory, called iteration of structure, in which the same abstract type of structure, expressed in terms of directed relationships, exists at both the classical and quantum levels of the theory. At the classical level, these directed relationships are represented by relations between pairs of events, while at the quantum level, they are represented by relationships between pairs of classical histories, called co-relative histories. Section 6.4 examines the theory of co-relative histories in more detail, and explains why multidirected structure arises at the quantum level. This represents a “higher-level application” of Grothendieck’s relative viewpoint, in which the “objects” are directed sets, and the “natural relationships” are co-relative histories. Section 6.5 explains how the abstract theory of path summation over a multidirected set may be used to define quantum theories in a variety of different ways. Section 6.6 revisits the construction of Feynman’s path integral, in order to motivate the discrete causal constructions examined in the remainder of the chapter. Section 6.7 adapts relevant notions from conventional path integration to construct a path summation version of discrete quantum causal theory in relation space. Section 6.8 reviews how Feynman re-derived Schrödinger’s equation, in order to motivate the derivation of analogous equations in the discrete causal context. Section 6.9 presents the derivation of causal Schrödinger-type equations, which serve as dynamical laws for discrete quantum causal theory. Section 6.10 describes how generalized quantum amplitudes may be computed via causal path algebras.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In particular, the hyperlink for iteration of structure (IS) in the electronic version of the book links to a formalization of this special case, even though the underlying idea is more general.
- 2.
A recent article of Bassi, Singh, and Ulbricht [BSU15] describes some nonlinear generalizations of quantum theory.
- 3.
Isham mentions Raptis’ incidence algebras in the preprint version of [IS05].
- 4.
See pp. 293–296 of [IS05].
- 5.
- 6.
Actually, the mathscript symbol \(\mathcal {L}\), which I use for the Lagrangian, is often used to denote a Lagrangian density in the context of field theory. In this book, however, mathscript symbols generally appear at the “category-functor level of hierarchy,” which is appropriate for the Lagrangian. In any case, densities are generally irrelevant in the discrete context.
- 7.
As mentioned at the end of Section 5.9, the notion of directed product extends to settings in which the target object is not a discrete multidirected set. Here, it means essentially the same thing as in Definition 5.9.7: a pair of paths in a real manifold has a directed product “joining the two paths together” if and only if “the terminal element of the first path coincides with the initial element of the second path.” This product is familiar in homotopy theory, where it descends to multiplication in the fundamental groupoid.
- 8.
This may be formalized via the theory of transitions, developed in Section 6.3, since the image of \(\gamma \) embeds as a full originary subobject of the image of \(\delta \).
- 9.
As stated in the first footnote in Section 2.3, the word “curve” means simply “locus of points” in this setting; it does not mean a map into a manifold.
- 10.
As mentioned in the last footnote in Section 2.9, loop quantum gravity goes some way toward achieving a “common language” in this context.
- 11.
See Anderson [AN09] for details.
- 12.
For example, consider the automorphism \(\beta \) of \(D_4\) in Figure 6.3.6 interchanging \(u_4,w_4\), and \(x_4\) with \(v_4, y_4,\) and \(z_4,\) respectively. It is easy to see that \(\beta ^{-1}F_{\tau _3}\beta \ne F_{\tau _3}\).
- 13.
Via private communication.
- 14.
The theory of pseudosimilarity is of interest in the graph reconstruction problem. McKay tells me that little work has been done on pseudosimilarity in the case of directed graphs.
- 15.
The category-theoretic meaning of the term covariant has only loose relationships with the various physics-related meanings of this term. Unfortunately, both of the terms covariant and contravariant carry many different and mutually contradictory meanings in both mathematics and physics.
- 16.
In particular, one is generally interested in how a given initial history \(D_i\) may evolve into a given terminal history \(D_t\), and this leads to the consideration of evolutionary pathways. Also, S-matrix-like approaches require compatibility conditions for consistency purposes, and such conditions are easier to formulate using the path summation approach.
- 17.
In fact, several different accessibility conditions are worth considering. These details are also discussed in Section 7.4.
- 18.
In the paper On the Axioms of Causal Set Theory [DR13], which provides much of the background for this book, I denote Euclidean spacetime by \(\mathbb {R}^{3+1}\), to highlight the distinguished time direction. Here, of course, I use \(\mathbb {R}^{3+1}\) to denote Minkowski spacetime. The latter convention is preferable, because there is nothing intrinsic to the geometry of Euclidean spacetime that distinguishes a particular time direction.
- 19.
Adapted from [FE48], p. 8.
- 20.
Adapted from [FE48], p. 9.
- 21.
For example, the kinematic schemes describing generational growth of finite acyclic directed sets in Section 7.8 are not star finite, since each generation may be of arbitrary finite size.
- 22.
From the algebraic viewpoint, the concatenation product of any pair of elements of \(\varGamma \) is zero, since these elements are maximal chains. Similarly, from the order-theoretic viewpoint, no element of \(\varGamma \) directly precedes any other element of \(\varGamma \) under the causal concatenation relation. Hence, it makes no difference whether \(\varGamma \) is called a “subset,” a “subsemicategory,” a “subobject,” or a “subspace” in this context. I choose the first option.
- 23.
Of course, \(M'\) itself may be regarded as a closer analogue of \(\mathbb {R}^4\) in Feynman’s setup, in the sense that “it is an element space, instead of a relation space.” In the cases of principal interest, however, \(M'\) represents the abstract structure of a kinematic scheme, so neither \(M'\) nor \(\mathcal {R}(M')\) are particularly close analogues of \(\mathbb {R}^4\). Ultimately, this is because path integration over subsets of \(\mathbb {R}^4\) yields only a very primitive theory.
- 24.
See, for example, [IS11], p. 6.
- 25.
- 26.
Again, this simplification does not apply to general causal paths in R.
- 27.
There is nothing to prevent consideration of such precursor functionals in continuum-based theories, but they are a priori cumbersome as formal sums, due to the large cardinalities of their sets of summands. However, they represent a radical method of treating certain divergence issues.
- 28.
It is not feasible to undertake here an explanation of the relationships between Lagrangian and Hamiltonian methods, or to describe the circumstances under which the two are equivalent. In the discrete causal context, the motivations for working with relation functions, and the manner in which they resemble Lagrangians, are both quite clear.
- 29.
The reason for using the notation \(J_0^-(R;r)\), rather than merely \(J_0^-(r)\), is to make it absolutely clear that only direct predecessors of r belonging to R are considered here. Other direct predecessors of r in the “ambient relation space” \(\mathcal {R}(M')\) are excluded.
References
Angelo Bassi, Tejinder Singh, and Hendrik Ulbricht. Is Quantum Linear Superposition an Exact Principle of Nature? Questioning the Foundations of Physics, The Frontiers Collection, Springer, pp. 131–138, 2015.
Christopher Isham. Quantising on a Category. Foundations of Physics, 35, 2, pp. 271–297, 2005. arXiv preprint: http://arxiv.org/pdf/quant-ph/0401175v1.pdf.
Rafael Sorkin. Toward a Fundamental Theorem of Quantal Measure Theory. Mathematical Structures in Computer Science, 22, 05 (special issue), pp. 816–852, 2012. arXiv preprint: http://arxiv.org/pdf/1104.0997v2.pdf.
Ioannis Raptis. Algebraic Quantization of Causal Sets. International Journal of Theoretical Physics, 39, 5, pp. 1233–1240, 2000. arXiv preprint: http://arxiv.org/pdf/gr-qc/9906103v1.pdf.
David Rideout and Rafael Sorkin. Classical sequential growth dynamics for causal sets. Physical Review D, 61, 2, 024002, 2000. arXiv preprint: http://arxiv.org/pdf/gr-qc/9904062v3.pdf.
Michael T. Anderson. A Survey of Einstein Metrics on 4 -Manifolds. Preprint, 2009. arXiv preprint: http://arxiv.org/pdf/0810.4830v2.pdf.
Edward A. Bender and Robert W. Robinson. The Asymptotic Number of Acyclic Digraphs II. Journal of Combinatorial Theory, Series B, 44, pp. 363–369, 1988.
Carlo Rovelli and Simone Speziale. Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction. Physical Review D, 67, 6, 064019, 2003. arXiv preprint: http://arxiv.org/pdf/gr-qc/0205108v1.pdf.
Christopher Isham. Topos Methods in the Foundations of Physics. In Deep Beauty: Understanding the Quantum World through Mathematical Innovation, edited by Hans Halvorson. Cambridge University Press, 2011. arXiv preprint: http://arxiv.org/pdf/1004.3564v1.pdf.
Richard Feynman. Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics, 20, 2, pp. 367–387, 1948.
P. A. M. Dirac. The Principles of Quantum Mechanics, Second Edition. Oxford University Press. 1935.
Benjamin F. Dribus. On the Axioms of Causal Set Theory. Preprint. arXiv link: http://arxiv-web3.library.cornell.edu/pdf/1311.2148v3.pdf.
Seth A. Major, David Rideout, and Sumati Surya. Spatial Hypersurfaces in Causal Set Cosmology. Classical and Quantum Gravity, 23, 14, pp. 4743–4751, 2006. arXiv preprint: http://arxiv.org/pdf/gr-qc/0506133v2.pdf.
Dionigi M. T. Benincasa and Fay Dowker. Scalar Curvature of a Causal Set. Physical Review Letters, 104, 18, 181301, 2010. arXiv preprint: http://arxiv.org/pdf/1001.2725v4.pdf.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Dribus, B.F. (2017). Quantum Spacetime. In: Discrete Causal Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-50083-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-50083-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50081-2
Online ISBN: 978-3-319-50083-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)