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Carl M Bender - One of the best experts on this subject based on the ideXlab platform.

  • making sense of non hermitian hamiltonians
    Reports on Progress in Physics, 2007
    Co-Authors: Carl M Bender
    Abstract:

    The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A Standard Axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical Axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

  • making sense of non hermitian hamiltonians
    arXiv: High Energy Physics - Theory, 2007
    Co-Authors: Carl M Bender
    Abstract:

    The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A Standard Axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical Axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

Schatz, Jeffrey Robert - One of the best experts on this subject based on the ideXlab platform.

  • Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms
    eScholarship University of California, 2019
    Co-Authors: Schatz, Jeffrey Robert
    Abstract:

    This dissertation explores the justification of strong theories of sets extending Zeremelo-Fraenkel set theory with choice and large cardinal Axioms. In particular, there are two noted program providing Axioms extending this theory: the inner model program and the forcing Axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred Axiom candidate on the basis of its supposed maximization potential. Since the maxim of ‘maximize’ proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This dissertation takes up just this project.The first chapter of this dissertation describes the history of Axiom selection in set theory, focusing on developments since 1980 which have led to the two Standard Axiom candidates for extending ZFC+LCs: V = Ult(L) and Martin’s Maximum. The second chapter explains the justification of the methodological maxim of ‘maximize’ as an informal principle, and presents two formal explications of the notion: one due to John Steel, the other to Penelope Maddy. The third chapter directly examines whether either approach to Axioms can be truly said to maximize over the other. It is shown that the Axiom candidates are equivalent in Steel’s sense of ‘maximize’, while in Maddy’s sense of ‘maximize’, Martin’s Maximum is found to maximize over V = Ult(L). Given the strong justification of Maddy’s explication in terms of the goals of set theory as a foundational discipline, it is argued that this result raises a serious justificatory challenge for advocates of the inner model program. The fourth chapter considers future directions of research, focusing on possible responses to the justificatory challenge, and highlighting issues that must be overcome before a full justificatory story of forcing Axioms can be developed

Jeffrey Robert Schatz - One of the best experts on this subject based on the ideXlab platform.

  • Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms
    2019
    Co-Authors: Jeffrey Robert Schatz
    Abstract:

    Author(s): Schatz, Jeffrey Robert | Advisor(s): Maddy, Penelope | Abstract: This dissertation explores the justification of strong theories of sets extending Zeremelo-Fraenkel set theory with choice and large cardinal Axioms. In particular, there are two noted program providing Axioms extending this theory: the inner model program and the forcing Axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred Axiom candidate on the basis of its supposed maximization potential. Since the maxim of ‘maximize’ proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This dissertation takes up just this project.The first chapter of this dissertation describes the history of Axiom selection in set theory, focusing on developments since 1980 which have led to the two Standard Axiom candidates for extending ZFC+LCs: V = Ult(L) and Martin’s Maximum. The second chapter explains the justification of the methodological maxim of ‘maximize’ as an informal principle, and presents two formal explications of the notion: one due to John Steel, the other to Penelope Maddy. The third chapter directly examines whether either approach to Axioms can be truly said to maximize over the other. It is shown that the Axiom candidates are equivalent in Steel’s sense of ‘maximize’, while in Maddy’s sense of ‘maximize’, Martin’s Maximum is found to maximize over V = Ult(L). Given the strong justification of Maddy’s explication in terms of the goals of set theory as a foundational discipline, it is argued that this result raises a serious justificatory challenge for advocates of the inner model program. The fourth chapter considers future directions of research, focusing on possible responses to the justificatory challenge, and highlighting issues that must be overcome before a full justificatory story of forcing Axioms can be developed.

Robert F. Nau - One of the best experts on this subject based on the ideXlab platform.

  • Indeterminate Probabilities on Finite Sets
    The Annals of Statistics, 1992
    Co-Authors: Robert F. Nau
    Abstract:

    This paper presents a quasi-Bayesian model of subjective uncertainty in which beliefs which are represented by lower and upper probabilities qualified by numerical confidence weights. The representation is derived from a system of Axioms of binary preferences which differs from Standard Axiom systems insofar as completeness is not assumed and transitivity is weakened. Confidence-weighted probabilities may be elicited through the acceptance of bets with limited stakes, a generalization of the operational method of de Finetti. The model is applicable to the reconciliation of inconsistent probability judgments and to the sensitivity analysis of Bayesian decision models.

Zuhair Aljohar - One of the best experts on this subject based on the ideXlab platform.

  • zf extensionality interprets full zf
    arXiv: Logic, 2020
    Co-Authors: Zuhair Aljohar
    Abstract:

    Dana Scott had shown that removing Extensionality from ZF set theory formalized in the customary manner would weaken it down to Zermelo set theory. The following proof is my personal attempt to solve the question of whether we can have a version of replacement that can withstand removal of Extensionality when ZF is formalized with it instead of the Standard Axiom schema of replacement. Dana Scott had investigated that, the general lines of the proof here are similar to his; however, the form of Replacement suggested here is different.