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

PHYSICAL REVIEW D 102, 025002 (2020)

Ladder operators in repulsive harmonic oscillator with application to the Schwinger effect

† ‡ Kenichi Aouda ,1,* Naohiro Kanda,1, Shigefumi Naka,1, and Haruki Toyoda2,§ 1Department of Physics, College of Science and Technology, Nihon University, Tokyo 101-8308, Japan 2Junior College, Funabashi Campus, Nihon University, Chiba 274-8501, Japan

(Received 27 December 2019; accepted 15 June 2020; published 6 July 2020; corrected 6 October 2020)

The ladder operators in harmonic oscillators 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 oscillators 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.

DOI: 10.1103/PhysRevD.102.025002

I. INTRODUCTION however, have been tried from a few different viewpoints: the dynamical groups including RHOs [3,4], the analytic The algebraic approaches to the potential problems in continuation of angular velocity ω → iω in HOs [5–7], quantum mechanics are commonly used ways from the the Bose systems in SUSY quantum mechanics [8,9],and early state of those fields [1]. In particular, the harmonic so on. oscillators (HOs) give a good operative example of an On the other hand, it is known that the eigenvalue algebraic approach to the eigenvalue problems in terms of problems of the RHO Hamiltonian are reduced to solve the ladder operators, the annihilation and creation operators ’ ˆ ˆ † ˆ ˆ † 1 Weber s equation, which has analytic solutions, so-called ða; a Þ characterized by ½a; a ¼ . In such a dynamical parabolic cylinder functions, or the Weber functions system, the eigenvalue problem of Hamiltonian can be [10,11]. The relation between the algebraic approaches solved exactly by use of those ladder operators without to RHOs and the analytic solutions, however, is not always depending on the representation of the eigenstates [1,2] clear. It is also important to study the completeness of the and, if we take the coordinate representation of those states, states constructed out of the algebraic approaches, since the the eigenstates will be reduced to the well-known analytic trace calculations in physical applications require such a solutions expressed in terms of Hermite polynomials. The property of those states. The purpose of this paper is, thus, use of ladder operators also provides necessary tools in the to give a simple algebraic approach to the eigenvalue field theories, since the dynamical degrees of freedom of problems of RHOs by introducing Hermitian ladder oper- bosonic-free fields are decomposed into those of infinite ators ðA; A¯ Þ characterized by ½A; A¯ ¼i. harmonic oscillators. We can show that the dynamical variables of RHOs In comparison with HOs, the physical applications of can be represented in the functional spaces constructed out the repulsive harmonic oscillators (RHOs)1 are limited, since ¯ ¯ of ðA; AÞ with two cyclic states ðϕ0; ϕ0Þ satisfying Aϕ0 ¼ the Hamiltonian of RHOs is parabolic and its eigenstates ¯ ¯ ¯ n n ¯ Aϕ0 0 [12]. Here, the A ϕ0 and A ϕ0 n ∈ N are are scattering states. The algebraic approaches to RHOs, ¼ f g f gð Þ conjugate states which form orthonormal pairs, though those themselves are not square integrable. Those pairs become complex conjugate of each other in the x repre- *[email protected] † sentation. As the result, those states form a discrete basis of [email protected] × ‡ a space of functionals Φ , which includes the Hilbert space [email protected] H §[email protected] for the RHO. It is also shown that there exist continuous ¯ × 1The inverted oscillator or reversed oscillator, in other words. bases fϕσ; ϕσgðσ ∈ RÞ in Φ , which are respective eigenstates of A and A¯ . Published by the American Physical Society under the terms of In the next section, we study those continuous and the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to discrete bases given in terms of the ladder operators with the author(s) and the published article’s title, journal citation, their cyclic states. In that place, the completeness of those and DOI. Funded by SCOAP3. bases is discussed carefully. The discussions are also made

2470-0010=2020=102(2)=025002(10) 025002-1 Published by the American Physical Society AOUDA, KANDA, NAKA, and TOYODA PHYS. REV. D 102, 025002 (2020) 1 on the eigenvalue problems of the RHO Hamiltonian by ˆ Φ ℏω Φ 0 1 2 3 … considering the relation between the ladder operator for- H n ¼ n þ 2 n ðn ¼ ; ; ; ; Þ; ð4Þ malism and the well-known analytic solutions.

In Sec. III, we discuss the applications of the present and the normalization hΦnjΦmi¼δn;m. The importance is ladder operator formalism to two topics: one is a problem that the states fΦng really form a complete basis of the of charged particles under a constant electric field, the functional space V, in which the canonical operators ðx;ˆ pˆ Þ problem of the Schwinger effect [13]. This dynamical are represented. Namely, in terms of the bra and the ket system is equivalent to RHO and the discrete basis in the states, the operator ladder operator formalism is shown to be useful to evaluate that effect. As another topic, we study an extension of X∞ ˆ Φ Φ RHOs to a model of supersymmetry (SUSY) quantum I ¼ j nih njð5Þ 0 mechanics by taking the advantage of the ladder operator n¼ formalism, though such an extension has been discussed is the unit operator in the functional space V and one can from the early stages of RHOs. We focus our attention on verify the fact that the Schwinger effect for fermions is closely related to such an extended model. hxjIˆjx0i¼δðx − x0Þ; ð6Þ Section IV is devoted to the summary of our results. In the Appendixes, some mathematical problems used in the where fjxig are the eigenstates of xˆ characterized by xˆjxi¼ text are discussed: the analytic solutions of Hamiltonian xjxi and hxjx0i¼δðx − x0Þ; ðx; x0 ∈ RÞ. Furthermore, if it is Φ eigenstates, a proof of completeness, and the evaluation of necessary, the x representation of n can be written explicitly in terms of the Hermitian polynomial H ðxÞ so that the Schwinger effect for fermions. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi Φ Φ 1 mω −mωx2=2ℏ ω ℏ nðxÞ¼hxj ni¼ 2nn! πℏ e Hnðx m = Þ. II. LADDER OPERATORS IN REPULSIVE HARMONIC OSCILLATORS B. The case of repulsive harmonic oscillators A. Summary of standard harmonic oscillators Now, for a repulsive harmonic oscillator, the To begin with, we summarize the ladder operator ˆ ˆ Hamiltonian operator Hr is given from H in Eq. (1) by approach to the problems of the usual harmonic oscillator, mω2 ˆ2 to which the Hamiltonian operator of a mass m particle with changing the sign of 2 x ; and, a complete basis in the the characteristic frequency ω of the oscillation in one- same functional space Vr by means of new ladder operators – dimensional space is given by can be constructed in roughly parallel with Eqs. (1) (6). Namely, one can start with the expression 1 ω2 ℏω 1 ˆ 2 m 2 † † ˆ H ¼ pˆ þ xˆ ¼ ðaˆ aˆ þ aâˆ Þ¼ℏω N þ ; 1 mω2 ℏω 2m 2 2 2 Hˆ ¼ pˆ 2 − xˆ2 ¼ − ðAA¯ þ AA¯ Þ; ð7Þ r 2m 2 2 ð1Þ where where Nˆ ¼ aˆ †aˆ and rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi mω 1 ω A ¼ xˆ − pffiffiffiffiffiffiffiffiffiffiffiffiffi p;ˆ ˆ m ˆ ffiffiffiffiffiffiffiffiffiffiffiffiffii ˆ 2ℏ 2mℏω a ¼ 2ℏ x þ p p; rffiffiffiffiffiffiffi 2mℏω ω 1 rffiffiffiffiffiffiffi ¯ m ˆ ffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ mω i A ¼ 2ℏ x þ p p: ð8Þ aˆ † ¼ xˆ − pffiffiffiffiffiffiffiffiffiffiffiffiffi p:ˆ ð2Þ 2mℏω 2ℏ 2mℏω By definition, A and A¯ ð≠ A†Þ are Hermitian operators ˆ 2 By definition hΦjNjΦi¼kaˆΦk ≥ 0; then, because of themselves; however, they satisfy a similar algebra as that † ˆ † † ˆ † ½a;ˆ aˆ ¼1, one can verify that ½H; aˆ ¼ℏωaˆ ; ½H; aˆ¼ of ða;ˆ aˆ Þ such as ½A; A¯ ¼−½A;¯ A¼i. Further, in terms of −ℏωˆ ˆ Φ ≥ ℏω Φ ¯ ˆ 3 a, and kH k 2 on a state normalized so that ðA; AÞ, the Hamiltonian operator Hr can be written as kΦk2 ¼ 1.2 This means that starting from the ground state 2 Φ ˆΦ 0 Φ 1 3 † 0 defined by a 0 ¼ with k 0k ¼ , the states In terms of the ladder operator ða;ˆ aˆ Þ defined in Eq. (1), ˆ 1 the Hamiltonian operator (7) can be represented as Hr ¼ ffiffiffiffiffi †n − ℏω ˆ †2 ˆ 2 Φn ¼ p aˆ Φ0 ðn ¼ 0; 1; 2; 3; …Þð3Þ 2 ða þ a Þ. From this expression, carrying out the successive ! iaˆ †2 n canonical (≠ unitary) transformations by U1 ¼ e2 and i 2 −4aˆ ˆ ˆ U2 ¼ e , one can find the relation between Hr and H such satisfy the eigenvalue equations ˆ −1 −1 ˆ ˆ that U2U1HrU1 U2 ¼ iH. The eigenvalue problem of Hr, thus, can also be solved in terms of ða;ˆ aˆ †Þ and these canonical 2 ˆ 2 ˆ 2 ˆ 2 2 ˆ 1 2 kHΦk ¼hΦjH jΦi ≥ hΦjHjΦi ¼ðℏωÞ ðhΦjNjΦiþ2Þ . transformations.

025002-2 LADDER OPERATORS IN REPULSIVE HARMONIC … PHYS. REV. D 102, 025002 (2020) 1 1 H ˆ − ℏω Λ − ℏω Λ¯ − which includes the Hilbert space for the RHO in the Hr ¼ i þ 2 ¼ i 2 ; ð9Þ framework of the Rigged Hilbert space.5 ¯ In those continuous complete bases, fϕσg and fϕσg, ¯ × where the aspect of the states ðϕ0; ϕ0Þ ∈ Φ satisfying Aϕ0 ¼ ¯ ¯ ¯ Aϕ0 ¼ 0 are characteristic. First, the ðϕ0; ϕ0Þ should be Λ ¼ −iAA¯ and Λ¯ ¼ −iAA¯ ð¼ Λ þ 1Þ: ð10Þ regarded as the counterparts of the ground state Φ0 in the Φ× Since Λ† ¼ −Λ − 1ðΛ¯ † ¼ −Λ¯ þ 1Þ, the Hermiticity of the HO. Second, those states become cyclic states of in the ϕ ϕ¯ ϕ ϕ¯ ˆ following sense: Writing ð ð0Þ; ð0ÞÞ¼ð 0; 0Þ, one can operator Hr given in Eq. (9) is formally guaranteed. ˆ verify that the states defined by The eigenvalue problem of Hr is, thus, reduced to those Λ Λ¯ ≠ Λ† of the operators and ð Þ, which are commutable ¯ n ¯ n ¯ ϕ ¼ A ϕ 0 ðϕ ¼ A ϕ 0 Þ; ðn ¼ 0; 1; 2; …Þ with each other. ðnÞ ð Þ ðnÞ ð Þ In order to solve the eigenvalue problem of Λ and Λ¯ , let ð16Þ ¯ us introduce eigenstates ðϕσ; ϕσÞ defined by rffiffiffiffiffiffiffi satisfy the eigenvalue equations ω 1 ϕ m ˆ − ffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ ϕ σϕ A σ ¼ x p p σ ¼ σ; ð11Þ Λϕ ϕ Λ¯ ϕ¯ − ϕ¯ 0 1 2 … 2ℏ 2mℏω ðnÞ ¼ n ðnÞ ð ðnÞ ¼ n ðnÞÞ; ðn ¼ ; ; ; Þ: rffiffiffiffiffiffiffi ¯ ¯ mω 1 ¯ ¯ ð17Þ Aϕσ ¼ xˆ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pˆ ϕσ ¼ σϕσ; ð12Þ 2ℏ 2 ℏω m ϕ ϕ¯ Namely, on the states ð ðnÞ; ðnÞÞ, the Hamiltonian operator σ where the is a real parameter. Then, the particular states Hˆ takes discrete eigenvalues (Fig. 1) such that ϕ ϕ¯ ϕ ¯ ϕ¯ 0 r ð 0; 0Þ defined by A 0 ¼ A 0 ¼ should be regarded as ) the counterparts of Φ0 in the HO. It should be noticed that ˆ ϕ − ℏω 1 ϕ Hr ðnÞ ¼ i ðn þ 2Þ ðnÞ in spite of the similarity of Eq. (11) to the coherent state ; ðn ¼ 0; 1; 2; …Þ; ð18Þ ˆ ϕ¯ ℏω 1 ϕ¯ equation in the HO, the index σ of ϕσ runs over the real Hr ðnÞ ¼ i ðn þ 2Þ ðnÞ continuous spectrum due to the Hermiticity of A and the ¯ ˆ same is true for ϕσ. which means that there are no ground states for Hr as In the x representation, Eqs. (11) and (12) can be solved expected from its nonpositive structure. ϕ ϕ¯ explicitly, and we obtain Those f ðnÞ; ðnÞg are the generalized eigenstates rffiffiffiffiffiffiffiffiffiffi belonging to Φ× instead of the Hilbert space for the ω pffiffiffiffiffi 4 m imωx2−i 2mωσx RHO. What is important is that the states fϕ g and their ϕσðxÞ¼ e 2ℏ ℏ ; ð13Þ ðnÞ 2ℏπ2 ϕ¯ rffiffiffiffiffiffiffiffiffiffi conjugate f ðnÞg are orthogonal each other under the inner ω pffiffiffiffiffi product, which can be determined from the algebra of ¯ 4 m −imωx2þi 2mωσx qffiffiffiffi ϕσðxÞ¼ e 2ℏ ℏ ; ð14Þ 2ℏπ2 ¯ ϕ¯ ϕ ≡ i ðA; AÞ and the normalization h ð0Þj ð0Þi N0 ¼ 2π only. 0 where the normalizations of those states are hϕσjϕσ0 i¼ Indeed for m ¼ n þ lðl> Þ, one can verify ¯ ¯ 0 ϕσ ϕσ0 δ σ − σ x h j i¼ ð Þ. In this representation, because of ¯ ¯ l n ¯ n ¯ ¯ hϕ jϕ i¼hϕ 0 jA A A jϕ 0 i ϕσðxÞ¼hxjϕσi¼ϕσðxÞ ¼hxjϕσi, the “bar” becomes ðmÞ ðnÞ ð Þ ð Þ ϕ¯ l n−1 ¯ n−1 ϕ simply complex conjugation, and the functional space of ¼ inh ð0ÞjA A A j ð0Þi ϕ ϕ¯ f σg coincides with that of f σg in the aggregate, though n ! ϕ¯ l ϕ ¯ ¼¼i n h ð0ÞjA j ð0Þi; ð19Þ ϕσ and ϕσ are independent states. Further, one can find the ϕ ϕ¯ completeness of ð σ; σÞ in the form which leads to ϕ¯ ϕ 0 m>n; the same is true for Z Z h ðmÞj ðnÞi¼ ð Þ the case m

ð15Þ ϕ¯ ϕ δ ≡ n ! h ðmÞj ðnÞi¼ m;nNnðNn i n N0Þ; ð20Þ ¯ Thus, the states fϕσg and their conjugate fϕσg are 4 Φ× 5 continuous complete bases of the functional space , For the quantum mechanics dealing with a continuous spectrum, the rigged Hilbert space [14,15] Φ ⊂ H ⊂ Φ× is qffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi −1 ℏ p useful to include continuous bases in the framework. Here, H 4Because of U ðx;ˆ pˆ ÞU ¼ A; ℏmωA¯ with U ¼ A A mω A is the Hilbert space with a countable orthonormal basis such as π 2 ¯ 2 −i8ðA þA Þ ¯ Φ Φ H e , the states jϕσi and jϕσi are unitary equivalents to the f ng in HOs. The is a dense subspace of associated with qffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi H Φ× Φ ℏ p a topology finer than that of : and the is the dual space of . x σ and p ℏmωσ , respectively. ¯ × j ¼ mω i j ¼ i The fjϕσig; fjϕσig, and fjxig are continuous bases of Φ .

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X∞ 1 Iˆ ¼ jϕ¯ ihϕ j; ð24Þ r N ðnÞ ðnÞ n¼0 n ˆ from which one can write the spectral decomposition of Hr so that

∞ X −iℏωðn þ 1Þ Hˆ ¼ 2 jϕ ihϕ¯ jð25Þ r N ðnÞ ðnÞ n¼0 n ∞ X iℏωðn þ 1Þ ¼ 2 jϕ¯ ihϕ j: ð26Þ N ðnÞ ðnÞ n¼0 n FIG. 1. There are many types of complete bases in the ˆ ϕ ϕ¯ – representation space of Hr. On the discrete bases ð ðnÞ; ðnÞÞ, The resultant equations, (24) (26), also have the meaning the Hˆ takes the eigenvalues shown in the figure on the left of the independent of the representation equation (21). Since r ˆ † ˆ vertical axis. Hr ¼ Hr, two types of spectral decomposition, (25) and ϕ ϕ¯ (26), are consistent and ð ð0Þ; ð0ÞÞ are not ground states which gives the meaning of ϕ¯ without depending on ˆ f ðnÞg corresponding to any lower bounds of Hr but rather, to the × the representation. Here, the complexity of Nn ’s again cyclic states of Φ . ϕ ϕ¯ ϕ ϕ¯ Φ× implies that the f ðnÞ; ðnÞg are not bases in a Hilbert space The states f ðnÞ; ðnÞg form a discrete basis of in in spite of the resemblance between those states and fΦng pairs in addition to those that are generalized eigenstates of in the HO. ˆ ˆ Hr. The eigenstates of Hr are not limited to those states; Nevertheless, Eq. (20) suggests that the operator ϕ ϕ¯ we emphasize that the discrete basis f ðnÞ; ðnÞg is closely X∞ related to Weber’s functions, which are continuous eigen- ˆ 1 ¯ I ≡ jϕ ihϕ jð21Þ value solutions for an eigenvalue equation of Hˆ , by means r N ðnÞ ðnÞ r n¼0 n of the analytic continuation with respect to n. In order to verify this, we take notice of the formula for a complex λ: plays the role of a unit operator in fϕ g space. The ðnÞ Z ˆ 1 ∞ expectation Ir ¼ , can be confirmed through the equation 1 ¯ A¯ λ ¼ dte−Att−ðλþ1Þ ð27Þ X∞ Γð−λÞ 0 ˆ in ¯ Z AIr ¼ jϕ n−1 ihϕ n j ffiffiffiffi ð Þ ð Þ 1 ∞ p ω 1 N − λ 1 −i 2 −t m xˆ −tpffiffiffiffiffiffiffipˆ n¼1 n ð þ Þ 4t 2ℏ 2mℏω ¼ Γ −λ dtt e e e : ð28Þ X∞ 1 ð Þ 0 iðn þ Þ ϕ ϕ¯ ¼ j ðnÞih ðnþ1Þj N 1 Here, Eq. (27) seems to hold on to the states such as n¼0 nþ ϕ¯ ; σ 0 ¯ X∞ 1 f σ > g, on which A becomes an operator with positive ¯ ˆ ϕ ¼ jϕ ihϕ jA ¼ I A; ð22Þ eigenvalues. Applying Eq. (28) to ð0ÞðxÞ, such a constraint N ðnÞ ðnÞ r n¼0 n will fade away in the sense of analytic continuation and we 1 obtain the expression which can be verified using iðnþ Þ ¼ 1 . In a similar way, Nnþ1 Nn rffiffiffiffiffiffiffiffiffiffi Z ¯ ˆ ˆ ¯ ¯ ∞ 4 mω 1 one can derive AIr ¼ IrA. Since A and A are composing ¯ λϕ A ð0ÞðxÞ¼ 2 dt elements of dynamical variables in RHOs, one can say 2ℏπ Γð−λÞ 0 ˆ pffiffiffiffi pffiffiffiffiffi I c1; c const in the sense of Schur’s lemma. Here, i 2 − mω mω ℏ 2 r ¼ ð ¼ Þ −ðλþ1Þ − t t 2ℏ x i ðxþit Þ 1 × t e 4 e e 2ℏ 2mω the constant in the right-hand side is necessary to be c ¼ ffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi −iπ p Z ˆ −1 4 2mω 2 because of I ϕ 0 ϕ 0 by Eq. (20). In Appendix B,we 4ðe ℏ xÞ ∞ rj ð Þi¼j ð Þi iπ 4 mω e 4 λ ¯ will show directly ¼ e 2 dt 2ℏπ Γð−λÞ 0 π ffiffiffiffiffi −i p2 ω ˆ −ðλþ1Þ −1t¯2 −t¯ðe 4 m xÞ hxjI jx0i¼δðx − x0Þ; ð23Þ × t¯ e 2 e ℏ r rffiffiffiffiffiffiffiffiffiffi ω iπλ 4 m e 4 Dλ z ; which says that the imaginary parts of each term in the ¼ 2ℏπ2 ð Þ ð29Þ right-hand side of Eq. (23) are cancelled out by the summation with respect to n. Thus, by taking into account qffiffiffiffiffiffiffi iπ −iπ 2 ω ϕ ϕ¯ † ϕ¯ ϕ 1† 1 where t¯ ¼ e 4 t and z ¼ e 4 m x. The last equality in ðj ðnÞih ðnÞjÞ ¼j ðnÞih ðnÞj and ¼ , Eq. (21) is equiv- ℏ ¯ λϕ alently represented as Eq. (29) shows the relationship [16] between A ð0ÞðxÞ and

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’ ˆ ˆ μ 2 Weber s function DλðzÞ (Appendix A). In a similar manner, ½ΠμðAÞΠ ðAÞþðmcÞ ΦðxÞ¼0; ð32Þ one can verify that Πˆ μ ˆ μ − g μ rffiffiffiffiffiffiffiffiffiffi where ðAÞ¼p c A and g ¼jej. We here set up the ω 0 ρ −iπρ 4 m A ϕ¯ 4 gauge potentials in such a way that ðAcðxÞ; cðxÞÞ ¼ A 0 ðxÞ¼e 2DρðizÞ; ð30Þ ð Þ 2ℏπ ð−Ex1; 0ÞðE ¼ const > 0Þ, which produces the uniform electric field E along the x1 direction. Then, which can be regarded as the analytic continuation of the ϕ¯ ϕ Πˆ 2 2 ˆ ˆ 2 relation ðnÞðxÞ¼ ðnÞðxÞ with respect to n. We note that if ðAcÞ ¼ mH01 þ p⊥; ð33Þ the λ in Eq. (29) and the ρ in Eq. (30) give the same iHˆ 1 1 where pˆ ⊥ ¼ðpˆ 2; pˆ 3Þ and eigenvalue of r, then λ þ ¼ −ðρ þ Þ or ρ ¼ −ðλ þ 1Þ. ℏω 2 2 Therefore, DλðzÞ and D−ðλþ1ÞðizÞ are independent eigen- 1 1 2 2 ˆ 2 jejE 1 c 0 ˆ 1 ˆ − ˆ iHr λ H01 ¼ p1 x þ p : ð34Þ states of ℏω belonging to the same eigenvalue þ 2. This is 2m 2m c gE a well-known result of discrete eigenstates in the eigenvalue problem of RHOs [10,11]. In terms of Weber’s D function, Further, in terms of the canonical variables defined by the i c 0 1 ℏð Þpˆ pˆ the completeness condition (23) can also be represented as unitary transformation, UE ¼ e gE so that X∞ 1 c c ˆ 0 ϕ ϕ¯ 0 μ μ −1 0 − ˆ 1 1 ˆ 0 2 3 hxjIrjx i¼ ðnÞðxÞ ðnÞðx Þ ðX Þ¼ðUEx UE Þ¼ x p ;x þ p ;x ;x ; 0 Nn gE gE n¼ X∞ n 1 i mω 2 ð35Þ ¼ D ðzÞD ðiz0Þ: ð31Þ N 2ℏπ2 n n n¼0 n ˆ μ ˆ μ −1 ˆ μ ðP Þ¼ðUEp UE Þ¼ðp Þ; ð36Þ ¯ In summary, the complete bases fϕσðxÞg and fϕσðxÞg the Hamiltonian operator (34) can be written as are respective eigenstates of A and A¯ belonging to continuous eigenvalues fσ ∈ Rg, but those are not eigenstates 1 ω2 ˆ ˆ 2 − m 2 ˆ ϕ ϕ¯ H01 ¼ P1 X1; ð37Þ of Hr. On the other hand, the eigenstates f ðnÞðxÞ; ðnÞðxÞg 2m 2 are eigenstates of Hˆ with discrete eigenvalues correspond- r ω jejE ing to the analytic continuation ω → iω of the eigenval- where the angular frequency is defined by ¼ mc. This ¯ ues in Eq. (4). The fϕ ðxÞϕ ðxÞg form a discrete basis of means that the H01 is just the Hamiltonian of the RHO ðnÞ ðnÞ μ μ × Φ in pairs. The fDλðzÞ; λ ∈ Rg are analytic solutions of defined in the phase space ðX ;P Þ. ˆ Now, the classical actionR of the gauge field under an eigenvalue equation for Hr; another aspect of DλðzÞ is an 1 4 2 ϕ consideration is SG½Ac¼2 d xE and the one loop analytic continuation of ðnÞðxÞ with respect to n. The ¯ correction due to the scalar field Φ adds the quantum effect eigenstates fϕ n ðxÞ; ϕ n ðxÞg and fDλðzÞ;DρðizÞg stand ð Þ ð Þ S ½A ¼−iℏ logfdetðΠˆ ðA Þ2 þðmcÞ2Þg−1 to S ½A .7 on the same footing as scattering states of Hˆ unless any Q c c G c r The resultant effective action of gauge fields S ½A ¼ boundary conditions are added. eff c SG½AcþSQ½Ac becomes Z ∞ III. TOPICS RELATED TO THE PRESENT dτ − τ 2− ϵ − τΠˆ 2 − ℏ i ððmcÞ i Þ i ½Ac Þ Seff½Ac¼SG½Ac i e trðe Þ RHO FORMALISM 0 τ ϕ ϕ ϕ¯ The complete bases f σg or f ðnÞ; ðnÞg based on ladder ð38Þ operator ðA; A¯ Þ give us useful ways to handle the problems disregarding unimportant additional constant. Namely, related to RHOs; in what follows, we exhibit two simple μ examples. under the classical background gauge field Ac, the scalar 0 0 ∼ QED gives rise to the transition amplitude h inj outi i 2 NeℏSeff ½AcðjNj ¼ 1Þ, which defines an unitary S-matrix A. Schwinger effect element for a real Seff½Ac. If the S matrix contains pair We note that the RHO is effectively realized by a particle productions, under which the state of the electric field is ≠ 0 interacting with a specific gauge field. Let us consider the constant in time, then ImSeff½Ac and we have scalar field Φ in four-dimensional spacetime for mass m μ 6 particles under gauge fields A satisfying 7 In the expression of SQ½Ac, use has been made of the −1 − well-knownR formulas fdetðMÞg ¼ e tr log M and tr log M ¼ 6 ∞ dτ −iτðM−iϵÞ diagðημνÞ¼ð− þ þþÞ. −tr 0 τ e ðþconstÞ.

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2 −2 0 0 ∼ ℏImSeff ½Ac ≠ 1 2πℏ δ 0 ∼ jh inj outij e . This ratio, the Schwinger which allows another possible choice such as ð Þ ð Þ “ ” ℏω ∼ kℏ effect, can be evaluated by calculating the trace in Eq. (38). c and V1 mc. For this purpose, it is convenient to use fjX0i ⊗ ϕ 1 ⊗ B. Extension to SUSY quantum mechanics j ðnÞðX Þi jX⊥ig as the baseP states in the trace calcu- 1 ∞ 1 ϕ ϕ¯ lation. Then, by taking ¼ 0 j ðnÞih ðnÞj and The present ladder operator formalism of RHOs is easily R n¼ Nn 1ϕ¯ 1 ϕ 1 extended to one of SUSY quantum mechanics [18–21].To dX ðnÞðX Þ ðnÞðX Þ¼Nn into account, we obtain Z this end, let us introduce the Fermi oscillators characterized ∞ † 2 †2 1 dτ 2 1 0 − −iτððmcÞ −iϵÞ by fb;b g¼ ;b ¼b ¼ , which can be represented in ImSeff½Ac¼ Re e ℏ 0 τ two-dimensional vector space so that − τ 2 ˆ ˆ 2 ×trðe i ð mH01þP⊥ÞÞ 00 01 Z † ∞ X∞ b ¼ ;b¼ : ð43Þ dτ − τ 2− ϵ −τ2 ℏω 1 10 00 − i ððmcÞ i Þ m ðnþ2Þ ¼ Re τ e e 0 0 Z Z n¼ In terms of ðb; b†Þ, the supersymmetric extension of Hˆ π r 0 2 should be × dX δð0Þ d X⊥ ð2πℏÞ2iτ pffiffiffiffiffiffiffiffiffiffi Λ¯ 0 mℏω Hˆ ¼ −iℏωð−iAA¯ þ b†bÞ¼−iℏω : ð44Þ × δð0Þ ∼ ; ð39Þ r 0 Λ ð2πℏÞ R −1 0 Then the generators of SUSY transformation defined by where the integral in δð0Þ¼ð2πℏÞ dP has been cut by the typical momentum scale in the X0;X1 so that ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi 0 − R ffiffiffiffiffiffiffiffiffiffi R ð Þ R p † p iA 0 p 0 2 Q ¼ −i ℏωAb ¼ ℏω ; dP ∼ mℏω. Putting here V0 ∼ dX , V⊥ ∼ d X⊥ 00 0 as cutoff volumes in X ;X⊥ spaces, respectively, the right- pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 00 hand side of this equation becomes, with ϵ 0, ¯ − ℏω ¯ ℏω ¼þ Q ¼ i Ab ¼ ¯ ; ð45Þ Z −iA 0 π ∞ τ −iτðmcÞ2 δ 0 i d e rhs ¼ ð ÞV0V⊥Re 2 2 are characterized by the algebras ð2πℏÞ ϵ τ 2 sinhðτmℏωÞ Z π 2 ∞ −iz † 0 δ 0 i ðmcÞ dz e ½Q; A¼fQ; b g¼ ; ¼ ð ÞV0V⊥ 4 2πℏ 2 P 2 ℏω pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð Þ −∞ z sinh ðz 2Þ ¯ ℏω † − ℏω mc ½Q; A¼ b ; fQ; bg¼ i A; ð46Þ ∞ 1 ℏω 2 X −1 nþ π 2 2 ðm Þ jejE ð Þ −n c ðm cÞ ¯ ¯ ¯ ¼ δð0ÞV0V⊥ e ℏ jejE ; ½Q; A¼fQ; bg¼0; 2 2πℏ 2 2 2 ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi ð Þ m c n¼1 n p p ½Q;¯ A¼− ℏωb; fQ;¯ b†g¼−i ℏωA;¯ ð47Þ ð40Þ and where the P denotes the principal value in the z τ mc 2 pð¼ffiffiffiffiffiffiffiffiffiffið Þ Þ ℏω Hˆ ¯ Hˆ 0 ¯ Hˆ integral.R In terms of the momentum scale m , the ½Q; r¼½Q; r¼ ; fQ; Qg¼ r: ð48Þ volume dX1 should also be given by 1 pffiffi ¯ piffiffi ¯ − pffiffiffiffiffiffiffiffiffiffi Z If we introduce Q1 ¼ 2 ðQ þ QÞ and Q2 ¼ 2 ðQ QÞ, ℏω m 1 the last equations can also be written as V1 ∼ 1 V1 ∼ dX ð41Þ kℏ ˆ fQi;Qjg¼δijHr; ði; j ¼ 1; 2Þ: ð49Þ with a dimensionless constant k. Thus, with Vð4Þ ¼ 2 ω jejE Those algebras should be compared with that of N ¼ V0V1V⊥ and ¼ m m2c , we arrive at the expression SUSY quantum mechanics, though Qi (i ¼ 1, 2) are not Hˆ 4 ∞ 1 Hermitian operators. The zero-point oscillation of r is 1 1 2 X −1 nþ π 2 2 Vð4Þm jejE ð Þ −n c ðm cÞ ImS ½A ∼ e ℏ jejE : removed by this supersymmetry. ℏ eff c k 16π3ℏ2 2 2 m c n¼1 n In spite of the formal resemblance of the present dynamical system to SUSY quantum mechanics of HOs, ð42Þ the true nature of both dynamical systems are fairly ¯ † For k ¼ 1, the result coincides with the formula of the pair different as can be seen from Q ≠ Q , the nonpositive ˆ creation given by Schwinger for scalar QED [13,17].It structure of Hr, and so on. On the discrete complete basis ϕ ϕ¯ Hˆ ϕ ϕ should also be noticed that one can replace the constraints f ðnÞ; ðnÞg, the eigenvalue equation rj Ei¼Ej Ei can 2πℏ δ 0 on the cutoff parameters by a weaker condition ð Þ ð Þ ∼ 1, be solved easily: for n 1; 2; …, mωV1 k ¼

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þ En ¼ −iℏωn doublet 0 ϕ− − ⊗ ϕ 2 j n i¼ ϕ ¼j i j ðnÞi ðnÞ 1 Q − piffiffi Aϕ þ ffiffiffiffiffiffiffiffiffi − n ðnÞ 0 jϕn i¼p jϕn i¼ ð50Þ ℏωn 0 -1

E− ¼ iℏωn doublet -2 n ϕ¯ ¯ ðnÞ ¯ jϕþi¼ ¼jþi⊗ jϕ i FIG. 2. The eigenvalues of Hˆ in an extended SUSY quantum n 0 ðnÞ r mechanics are illustrated. The states with the superscript are ¯ − ¯ þ ¯ 0 pair states mapped by Q or Q. The states jϕ0 i and jϕ0 i are fixed ¯ − Q ¯ jϕ i¼pffiffiffiffiffiffiffiffiffi jϕþi¼ ; ð51Þ states under those mapping. n n − piffiffi ¯ ϕ¯ ℏωn n A ðnÞ ℏ 0 1 − γ Πˆ 2 Πˆ 2 − g σ μν − ð · ðAcÞÞ ¼ ðAcÞ μνF where j i¼ 0 and jþi ¼ 0 . Here, the mapping 2c − ℏg jϕþi¼pffiffiffiffiffiffiffiQ jϕ i, ðn ≥ 1Þ can be inverted by Πˆ 2 − σ01 n ℏωn n ¼ ðAcÞ E ¯ c − ffiffiffiffiffiffiffiQ þ jϕn i¼i p jϕn i, and so the states jϕn i, ðn ¼ 1; 2; …Þ ℏ ℏωn ˆ 2 g ¼ 2mH01 þ pˆ − ðiσ1 ⊗ σ1ÞE: ð54Þ form a tower of super pairs. In the same sense, the states ⊥ c ¯ jϕn i; ðn ¼ 1; 2; …Þ form another tower of super pairs. − ¯ þ In contrast, the states jϕ0 i and jϕ0 i belonging to the Carrying out the unitary transformation in 4-spinor space iπσ iπσ 4 2 ⊗ 4 2 same eigenvalue E0 ¼ 0 are two super singlets, which by U ¼ e e , Eq. (54) becomes − ¯ þ satisfy Qijϕ0 i¼Qijϕ0 i¼0; (i ¼ 1, 2). Therefore, in the 2 − − γ Πˆ † ϕ ; ϕ Uð · ðAcÞÞ U space of states fj 0 i fj n igg, the supersymmetry is realized as a good symmetry; that is, SUSY is not broken. 1 ¯ 2 ϕ¯ þ ϕ¯ ¼ 2m −iℏω −iAA þ − mℏωiσ3 ⊗ σ3 þ pˆ ⊥ The same is true for the space of states fj 0 i; fj n igg.In 2 each space, the operators Q work as the generators of i −iAA¯ þ b†b 0 supersymmetry; however, there arises no mapping between ¼ 2m −iℏω þ pˆ 2 ; 0 − ¯ † ⊥ those two spaces by Qi (Fig. 2). iAA þ bb In the context of this SUSY quantum mechanics, we ð55Þ emphasize the following: in the Schwinger effect for fermions, the SUSY quantum mechanics of the RHO plays † jejE σ3 ⊗ σ3 σ3 ⊗ b ;b ω an effective role in its background; that is, topics III.A and where we have used ¼ ½ and ¼ mc III.B are not independent in this effect. as before. The result implies that the spectra of upper ˜ The Dirac field Ψ interacting with an external gauge field components of Ψ ¼ UΨ are those of the supersymmetric μ 8 ˆ A obeys the Uð1Þ symmetry field equation Hamiltonian Hr; on the other side, the spectra of lower ˜ ˆ 0 components of Ψ are governed by Hr, which is obtained ˆ μ ˆ † ðγ · ΠðAÞþmcÞΨ ¼ 0 ðγ · ΠðAÞ¼γμΠðAÞ Þ: ð52Þ from Hr changing the role of ðb; b Þ. Thus, one can evaluate the Schwinger effect for fermions again according When we multiply this equation by −ðγ · Πˆ ðAÞ − mcÞ from to the procedure of Eqs. (39) and (40) (Appendix C). the left, the field equation becomes the second order form such that IV. SUMMARY In this paper, we have discussed the eigenvalue problems − γ Πˆ 2 2 Ψ 0 ½ ð · ðAÞÞ þðmcÞ ¼ : ð53Þ of RHOs in terms of ladder operators ðA; A¯ Þ introduced by an analogous way to the ladder operator ða;ˆ aˆ †Þ in HOs. Here, if we use the configuration of gauge potentials The nonpositive property of the Hamiltonian operator Hˆ in 0 A − 1 0 σμν r ðAcðxÞ; cðxÞÞ¼ð Ex ; Þ as in Eq. (33), then with ¼ RHOs is a result of the property of ladder operators such as i γμ γν ∂ † ¯ † ¯ ¯ 2 ½ ; and Fμν ¼ ½μðAcÞν, we obtain A ¼ A, A ¼ A, and ½A; A¼i. Then, the eigenstates ¯ ¯ ¯ Aϕσ ¼ σϕσ and Aϕσ ¼ σϕσðσ ∈ RÞ are able to normalize 8 μ ν μν ¯ ¯ 0 The gamma matrices are normalized so that fγ ; γ g¼−2η . so that hϕσjϕσ0 i¼hϕσjϕσ0 i¼δðσ − σ Þ. Namely, the

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¯ × fϕσ; ϕσg are continual bases of the space of functionals Φ various problems other than the topics discussed in this paper. including the Hilbert space H of the RHO in the framework For example, the Hamiltonian Hˆ is able to take continuous ¯r of rigged Hilbert space. Those continual bases are not eigenvalues on the states ðϕσ; ϕσÞ; in the space of those ˆ eigenstates of Hr, but rather the states related to fjxi; jpig eigenstates, the SUSY may show a different feature from the by a unitary transformation. standard analysis. Those are interesting future problems. ϕ ¯ nϕ ϕ¯ On the other hand, the states ðnÞ ¼ A ð0Þ and ðnÞ ¼ nϕ¯ ∈ N ϕ ϕ¯ ϕ ϕ¯ ACKNOWLEDGMENTS A ð0Þ ðn Þ with ð ð0Þ; ð0ÞÞ¼ð 0; 0Þ are eigenstates ˆ 1 of Hr belonging to the eigenvalues iℏωðn þ 2Þ. The authors wish to thank the members of the theoretical Since those states satisfy the normalization of the form group at Nihon University for their hospitality. They also ϕ¯ ϕ δ ϕ ϕ¯ h ðmÞj ðnÞi¼ m;nNn, it can be shown that the f ðnÞ; ðnÞg appreciate one of referees concerning improvements to the form a discrete complete basis of Φ× in pairs. Contrary to descriptions of Sec. II. this, the discrete eigenstates fΦng of the Hamiltonian for a HO are the basis of a Hilbert space. APPENDIX A: WEBER’S FUNCTIONS We can also show that Weber’s D functions, the special AS THE ENERGY EIGENVALUE FUNCTIONS functions known as analytic solutions of the eigenvalue FOR THE RHO ˆ equation for Hr with continuous eigenvalues, are obtained We here summarize the standard way to make the ϕ ϕ¯ by means of the analytic continuation of f ðnÞ; ðnÞg with eigenvalue functions of the RHO reduce to Weber’s functions. ¯ ∂ respect to n. The D functions and fϕ n ; ϕ n g stand on the ˆ − ℏ ð Þ ð Þ In the x representation with p ¼ i ∂x, the eigenvalue ˆ ˆ same footing as the scattering states of Hr unless any equation of Hr can be written as boundary conditions are added. 2 2 2 As good applications of this ladder operator formalism, ℏ d mω 2 − − x − E ψ ðxÞ¼0: ðA1Þ we have shown two topics: the Schwinger effect in scalar 2m dx2 2 E QED and an extension of RHO to SUSY quantum mechanics. In the first, the Hamiltonian of particles Introducing here the variable z defined by rffiffiffiffiffiffiffiffiffiffi interacting with a constant electric field is shown to be 2 2 iπ ℏ d 2mω d canonically equivalent to one of RHOs and so the knowl- 4 x ¼ e 2 ωz; 2 ¼ ℏ 2 ; ðA2Þ edge of RHOs is useful to handle the problem of pair m dx i dz production by the electric field. Indeed, it has been shown Eq. (A1) with ψ ðxðzÞÞ ¼ w ðzÞ gives rise to ϕ ϕ¯ E E that the discrete complete bases f ðnÞ; ðnÞg characterized 2 by Eq. (21) give a simple way to evaluate such a production i d iE 1 2 − ×Eq:ðA1Þ¼ þ − z w ðzÞ¼0: ðA3Þ rate within the framework of quantum mechanics. ℏω dz2 ℏω 4 E Second, we have tried to extend the present RHO system iE λ 1 to a supersymmetric dynamical system; the extended Writing ℏω ¼ þ 2 and wEðzÞ¼wλðzÞ, Eq. (A3) becomes ˆ the standard form of Weber’s equation Hamiltonian Hr is again a nonpositive Hermitian operator † constructed out of fermionic oscillators ðb; b Þ and ladder 2 1 2 ¯ d wλðzÞ λ − z 0 operators ðA; AÞ. The ladder operator formalism gives rise 2 þ þ 2 4 wλðzÞ¼ : ðA4Þ ¯ dz to two towers of super-pair states jϕn i and jϕn i 1 2 … þ 1z2 ðn ¼ ; ; Þ, which belong to the eigenvalues En ¼ w˜ λ z e4 wλ z − For ð Þ¼ ð Þ, Eq. (A4) can also be written as −iℏωn and En ¼ iℏωn, respectively. In addition to this, − ¯ þ 2 the n ¼ 0 states jϕ0 i and jϕ0 i exist as two singlet states, d d − λ ˜ λ 0 − ¯ þ 2 z þ w ðzÞ¼ : ðA5Þ which satisfy Qijϕ0 i¼Qijϕ0 i¼0; (i ¼ 1, 2). Namely, in dz dz each space of super-pair tower states, SUSY is realized as a To solve Eq. (A5), let us use the Fourier-Laplace good symmetry, though the SUSY in this model is an representation extended concept from the standard one as can be seen Z † ≠ from Qi Qi. −zt w˜ λðzÞ¼ dte fλðtÞ; ðA6Þ Furthermore, we have brought up the following: if we Γ consider the Dirac fields interacting with an external electric Γ field, then the supersymmetric structure of RHOs will be where is a path from a to b in the complex t plane. Then implicitly included in a loop effect of those Dirac fields. under the integration by parts with respect to t, Eq. (A5) According to this line of approach, we have shown the way with Eq. (A4) gives to evaluate the Schwinger effect for fermions in Appendix C. d λ The knowledge on the complete bases in RHOs under the ftfλðtÞg þ t þ ftfλðtÞg ¼ 0 ðA7Þ ladder operator formalism is expected to give useful tools in dt t

025002-8 LADDER OPERATORS IN REPULSIVE HARMONIC … PHYS. REV. D 102, 025002 (2020) rffiffiffiffiffiffi Z ∞ i 0 2 i −i 2 0 e2ðA Þ ¼ dke 2k þikA ðB2Þ 2π −∞

0 aˆ bˆ aˆ bˆ −1 a;ˆ bˆ and using again e þ ¼ e e e 2½ , we arrive at rffiffiffiffiffiffiffiffiffiffi Z ∞ pffiffiffiffiffi mω mω 2 i 2 − 2mω − 0 mω 02 C ˆ 0 i 2ℏ x −2k ið ℏ x kÞA i 2ℏ x hxjIrjx i¼ 2e dke e e 2ℏπ −∞ rffiffiffiffiffiffiffiffiffiffi Z FIG. 3. Contour C in a complex plane. ∞ pffiffiffiffiffi pffiffiffiffi mω mω 2 i 2 − 2mω − mω 0 i 2ℏ x −2k ið ℏ x kÞ 2ℏ x ¼ 2ℏπ2e dke e −zt b −∞ on the condition that e tfλ t 0. Equation (A7) ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi ½ f ð Þga ¼ p 2 p p 2 1 2 i 2mω − mω 0− 2mω − ℏ − t −ðλþ1Þ 4ð ℏ x kÞ i 2ℏ fx ð ℏ x kÞ 2 ωg can be solved easily so that fλðtÞ¼conste 2 t ; since ×e e m rffiffiffiffiffiffiffiffiffiffi the boundary conditions are satisfied by ða; bÞ¼ð0; ∞Þ for ω m mω 2 − 2mω 0 mω 2 mω 02 Reλ < 0 on the real t axis, and we finally obtain the integral ei 2ℏ x e i ℏ xx þi ℏ x þi 2ℏ x ¼ 2ℏπ2 × −1z2 representation for wλ z e 4 w˜ λ z Dλ z in such a Z ð Þ¼ ð Þð¼ ð ÞÞ ∞ pffiffiffiffiffi 2mω − 0 form as [16,22] × dkeik ℏ ðx x Þ ¼ δðx−x0Þ: ðB3Þ Z −∞ −1z2 ∞ e 4 − −1 2 − λ 1 zt 2t ð þ Þ λ 0 ˆ DλðzÞ¼Γ −λ dte t ðRe < ÞðA8Þ Therefore, Ir is nothing but the unit operator for the present ð Þ 0 RHO system. Z Γðλ þ 1Þ −1 2 − −1 2 − λ 1 ¼ − e 4z dte zt 2t ð−tÞ ð þ Þ; ðA9Þ 2πi C APPENDIX C: THE SCHWINGER EFFECT FOR FERMIONS where C is the contour given in Fig. 3. It is not difficult to Ψ rewrite the contour integral in Eq. (A9) to the path integral The action of the Dirac field R obeying Eq. (52) with the μ Ψ 4 Ψ¯ γ Πˆ Ψ in Eq. (A8) by taking into account Γðλ þ 1Þ sinð−πλÞ¼ gauge fields Ac is SD½ ;Ac¼ d x ð · ðAcÞþmcÞ ; π . Ψ¯ Ψ†γ0 Γð−λÞ ð ¼ R Þ. Then the path integral result SQ½Ac¼ 9 i ’ ¯ ℏSD ˆ The function DλðzÞ is Weber s D function (Parabolic −iℏ log DΨDΨe ¼ −iℏTr logðγ · Π þR mcÞþconst is 1 4 2 cylinder function) [22], by which the independent solutions the quantum correction to SG½Ac¼2 d xE so that iE λ 1 of Eq. (A1) for ℏω ¼ þ 2 are given as DλðzÞ and Seff½Ac¼SG½AcþSQ½Ac becomes the effective action D−λ−1ðizÞ. of Ac. Here, the “Tr” involves the trace over four-component spinor space. To evaluate the Tr in S ½A , we notice that ˆ Q c APPENDIX B: ANOTHER PROOF OF Ir =1 d ˆ By the definitions of ϕ and ϕ¯ , we obtain the Tr log½γ · Π þðmcÞþa ðnÞ ðnÞ da expression −ðγ · Πˆ ÞþfðmcÞþag ¼ Tr X∞ 1 −ðγ · Πˆ Þ2 þfðmcÞþag2 hxjIˆ jx0i¼ ϕ ðxÞϕ¯ ðx0Þ r N ðnÞ ðnÞ fðmcÞþag n¼0 n ¼ Tr rffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffi − γ Πˆ 2 2 2π ω X∞ 1 ð · Þ þfðmcÞþag m iπ mω 2 mω 02 Z − 2 ¯ 0 n i 2ℏ x i 2ℏ x ¼ 2 ðe AA Þ e e 1 ∞ τ i 2ℏπ n! d d −iτ½−ðγ·Πˆ Þ2þfðmcÞþag2−iϵ n¼0 ¼ − Tr e ; ðC1Þ rffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffi da 2 0 τ 2π ω −iπ m 2 ¯ 0 mω 2 mω 02 e AA i 2ℏ x i 2ℏ x ¼ 2e e e γ i 2ℏπ in consideration of which the trace of odd powers of rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi ffiffiffiffi matrices vanishes. Integrating this equation with respect to a 2π ω p ω m mω 2 −2iA0 m x i 0 2 mω 02 ¼ ei 2ℏ x e 2ℏ e2ðA Þ ei 2ℏ x : ðB1Þ from a1 to a2, we obtain i 2ℏπ2 γ Πˆ − γ Πˆ ˆ ˆ ˆ ˆ −1 ˆ ˆ Tr log ½ · þðmcÞþa2 Tr log ½ · þðmcÞþa1 eaþb eaebe 2½a;b Z Here, we have used the formula ¼ pffiffiffiffiffi for ∞ −iπ ω 1 dτ ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ 2 0 m iτfðγ·ΠÞ þiϵg ½½a; b; a¼½½a; b; b¼ , with a ¼ e A 2ℏ x and ¼ − Tr e −iπ −1 2 0 τ ˆ 2 0 pffiffiffiffiffiffiffiffiffi ˆ b ¼ e A 2 ℏω p. Remembering, further, 2 2 m × ½e−iτfðmcÞþa2g − e−iτfðmcÞþa1g : ðC2Þ 9 The D function is normalized so that DnðzÞ; ðn ¼ 0; 1; …Þ −1 2 ˆ 4z Setting a2 ¼ 0 and a1 ¼ −ðγ · Π þ mcÞþ1, we get the reduces to e Hen ðzÞ, where fHen ðzÞg are the Chebyshev- Hermite polynomials. expression

025002-9 AOUDA, KANDA, NAKA, and TOYODA PHYS. REV. D 102, 025002 (2020) Z ∞ γ Πˆ 1 dτ ˆ 2 2 Tr logð · þ mcÞ Tr logðγ · Πˆ þ mcÞ¼− Tr eiτðγ·ΠÞ e−iτfðmcÞ −iϵg Z 2 τ 2 2 π ∞ −iz 0 −δ 0 ðmcÞ dz e ¼ ð ÞV0V⊥ × 2 2 ℏω ; ðC5Þ 2πℏ i ϵ 2 ðC3Þ ð Þ z tanh ðz mc Þ 2 disregarding an unimportant additional constant. Then where z ¼ τðmcÞ . The integration with respect to z in τℏg σ01 ℏgE Eq. (C5) can be carried out in the same manner as Eq. (40) i c E τ remembering Eq. (54) and using e ¼ cosh ð c Þþ n 2 ℏ except replacing the residue ð−1Þ of 1= sinhðzℏω=mc Þ by iσ01 sinh ðτ gEÞ, Eq. (C3) becomes pffiffiffiffiffiffiffi c 1 ℏω 2 mℏω ∼ 1 1of = tanhðz =mc Þ. Using further V1 kℏ with k ¼ 1, the resultant formula corresponding to Eq. (42) in the Trlogðγ ·Πˆ þmcÞ Z case of Dirac fields becomes ∞ 1 dτ ˆ 2 2 ℏgE − 4 −iτðΠ þðmcÞ −iϵÞ τ 1 ¼ 2× tr τ e cosh ðC4Þ − γ Πˆ 0 c ℏ ImSeff½Ac¼ ReTr log ½ · ðAcÞþmc 4 ∞ 2 X 1 π 2 2 01 m jejE −n c ðm cÞ by virtue of the trace of σ vanishes. Here, the “tr” ∼ V 4 e ℏ jejE : ð Þ 8π3ℏ2 m2c n2 denotes the trace in the functional space, which yields n¼1 −iτΠˆ 2 δ 0 1 1 ω jejE tre ¼ ð ÞV0V⊥ð4πiℏ2τÞ 2 sinhðτmℏωÞ with ¼ mc and ðC6Þ pffiffiffiffiffiffiffi δ 0 mℏω ð Þ ð2πℏÞ as in the case of scalar QED. Therefore, we arrive This formula is nothing but the one given originally by at the expression with ϵ ¼þ0 Schwinger [13].

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