Gauge transformations and conserved quantities in classical and quantum mechanics

Gauge invariance in both quantum and classical mechanics is investigated using the Feynman path integral formulation of quantum mechanics. The Feynman path integral expresses the wave function in terms of the Lagrangian. A gauge transformation in classical mechanics is the addition of a total derivative of a gauge function to the Lagrangian. A gauge transformation of the second and first kind in quantum mechanics is obtained by adding and subtracting, respectively, a total derivative to the Lagrangian in the Feynman path integral.

Introduction. The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be hypermaximal(l) or self-adjoint(2) in the strict sense employed in the theory of Hilbert space, which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, that there exists a complete set (discrete or continuous) of eigenfunctions. It will, therefore, be one of the important problems of mathematical physics to show that this is actually the case for all operators appearing in current structure of quantum mechanics, for the self-adjointness of an operator is a property requiring careful separate consideration and by no means a direct consequence of its Hermitian symmetry [cf. (S), chap. X]. The problem has, of course, been solved in the case of operators for which the eigenvalue problem is explicitly solved by separation of variables or other methods, but it seems not to have been settled in the general case of manyparticle systems('). The main purpose of the present paper is to show that the Schrodinger Hamiltonian operator(4) of every atom, molecule, or ion, in short, of every system composed of a finite number of particles interacting with each other through a potential energy, for instance, of Coulomb type, is essentially selfadjoint(5) (6) . Thus our result serves as a mathematical basis for all theoretical works concerning nonrelativistic quantum mechanics, for they always pre-

A review of Wigner's time reversal is presented and some important aspects are emphasized. The subject is introduced via classical mechanics. Non-physical statements as “time running backwards” are avoided. Comments are made on the roles of time and of the operatoriħ(∂/∂t) in quantum mechanics. The role of symmetries and conservation laws and some properties of the time-reversed states are discussed.

We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given.

Physically measurable quantities in quantum mechanics are associated with expectation values or matrix elements of the corresponding operators. It is therefore of fundamental importance to elaborate efficient methods for the calculation of such matrix elements. One of the most attractive features of the second quantized approach is its simplicity in evaluating matrix elements. To appreciate this, one must get some practice in the formalism.

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se, suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces.

This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether’s theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics, this mapping is multiplication by the square root of ?1. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a ‘dynamical correspondence’ and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.

Two types of prediction in Newtonian and quantum mechanics

Different approaches to the study of open linear freely-jointed (unhindered) threedimensional macromolecular chains at thermal equilibrium, based upon either Classical Hamiltonian Dynamics with constraints (namely, a formulation equivalent to Kramers' one) or Quantum Mechanics (with stiff harmonic springs), are reviewed and discussed in detail. Results from those approaches are compared to those arising from the standard Gaussian model. Fraenkel's approach, based upon Classical Mechanics (with stiff harmonic springs) and its comparison with the Gaussian model are also treated. It is argued that the quantum formulation is required in order that the treatment of the high-frequency vibrations be consistent. The role of the uncertainty principle is discussed, in support of the quantum-mechanical approach. At a later stage, internal rotations can be treated through Classical Statistical mechanics, under certain conditions. The quantum-Mechanics-based model appears to be consistent with the Gaussian model, while the one based upon constrained Hamiltonian Dynamics yields certain discrepancies with the latter. Classical and quantum-mechanical approaches to open linear freely-rotating (hindered) threedimensional macromolecular chains are also treated: the consistency of the quantum-mechanics-based model is displayed by deriving approximately the existence of a persistence length. It is argued that the possible advantages of the quantum-mechanical formulation should be expected not at large scales but at shorter ones. The specific topics to be treated are described in the summary of contents given below.

A simple $U(1)$ gauge transformation, which is equivalent to the inclusion of the Aharonov-Bohm type potential, is introduced and applied to superintegrable systems in classical and quantum mechanics. It produces an inverse-square potential with two- and three-particle interactions which is identical to the potential of the recently studied truncated Calogero model.

One of the most striking features of quantum mechanics is the profound effect exerted by measurements alone. Sophisticated quantum control is now available in several experimental systems, exposing discrepancies between quantum and classical mechanics whenever measurement induces disturbance of the interrogated system. In practice, such discrepancies may frequently be explained as the back-action required by quantum mechanics adding quantum noise to a classical signal. Here, we implement the "three-box" quantum game [Aharonov Y, et al. (1991) J Phys A Math Gen 24(10):2315-2328] by using state-of-the-art control and measurement of the nitrogen vacancy center in diamond. In this protocol, the back-action of quantum measurements adds no detectable disturbance to the classical description of the game. Quantum and classical mechanics then make contradictory predictions for the same experimental procedure; however, classical observers are unable to invoke measurement-induced disturbance to explain the discrepancy. We quantify the residual disturbance of our measurements and obtain data that rule out any classical model by ≳7.8 standard deviations, allowing us to exclude the property of macroscopic state definiteness from our system. Our experiment is then equivalent to the test of quantum noncontextuality [Kochen S, Specker E (1967) J Math Mech 17(1):59-87] that successfully addresses the measurement detectability loophole.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.