Abstract The purpose of this paper show how mathematical concepts not designed for physics can muddle understanding and/or prevent progress. We provide the following examples to justify our position: 1. A proof that Minkowski’s (third) postulate is incompatible with Einstein’s first postulate for two or more particles. 2. A proof that the Dirac equation is not physically equivalent to the square root equation but is related by a unitary transformation. 3. The design of a Hilbert space for Feynman’s path integral formulation of quantum theory KS 2[ℝ n ], which is not physically equivalent to L 2[ℝ n ], the Hilbert space for the Schödinger formulation, but is related by a unitary transformation. A major contribution of this paper is that mathematical equivalence is not the same as physical equivalence and that unbiased physical analysis and justification of mathematical concepts should be required before acceptance into physical theory.
Abstract The purpose of this paper show how mathematical concepts not designed for physics can muddle understanding and/or prevent progress. We provide the following examples to justify our position: 1. A proof that Minkowski’s (third) postulate is incompatible with Einstein’s first postulate for two or more particles. 2. A proof that the Dirac equation is not physically equivalent to the square root equation but is related by a unitary transformation. 3. The design of a Hilbert space for Feynman’s path integral formulation of quantum theory KS 2[ℝ n ], which is not physically equivalent to L 2[ℝ n ], the Hilbert space for the Schödinger formulation, but is related by a unitary transformation. A major contribution of this paper is that mathematical equivalence is not the same as physical equivalence and that unbiased physical analysis and justification of mathematical concepts should be required before acceptance into physical theory.
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.
Synopsis: The mathematics behind the deceptively simple ideas encountered in the spin measurements from the first chapter is further developed. Technically, we limit ourselves to finite dimensional Hilbert spaces, although exceptions are pointed out along the way. The operator/matrix analogy is developed, and important physical and mathematical concepts are defined and refined. These concepts include operator compatibility, uncertainty relations, and unitary transformations. Unitary transformations are applied to coordinate transformations and used to develop the Heisenberg time-development picture in quantum mechanics.
Quantum theory (QT) is usually formulated in terms of abstract mathematical postulates involving Hilbert spaces, state vectors and unitary operators. In this paper, we show that the full formalism of QT can instead be derived from five simple physical requirements, based on elementary assumptions regarding preparations, transformations and measurements. This is very similar to the usual formulation of special relativity, where two simple physical requirements—the principles of relativity and light speed invariance—are used to derive the mathematical structure of Minkowski space–time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three dimensionality) and suggests several natural possibilities to construct consistent modifications of QT.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.
In this paper, we argue that there are foundational dilemmas in theoretical physics related to the concept of reality and the nature of mathematics in physics. Physical theory is treated as a conceptual organism which develops under the weight of its internal contradictions. The paper discusses in depth the problem of objective reality in physics and its relation to scientific practice. Then, it explores the problematic relation between physical meaning and mathematics in modern physical theory, followed by a discussion of the trend of contemporary physics to replace physical principles with pure mathematical principles. Finally, it discusses the problem of logical coherence in modern physical theory. The paper emphasizes the importance of resolving these dilemmas to the proper practice of theoretical physics.