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Theoretical Physics Astrophysics & Cosmology

Symmetries of harmonic oscillator generating the zero-point and negative energies

Zahid Zakir

Published Sep 22, 2006• 0 citations
Publisher

Abstract

The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry.The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zeropoint energy.Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise.A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle.It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics.In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time.For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.

DOI
10.9751/tpac.2091-001

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The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry.The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zeropoint energy.Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise.A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle.It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics.In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time.For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.
The system of two harmonic oscillators with different sign frequencies is presented as a positive-frequency oscillator with a complex generalized coordinate where there are a global U(1) symmetry and a charge conjugation symmetry (Csymmetry).It is shown that two pairs of ladder operators, appearing at the frequency decomposition of canonical variables, are not mutually charge-conjugate and their standard interpretation as operators of charge-conjugate quanta breaks the Csymmetry.Bilinear operator identities for the ladder operators are found allowed the theory to obey the C-symmetry restrictions and to express observables through the mutually charge-conjugate operators.These identities hold for C-symmetric interactions also.It is shown that C-symmetry eliminates a ground state zero-point charge and a zero-point energy.The uncertainty relations are generalized for nonhermitian canonical variables and it is shown that the charge-conjugation symmetric ground states do not quantized.It is shown that the C-symmetric interactions do not contribute to the ground state energy in all orders of perturbation theory.
Zisman-Stueckelberg-Feynman (ZSF) interpretation, by describing antiparticles as negative-energy particles evolving backward in ordinary time, makes relativistic quantum mechanics (RQM) covariant, but relativistic theory was not agreed with it. Consistent agreement with ZSF interpretation led to the time-symmetric relativity theory (TSRT) (Article 1). In TSRT, relativity principle is extended to rest frames of negative-energy particles going into lower light cone, and symmetry group is extended to the general Lorentz group O(1,3) with 4-inversion. In this article, a time-symmetric RQM (TS RQM) based on TSRT is formulated. The states of negative-energy particles in their rest frames are transformed into usual frames with reflecting time axis. In Hamilton-Jacobi formalism, translational symmetry leads to a general relativistic wave equation for components of wave function of any free particle with a quadratic Hamiltonian. Using in it relativistic relationship between energy and momentum gives Klein-Gordon equation, which yields a general average of 4-momentum and general probability current, time components of which describe currents in both directions of time axis and are negative at negative energy. Special wave equations with first time derivative (Duffin-Kemmer-Petiou for spins 0 and 1, and Dirac for spin 1/2) account for mixing of wave function components due to spin and interactions. In TSRT, 4-vector currents change sign upon 4-inversion, which improves theories of bosons and fermions, eliminating former paradoxes and errors. Time-symmetric transition amplitudes in propagator approach lead to causal propagators and usual diagram technique. Total energy of system of particles in loop diagrams creates external gravitational field, which slows down proper times relative to world time until they freeze at Planck length, gravitational radius of system. This leads to gravitational regularization of loop diagrams, making them finite, and fields of Standard Model (except for scalar field) and quantum gravity renormalizable. Quantization of fields in TSRT is described in Article 3.
The states of a planar oscillator are separated to a vibrational mode, containing a zero-point energy, and a rotational mode without the zero-point energy, but having a conserved angular momentum.On the basis of the analysis of properties of models of rigid and semirigid rotators, the theory of soft rotators is formulated where the harmonic attractive force is balanced only by the centrifugal force.As examples a Coulomb rotator (the Bohr model) and a magneto-harmonic rotator (the Fock-Landau levels) are considered.Disappearance of the radial speed in the model of a magneto-harmonic rotator is taken as a defining property of a pure rotational motion in the harmonic potential.After the exception of energies of the magnetic and spin decompositions, specific to magnetic fields, one turns to a simple and general model of a plane harmonic rotator (circular oscillator without radial speed) where kinetic energy is reduced to the purely rotational energy.Energy levels of the harmonic rotator have the same frequency and are twice degenerate, the energy spectrum is equidistant.In the ground state there is no zeropoint energy from rotational modes, and the zero-point energy of vibrational modes can be compensated by spin effects or symmetries of the system.In this case the operators of observables vanish the ground state, i.e. are "strongly" normally ordered.In a chain of harmonic rotators collective rotations around a common axis lead to transverse waves, at quantization of which there appear quasi-particles and holes carrying an angular momentum.In the chain SU (2) appears as a group of symmetry of a rotator.
The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that aaa annihilates, but is instead built on the vacuum that a†aa^{\dagger} aa†a annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear PTPTPT symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of PTPTPT symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.
Factorization in normal order form provides positive energy solutions, while the alternative anti-normal order form provides negative energy solutions of the time-independent Schroedinger equation for a linear harmonic oscillator. The positive and negative energy state spaces are related by quantum conjugation effected through sign-reversal of Planck's quantization constant . � Photons occupying negative energy states have the same displacement xand Hamiltonian H, but opposite momentum p ˆ - compared to photons occupying the corresponding positive energy partner states. Emission of positive energy photons from a positive energy quantum state is equivalent to absorption of negative energy photons into a negative energy quantum state, leading to state lowering, while absorption of positive energy photons into a positive energy quantum state is equivalent to emission of negative energy photons from a negative energy quantum state, leading to state raising. The complete positive- negative energy spectrum of the quantized oscillator may then be
We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers of the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ω to iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-PT-symmetric harmonic oscillator, we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wavefunctions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.