# A new look at the harmonic oscillator problem in a finite-dimensional hilbert space

TL;DR: In this paper, the basic properties of a truncated oscillator are studied and new parasupersymmetric schemes are constructed by using finite-dimensional representation matrices of the oscillator.

Abstract: In this Letter some basic properties of a truncated oscillator are studied. By using finite-dimensional representation matrices of the truncated oscillator we construct new parasupersymmetric schemes and remark on their relevance to the transition operators of the non-interacting N-level system endowed with bosonic modes.

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TL;DR: In this paper, the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e., all of which obey Q/2 = 0).

Abstract: We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e. all of which obey Q/2 = 0). However, the Hamiltonian cannot be expressed in a simple form in terms of the p supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, the p supercharges, the p conformal supercharges along with Hamiltonian H, conformal generator K and dilatation generator D form a closed algebra.

17 citations

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TL;DR: In this paper, an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems was proposed without recourse to any specific matrix representation of paraffermionic operators and any kind of deformed algebra.

Abstract: We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed algebra. Within our formulation, we show generically that every parasupersymmetric quantum system of order p consists of N -fold supersymmetric pairs with N ⩽ p and thus has weak quasi-solvability and isospectral property. We also propose a new type of non-linear supersymmetries, called quasi-parasupersymmetry, which is less restrictive than parasupersymmetry and is different from N -fold supersymmetry even in one-body systems though the conserved charges are represented by higher-order linear differential operators. To illustrate how our formulation works, we construct second-order parafermionic algebra and three simple examples of parasupersymmetric quantum systems of order 2, one is essentially equivalent to the one-body Rubakov–Spiridonov type and the others are two-body systems in which two supersymmetries are folded. In particular, we show that the first model admits a generalized 2-fold superalgebra.

15 citations

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TL;DR: In this article, the authors build the coherent states for a family of solvable singular Schrodinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator and study basic properties of such states like continuity in the complex parameter, resolution of the identity, probability density, time evolution and possibility of entanglement.

Abstract: We build the coherent states for a family of solvable singular Schrodinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator. The main feature of such systems is the fact that their eigenfunctions are not completely connected by their natural ladder operators. We find a definition that behaves appropriately in the complete Hilbert space of the system, through linearised ladder operators. In doing so, we study basic properties of such states like continuity in the complex parameter, resolution of the identity, probability density, time evolution and possibility of entanglement.

12 citations

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TL;DR: In this article, the parafermion algebra of order p is expressed in terms of bilinear relation (commutator form) and the correct Hamiltonian is derived by using the parasuper...

Abstract: In this paper, we find the parafermion algebra of order p expressed in terms of bilinear relation (commutator form). We find the correct Hamiltonian of parafermion of order p by using the parasuper...

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TL;DR: In this article, general conditions for dynamical supersymmetry breaking are discussed and examples are given (in 0 + 1 and 2 + 1 dimensions) in which such a program in four dimensions is possible.

Abstract: General conditions for dynamical supersymmetry breaking are discussed. Very small effects that would usually be ignored, such as instantons of a grand unified theory, might break supersymmetry at a low energy scale. Examples are given (in 0 + 1 and 2 + 1 dimensions) in which dynamical supersymmetry breaking occurs. Difficulties that confront such a program in four dimensions are described.

3,164 citations

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TL;DR: This paper investigates the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and finds them to be well behaved.

Abstract: The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

694 citations

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TL;DR: In this paper, a unitary phase operator whose eigen states are well-defined phase states and whose properties coincide with those normally associated with a phase is presented, and a natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.

Abstract: The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.

499 citations

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TL;DR: In this article, a model of conformal parasupersymmetric quantum mechanics of one boson and one parafermion of (arbitrary) order p is discussed.

Abstract: The various aspects of parasupersymmetric quantum mechanics of one boson and one parafermion of (arbitrary) order p are discussed in some detail. In particular it is shown that the parasupersymmetry algebra is given by Q1pQ1++Q1p−1Q1+Q1+...+Q1Q1+ Q1p−1+Q1+Q1p =2pQ1p−1H, [H,Q1]=0; Q1p+1=0 and the Hermitian conjugated relations where Q1 is the parasupercharge and H the Hamiltonian. It is also shown that such a system always possesses (p−1) other conserved parasupercharges and p bosonic constants. Further, a special case is pointed out when the above algebra takes a very simple form as given by Q1Q1+Q1=2Q1H, Q1pQ1++Q1+Q1p= 2Q1p−1H. A model of conformal parasupersymmetry of degree p is discussed and it is shown that in this case one has p supercharges, p conformal supercharges, and p bosonic constants which along with H, dilatation generator D, and conformal generator K form a closed algebra. A model of parasupersymmetry of degree p is discussed which is not conformal invariant and yet whose spectrum can be a...

59 citations

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