Time: And time again

Finding ‘‘real’’ time in quantum mechanics

The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.

Time, classical and quantum

called the special theory of relativity, no doubt one of the boldest and most interesting theories ever designed.' Generations of physicists, mathematicians and philosophers were inspired by it. Bitter battles were fought over its principles and deductions which clashed with many an honourable idea backed up by honourable philosophical and scientific traditions. Yet all the knocks and bites from left and right seem to have left Einstein's theory unshaken. Today the battle-grounds of yesterday look rather deserted. The critic is likely to be silenced in the face of the theory's spectacular success. In the following I shall try to throw some new light on one or two old problems which Einstein's theory gives rise to. Section z is concerned with the obstacles we are faced with when we try to combine Einstein's special theory of relativity with a causal account of relativistic phenomena. The argument does not show, of course, that Einstein's theory is false, but it points to a possible incompleteness and to certain limitations inherent in the special theory of relativity. In section 3 a substratum theory of relativity is outlined which seems to be as simple and coherent as Einstein's special theory of relativity; the Lorentz transformations are deduced from two basic principles without making use of the principle of relativity. Finally, in section 4, the two theories are compared and some general conclusions are drawn. But let us first have a short look at the ideas underlying Einstein's theory. To begin with, the special theory of relativity is a transformation theory. This is its most fundamental aspect. Suppose we know the space and time co-ordinates of an event particle e (event with negligible spatiotemporal extension) in an inertial frame of reference F; then the theory

The formulation of the wave function of a single photon has always been in a con ict with the spatial localization of a photon. Lack of the correct treatment for the probability and current density of a single photon resisted self-consistent treatment of the wave function of a single photon. Here we proceed with the construction of the wave function for a single photon that allows us to introduce probability and current density of a single photon. The wave function of a single photon is constructed from the matrix elements of the electric and magnetic elds that couple vacuum state with the Fock state occupied by a single photon. Further we show that the spin of a photon, being projected on the direction of the photon ight propagation, de nes the time evolution of the photon wave function. As the result it can be identi ed with the Hamiltonian of a single photon that establishes Schrodinger equation for a single photon. The Schrodinger description naturally leads to the current and the probability density for a single photon that satis es the continuity equation. The Schrodinger formulation of the Maxwell equations provides a clear physical meaning for the spin of a photon in a similar way to the Dirac equation provides a natural explanation for the spin of an electron.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Time in quantum mechanics

Time has always been a fundamental yet enigmatic concept in physics. From the arrow of time in thermodynamics to the timeless nature of quantum mechanics, understanding time's essence has challenged the foundations of modern physics. In this paper, we propose a novel interpretation of time in quantum mechanics, distinguishing between real time as observable events (wave function collapse or address changes) and imaginary time as the continuous evolution of the quantum state. By connecting this framework with the holographic principle, we aim to provide insights into the nature of time, its arrow, and its relationship with quantum mechanics. Keywords Thermodynamics; Wave function; Quantum mechanics; Holographic principle

Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. The lectures covered quantum mechanics for closed systems like the universe, generalized quantum mechanics, time in quantum mechanics, the quantum mechanics spacetime, and practical quantum cosmology. References have not been updated.

مقدمه: اصول و قوانین، نقش بیان اساسی‌ترین ایده‌ها در یک علم و نیز ایجاد چارچوب و روش‌شناسی برای حل مسائل آن علم را بازی می‌کنند. این مقاله بر آن است تا با گذری بر مفاهیم و مراتب چهارگانه سیبرنتیک، به بررسی اصول حاکم بر این علم بر اساس رویکردهای فیزیک جدید و قدیم بپردازد. روش‌: در نگارش مقاله رویکرد مروری‌تحلیلی مورد استفاده قرار گرفته است. این مقاله سعی دارد با بررسی متون علمی این حوزه از طریق روش کتابخانه‌ای، ابتدا گذری بر مفاهیم و مراتب چهارگانه سیبرنتیک داشته باشد، سپس اصول حاکم بر این علم از دو دیدگاه پوزیتیویستی و هرمنوتیکی بررسی نماید. یافته ها: نحوه نگرش به جهان هستی موجب شکل‌گیری دیدگاه‌های مختلف در علم و در عین حال پیشرفت شناخت علمی شده‌است. در ابتدا دیدگاه پوزیتویستی بر اساس تئوری نیوتون و دیدگاه مکانیکی شکل گرفت. سپس پسا اثبات‌گرایان با توجه به دو اصل ضرورت و تصادف قوانین زیست‌شناسی را پی‌ریزی کردند. با ورود تئوری کوانتوم و نسبیت به عرصه فیزیک و آشکار شدن ناتوانی‌های فیزیک کلاسیک در چند مورد، برداشت از جهان و پدیده‌های آن به‌کلی متحول شد. گذر از فیزیک نیوتنی (فیزیک قدیم) به فیزیک کوانتومی (فیزیک جدید)، تحول تئوریک و فلسفی را موجب گردید. در سایه این چرخش، مبانی تئوریک علوم مختلف نیز دگرگون گردید. سیبرنتیک نیز از این نگاه متحولانه بهره‌مند شده‌است. تحلیل متون نشان می‌دهد، بن‌مایه فکری نسل‌ها و اصول سیبرنتیک آمیزه‌ای از رویکردهای پوزیتویستی، ارگانیستی، جهان‌بینی سیستمی، و فیزیک جدید است. بحث و نتیجه گیری: راه‌حل‌های معمول سنتی دیگر جواب‌گوی آشفتگی‌های موجود در دنیای کنونی نیست و به راه‌حل‌های جدیدی نیازمندیم تا بتوانیم پیچیدگی‌های موجود در این دنیا و عصر حاضر را کنترل نماییم. سیبرنتیک یکی از این راه‌حل‌هاست. تفاوت در نوع نگاه به مفهوم مشاهده و مشاهده‌گر است که باعث شکل‌گیری رویکردهای متفاوت و جهان‌بینی‌های مختلف در علوم و متعاقب آن سیبرنتیک شده است. تا جایی که نقطه عطف شکل‌گیری فیزیک جدید را می‌توان مداخله دادن مشاهده‌گر در رویداد دانست. جهانی که در آن مشاهده‌کننده جدای از آنچه مشاهده می‌کند نیست، بلکه قسمتی از همان پدیده‌ است در حالی که در فیزیک کلاسیک تمام پدیده‌ها مستقل از مشاهده‌گر همچون ساعتی دقیق و قانون‌مدار با نظم و ترتیب خاص رخ می‌دهند. بنابراین می توان گفت بن‌مایه فکری نسل‌ها و اصول سیبرنتیک آینده‌ای از رویکردهای پوزیتویستی، ارگانیستی، جهان‌بینی سیستمی و فیزیک جدید است. بر این اساس رویکردی سیبرنتیکی در طی عمر کوتاه آن متکی بر دیدگاه‌های حاکم متحول شده است.

Since the advent of Modern Physics in 1905, we observe an increasing activity of “interpreting” the new theories. We mention here the theories of Special Relativity, General Relativity and Quantum Mechanics. However, similar activities for the theories of Classical Physics were not known. We ask for the reasons for the different ways to treat classical physics and modern physics. The answer, that we provide here is very surprising: the different treatments are based on a fundamental misunderstanding of the theories of classical physics.

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc. In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz. In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time. In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements. For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.

Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.

While direct-interaction particle theories are generally thought to be incompatible with relativity in classical physics, such relativistic theories in quantum mechanics have recently attracted attention. The reasons for rejecting these theories in classical physics are based on the consideration of world lines, while relativistic quantum mechanics has no covariant position operator so that the classical refuting argument cannot be adapted.

In Hawking’s view physics has been broken up into many partial theories, while the ultimate goal of physicists is to unify them. The two basic theories of 20th-century physics, relativity theory and quantum theory, are based on completely different logical prerequisites and exactly separate: matter is described as particles in relativity theory and as waves in quantum mechanics. Here, based on the identical logical prerequisites, we unify Einstein’s special relativity (SR) and de Broglie’s matter-wave theory (MWT) into the theory of dualistic relativity (DR), taking a significant step toward the unification of relativity and quantum mechanics. From the definition of time, we derive the Lorentz transformation in differential form and establish the theory of DR, which generalizes the wave-particle duality of matter motion, and uniformly derives Einstein’s formula E=mc2, Planck’s equation E=hf, and de Broglie’s relation λ=h/p. From the logical prerequisite completely different from Einstein’s hypothesis of the invariance of light speed and along the logical path completely different from Einstein’s SR, we have deduced the whole theoretical system of Einstein’s SR and de Broglie’s MWT. In the theory of DR, the two great formulae originally separated, Einstein’s formula E=mc2 and Planck’s equation E=hf, become a pair of twin formulae unified in an identical theoretical system.