Measure and Integration in Quantum Set Theory

We perform a dynamical symmetry analysis (DSA) of the high-order harmonic generation (HHG) spectrum of an atom interacting with a bichromatic laser field. Within the framework of the conventional Hermitian quantum mechanics (QM), the HHG spectrum calculated using a single Floquet state, or any finite number of Floquet states, is invariant under the inversion of the relative phase of the two-frequency components, $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$. The asymmetry with respect to the phase inversion seen in the simulated HHG spectra is obtained in the conventional QM only when the Floquet spectrum is continuous and ionization is taken into consideration. However, when the Hamiltonian is complex scaled the description is different. Even a single eigenstate of the complex scaled Floquet operator is enough to describe the breaking of the $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$ symmetry in the HHG spectra. We find that there is a direct correlation between the strength of the asymmetry with respect to the relative phase inversion and the magnitude of the ionization rate. For illustration purposes, the DSA is accompanied by the results obtained for a one-dimensional effective single-electron model Hamiltonian mimicking xenon atom interacting with strong laser field.

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.

The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under D-dimensional Mobius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global Mobius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the Möbius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrodinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L 2(Rr,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field operators. We shall show that the isomorphism between the exponential ofV andL 2(Rr,dx) can be understood as the decomposition by generalized eigenvectors of field operators (“Fourier transform”).

The paper shows that the past, “history” in a non-technical sense, can be changed in quantum mechanics. The first part of the paper reviews Deutsch's analysis in his paper of 1991. It is demonstrated that Deutsch assumes the existence of a multiplicity of essentially classical worlds. Such a multiplicity of worlds would allow the past to be changed in classical mechanics. It is argued that the existence of multiple “classical” worlds is not required by quantum mechanics. It is then shown that it is possible to change the past in conventional quantum mechanics even without the assumption of a multiplicity of worlds.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Carl M. Bender; The complex pendulum. AIP Conf. Proc. 15 December 1998; 453 (1): 167–176. https://doi.org/10.1063/1.57117 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

Realism and time symmetry in quantum mechanics

A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian (J-self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨n, n⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J-self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.

Matrix theory [1] is a nonperturbative theory of fundamental processes which evolved out of the older perturbative string theory. There are two well-known formulations of string theory, one covariant and one in the so-called light cone frame [2]. Each has its advantages. In the covariant theory, relativistic invariance is manifest, a euclidean continuation exists and the analytic properties of the S matrix are apparent. This makes it relatively easy to derive properties like CPT and crossing symmetry. What is less clear is that the theory satisfies all the rules of conventional unitary quantum mechanics. In the light cone formulation [3], the theory is manifestly hamiltonian quantum mechanics, with a distinct non-relativistic flavor. There exists a space of states and operators which is completely gauge invariant and there are no ghosts or unphysical degreees of freedom. Among other things, this makes it easy to identify the spectrum of states. Matrix theory is also formulated in the light cone frame and enjoys the same advantages and disadvantages as LCF string theory. Unlike perturbative string theory, matrix theory is capable of addressing many of the nonperturbative questions raised by string theory and quantum gravity.

Abstract

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

We derive the Bell–Clauser–Horne–Shimony–Holt inequalities for two-particle mixed spin states both in the conventional quantum mechanics and in the hidden-variables theory. We consider two cases for the vectors $$\vec a,\vec b,\vec c$$ , and $$\vec d$$ specifying the axes onto which the particle spins of a correlated pair are projected. In the first case, all four vectors lie in the same plane, and in the second case, they are oriented arbitrarily. We compare the obtained inequalities and show that the difference between the predictions of the two theories is less for mixed states than for pure states. We find that the inequalities obtained in quantum mechanics and the hidden-variables theory coincide for some special states, in particular, for the mixed states formed by pure factorable states. We discuss the points of similarity and difference between the uncertainty relations and Bell's inequalities. We list all the states for which the right-hand side of the Bell–Clauser–Horne–Shimony–Holt inequality is identically equal to zero.