## 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.

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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/... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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 ... |

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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/... |

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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... |

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