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Quantum mechanics in classical dynamics
1.See, for example, B. H. Bransden and C. J. Joachain, Introduction to Quantum Mechanics (Longman Scientific and Technical, Harlow, 1989).
2.For a survey of two approaches other than that presented here, see S. Bugajski, “Classical frames for a quantum theory—A bird’s-eye view,” Int. J. Theoret. Phys. 32, 969 (1993).
2.A recent development in one of these approaches is given in E. G. Beltrametti, and S. Bugajski, “A classical extension of quantum mechanics,” J. Phys. A 28, 3329(1995).
3.F. Strocchi, “Complex coordinates and quantum mechanics,” Rev. Mod. Phys. 38, 36 (1966).
4.D. J. Rowe, A. Ryman, and G. Rosensteel, “Many-body quantum mechanics as a symplectic dynamical system,” Phys. Rev. A 40, 2362 (1980).
5.A. Heslot, “Quantum mechanics as a classical theory,” Phys. Rev. D 31, 1341 (1985).
6.K. R. W. Jones, “Classical mechanics as an example of generalised quantum mechanics,” Phys. Rev. D 45, 2590 (1992).
7.In the interest of brevity, and since this material is not the subject of this paper, the terms used will not be defined here, but the reader is referred to, for instance, S. MacLane and G. Birkhoff, Algebra (MacMillan, New York, 1979). Unlike some definitions of rings, here it will not be assumed at the outset that the algebra’s multiplication is associative.
8.A good introduction to Clifford algebras and their many applications in physics is contained in B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, Singapore, 1989).
9.The term real is used instead of self-conjugate for the sake of brevity and because it is already familiar in this meaning. It is important to note that real is not the same as real-valued, a term which will be introduced shortly.
10.With this definition, if a complex number , with x and y elements of R, the imaginary part is iy, not just y.
11.A set of states is separable if there is a dense sequence of states that arbitrarily closely approximates any state. A set of states is complete if every Cauchy sequence of states has a limit which is also a state.
12.S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995).
13.This paper will only concern itself with such ring linear operators, and so “ring linear” will not be explicitly written every time.
14.For a commutative algebra, transposition is also an antiautomorphism.
15.Adler (Ref. 12) discusses this in the case of quantum mechanics using quaternions.
16.It should be noted that Greek indices will refer to the real-valued coordinates and will obey the Einstein summation convention. Indices from the second half of the Roman alphabet will refer to the orthonormal basis being used to expand a general wave function, while indices from the first half will refer to the ring generators.
17.The converse statement is not generally true. For example, the Clifford algebra in , generated from , where and , is not a division algebra— and square to zero—even though each generator has an inverse.
18.In this treatment of quantum mechanics, it is nowhere necessary to normalize wave functions, since it is the dynamical aspects that are of primary concern and not the issue of measurement via the usual statistical interpretation. It is not, therefore, a problem that, in cases where the ring is not a division algebra, the norm of a nonzero state is not necessarily nonzero.
19.Rowe, Ryman, and Rosensteel (Ref. 4) obtain the analogous result (in quantum mechanics using complex numbers) for a self-adjoint Hamiltonian, using a variational principle due to Dirac, that also leads to Eq. (4).
20.See, for example, L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics, Vol. 1 (Pergamon, Oxford, 1976).
21.See, for example, V. I. Arnold, Mathematical Methods of Classical Physics (Springer-Verlag, New York, 1978).
22.It is generally assumed that the quantum Poisson bracket is (up to a constant factor) the commutator as part of the correspondence principle—see, for example, P. A. M. Dirac, “Fundamental equations of quantum mechanics,” Proc. R. Soc. London, Ser. A 114, 642 (1925).
23.Note that, using the form of the Poisson bracket given in Eq. (16), the derivatives are, respectively, and .
24.S. Weinberg, “Testing quantum mechanics,” Ann. Phys. 194, 336 (1989).
25.See, for example, A. Sudbery, “Quaternionic analysis,” Math. Proc. Camb. Philos. Soc. 85, 199 (1979).
26.K. R. W. Jones, “The Schrödinger equation from three postulates,” Mod. Phys. Lett. A (to appear).
27.Heslot (Ref. 5) obtains the analogous result (in quantum mechanics using complex numbers) for a real function G arising from a self-adjoint operator and uses this to define observables as real “regular functions of the state whose canonical transformations they generate are automorphisms of the whole quantum structure, i. e., are unitary transformations. ”
28.The transformation corresponding to the flow must be linear (the matrix elements could still depend on time, however). L. P. Horwitz (private communication) raises the possibility of the matrix elements being equal and coordinate independent along some Hamiltonian flow but depending on the elsewhere.
29.Substituting x for , for , y for and for , these are times the constants of motion for the two-dimensional isotropic harmonic oscillator—see, for example, J. Pollett, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28, 7287 (1995).
30.This can be thought of as the set of cube roots of unity; however, no formal identification of the elements of the ring generated by should be made with the complex numbers.
31.Adler (Ref. 12) shows that (in quantum mechanics using quaternions) the eigenstates of an anti-self-adjoint operator divide into mutually orthogonal eigenclasses, such that all of the eigenstates in a given eigenclass are related by reraying, that is, by multiplying on the right by a quaternionic phase. Since, for the quaternions, all unit imaginaries are in the same automorphism class, it is possible to pick, for a given anti-self-adjoint operator, a particular ray representative such that the eigenvalues are all dependent on the same unit imaginary.
32.An element of the Clifford algebra generated by a set of basis vectors which can be written in the form is known as a blade, where n is its grade. A general element of the Clifford algebra, or cliffor, is then a sum of blades with various grades. Given that the wedge product is associative and anti-commutative, a blade of grade n picks up a factor of under reversion.
33.In Clifford algebras generated by more than two basis vectors, all blades of odd grade have a sign change in addition to that from reversing the order of products.
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