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Physics of Unitary Transformations

K. Kong Wan

Published Jul 9, 2019• 0 citations
Publisher

Abstract

A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.

DOI
10.1201/b22504-23

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A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.
Vectors and operators in a Hilbert space are independent quantities and there is no obvious link between them. The mathematical description of the states and observables of a quantum system given by Postulates 25.1(PS) and 26.1(OV) in terms of vectors and operators does not tell us how a given state would determine the value of an observable. Physically the relationship between states and observables is about how the measurable values of an observable is related to a state. This relationship should lead to the quantum properties QMP5.3(1) to QMP5.3(3). First we must know how a state can give rise to a probability distribution of measurable values of an observable. This would require a prescription to relate a unit vector to the spectrum of selfadjoint operators. The discussion in Chapter 22 tells us how to obtain such a prescription. Because of the intricacy of the relationship both physically and mathematically we shall set out the relationship for discrete and continuous observables separately.
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
In this thesis we explore the mathematical foundations that unite physics at a quantum scale, quantum mechanics, with a macroscopic scale, classical mechanics. We seek to understand the mathematical motivation behind the quantum-classical correspondence and how it unites two seemingly different theories of the physical world. We show how this correspondence binds the Hamiltonian theory of classical physics to the Hilbert space theory in quantum mechanics, and establish a way to translate between classical observables and quantum operators, using the Fourier transform. These approaches to “quantizing” a physical state can be applied generally to a wide variety of observable quantities in classical mechanics.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.