Skip to main content
SpringerLink
Log in

The computational status of physics

A computable formulation of quantum theory

  • Published:
Natural Computing Aims and scope Submit manuscript

Abstract

According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Several idealised computational models are known which suggest the possibility of hypercomputation, some Newtonian, some based on cosmology, some on quantum theory. While these models’ physicality is debatable, they nonetheless throw into question the validity of extending CTT to include all physical systems. We consider the physicality of hypercomputational behaviour from first principles, by showing that quantum theory can be reformulated in a way that explains why physical behaviours can be regarded as ‘computing something’ in the standard computational state-machine sense. While this does not rule out the physicality of hypercomputation, it strongly limits the forms it can take. Our model also has physical consequences; in particular, the continuity of motion and arrow of time become theorems within the basic model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Andréka et al. (2008) argue that the physical variant of CTT was first considered as far back as the 1930s.

  2. There is as yet no empirical evidence that Hawking radiation, the mechanism by which evaporation takes place, exists in Nature. However, the final stages of a primordial micro black hole’s evaporation should theoretically result in a burst of gamma-rays; one of the goals of the GLAST satellite, launched by NASA on 11th June 2008, is to search for such flashes.

  3. Integrating over a union of disjoint rectangles is the same as summing the component integrals: given any integrable function f(x, t) defined on a disjoint union \(R = \bigcup_{\alpha}{R_\alpha},\) we have \(\int_{R}{f} = \sum_{\alpha}{\int_{R_\alpha}{f}}.\)

  4. Strictly, only the internal points of the trajectory are required to lie in R. Either (or both) of the endpoints q I and q F can lie outside R, provided they are on its boundary.

  5. As explained in his 1965 Nobel Prize address, Feynman 1965 subsequently described anti-particles as particles moving ‘backwards in time’. In effect, our own approach adopts this temporal bi-directionality, and places it centre-stage.

  6. For example, suppose we know (from wave-equation methods, say) that P has amplitude η(x) to be at location \(x^\dagger = (x,t^\dagger),\) for each \(x {\in}X.\) A more intuitive solution might then be to take \(\langle{x^\dagger|y^\dagger }\rangle_{h}= {\eta(x^\dagger)}/{\eta(y^\dagger)}.\) This gives \(\langle{x^\dagger|x^\dagger}\rangle_{h}=1\) in agreement with the ‘classical amplitude’, but also provides information about the relative amplitudes of all other spatial locations at time \(t^\dagger.\)

References

  • Andréka H, Németi I, Németi P (2008) General relativistic hypercomputing and foundation of mathematics. In: Calude CS, Costa JF (eds) Physics and computation (renaissance) international workshop, Vienna, Austria, August 25–28, 2008: Pre-proceedings, vol 327 of CDMTCS Research Report Series, pp 210–229, University of Auckland, New Zealand, July 2008. Centre for Discrete Mathematics and Theoretical Computer Science

  • Beggs E, Costa JF, Loff B, Tucker JV (2008) Computational complexity with experiments as oracles. Proc R Soc A 464:2777–2801

    Article  MATH  MathSciNet  Google Scholar 

  • Bekenstein JD (1972) Black holes and the second law. Lettere al Nuovo Cimento 4:737

    Article  Google Scholar 

  • Bekenstein JD (1973) Black holes and entropy. Phys Rev D 7:2333

    Article  MathSciNet  Google Scholar 

  • Copeland BJ (2002) The Church-Turing Thesis. In: Zalta EN (ed) The Stanford encyclopedia of philosophy. Stanford University, Stanford, CA, Fall 2002. Online publication, http://plato.stanford.edu/archives/fall2002/entries/church-turing/. Accessed 4 July 2008

  • Deutsch D (1985) Quantum theory, the Church-Turing principle and the universal quantum computer. Proc R Soc Ser A 400:97–117

    Article  MATH  MathSciNet  Google Scholar 

  • Earman J, Norton J (1993) Forever is a day: supertasks in Pitowsky and Malament–Hogarth spacetimes. Philos Sci 5:22–42

    Article  MathSciNet  Google Scholar 

  • Earman J, Norton J (1996) Infinite pains: the trouble with supertasks. In: Morton A, Stich S (eds) Benacerraf and his critics. Blackwell, Cambridge, pp 231–261

    Google Scholar 

  • Eilenberg S (1974) Automata, languages, and machines, vol A. Academic Press, London

    MATH  Google Scholar 

  • Etesi G, Németi I (2002) Non-Turing computability via Malament–Hogarth space-times. Int J Theor Phys 41(2):341–370

    Article  MATH  Google Scholar 

  • Feynman RP (1948) Space–time approach to non-relativistic quantum mechanics. Rev Mod Phys 20(2):367–387

    Google Scholar 

  • Feynman RP (1965) The development of the space-time view of quantum electrodynamics. In: Les Prix Nobel 1965, Stockholm. Imprimerie Royale P.A. Norstedt & Soner, pp 172–191

  • Gandy R (1980) Church’s thesis and principles for mechanisms. In: Barwise J, Keisler HJ, Kunen K (eds) The Kleene symposium. North-Holland, Amsterdam, pp 123–148

  • Goldreich O (2008) Computational complexity: a conceptual perspective. Cambridge University Press, Cambridge

  • Hawking SW (1974) Black hole explosions? Nature 248:30–31

    Google Scholar 

  • Hogarth M (1992) Does general relativity allow an observer to view an eternity in a finite time? Found Phys Lett 5:173–181

    Article  MathSciNet  Google Scholar 

  • Hogarth M (2004) Deciding arithmetic using SAD computers. Br J Philos Sci 55(4):681–691

    Article  MATH  MathSciNet  Google Scholar 

  • Horgan J (1991) Profile: physicist John A. Wheeler, Questioning the “It from Bit”. Scientific American, June 1991, pp 36, 37

  • Kieu T (2009) Hypercomputability of quantum adiabatic processes. Int J Unconv Comput (Special Issue on Hypercomputation) 4 (In press)

  • Kleene SC (1952) Introduction to metamathematics. North Holland, Amsterdam

  • Lloyd S (2006) A theory of quantum gravity based on quantum computation. Preprint, arXiv:quant-ph/0501135. Accessed 4 July 2008

  • Matiyasevich Y (1993) Hilbert’s tenth problem. MIT Press, Cambridge, MA

    Google Scholar 

  • Milner R (1999) Communicating and mobile systems: the π-calculus. Cambridge University Press, Cambridge

    Google Scholar 

  • Németi I, Andréka H (2006) Can general relativistic computers break the Turing barrier? In: Beckmann A, Berger U, Löwe B, Tucker JV (eds) Logical approaches to computational barriers (Proc Conf CiE 2006, Swansea, UK, July 2006), vol 3988 of Lecture notes in computer science, Springer-Verlag, Berlin, pp 398–412

  • Németi I, Dávid Gy (2006) Relativistic computers and the Turing barrier. J Appl Math Comput 178:118–142

    Article  MATH  Google Scholar 

  • Penrose R (1990) The Emperor’s new mind. Oxford University Press, Oxford

    Google Scholar 

  • Penrose R (1999) The question of cosmic censorship. In: Wald RM (ed) Black holes and relativistic stars. University of Chicago Press, Chicago, IL. Reprinted in J Astrophys Astron 20:233–248 (1998)

  • Sangiorgi D, Walker D (2001) The π calculus: a theory of mobile processes. Cambridge University Press, Cambridge

    Google Scholar 

  • Schmidhuber J (1997) A computer scientist’s view of life, the universe, and everything. In: Freksa C (ed) Foundations of computer science: potential–Theory–Cognition, vol 1337 of Lecture notes in computer science, Springer, Berlin, pp 201–208

  • Schmidhuber J (2000) Algorithmic theories of everything. Technical Report IDSIA–20–00, IDSIA, Galleria 2, 6928 Manno (Lugano), Switzerland, December 2000. Available Online, arXiv:quant-ph/0011122. Accessed 4 July 2008

  • Shor P (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In: Goldwasser S (ed) Proceedings of 35th annual symposium on foundations of computer science. IEEE Computer Society Press, Los Alamitos, CA, p 124, 1994. Revised version (1995), available Online, arXiv:quant-ph/9508027v2. Accessed 4 July 2008. Subsequently published as SIAM J Sci Stat Comput 26:1484

  • Stannett M (2003) Computation and hypercomputation. Minds Mach 13(1):115–153

    Article  MATH  Google Scholar 

  • Stannett M (2006) The case for hypercomputation. Appl Math Comput 178:8–24

    Article  MATH  MathSciNet  Google Scholar 

  • Stannett M (2007) Physical hypercomputation. In: Adamatzky A, Bull L, De Lacy Costello B, Stepney S, Teuscher C (eds) Unconventional computing. Luniver Press, Beckington, UK, pp 265–288

  • Susskind L, Lindesay J (2005) An introduction to black holes, information and the string theory revolution—the holographic universe. World Scientific, Singapore

  • Tegmark M (2008) The mathematical universe. Found Phys 38(2):101–150. Preprint available Online, arXiv:0704.0646 [gr-qc]. Accessed 4 July 2008

    Google Scholar 

  • Wheeler J (1990) Recent thinking about the nature of the physical world. It from Bit. In: Mendell RB, Mincer AI (eds) Frontiers in cosmic physics; symposium in memory of Serge Alexander Korff, held on September 13, 1990, in New York, NY, vol 655 of Annals of the New York Academy of Sciences. Academy of Sciences, New York, NY, p 349

  • Wolfram S (2002) A new kind of science. Wolfram Media, Champaign, IL. ISBN 1-57955-008-8

  • Xia Z (1992) The existence of non-collision singularities in the n-body problem. Ann Math 135(3):411–468

    Article  Google Scholar 

  • Zizzi P (2004) Spacetime at the Planck scale: the quantum computer view. In: Garola C, Rossi A, Sozzo S (eds) The foundations of quantum mechanics: historical analysis and open questions—Cesena 2004, pp 345–358. World Scientific, Singapore. Preprint available Online, arXiv:gr-qc/0304032. Accessed 4 July 2008

  • Zuse K (1967) Rechnender Raum. Elektronische Datenverarbeitung 8:336–344. Available Online, ftp://ftp.idsia.ch/pub/juergen/zuse67scan.pdf. Accessed 4 July 2008. This is a preliminary summary of Zuse (1969)

  • Zuse K (1969) Rechnender Raum. Schriften zur Datenverarbeitung, Band 1. Friedrich Vieweg & Sohn, Braunschweig. English translation: Calculating Space, MIT Technical Translation AZT-70-164-GEMIT, MIT (Proj. MAC), Cambridge, MA. 02139, Feb. 1970. Available Online, ftp://ftp.idsia.ch/pub/juergen/zuserechnenderraum.pdf. Accessed 4 July 2008. A German-language summary appeared in 1967 as Zuse (1967)

Download references

Acknowledgements

This research was supported in part by the EPSRC HyperNet project (Hypercomputation Research Network, grant number EP/E064183/1).

Author information

Authors and Affiliations

  1. Department of Computer Science, Regent Court, 211 Portobello Street, S1 4DP, Sheffield, UK

    Mike Stannett

Authors
  1. Mike Stannett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mike Stannett.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stannett, M. The computational status of physics. Nat Comput 8, 517–538 (2009). https://doi.org/10.1007/s11047-009-9115-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11047-009-9115-2

Keywords

Search

Navigation