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arXiv:0704.1613 · Journal of Physics A Mathematical and Theoretical

Reply to ‘Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’’

Rafael de la Madrid

Published Apr 11, 2007• 3 citations
arXivHTMLPDF

Abstract

In this reply, we show that when we apply standard distribution theory to the Lippmann–Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. Also, we point out several differences between the ‘standard method’ of constructing rigged Hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory.

DOI
10.1088/1751-8113/40/17/n02

Related papers

On the inconsistency of the Bohm–Gadella theory with quantum mechanics

Rafael de la Madrid

The Bohm–Gadella theory, sometimes referred to as the time asymmetric quantum theory of scattering and decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann–Schwinger equation are functionals over spaces of Hardy functions. The preparation–registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann–Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation–registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann–Schwinger equation nor should they appear. It is also shown that the Bohm–Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm–Gadella theory must therefore be abandoned.

arXiv:quant-ph/0606186•Journal of Physics A Mathematical and General•Jul 5, 2006•5 citations

Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’

Manuel Gadella, S. Wickramasekara

Rigged Hilbert spaces of Hardy functions lead to a consistent theory of resonance scattering and decay. Contrary to the claims of a recent article [8], the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances.

Journal of Physics A Mathematical and Theoretical•Apr 11, 2007•2 citations

On the inconsistency of the Bohm-Gadella theory with quantum mechanics

De la Madrid, R

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.

•Jun 22, 2006•4 citations

Remarks related to the paper of Rafael de la Madrid: "On the inconsistency of the Bohm-Gadella theory with quantum mechanics", JPhysA 39, No. 29, 9255-9268 (2006)

Hellmut Baumgaertel

The paper contains critical comments to the paper mentioned in the title from the mathematical point of view

arXiv:0704.3529•Apr 26, 2007•1 citation

Reply to Comment on "Surrealistic Bohm Trajectories"

Berthold‐Georg Englert, Marlan O. Scully, Georg Süssmann +1 more

Let us begin with a response to the summary of the Comment by D. Durr, W. Fusseder, S. Goldstein, and N. Zanghi [1]. We are accused to not stay inside orthodox quantum but to employ a naive, largely incoherent operationalism instead. We think that this charge is not justified, because we did apply standard quantum theory correctly and also followed the rules of Bohmian Mechanics (BM) carefully. In the following we react to the more essential points of the Comment in some detail. The Comment suggests that we have conceded many virtues to BM and that we have admitted that standard quantum theory is plagued by notorious conceptual difficulties. This is not the case. On the contrary, we maintain that BM is not needed to have the Schrodinger equation embedded into a physical theory. Standard quantum theory has already clarified the significance of Schrodinger's wave function as a tool used by theoreticians to arrive at probabalistic predictions. It is quite unnecessary, and indeed dangerous, to attribute any additional meaning to the (/^-function. The semantic difference between inconsistent and surrealistic is not the issue. It is the purpose of our paper to show clearly that the interpretation of the Bohm trajectory as the real retrodicted history of the atom observed on the screen is implausible, because this trajectory can be macroscopically at variance with the detected, actual way through the interferometer. And yes, we do have a framework to talk about path detection; it is based upon the local interaction of

Zeitschrift für Naturforschung A•Dec 1, 1993•24 citations

Reply to Comment on Experimental realization of a first test of de Broglie–Bohm theory

G. Brida, E. Cagliero, Marco Genovese +1 more

We reply to Akhavan and Golshani's comment on our double-slit experiment addressed to test standard quantum mechanics versus de Broglie–Bohm theory. We show that their results, albeit representing a relevant contribution to the discussion on the realizability of a test of de Broglie–Bohm theory, do not allow us to draw any conclusion about our experiment since their wavefunction does not describe well the photon states produced by our set-up. On the other hand, we give arguments of why our experiment can be considered a good realization of the theoretical proposal of Ghose (1999 Foundations of Quantum Theory and Quantum Optics ed S M Roy), Ghose et al (2001 Phys. Lett. A 290 205) and Golshani and Akhavan ( 2001 J. Phys. A: Math. Gen. 34 5259 ).

Journal of Physics B Atomic Molecular and Optical Physics•Sep 15, 2004•6 citations
In this reply, we show that when we apply standard distribution theory to the Lippmann–Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. Also, we point out several differences between the ‘standard method’ of constructing rigged Hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory.
The Bohm–Gadella theory, sometimes referred to as the time asymmetric quantum theory of scattering and decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann–Schwinger equation are functionals over spaces of Hardy functions. The preparation–registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann–Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation–registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann–Schwinger equation nor should they appear. It is also shown that the Bohm–Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm–Gadella theory must therefore be abandoned.
The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.
Let us begin with a response to the summary of the Comment by D. Durr, W. Fusseder, S. Goldstein, and N. Zanghi [1]. We are accused to not stay inside orthodox quantum but to employ a naive, largely incoherent operationalism instead. We think that this charge is not justified, because we did apply standard quantum theory correctly and also followed the rules of Bohmian Mechanics (BM) carefully. In the following we react to the more essential points of the Comment in some detail. The Comment suggests that we have conceded many virtues to BM and that we have admitted that standard quantum theory is plagued by notorious conceptual difficulties. This is not the case. On the contrary, we maintain that BM is not needed to have the Schrodinger equation embedded into a physical theory. Standard quantum theory has already clarified the significance of Schrodinger's wave function as a tool used by theoreticians to arrive at probabalistic predictions. It is quite unnecessary, and indeed dangerous, to attribute any additional meaning to the (/^-function. The semantic difference between inconsistent and surrealistic is not the issue. It is the purpose of our paper to show clearly that the interpretation of the Bohm trajectory as the real retrodicted history of the atom observed on the screen is implausible, because this trajectory can be macroscopically at variance with the detected, actual way through the interferometer. And yes, we do have a framework to talk about path detection; it is based upon the local interaction of
We reply to Akhavan and Golshani's comment on our double-slit experiment addressed to test standard quantum mechanics versus de Broglie–Bohm theory. We show that their results, albeit representing a relevant contribution to the discussion on the realizability of a test of de Broglie–Bohm theory, do not allow us to draw any conclusion about our experiment since their wavefunction does not describe well the photon states produced by our set-up. On the other hand, we give arguments of why our experiment can be considered a good realization of the theoretical proposal of Ghose (1999 Foundations of Quantum Theory and Quantum Optics ed S M Roy), Ghose et al (2001 Phys. Lett. A 290 205) and Golshani and Akhavan ( 2001 J. Phys. A: Math. Gen. 34 5259 ).