This thesis concerns the following topics: geometric phase in the context of Galilean invariance and quantum measurements, Rydberg states of hydrogen atoms, vibronic coupling in the E A e Jahn-Teller system and realism in quantum computations. In the analyses the de Broglie-Bohm pilot-wave formulation of quantummechanics is mainly used.It is shown that geometric phase is not Galilean invariant. Experimental implications are discussed and it is found that the experiments performed to date are frame independent. An experiment which is in principle able to detect the noninvariance is sketched. By adopting the measurement theory of the pilot-wave formulation it is shown how the measurement induced geometric phase continuously emerges. The Samuel-Bhandari geometric phase is identified as the nonrandom part of the total geometric phase induced in the measurement.Ensembles of particles for a circular Rydberg wave packet are studied. The trajectories of pilot-wave particles are shown to accurately imitate the behaviour of the wave packet in the high quantum number limit. The nonclassical features of the wave packet are intuitively explained by the nonvanishing quantum potential.Vibronic coupling in the Longuet-Higgins model of the E A e Jahn-Teller system is investigated by means of quantal trajectories. The pilot-wave picture provides an intuitive tool for discussing time-scales. An argument based on ergodicity leads to an understanding of the averaging procedure over the electronic motion which provides the approximate nuclear motion.The existence of efficient quantum algorithms triggers questions on Natures ability of storing and processing information during quantum computations. The role of elements of reality in quantum computations is addressed using quantal trajectories. It is found that there is a many-to-one relationship between quantal trajectories and performed computations when quantum parallelism is utilized.This thesis concerns the following topics: geometric phase in the context of Galilean invariance and quantum measurements, Rydberg states of hydrogen atoms, vibronic coupling in the E A e Jahn-Teller system and realism in quantum computations. In the analyses the de Broglie-Bohm pilot-wave formulation of quantummechanics is mainly used.It is shown that geometric phase is not Galilean invariant. Experimental implications are discussed and it is found that the experiments performed to date are frame independent. An experiment which is in principle able to detect the noninvariance is sketched. By adopting the measurement theory of the pilot-wave formulation it is shown how the measurement induced geometric phase continuously emerges. The Samuel-Bhandari geometric phase is identified as the nonrandom part of the total geometric phase induced in the measurement.Ensembles of particles for a circular Rydberg wave packet are studied. The trajectories of pilot-wave particles are shown to accurately imitate the behaviour of the wave packet in the high quantum number limit. The nonclassical features of the wave packet are intuitively explained by the nonvanishing quantum potential.Vibronic coupling in the Longuet-Higgins model of the E A e Jahn-Teller system is investigated by means of quantal trajectories. The pilot-wave picture provides an intuitive tool for discussing time-scales. An argument based on ergodicity leads to an understanding of the averaging procedure over the electronic motion which provides the approximate nuclear motion.The existence of efficient quantum algorithms triggers questions on Natures ability of storing and processing information during quantum computations. The role of elements of reality in quantum computations is addressed using quantal trajectories. It is found that there is a many-to-one relationship between quantal trajectories and performed computations when quantum parallelism is utilized.

Jan 1, 19980 citations

Foundations of Statistical Mechanics and the Status of the Born Rule in de Broglie-Bohm Pilot-Wave Theory

Antony Valentini

We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest 'empirical approach' to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest 'empirical approach' to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.

arXiv:1906.10761WORLD SCIENTIFIC eBooksApr 21, 202034 citations

Pilot-Wave Theory Without Nonlocality

Paul Tappenden

arXiv:2209.05159Foundations of PhysicsOct 1, 20223 citations

Local Scale Invariance in Quantum Theory: A Non-Hermitian Pilot-Wave Formulation

Indrajit Sen, Matthew Leifer

We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (de Broglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|ψ|^2$ to the local scale invariant, trajectory-dependent ratio $|ψ|^2/ \mathbf 1^2[\mathcal C]$ , where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (de Broglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|ψ|^2$ to the local scale invariant, trajectory-dependent ratio $|ψ|^2/ \mathbf 1^2[\mathcal C]$, where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.

quant-pharXiv:2601.03567Jan 8, 20260 citations

|ψ|^2

to the local scale invariant, trajectory-dependent ratio

|ψ|^2/ \mathbf 1^2[\mathcal C]

, where

\mathbf 1[\mathcal C]

is a scale factor that depends on the pilot-wave trajectory

\mathcal C

in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.