Ladder Operators in Repulsive Harmonic Oscillator with Application to the Schwinger Effect

The ladder operators in harmonic oscillator are a well-known strong tool for various problems in physics. In the same sense, it is sometimes expected to handle the problems of repulsive harmonic oscillator in a similar way to the ladder operators in harmonic oscillators, though their analytic solutions are well known. In this paper, we discuss a simple algebraic way to introduce the ladder operators of the repulsive harmonic oscillators, which can reproduce well-known analytic solutions. Applying this formalism, we discuss the charged particles in a constant electric field in relation to the Schwinger effect; the discussion is also made on a supersymmetric extension of this formalism.


Introduction
The algebraic approaches to the potential problems in quantum mechanics are commonly used ways from the early state of those fields [1]. In particular, the harmonic oscillators (h.o.) give a good operative example of algebraic approach to the eigenvalue problems in terms of the ladder operators, the annihilation and creation operators (â,â † ) characterized by [â,â † ] = 1. In such a dynamical system, the eigenvalue problem of Hamiltonian can be solved exactly by use of those ladder operators without depending on the representation of the eigenstates [1,2]; and, if we take the coordinate representation of those states, then the eigenstates will be reduced to the well-known analytic solutions expressed in terms of Hermite polynomials. The use of ladder operators also provides necessary tools in the field theories, since the dynamical degrees of freedom of bosonic-free fields are decomposed into those of infinite harmonic oscillators.
In comparison with h.o., the physical applications of the repulsive harmonic oscillators (r.h.o.) 1 are limited, since the Hamiltonian of r.h.o. is parabolic; and, its eigenstates are scattering states. The algebraic approaches to r.h.o., however, have been tried from a few different viewpoints: the dynamical groups including r.h.o. [3,4], the analytic continuation of angular velocity ω → ±iω in h.o. [5,6], the Bose systems in SUSY quantum mechanics [7][8][9], and so on.
On the other hand, it is known that the eigenvalue problems of r.h.o.-Hamiltonian are reduced to solve Weber's equation, which has analytic solutions so-called parabolic cylinder functions or the Weber functions [10,11]. The relation between the algebraic approaches to r.h.o and the analytic solutions, however, is not always clear. It is also important to study the completeness of the states constructed out of the algebraic approaches, since the trace calculations in physical applications require such a property of those states.
The purpose of this paper is, thus, to give a simple algebraic approach to the eigenvalue problems of r.h.o. by introducing Hermitian ladder operators (A,Ā) characterized by [A,Ā] = i. We can show that the dynamical variables of r.h.o. can be represented in the functional spaces constructed out of (A,Ā) with two ground states (φ 0 ,φ 0 ) satisfying Aφ 0 =Āφ 0 = 0 [12]. The states generated out of φ 0 andφ 0 form dual spaces each other so that the complete basis can be constructed based on the inner products between those states belonging to respective dual spaces.
In the next section, we study some types of complete bases, a discrete basis and continuous bases, representing r.h.o. in terms of the ladder operators with their ground states; there, the completeness of those bases is discussed carefully. The discussions are also made on the eigenvalue problems of r.h.o-Hamiltonian by considering the relation between the ladder operator formalism and the well-known analytic solutions.
In section 3, we discuss the applications of the present ladder operator formalism to two topics: one is a problem of charged particles under a constant electric field, the problem of the Schwinger effect [13]. This dynamical system is equivalent to r.h.o.; and, the discrete basis in the ladder operator formalism is shown to be a useful to evaluate that effect. As another topic, we study an extension of r.h.o. to a model of SUSY quantum mechanics by taking the advantage of the ladder operator formalism, though such an extension has been discussed from the early stages of r.h.o.. We focus our attention on that the Schwinger effect for fermions is closely related to such an extended model. Section 4 is devoted to the summary of our results. In appendices, some mathematical problems used in the text are discussed: the analytic solutions of Hamiltonian eigenstates, a proof of completeness, and the evaluation of the Schwinger effect for fermions.

Summary of standard harmonic oscillators
To begin with, we summarize the ladder operator approach to the problems of the usual harmonic oscillator, to which the Hamiltonian operator of a mass m particle with the characteristic frequency ω of the oscillation in one-dimensional space is given bŷ Then, because of [â,â † ] = 1, one can verify that [Ĥ,â † ] = ωâ † , [Ĥ,â] = − ωâ, and Ĥ Φ ≥ ω 2 on a state Φ normalized so that Φ 2 = 1. This means that starting from the ground state Φ 0 defined bŷ aΦ 0 = 0 with Φ 0 2 = 1, the states Φ n = 1 √ n!â †n Φ 0 (n = 0, 1, 2, 3, · · · ) (2.3) satisfy the eigenvalue equationŝ HΦ n = ω n + 1 2 Φ n (n = 0, 1, 2, 3, · · · ), (2.4) and the normalization Φ n |Φ m = δ n,m . The importance is that the states {Φ n } really form a complete basis of the functional space V , in which the canonical operators (x,p) are represented. Namely, in terms of the bra and the ket states, the operator is the unit operator in the functional space V ; and, one can verify where {|x } are the eigenstates ofx characterized byx|x = x|x and Furthermore, if it is necessary, the x-representation of Φ n can written explicitly in terms of the Hermitian polynomial H n (x) so that Φ n (x) = x|Φ n = 1 2 n n! mω π e −mωx 2 /2 H n (x mω/ ).

The case of repulsive harmonic oscillators
Now, for a repulsive harmonic oscillator, the Hamiltonian operatorĤ r is given fromĤ in eq.(2.1) by changing the sign of mω 2 2x 2 ; and, a complete basis in the same functional space V r by means of new ladder operators can be constructed in roughly parallel with equations (2.1) ∼ (2.6). Namely, one can start with the expressionĤ (2.8) By definition, A andĀ ( = A † ) are Hermitian operators themselves; however, they satisfy a similar algebra as that of (â,â † ) such as [A,Ā] = −[Ā, A] = i. Further, in terms of (A,Ā), the Hamiltonian operatorĤ r can be written as 2 where Λ = −iĀA andΛ = −iAĀ ( = Λ + 1 ) .
In order to solve the eigenvalue problem of Λ andΛ, let us introduce eigenstates (φ σ ,φ σ ) defined by where the σ is a real parameter. Then, the particular states (φ 0 ,φ 0 ) defined by Aφ 0 =Āφ 0 = 0 should be regarded as the counterparts of Φ 0 in the h.o.. It should be noticed that in spite of the similarity of eq.(2.11) to the coherent state equation in h.o., the index σ of φ σ runs over real continuous spectrum due to the Hermiticity of A; and, the same is true forφ σ .
In the x-representation,eqs.(2.11) and (2.12) can be solved explicitly, and we obtain where the normalizations of those states are φ σ |φ σ ′ = φ σ |φ σ ′ = δ(σ − σ ′ ). In this representation, because ofφ σ (x) = φ σ (x) * , the "bar"becomes simply complex conjugation, and one can find the completeness of (φ σ ,φ σ ) in the form The meaning of the "bar", the definition of dual states, however, depends on the representation. In the ladder operator formalism for r.h.o., the meaning of dual spaces can be given without depending on the representation as in the case of ladder operator formalism for h.o.. To make clear this point, let us write the (φ 0 ,φ 0 ), the ground states in the sense of Aφ 0 =Āφ 0 = 0, as (φ (0) ,φ (0) ). Then, the discrete states defined by are able to satisfy the eigenvalue equations Thus, on the states (φ (n) ,φ (n) ), the Hamiltonian operatorĤ r takes discrete eigenvalues (figure 1) such thatĤ Now, the inner product of those states can be determined from the algebra of (A,Ā) and the  which leads to φ (m) |φ (n) = 0 (m > n); the same is true for the case m < n. Thus, the inner products between any m, n states can be represented as plays the role of a unit operator in {φ (n) } space. The expectationÎ r = 1, can be confirmed through the equation which can be verified using i(n+1) Nn+1 = 1 Nn . In a similar way, one can deriveĀÎ r =Î rĀ . Since A and A are composing elements of dynamical variables in r.h.o., one can sayÎ r = c1, (c = const.) in the sense of Schur's lemma. Here, the constant in the right-hand side is necessary to be c = 1 because of I r |φ (0) = |φ (0) by eq.(2.20). Another proof ofÎ r = 1 is to show directly which will be given in appendix B. It should also be emphasize that the discrete basis (φ (n) ,φ (n) ) are closely related to Weber's functions as the energy eigenstates ofĤ r by means of the analytic continuation with respect to n. In order to verify this, we take notice the formula for a complex λ: (2.25) Here, eq.(2.24) seems to hold on the states such as {φ σ ; σ > 0}, on whichĀ becomes an operator with positive eigenvalues. Applying eq.(2.25) to φ (0) (x), such a constraint will fade away in the sense of analytic continuation; and, we obtain the expression x. The last equality in eq.(2.26) shows the relationship [14] between In a similar manner, one can verify that which can be regarded as the analytic continuation of the relationφ (n) (x) = φ * (n) (x) with respect to n. We note that if the λ in eq.(2.26) and the ρ in eq.(2.27) give the same eigenvalue of iĤr ω , then λ + 1 2 = −(ρ + 1 2 ) or ρ = −(λ + 1). Therefore, D λ (z) and D −(λ+1) (iz) are independent eigenstates of iĤr ω belonging to the same eigenvalue λ + 1 2 . This is a well-known result of discrete eigenstates in the eigenvalue problem of r.h.o. [10,11].
Furthermore, in terms of Weber's D-function, the completeness condition (2.23) can also be represented as In summary, the complete basis {φ σ (x)} are eigenstates of A belonging to continuous eigenvalues {σ ∈ R}, but those are not eigenstates ofĤ r . On the other hand, the complete basis {φ (n) (x)} are eigenstates ofĤ r belonging to discrete eigenvalues labeled by the non-negative integers n (= 0, 1, 2, · · · ). The D λ (z) is an analytic continuation of φ (n) (x) with respect to n, on whichĤ r takes the continuous eigenvalue −i ω λ + 1 2 .
3 Topics related to the present r.h.o. formalism

Schwinger effect
We note that the r.h.o. is effectively realized by a particle interacting with a specific gauge field. Let us consider the scalar field Φ in 4-dimensional spacetime for a mass m particles under gauge fields A µ satisfying 3 whereΠ µ (A) =p µ − g c A µ and g = ±|e|. We, here, setup the gauge potentials in such a way that , which produces the uniform electric field E along x 1 direction. Then,Π Further, in terms of the canonical variables defined by the unitary transformation U E = e i ( c gE )p 0p1 so that where the angular frequency is defined by ω = |e|E mc . This means that the H 01 is just the Hamiltonian of r.h.o. defined in the phase space (X µ , P µ ). Now, the classical action of gauge field under consideration is S G [A c ] = 1 2 d 4 xE 2 ; and, the one loop correction due to the scalar field Φ adds the quantum effect For this purpose, it is convenient to use {|X 0 ⊗ |φ (n) (X 1 ) ⊗ |X ⊥ } as the base states in the trace calculation. Then by taking 1 = ∞ n=0 1 Nn |φ (n) φ (n) | and dX 1φ (n) (X 1 ) * φ (n) (X 1 ) = N n into account, we obtain (3.8) Putting, here, V 0 ∼ dX 0 , V ⊥ ∼ d 2 X ⊥ as cutoff volumes in X 0 , X ⊥ spaces respectively, the righthand side of this equation becomes where the P denotes the principal value in z integral. Further, since mc and mc play respectively the roles of typical momentum and the length in this system, we may put ; and so, δ(0)V 0 V ⊥ ∼ V (4) (mc) 2 2π 2 with V (4) = V 0 V 1 V ⊥ . Therefore, we finally arrive at the expression The result just coincides with the formula of pair creation given by Schwinger for scalar QED [13,16].

Extension to SUSY quantum mechanics
The present ladder operator formalism of r.h.o. is easily extended to one of SUSY quantum mechanics [17][18][19][20] . To this end, let us introduce the Fermi oscillators characterized by {b, b † } = 1, b 2 = b †2 = 0, which can be represented in 2-dimensional vector space so that In terms of (b, b † ), the supersymmetric extension ofĤ r should bê Then the generators of SUSY transformation defined by are characterized by the algebras , the last equations can also be written as Those algebras should be compared with that of N = 2 SUSY quantum mechanics, though Q i (i = 1, 2) are not Hermitian operators. The zero-point oscillation ofĤ r is removed by this supersymmetry.
In the context of this SUSY quantum mechanics, we emphasize the following: in the Schwinger effect for fermions, the SUSY quantum mechanics of the r.h.o. plays an effective role in its background; that is, the topics 3.1 and 3.2 are not independent in this effect.
The Dirac field Ψ interacting with an external gauge fields A µ obeys the U (1) symmetry field equation 5 Here, if we use the configuration of gauge potentials and ω = |e|E mc as before. The result implies that the spectra of upper components ofΨ = U Ψ are those of the supersymmetric HamiltonianĤ r ; on the other side, the spectra of lower components ofΨ are governed byĤ ′ r , which is obtained fromĤ r changing the role of (b, b † ). Thus, one can evaluate the Schwinger effect for fermions again according to the procedure of eq.(3.8)-eq.(3.9) (appendix C).

Summary
In this paper, we have discussed the eigenvalue problems of r.h.o. in terms of ladder operators (A,Ā) introduced by an analogous way to the ladder operator (â,â † ) in h.o.. The non-positive property of the Hamiltonian operatorĤ r in r.h.o. is a result of the property of ladder operators such as A † = A, Ā † =Ā, andĀ = A † . Then the states defined by Aφ 0 = 0 andĀφ 0 = 0 play the role of different ground states, which can be normalized by φ 0 |φ 0 = iπ 2 in spite of φ 0 |φ 0 = φ 0 |φ 0 = ∞. The complete bases representing r.h.o. can be constructed by the combination of (A, φ 0 ) or that of (Ā,φ 0 ); the aspects of the complete basis are various depending on the ways of construction. For instance, the {φ σ (x)} are eigenstates of A with continuous eigenvalues {σ ∈ R}, though those are not eigenstates ofĤ r . On the other hand, the {φ (n) (x)} are eigenstates ofĤ r with discrete eigenvalues −i ω n + 1 2 ; (n = 0, 1, 2, · · · ) ; the Hamiltonian operatorĤ r is able to have continuous eigenvalues {−i ω λ + 1 2 ; λ ∈ R} on the states of Weber's D-function D λ (z), where the D-function can be shown to be an analytic continuation of φ (n) (x) with respect to n.
As good applications of this ladder operator formalism, we have shown two topics: the Schwinger effect in scalar QED and an extension of r.h.o. to SUSY quantum mechanics. In the first, the Hamiltonian of particles interacting with a constant electric field is shown to be canonically equivalent to one of r.h.o.; and so, the knowledge of r.h.o. is useful to handle the problem of pair production by the electric field. Indeed, it has been shown that the discrete complete bases {φ (n) ,φ (n) } characterized by eq.(2.21) gives a simple way to evaluate such a production rate within the framework of quantum mechanics.
Secondly, we have tried to extend the present r.h.o. system to a supersymmetric dynamical system; the extended HamiltonianĤ r is again a non-positive Hermitian operator constructed out of fermionic oscillators (b, b † ) and ladder operators (A,Ā). The ladder operator formalism gives rise to two towers of super-pair states |φ ± n and |φ ± n (n = 1, 2, · · · ), which belong to the eigenvalues E + n = −i ωn and E − n = i ωn respectively. In addition to this, the n = 0 ground states exist as two singlets |φ − 0 and |φ + 0 , which satisfy Q i |φ − 0 = Q i |φ + 0 = 0, (i = 1, 2). Namely, in each space of super-pair tower states, SUSY is realized as a good symmetry, though the SUSY in this model is an extended concept from the standard one as can be seen from Q † i = Q i . Furthermore, we have brought up the following: if we consider the Dirac fields interacting with an external electric field, then the supersymmetric structure of r.h.o. will be implicitly included in a loop effect of those Dirac fields. According to this line of approach, we have shown the way to evaluate the Schwinger effect for fermions in appendix C.
The knowledge on the complete bases in r.h.o. under the ladder operator formalism is expected to give useful tools in various problems other than the topics discussed in this paper. For example, the HamiltonianĤ r is able to take continuous eigenvalues on the states (φ σ ,φ σ ); in the space of those eigenstates, the SUSY may show a different feature from the standard analysis. Those are interesting future problems.
In the x-representation withp = −i ∂ ∂x , the eigenvalue equation ofĤ r can be written as Introducing, here, the variable z defined by Writing iE ω = λ + 1 2 and w E (z) = w λ (z), the eq.(A.3) becomes standard form of Weber's equation Forw λ (z) = e 1 4 z 2 w λ (z), eq.(A.4) can also be written as To solve eq.(A.5), let us use the Fourier-Laplace representatioñ where Γ is a path from a to b in complex t plane. Then under the integration by parts with respect to t, the eq.(A.5) with eq.(A.4) gives on condition that [e −zt {tf λ (t)}] b a = 0. The eq.(A.7) can be solved easily so that f λ (t) = const.e − 1 2 t 2 t −(λ+1) ; since the boundary conditions are satisfied by (a, b) = (0, ∞) for Reλ < 0 on the real t axis, we finally obtain the integral representation for w λ (z) = e − 1 4 z 2w λ (z) (= D λ (z)) in such a form as [14,15] where C is the contour given in ( figure 3). It is not difficult to rewrite the contour integral in eq.(A.9) to the path integral in eq.(A.8) by taking into account Γ(λ + 1) sin(−πλ) = π Γ(−λ) . The function D λ (z) is Weber's D-function 6 (Parabolic cylinder function) [15], by which the independent solutions of eq.(A.1) for iE ω = λ + 1 2 are given as D λ (z) and D −λ−1 (iz). 6 The D-function is normalized so that Dn(z), (n = 0, 1, · · · ) reduces to e − By definition of φ (n) andφ (n) , we obtain the expression Here, we have used the formula and using again eâ +b = eâebe − 1 2 [â,b] , we arrive at Therefore,Î r is nothing but the unit operator for the present r.h.o. system.

C The Schwinger effect for fermions
The action of the Dirac field Ψ obeying the eq. where z = τ (mc) 2 . The integration with respect to z in eq.(C.5) can be carried out in the same manner as eq.(3.9) except replacing the residue (−1) n of 1/ sinh(z ω/mc 2 ) by 1 of 1/ tanh(z ω/mc 2 ). The resultant formula corresponding to eq. This formula is nothing but the one given originally by Schwinger [13].