## On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory (2009)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Svozil09onthe,

author = {Karl Svozil},

title = {On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory },

year = {2009}

}

### OpenURL

### Abstract

Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper-) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the self-contradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.

### Citations

976 |
Quantum Field Theory
- Itzpkson, Zuber
- 1980
(Show Context)
Citation Context ...ns made) used divergent power series for calculations of planetary motion. Thus it is not totally unreasonable to suspect that the perturbation series solutions obtained via diagrammatical techniques =-=[34,35,36]-=- for the differential equations of quantum electrodynamics are of this “well behaved” type. Therefore, at least in principle, the Thomson lamp could be perceived as a physical process governed by some... |

837 | Theory of recursive functions and effective computability - Rogers - 1967 |

308 |
Classical Recursion Theory
- Odifreddi
- 1989
(Show Context)
Citation Context ...mpossible to operationalize and construct some process “on top of” or after a non-terminating, infinite process [20]. The method of diagonalization presents an important technique of recursion theory =-=[21,22,23]-=-. Some aspects of its physical realizability have already been discussed by the author [24,25,26,27]. In what follows we shall investigate some further physical issues related to “accelerated” agents ... |

301 | Mathematical Methods for Physicists - Arfken - 1985 |

225 |
Optical Coherence and Quantum Optics
- Mandel, Wolf
- 1995
(Show Context)
Citation Context ...here ν and λ refer to the frequency and wavelength of the associated field mode. We shall also not be concerned with the photon statistics, which would require a detailed analysis of the light source =-=[41]-=-. In what follows, the notation of Mermin [42] will be used. Let |0〉 = (1, 0) and |1〉 = (0, 1) be the representations of the “off” and “on” states of the Thomson lamp, respectively. ( ) Then, the swit... |

186 |
Computability and logic
- Boolos, Burgess, et al.
- 2007
(Show Context)
Citation Context ...mpossible to operationalize and construct some process “on top of” or after a non-terminating, infinite process [20]. The method of diagonalization presents an important technique of recursion theory =-=[21,22,23]-=-. Some aspects of its physical realizability have already been discussed by the author [24,25,26,27]. In what follows we shall investigate some further physical issues related to “accelerated” agents ... |

167 |
Divergent Series
- Hardy
- 1949
(Show Context)
Citation Context ... who considers physical methods of producing infinity is, of course, in a state of sin. Such sinful physical pursuits qualify for Neils Henrik Abel’s verdict that (Letter to Holmboe, January 16, 1826 =-=[3, 4]-=-), “divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.” This, of course, has prevented no-one, in particular not Abel himself, from consi... |

76 |
Experimental realization of any discrete unitary operator
- Reck, Zeilinger, et al.
- 1994
(Show Context)
Citation Context ...e iλ ,e iλ ). Another example is U bs (π/2, 2λ, −π/2 − λ, 0) = diag(e iλ ,e −iλ ). For the physical realization of general unitary operators in terms of beam splitters the reader is referred to Refs. =-=[55,56,57,58]-=-. 6 Summary In summary we have discussed some physical aspects related to the limit operations of the Thomson lamp. This physical limit, regardless of the formally unbounded and even infinite number o... |

73 | Infinite time turing machines
- Hamkins
- 2002
(Show Context)
Citation Context ...t the use of Cantor-type diagonalization procedures on the basis that it is physically impossible to operationalize and construct some process “on top of” or after a non-terminating, infinite process =-=[20]-=-. The method of diagonalization presents an important technique of recursion theory [21–23]. Some aspects of its physical realizability have already been discussed by the author [24–27]. In what follo... |

60 |
Theory of recursive functions and effective computability
- Jr
- 1987
(Show Context)
Citation Context ...mpossible to operationalize and construct some process “on top of” or after a non-terminating, infinite process [20]. The method of diagonalization presents an important technique of recursion theory =-=[21,22,23]-=-. Some aspects of its physical realizability have already been discussed by the author [24,25,26,27]. In what follows we shall investigate some further physical issues related to “accelerated” agents ... |

43 | Does general relativity allow an observer to view an eternity in a finite time
- Hogarth
- 1992
(Show Context)
Citation Context ...fixed points [U2(ω, α, β, ϕ)] −1 diag(eiµ , eiλ ( )U2(ω, α, β, ϕ) = e = i µ cos2 ω + ei λ sin 2 1 ω 1 2e−i (α+ϕ) ( ei λ − ei µ) sin(2 ω) 2ei (α+ϕ) ( ei λ − ei µ) sin(2 ω) ei λ cos2 ω + ei µ sin 2 ) ω =-=(11)-=- with µ, λ ̸= 2nπ, n ∈ N0 gives rise to eigenvectors which are not fixed points, and which acquire non-vanishing phases µ, λ in the generalized diagonalization process. |0〉 |0〉 P3, ϕ S(T (ω)) ′ |1〉 ′ ... |

36 | Uber das Unendliche. Mathematische Annalen, 95:161-190,1926. [34] David Hilbert and Paul Bernays. Grundlagen der Mathematik. Grundlehrender Mathematischen Wissenschaften. Springer-Verlag, 2nd edition, 19681970. Volume I - Hilbert - 1987 |

35 |
The Church-Turing Thesis as a Guiding Principle for Physics
- Svozil
(Show Context)
Citation Context ...finite process [20]. The method of diagonalization presents an important technique of recursion theory [21,22,23]. Some aspects of its physical realizability have already been discussed by the author =-=[24,25,26,27]-=-. In what follows we shall investigate some further physical issues related to “accelerated” agents and the operationalizability of the diagonalization operations in general. 2 Classical Brightness of... |

33 |
Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations
- Balser
- 2000
(Show Context)
Citation Context ...ent sums could be valuable approximations of solutions of differential equations which might be even analytically solvable. One of the bestknown examples of this case is Euler’s differential equation =-=[32, 4, 33]-=- z 2 y ′ +y = z, which has both (i) a formal solution as a power series ˆ f(z) = − ∑ ∞ n=0 n! (−z)n , which diverges for all nonzero values of z; as well as (ii) an exact solution ˆf(z) = e 1/z Ei(−1/... |

31 |
The Unitary and Rotation Groups
- Murnaghan
- 1962
(Show Context)
Citation Context ...zation operator modeled by X. One could proceed a step further and allow non-classical diagonalization procedures. Thereby, one could allow the entire range of two-dimensional unitary transformations =-=[51]-=- −i β U2(ω, α, β, ϕ) = e ( i α −i ϕ e cos ω −e sin ω ei ϕ sin ω e−i α ) cos ω , (8) where −π ≤ β, ω ≤ π, − π π 2 ≤ α, ϕ ≤ 2 , to act on the quantum bit. A typical example of a non-classical operation ... |

31 |
The intersection of recurrence sequences
- SCHLICKEWEI, SCHMIDT
- 1995
(Show Context)
Citation Context ... and “off,” respectively, and the switching process with the concatenation of “+1” and “-1” performed so far, then the divergent infinite series associated with the Thomson lamp is the Leibniz series =-=[29,30,3,31]-=- ∞∑ s = (−1) n =1− 1+1− 1+1−··· A = 1 . (3) 2 n=0 Here, “A” indicates the Abel sum [3] obtained from a “continuation” of the geometric series, or alternatively, by s =1−s. As this shows, formal summat... |

31 | Über das Unendliche - Hilbert - 1925 |

29 | Enderton01] A Mathematical Introduction to Logic, Second Edition - Enderton - 2001 |

29 | Zur Quantenmechanik der Stoßvorgänge, Zeits. f. Physik 37 - Born - 1926 |

28 | Randomness and Undecidability - Svozil - 1993 |

25 |
A physicist’s second reaction to Mengenlehre
- Bridgman
- 1934
(Show Context)
Citation Context ...r overly prudent and displaced, stimulating even more caveats or outright rejection. Yet, one should keep in mind that, to rephrase a dictum of John von Neumann [1], from an operational point of view =-=[2]-=-, anyone who considers physical methods of producing infinity is, of course, in a state of sin. Such sinful physical pursuits qualify for Neils Henrik Abel’s verdict that (Letter to Holmboe, January 1... |

22 | Relativistic computers and the Turing barrier
- Németi, Dávid
(Show Context)
Citation Context ...ices in reverse order in which the quanta pass these elements [53, 54]; i.e., U bs ( ) ( ) ( ) ( iϕ e 0 i sin ω cos ω i(α+β) e 0 1 0 (ω, α, β, ϕ) = 0 1 cos ω i sin ω 0 1 0 eiβ ) ( ) i (α+β+ϕ) i (β+ϕ) =-=(13)-=- i e sin ω e cos ω = . e i (α+β) cos ω i e i β sin ω For this physical setup, the phases ω = π/2, β = λ − π/2 and ϕ = −α can be arranged such that U bs (π/2, α, λ − π/2, −α) = diag(e iλ , e iλ ). Anot... |

22 |
Multiparticle interferometry and the superposition principle
- Greenberger, Horne, et al.
- 1993
(Show Context)
Citation Context ... with equal phases in the diagonal terms; i.e., µ = λ, thus reducing to diag(e iλ , e iλ ). The generalized quantum optical beam splitter sketched in Fig. 1 can be described either by the transitions =-=[52]-=- P1 : |0〉 → |0〉e i(α+β) , P2 : |1〉 → |1〉e iβ , S : |0〉 → √ T |1 ′ 〉 + i √ R |0 ′ 〉, S : |1〉 → √ T |0 ′ 〉 + i √ R |1 ′ 〉, P3 : |0 ′ 〉 → |0 ′ 〉e iϕ , (12) where every reflection by a beam splitter S con... |

20 |
Philosophy of mathematics and natural sciences
- Weyl
- 1949
(Show Context)
Citation Context ... demonstration whatsoever.” This, of course, has prevented no-one, in particular not Abel himself, from considering these issues. Indeed, by reviving old eleatic ideas [5,6], accelerated computations =-=[7]-=- have been the main paradigm of the fast growing field of hypercomputations (e.g., Refs. [8,9,10]). For the sake of hypercomputation, observers have been regarded on their path toward black holes [11,... |

19 |
Maxwell’s Demon
- Leff, Rex
- 1990
(Show Context)
Citation Context ...ould be justified from the point of view of classical physics as follows. Suppose that the switching processes can be disregarded, an assumption not dissimilar to related concerns for Maxwell’s demon =-=[37]-=-. Suppose further that all measurements are finite in the sense that the temporal resolution δ of the observation of the Thomson lamp cannot be made “infinitely small;” i.e., the observation time is f... |

19 |
Handbook of Mathematical Logic
- Barwise
- 1977
(Show Context)
Citation Context ...cal case and for the formal Abel sum, they represent a fifty-fifty mixture of the “on” and “off” states. 4 Quantum Fixed Point of Diagonalization Operator In set theory, logic and in recursion theory =-=[21,44,45,46,22,23]-=-, the method of proof by contradiction (reductio ad absurdum) requires some “switch of the bit value,” which is applied in a self-reflexive manner. Classically, total contradiction is achieved by supp... |

16 |
Why there is no such discipline as hypercomputation
- Davis
(Show Context)
Citation Context ...al diagonalization procedures. Thereby, one could allow the entire range of two-dimensional unitary transformations [51] −i β U2(ω, α, β, ϕ) = e ( i α −i ϕ e cos ω −e sin ω ei ϕ sin ω e−i α ) cos ω , =-=(8)-=- where −π ≤ β, ω ≤ π, − π π 2 ≤ α, ϕ ≤ 2 , to act on the quantum bit. A typical example of a non-classical operation on a quantum bit is the “square root of not” ( √ X · √ X = X) gate operator √ X = 1... |

16 | The many forms of hypercomputation
- Ord
(Show Context)
Citation Context ...unitary transformations of the form [U2(ω, α, β, ϕ)] −1 diag(1, eiλ ( )U2(ω, α, β, ϕ) = cos = 2 ω + ei λ sin 2 1 ω 1 2e−i (α+ϕ)(ei λ − 1) sin(2 ω) 2ei (α+ϕ)(ei λ − 1) sin(2 ω) ei λ cos2 ω + sin 2 ) ω =-=(10)-=- for arbitrary λ have fixed points.Applying non-classical operations on quantum bits with no fixed points [U2(ω, α, β, ϕ)] −1 diag(eiµ , eiλ ( )U2(ω, α, β, ϕ) = e = i µ cos2 ω + ei λ sin 2 1 ω 1 2e−i... |

16 |
Tasks, supertasks, and the modern Eleatics
- Benacerraf
- 1962
(Show Context)
Citation Context ...all not be concerned with issues related to unbounded space or memory consumption discussed by Calude and Staiger [15], although we acknowledge their importance. In analogy to Benacerraf’s discussion =-=[16]-=- of Thomson’s proposal [17, 18] of a lamp which isswitched on or off at geometrically decreasing time delays, Shagrir [19] suggested that the physical state has little or no relation to the states in... |

15 | Even Turing Machines Can Compute Uncomputable Functions - Copeland - 1998 |

11 | Dynamical bias in the coin toss
- Diaconis, Holmes, et al.
(Show Context)
Citation Context ...ation. For the general case discussed, with regards to the question of whether or not a computer halts, the quantum “solution” fixed point state is equivalent to the throwing of a fair classical coin =-=[50]-=-. 5 Quantum diagonalization The above argument used the continuity of quantum bit states as compared to the discrete classical spectrum of just two classical bit states for a construction of fixed poi... |

11 |
Realizable higher-dimensional twoparticle entanglements via multiport beam splitters
- Zukowski, Zeilinger, et al.
- 1997
(Show Context)
Citation Context ...e iλ ,e iλ ). Another example is U bs (π/2, 2λ, −π/2 − λ, 0) = diag(e iλ ,e −iλ ). For the physical realization of general unitary operators in terms of beam splitters the reader is referred to Refs. =-=[55,56,57,58]-=-. 6 Summary In summary we have discussed some physical aspects related to the limit operations of the Thomson lamp. This physical limit, regardless of the formally unbounded and even infinite number o... |

11 | Noncontextuality in multipartite entanglement
- Svozil
- 2005
(Show Context)
Citation Context ...e iλ ,e iλ ). Another example is U bs (π/2, 2λ, −π/2 − λ, 0) = diag(e iλ ,e −iλ ). For the physical realization of general unitary operators in terms of beam splitters the reader is referred to Refs. =-=[55,56,57,58]-=-. 6 Summary In summary we have discussed some physical aspects related to the limit operations of the Thomson lamp. This physical limit, regardless of the formally unbounded and even infinite number o... |

11 | Advertisement for a paper I like - Landauer - 1994 |

10 |
Abstract geometrical computation for black hole computation (extended abstract
- Durand-Lose
- 2005
(Show Context)
Citation Context ...g. 1 can be described either by the transitions [52] P1 : |0〉 → |0〉e i(α+β) , P2 : |1〉 → |1〉e iβ , S : |0〉 → √ T |1 ′ 〉 + i √ R |0 ′ 〉, S : |1〉 → √ T |0 ′ 〉 + i √ R |1 ′ 〉, P3 : |0 ′ 〉 → |0 ′ 〉e iϕ , =-=(12)-=- where every reflection by a beam splitter S contributes a phase π/2 and thus a factor of eiπ/2 = i to the state evolution. Transmitted beams remain unchanged; i.e., there are no phase changes. Altern... |

10 |
Tasks and super-tasks, Analysis 15
- Thomson
- 1954
(Show Context)
Citation Context ...ssues related to unbounded space or memory consumption discussed by Calude and Staiger [15], although we acknowledge their importance. In analogy to Benacerraf’s discussion [16] of Thomson’s proposal =-=[17, 18]-=- of a lamp which isswitched on or off at geometrically decreasing time delays, Shagrir [19] suggested that the physical state has little or no relation to the states in the previous acceleration proc... |

9 |
On the computational power of physical systems, undecidability, the consistency of phenomena and the practical uses of paradoxa
- Svozil
- 1995
(Show Context)
Citation Context ...finite process [20]. The method of diagonalization presents an important technique of recursion theory [21,22,23]. Some aspects of its physical realizability have already been discussed by the author =-=[24,25,26,27]-=-. In what follows we shall investigate some further physical issues related to “accelerated” agents and the operationalizability of the diagonalization operations in general. 2 Classical Brightness of... |

9 | Halting probability amplitude of quantum computers
- Svozil
- 1995
(Show Context)
Citation Context ...finite process [20]. The method of diagonalization presents an important technique of recursion theory [21,22,23]. Some aspects of its physical realizability have already been discussed by the author =-=[24,25,26,27]-=-. In what follows we shall investigate some further physical issues related to “accelerated” agents and the operationalizability of the diagonalization operations in general. 2 Classical Brightness of... |

8 |
Summable Series and Convergence Factors
- Moore
- 1938
(Show Context)
Citation Context ... and “off,” respectively, and the switching process with the concatenation of “+1” and “-1” performed so far, then the divergent infinite series associated with the Thomson lamp is the Leibniz series =-=[29, 30, 3, 31]-=- ∞∑ s = (−1) n = 1 − 1 + 1 − 1 + 1 − · · · A = 1 . (3) 2 n=0 Here, “A” indicates the Abel sum [3] obtained from a “continuation” of the geometric series, or alternatively, by s = 1−s. As this shows, f... |

8 |
Recurrence sequences, volume 104
- Everest, Poorten, et al.
- 2003
(Show Context)
Citation Context ... and “off,” respectively, and the switching process with the concatenation of “+1” and “-1” performed so far, then the divergent infinite series associated with the Thomson lamp is the Leibniz series =-=[29, 30, 3, 31]-=- ∞∑ s = (−1) n = 1 − 1 + 1 − 1 + 1 − · · · A = 1 . (3) 2 n=0 Here, “A” indicates the Abel sum [3] obtained from a “continuation” of the geometric series, or alternatively, by s = 1−s. As this shows, f... |

8 |
Fourth-order interference of joint single-photon wave packets in lossless optical systems
- Campos, Saleh, et al.
- 1990
(Show Context)
Citation Context ...1, eiϕ) in twodimensional Hilbert space. The action of the entire device consisting of such elements is calculated by multiplying the matrices in reverse order in which the quanta pass these elements =-=[53, 54]-=-; i.e., U bs ( ) ( ) ( ) ( iϕ e 0 i sin ω cos ω i(α+β) e 0 1 0 (ω, α, β, ϕ) = 0 1 cos ω i sin ω 0 1 0 eiβ ) ( ) i (α+β+ϕ) i (β+ϕ) (13) i e sin ω e cos ω = . e i (α+β) cos ω i e i β sin ω For this phys... |

8 |
Quantum phase tracing of correlated photons in optical multiports
- Reck, Zeilinger
- 1994
(Show Context)
Citation Context |

7 |
Consistent use of paradoxes in deriving contraints on the dynamics of physical systems and of no-go-theorems
- Svozil
- 1995
(Show Context)
Citation Context |

7 |
SU(2) and SU(1,1) interferometers
- Yurke, McCall, et al.
- 1985
(Show Context)
Citation Context ...1,e iϕ) in twodimensional Hilbert space. The action of the entire device consisting of such elements is calculated by multiplying the matrices in reverse order in which the quanta pass these elements =-=[53,54]-=-; i.e., 2 The standard labeling of the input and output ports are interchanged. Therefore, sine and cosine functions are exchanged in the transition matrix.On the Brightness of the Thomson Lamp 243 U... |

6 |
Zeno’s Paradoxes
- Salmon
- 1970
(Show Context)
Citation Context ... is shameful to base on them any demonstration whatsoever.” This, of course, has prevented no-one, in particular not Abel himself, from considering these issues. Indeed, by reviving old eleatic ideas =-=[5, 6]-=-, accelerated computations [7] have been the main paradigm of the fast growing field of hypercomputations (e.g., Refs. [8–10]). For the sake of hypercomputation, observers have been regarded on their ... |

6 |
Asymptotics and Borel Summability
- Costin
- 2009
(Show Context)
Citation Context ...ent sums could be valuable approximations of solutions of differential equations which might be even analytically solvable. One of the bestknown examples of this case is Euler’s differential equation =-=[32, 4, 33]-=- z 2 y ′ +y = z, which has both (i) a formal solution as a power series ˆ f(z) = − ∑ ∞ n=0 n! (−z)n , which diverges for all nonzero values of z; as well as (ii) an exact solution ˆf(z) = e 1/z Ei(−1/... |

6 |
The Undecidable. Basic Papers on Undecidable, Unsolvable Problems and Computable Functions
- Davis
- 1965
(Show Context)
Citation Context ...cal case and for the formal Abel sum, they represent a fifty-fifty mixture of the “on” and “off” states. 4 Quantum Fixed Point of Diagonalization Operator In set theory, logic and in recursion theory =-=[21,44,45,46,22,23]-=-, the method of proof by contradiction (reductio ad absurdum) requires some “switch of the bit value,” which is applied in a self-reflexive manner. Classically, total contradiction is achieved by supp... |

6 |
A.: The message of the quantum. Nature 438
- Zeilinger
- 2005
(Show Context)
Citation Context ... into the classical states corresponding to |0〉 and |1〉. Any single measurement yields an indeterministic result: According to the Born rule (cf. [47, p. 804], English translation in [48, p. 302] and =-=[49]-=-), when measured “along” the classical basis (observable) {|0〉, |1〉}, the probability that the fixed point state |ψ+〉 returns at random one of the two classical states |0〉 (exclusive) or |1〉, is 1/2. ... |

6 |
A Mathematical Introduction to Logic 2nd edn
- Enderton
- 2001
(Show Context)
Citation Context ...cal case and for the formal Abel sum, they represent a fifty-fifty mixture of the “on” and “off” states. 4 Quantum Fixed Point of Diagonalization Operator In set theory, logic and in recursion theory =-=[21,44,45,46,22,23]-=-, the method of proof by contradiction (reductio ad absurdum) requires some “switch of the bit value,” which is applied in a self-reflexive manner. Classically, total contradiction is achieved by supp... |

5 | Zeno machines and hypercomputation - Potgieter - 2006 |