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Introduction

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Discrete Causal Theory

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Abstract

This chapter provides a concise qualitative overview of discrete causal theory. Section 1.1 illustrates how discrete causal theory models fundamental spacetime structure at the quantum level, and introduces a “discrete diffusion-type or wave-type equation,” called a causal Schrödinger-type equation, that serves as an example of the kind of dynamical law that arises in this context. Section 1.2 introduces the causal metric hypothesis, which states that the properties of the physical universe, and in particular, the metric structure of spacetime, emerges from causal structure at the fundamental scale. An important version of this hypothesis appears in Sorkin’s causal set theory. Section 1.3 describes, at a qualitative level, how spacetime geometry can emerge from fundamental causal structure, with motivation from general relativity. Section 1.4 illustrates some basic structural principles, in terms of “toy models” of discrete causal classical histories. Section 1.5 introduces Grothendieck’s relative viewpoint, which states that “objects should be studied together with their natural relationships, rather than in isolation.” An important embodiment of this idea is relation space, which is a space whose elements represent causal relationships between pairs of spacetime events. Relation space methods are crucial to the technical apparatus of discrete causal theory. Section 1.6 discusses the principle of background independence, which states that “all physical entities invoked by a theory should participate nontrivially in the dynamics of the theory.” In particular, as is known from general relativity, spacetime itself is dynamical, rather than static. Section 1.7 gives a brief overview of the subject of particles and fields in discrete causal theory. Section 1.8 explains the distinction between kinematics and dynamics, and describes how these topics are treated in the discrete causal context. Section 1.9 describes some qualitative ideas regarding discrete causal phenomenology, including a possible alternative to the inflationary hypothesis in the cosmology of the early universe. Section 1.10 poses a number of obvious questions raised by the foregoing material, and describes how the book attempts to address these questions.

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Notes

  1. 1.

    As noted in the preface, and repeated throughout the book, the term “directed set” does not have the same narrow meaning here as in category theory.

  2. 2.

    The purpose of the acronym (IS) following the term iteration of structure is to serve as a clickable hyperlink to the definition of the term in a later chapter. I generally do not use acronyms as mere abbreviations, since this would only confuse and annoy the reader.

  3. 3.

    An interesting historical detail is that Feynman’s paper [FE48] and Shannon’s landmark paper on information theory [SH48] both appeared in the year 1948.

  4. 4.

    In the present “quantum-level” context, the expression \(r^-\prec r\) represents a relationship between a pair of co-relative histories, i.e., a “relationship between a pair of evolutionary relationships,” in which the evolutionary relationship represented by r is a possible sequel to the evolutionary relationship represented by \(r^-\).

  5. 5.

    For example, the specific derivation of Equation (1.1.2) in Chapter 6 assumes that spacetime is acyclic (AC), although the overall development of the theory is much more general. It is worth noting that Einstein’s field equation, which appears as Equation (1.3.1) below, involves only a small part of the emergent structure that should arise from equations such as (1.1.2), since it takes for granted so much geometric information; for example, the integer dimensionality of spacetime. This topic is revisited in Section 8.2.

  6. 6.

    The fact that the word “discrete” begins with the same letter is a happy coincidence, but many of the results of this book also apply to non-discrete directed sets.

  7. 7.

    The appearance of the same precursor symbol \(\prec \) in the causal Schrödinger-type equation (1.1.2) is a reflection of iteration of structure (IS) .

  8. 8.

    The qualifier “generalized” is necessary because “true” or “strict” Hasse diagrams encode partially ordered sets, which are less general than directed sets under the conventions used in this book. See Section 2.2 or Chapter 3 for more details.

  9. 9.

    That is, in the most straightforward versions of discrete causal theory. Section 8.7 includes brief speculation about “subclassical structure.”

  10. 10.

    The reason why these references seem to appear in reverse chronological order is because the paper [DR15] was first presented in 2012; the reference dated 2015 includes it as an individual chapter. The link to the 2012 version appears in the list of references along with the bibliographical information for the 2015 version.

  11. 11.

    The scalar curvature has nothing to do with the relation space referred to in Equation (1.1.2) above, also denoted by R. The justification for using the same symbol twice is that the scalar curvature plays little direct role in this book, and the notation is standard.

  12. 12.

    The reason for the quotation marks here is that the term “cosmological constant” itself imposes a very specific and conventional physical interpretation on certain observed data.

  13. 13.

    This is a “classical” description of Einstein’s equation and its constituents. Mathematically, it is very old-fashioned, and even in conventional physics there exist more modern descriptions enjoying certain advantages. For example, in the tetrad formalism, Einstein’s equation may be written as

    $$\begin{aligned} \varepsilon _{IJKL}\big (e^I\wedge R^{JK}-\frac{2}{3}\varLambda e^I\wedge e^J\wedge e^K\big )=2\pi GT_L, \end{aligned}$$

    where \(\varepsilon _{IJKL}\) is the totally antisymmetric Levi-Civita symbol, e is a 1-form called the tetrad field, R is the curvature 2-form, and T is the energy-momentum 3-form. One advantage of this formalism is that it can incorporate fermionic fields. Rovelli gives a useful description of different formalisms for general relativity in the second chapter of his book [RO04].

  14. 14.

    The notion of “systematic” Lorentz invariance is described precisely in Chapter 4.

  15. 15.

    In general, a symmetry of a mathematical object means an automorphism of the object in the category to which it belongs. In this case, symmetries are isometries, i.e., self-maps preserving the Minkowski metric . See Chapter 2 for more details.

  16. 16.

    That is, the tangent spaces \(T_xX\) at each point \(x\in X\) of a relativistic spacetime manifold are isomorphic to \(\mathbb {R}^{3+1}\). This ignores singularities, such as those resulting from gravitational collapse, which are not considered to be part of the actual spacetime.

  17. 17.

    In particular, referring to the discussion below, this directed set looks nothing like a “sprinkled” causal set, in which most of the irreducible relations are “almost null.”

  18. 18.

    Torsten Asselmeyer-Maluga has built an interesting research program in fundamental physics by focusing on the special properties of four-dimensional real manifolds in the context of differential topology. In particular, four-dimensional manifolds may possess an infinite, or even uncountable, number of inequivalent smoothness structures, and it is natural to wonder if it is a coincidence that four is the “physical dimension.” Asselmeyer-Maluga [AS15] goes as far as to suggest that “the plethora of exotic smoothness structures in dimension four could be the cornerstone of quantum gravity.” This line of thought goes in a completely different direction from discrete causal theory, arguing that fundamental spacetime discreteness is simply the wrong idea!

  19. 19.

    This terminology can sometimes be useful for rendering “concrete” the physical interpretation of the mathematical structures involved, but the more neutral terminology of “elements and relations” is usually preferable.

  20. 20.

    It is worth noting that significant and possibly decisive progress on this problem has recently been claimed by László Babai [BA15], a mathematician and computer scientist at the University of Chicago. As of early 2016, this result remained under review by other experts.

  21. 21.

    One reason why the analogy is merely “rough” is because multiple distinct relationships may exist between a given pair of classical histories, even a pair differing by a single element. This subtlety is explained in Section 6.4. Also, the S-matrix was introduced to treat asymptotic behavior, “ignoring the details of the interaction,” while the relationships of principal interest in the discrete causal context are often “direct” or “immediate” relationships. Such relationships generally do not exist in the continuum-based setting, due to the interpolative property of the real numbers \(\mathbb {R}\).

  22. 22.

    These transitions have nothing to do with the graph-dynamical phase transitions discussed in Section 1.9.

  23. 23.

    The letters r, s, and t would be a bad choice, because t is used in Chapter 5 to denote a “terminal element map” in the same context. Even here, t would clash with the subscript in \(D_t\).

  24. 24.

    However, as described in Chapter 5, the mathematical construction of relation space may be applied to structures with physical interpretations different than models of classical spacetime. In fact, one may construct the relation space over any multidirected set.

  25. 25.

    This is one possible qualitative description of background independence. Others describe the principle quite differently; for example, Thiemann [TH07] equates it with general covariance in the relativistic context. In discrete causal theory, the terms “background independence” and “covariance” are assigned very different meanings.

  26. 26.

    Of course, the strong interpretation of the causal metric hypothesis (CMH) assumes “interactions,” or “forces,” to be ultimately structural in nature as well.

  27. 27.

    As expressed by Thiemann [TH07], p. 9.

  28. 28.

    The unfortunate modern prominence of the string theory landscape and the anthropic principle places a burden on new theories to offer assurances against the use of similarly non-explanatory mechanisms, under which “anything goes, somewhere.” In the present case, the incorporation of all “physically reasonable” structures means essentially the same thing as in Feynman’s path summation approach to ordinary quantum theory: structures favored by the dynamics of the theory are reinforced, while disfavored structures are damped out via destructive interference.

  29. 29.

    As explained in Section 4.3, reflexive relations \(x\prec x\) are counted twice in determining v(x), because such relations both begin and terminate at x. The examples considered in this section involve only acyclic directed sets, which do not exhibit such reflexive relations.

  30. 30.

    As in the case of the “cosmological constant,” the use of quotation marks is intended to highlight the fact that the term “dark matter” itself imposes a rather specific and conventional, though well-supported, interpretation on certain observed phenomena.

  31. 31.

    See the recent preprint of Saravani and Aslanbeigi [SA15] for an interesting discussion about the possible origin of dark matter due to nonlocal effects in discrete gravity.

  32. 32.

    This basic idea is not restricted to discrete causal theory; in particular, one may define kinematic schemes involving “classical histories” represented by other objects; for example, spin foams.

  33. 33.

    A closer analogy exists between frames of reference and refinements of the causal relation on \(D_t'\), as explained in Sections 7.1 and 7.2.

  34. 34.

    For historical reasons, the terms invariance, covariance, and gauge are all used in the study of extraphysical information, and these terms are imbued with many different shades of meaning in different contexts. I choose the term “covariance” to describe such considerations in the discrete causal setting. One reason for this choice is that the formation of the word “covariance” suggests the intended meaning that “varying one’s point of view demands a corresponding systematic variation in associated quantities.” In particular, the choice of phase map varies along with the choice of kinematic scheme in discrete causal dynamics.

  35. 35.

    In general, more than one co-relative history may exist between a given source and target, even in the case of sequential growth, but such details are postponed until Chapter 6.

  36. 36.

    Chapter 3 explains in detail the distinctions among direct, indirect, maximal, and minimal relationships in this context. In general, the version of discrete causal theory developed in this book is described principally in terms of direct relationships, whether or not these relationships are maximal or minimal.

  37. 37.

    In Equation (1.8.3), one considers the family of all co-relative histories with a fixed target \(D_t'\). It can also be interesting to consider such a family with a fixed source. For example, in the present classical stochastic context, the Shannon entropy associated with the family of co-relative histories \(\{D_i\Rightarrow D'\}\), for a fixed source set \(D_i\), is given by the formula

    $$\begin{aligned} H=-\sum _{D_i\prec D'}p(D_i\Rightarrow D')\log p(D_i\Rightarrow D'). \end{aligned}$$

    See Section 6 of Shannon’s paper [SH48] for an analogous discussion involving abstract probabilities. The subject of entropy returns in Section 8.2 in the context of entropic phase maps.

  38. 38.

    For a string theory viewpoint, see [BBS07], Section 10.7.

  39. 39.

    Of course, inflation solves more than just the horizon problem, and a serious discrete causal alternative would have to address its other successes in a suitable way. It is worth noting, however, that inflation has some very credible critics whose objections have nothing to do with discreteness hypotheses; these include Roger Penrose [PE10].

  40. 40.

    The qualifier “seems to” is added here in anticipation of Section 4.5, which discusses the tension between spatiotemporal locality and “approximate Lorentz invariance” in discrete causal theory.

  41. 41.

    Also, this naïve enumeration ignores isomorphisms between pairs of “physically equivalent” histories, but this does not alter the conclusions.

  42. 42.

    A “tamer” type of spatiotemporal nonlocality occurs in causal sets constructed via global sprinkling into Minkowski spacetime \(\mathbb {R}^{3+1}\), as discussed in Section 4.5. In this context, a given element is directly related to elements that are distant in both a spatial and temporal sense in a given frame of reference; the separation between such elements is therefore small under the Minkowski metric.

  43. 43.

    Or at least, locally finite.

  44. 44.

    Comparable to finding the Holy Grail and the two Arks in a single archaeological expedition.

References

  1. Richard Feynman. Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics, 20, 2, pp. 367–387, 1948.

    Google Scholar 

  2. P. A. M. Dirac. The Principles of Quantum Mechanics, Second Edition. Oxford University Press. 1935.

    Google Scholar 

  3. Claude Shannon. Space-Time Approach to Non-Relativistic Quantum Mechanics. The Bell System Technical Journal, 27, 3, pp. 379–423, 1948.

    Google Scholar 

  4. Benjamin F. Dribus. On the Foundational Assumptions of Modern Physics. Questioning the Foundations, The Frontiers Collection, Springer, pp. 45–60, 2015. Early version: http://www.fqxi.org/community/essay/winners/2012.1#dribus.

  5. Benjamin F. Dribus. On the Axioms of Causal Set Theory. Preprint. arXiv link: http://arxiv-web3.library.cornell.edu/pdf/1311.2148v3.pdf.

  6. S. W. Hawking, A. R. King, and P. J. McCarthy. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. Journal of Mathematical Physics, 17, 2, pp. 174–181, 1976.

    Google Scholar 

  7. David B. Malament. The class of continuous timelike curves determines the topology of spacetime. Journal of Mathematical Physics, 18, 7, pp. 1399–1404, 1977.

    Google Scholar 

  8. Keye Martin and Prakash Panangaden. A Domain of Spacetime Intervals in General Relativity. Communications in Mathematical Physics, 267, 3, pp. 563–586, 2006.

    Google Scholar 

  9. Carlo Rovelli. Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.

    Book  MATH  Google Scholar 

  10. Sabine Hossenfelder. Phenomenology of Space-time Imperfection II: Local Defects. Physical Review D, 88, 12, 124031, 2013. arXiv preprint: http://arxiv.org/pdf/1309.0314v2.pdf.

  11. Luca Bombelli, Joe Henson, Rafael Sorkin. Discreteness without symmetry breaking: a theorem. Modern Physics Letters A, 24, 32, pp. 2579–2587, 2009. arXiv preprint: http://arxiv.org/pdf/gr-qc/0605006v1.pdf

  12. Christopher Moore. Comment on “Space-Time as a Causal Set.” Physical Review Letters, 60, 7, pp. 655, 1988.

    Google Scholar 

  13. Luca Bombelli, Joohan Lee, David Meyer, and Rafael Sorkin. Bombelli et al. Reply to Comment on “Space-Time as a Causal Set.” Physical Review Letters, 60, 7, pp. 656, 1988.

    Google Scholar 

  14. Torsten Asselmeyer-Maluga. A Chicken-and-Egg Problem: Which Came First, the Quantum State or Spacetime? Questioning the Foundations of Physics, The Frontiers Collection, Springer, pp. 205–217, 2015.

    Google Scholar 

  15. László Babai. Graph Isomorphism in Quasipolynomial Time. arXiv preprint: https://arxiv.org/pdf/1512.03547v2.pdf

  16. 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.

    Google Scholar 

  17. 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.

  18. Thomas Thiemann. Modern Canonical Quantum General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.

    Book  MATH  Google Scholar 

  19. Fotini Markopoulou. Quantum Causal Histories. Classical and Quantum Gravity, 17, 10, pp. 2059, 2000. arXiv preprint: https://arxiv.org/pdf/hep-th/9904009v5.pdf.

  20. Eli Hawkins, Fotini Markopoulou, and Hanno Sahlmann. Evolution in Quantum Causal Histories. Classical and Quantum Gravity, 20, 16, pp. 3839, 2003. arXiv preprint: http://arxiv.org/pdf/hep-th/0302111.pdf.

  21. 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.

  22. 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.

  23. Olaf Dreyer. Not on, but of. Questioning the Foundations, The Frontiers Collection, Springer, pp. 131–138, 2015.

    Google Scholar 

  24. Mehdi Saravani and Siavash Aslanbeigi. Dark Matter from Spacetime Nonlocality. arXiv preprint: http://arxiv.org/pdf/1502.01655v2.pdf.

  25. John Archibald Wheeler and Kenneth W. Ford. Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton and Company, New York, 1998.

    Google Scholar 

  26. Maqbool Ahmed, Scott Dodelson, Patrick B. Greene, and Rafael Sorkin. Everpresent \(\Lambda \). Physical Review D, 69, 10, 103523, 2004. arXiv preprint: http://arxiv.org/pdf/astro-ph/0209274v1.pdf.

  27. Alan Guth. Inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D, 23, 2, pp. 347–356, 1981.

    Google Scholar 

  28. A. D. Linde. A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy, and primordial monopole problems. Physics Letters B, 108, 6, pp. 389–393, 1982.

    Google Scholar 

  29. Alexander Vilenkin. The birth of inflationary universes. Physical Review D, 27, 12, pp. 2848–2855, 1983.

    Google Scholar 

  30. A. D. Linde. Eternal chaotic inflation. Modern Physics Letters A, 1, 2, 1986.

    Article  Google Scholar 

  31. Katrin Becker, Melanie Becker, and John H. Schwarz. String Theory and M-Theory. Cambridge University Press, 2007.

    Google Scholar 

  32. George F. R. Ellis, Roy Maartens, and Malcolm A. H. MacCallum. Relativistic Cosmology. Cambridge University Press, 2012.

    Google Scholar 

  33. Alan H. Guth. The Inflationary Universe. Basic Books, 1997.

    Google Scholar 

  34. Roger Penrose. Cycles of Time. Vintage Books, New York, 2010.

    MATH  Google Scholar 

  35. Paul Erdös and Alfred Rényi. On random graphs I. Publ. Math. Debrecen, 6, pp. 290–297, 1959.

    Google Scholar 

  36. Béla Bollobás. Random Graphs. Cambridge Studies in Advanced Mathematics, 73, Second Edition. Cambridge University Press, 2001.

    Google Scholar 

  37. Jan Ambjørn, S. Jordan, J. Jurkiewica, and R. Loll. Second- and first-order phase transitions in causal dynamical triangulations. Physical Review D, 85, 12, 124044, 2012. arXiv preprint: http://arxiv.org/pdf/1205.1229.pdf.

  38. Tomasz Konopka, Fotini Markopoulou and Lee Smolin. Quantum Graphity. Preprint, 2006. arXiv preprint: http://arxiv.org/pdf/hep-th/0611197v1.pdf.

  39. Tomasz Konopka, Fotini Markopoulou, and Simone Severini. Quantum graphity: A model of emergent locality. Physical Review D, 77, 10, 104029, 2008.

    Google Scholar 

  40. Dmitri Krioukov, Maksim Kitsak, Robert S. Sinkovits, David Rideout, David Meyer, and Marián Boguñá. Network Cosmology. Nature Scientific Reports 2, 793, 2012.

    ADS  Google Scholar 

  41. Stan Gudder. Inflation and Dirac in the Causal Set Approach to Discrete Quantum Gravity. Preprint, 2015. arXiv preprint: http://arxiv.org/pdf/1507.01281v1.pdf.

  42. Stan Gudder. Discrete Quantum Gravity and Quantum Field Theory. Preprint, 2016. arXiv preprint: http://arxiv.org/pdf/1603.03471v1.pdf.

  43. Erik Verlinde. On the origin of gravity and the laws of Newton. Journal of High Energy Physics, 04, 29, 2011.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Julian Gonzalez-Ayala, Rubén Cordero, and F. Angulo-Brown. Is the \((3+1)-{\rm {d}}\) nature of the universe a thermodynamic necessity? Europhysics Letters, 113, 4, 2016.

    Google Scholar 

  45. Luca Bombelli, Joohan Lee, David Meyer, and Rafael Sorkin. Space-Time as a Causal Set. Physical Review Letters, 59, 5, 1987.

    Article  MathSciNet  Google Scholar 

  46. David Finkelstein. Space-Time Code. Physical Review, 184, 5, pp. 1261–1271, 1969.

    Google Scholar 

  47. J. Myrheim. Statistical Geometry. CERN preprint, 1978.

    Google Scholar 

  48. Gerard ’t Hooft. Quantum Gravity: A Fundamental Problem and some Radical Ideas. Recent Developments in Gravitation (Proceedings of the 1978 Cargese Summer Institute), edited by M. Levy and S. Deser.

    Google Scholar 

  49. J. Ambjorn, A. Dasgupta, J. Jurkiewicz, R. Loll. A Lorentzian cure for Euclidean troubles. Nuclear Physics B, Proceedings Supplements, 106-107, pp. 977–979, 2002. arXiv preprint: http://arxiv.org/pdf/hep-th/0201104v1.pdf.

  50. Giacomo Mauro D’Ariano and Paolo Perinotti. Derivation of the Dirac Equation from Principles of Information Processing. Physical Review A, 90, 6, 062106, 2014. arXiv preprint: http://arxiv.org/pdf/1306.1934v1.pdf.

  51. David Finkelstein. “Superconducting” Causal Nets. International Journal of Theoretical Physics, 27, 4, pp. 473–519, 1988.

    Google Scholar 

  52. Roger Penrose. Twistor Algebra. Journal of Mathematical Physics, 8, 2, 345–366, 1967.

    Google Scholar 

  53. Julian Barbour. Shape Dynamics: an Introduction. Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework, Edited by Felix Finster, Olaf Müller, Marc Hardmann, Jürgen Tolksdorf, and Eberhard Zeidler. arXiv preprint: http://arxiv.org/pdf/1105.0183v1.pdf.

  54. Jan Ambjørn, J. Jurkiewicz, and R. Loll. Spectral Dimension of the Universe. Physical Review Letters, 95, 171301, 2005. arXiv preprint: https://arxiv.org/pdf/hep-th/0505113v2.pdf.

  55. Rafael Sorkin. Does locality fail at intermediate length scales? Approaches to Quantum Gravity, edited by Daniele Oriti. Cambridge University Press, 2009.

    Google Scholar 

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    Benjamin F. Dribus

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Dribus, B.F. (2017). Introduction. In: Discrete Causal Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-50083-6_1

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