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Normalization of the Ground State of the Supersymmetric Harmonic Letters in High Energy Physics LHEP xx, xxx, 2018 Normalization of the Ground State of the Supersymmetric Harmonic Oscillator Ahmed Ayad School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg, South Africa Abstract Supersymmetry plays a main role in all current thinking about superstring theory. Indeed, many remark- able properties of string theory have been explained using supersymmetry as a tool. So far, there has been no unbroken supersymmetry observed in nature, and if nature is described by supersymmetry, it must be broken. Supersymmetry may be broken spontaneously at any order of perturbation theory or dynamically due to nonperturbative effects. To examine the methods of supersymmetry breaking, special attention is given to discuss the normalization of the ground state of the supersymmetric harmonic oscillator. This study explains that perturbation theory gives incorrect results for both the ground-state wave function and the energy spectrum and it fails to give an explanation to the supersymmetry breaking. Keywords: supersymmetry breaking, perturbation theory, fermionic as well as bosonic fields using the usual quantum nonperturbative effects mechanical operators method. Then, we consider supersym- metric classical mechanics and study generalized classical Pois- son brackets to Dirac brackets and quantization rules in or- 1. INTRODUCTION der to introduce the superalgebra. Furthermore, the formal- ism of supersymmetry in quantum mechanics is introduced. Supersymmetry, often abbreviated SUSY, was first introduced Subsequently, we give a general background in the concepts in 1971 by Gelfand and Likhtman, Raymond, and Neveu and and methods of supersymmetric quantum mechanics. This fol- Schwartz, and later, it was rediscovered by other groups [1, 2]. lowed by providing a more specific study of the ground state Supersymmetric quantum mechanics was developed by Witten of the supersymmetric harmonic oscillator. Finally, our conclu- in 1982 [3], as a toy model to test the breaking of supersymme- sion is given, and for completeness, some needed derivations try. In general, supersymmetry plays a main role in the mod- are broadly outlined in the appendix. ern understanding of theoretical physics, in particular, quan- tum field theories, gravity theories, and string theories [4]. Supersymmetry is a symmetry that connects particles 2. HARMONIC OSCILLATOR of integer spins (bosons) and particles of half-integer spins (fermions) [5]. In supersymmetry, it is possible to introduce op- Supersymmetric theories necessarily include both bosons and erators that change bosons which are commuting variables into fermions, so a system that includes both bosonic and fermionic fermions which are anticommuting variables and vice versa [6]. degrees of freedom is considered here. The supersymmetry al- Supersymmetric theories are theories which are invariant un- gebra is a mathematical formalism for describing the relation der those transformations [7]. The supersymmetry algebra in- between bosons and fermions, and we will discuss this later in volves commutators as well as anticommutators [8]. Section 4. In quantum mechanics, bosons are integer spin parti- In this study, the main mathematical structure which in- cles which obey Bose-Einstein statistics, and the bosonic opera- volves supersymmetric quantum mechanics is derived starting tors satisfy the usual commutation relations, while fermions are with explaining the basic idea of supersymmetry and followed half-integer spin particles characterized by FermiDirac statis- by introducing the necessary framework to make a supersym- tics, obey the Pauli exclusion principle, and can be described metric theory. We derive the basic formulation of supersym- by Grassmann variables, which are anticommuting objects; for metric quantum mechanics starting with introducing the con- review, see [9, 10, 11, 12]. cepts of supercharges and superalgebra. We show that if there The simplest system that consists of a combination of is a supersymmetric state, it is the zero-energy ground state. If bosonic and fermionic fields is the one-dimensional harmonic arXiv:2008.02080v1 [physics.gen-ph] 24 Jul 2020 such a state exists, the supersymmetry is unbroken; otherwise, oscillator. There are many excellent reviews on this subject; we it is broken. So far, there has been no unbroken supersymmetry refer the reader to them for more and complete expositions observed in nature, and if nature is described by supersymme- [2, 13, 14, 15]. We begin by considering the Lagrangian formu- try, it must be broken. In fact, supersymmetry may be broken lation for the bosonic and fermionic harmonic oscillator: spontaneously at any order of perturbation theory or dynam- 1 2 αβ 2 ically due to nonperturbative effects. To examine the methods L = q˙ + iψαδ ψ˙ β + F + ω (iψ1ψ2 + qF) , (1) of supersymmetry breaking, we study the normalization of the 2 ground state of the supersymmetric harmonic oscillator and where q(t) and F(t) are real bosonic fields, ψ (t) (with α α ∈ calculate the corrections to the ground-state energy using per- 1, 2 ) are real fermionic fields, ω is a real parameter, δαβ is the { } turbation theory. We found out that no perturbation effect can Kronecker delta, and we used the Einstein summation conven- lead to supersymmetry breaking and it must be due to the non- tion. perturbative effects. It is straightforward, using the classical equation of motion This work is structured as follows. After a brief intro- for F + ωq = 0, to eliminate F from Eq. (1) and obtain the equiv- duction, we study the problem of a harmonic oscillator with 1 Letters in High Energy Physics LHEP xx, xxx, 2018 alent Lagrangian: that the bosonic ladder operators aˆ and aˆ† satisfy the following commutation relations 1 L = q˙2 + iψ δαβψ˙ ω2q2 + iωψ ψ . (2) 2 α β − 1 2 [aˆ, aˆ†]= 1, [aˆ, aˆ]=[aˆ†, aˆ†]= 0 . (11) Then, using the Legendre transformation from the variables However, on the other hand, the lowering (annihilation) and q, q˙, ψ, ψ˙ to q, p, ψ, π , we obtain raising (creation) operators for fermions, bˆ† and bˆ, respectively { } { } are defined as αβ β 1 2 2 2 H = qp˙ + ψαδ π L = p + ω q iωψ ψ , (3) − 2 − 1 2 1 1 bˆ = ψˆ iψˆ , bˆ† = ψˆ + iψˆ . (12) 2¯h 1 − 2 2¯h 1 2 where πα are the momenta conjugate to ψ . r r α Later in this study, we shall see that the general proper- Also, as will be shown in Section 4, the fermionic operators ψα ties of supersymmetry have to be clear if the Hamiltonian is and ψβ satisfy the canonical quantum anticommutation con- built as a larger Hamiltonian consisting of two components, dition ψ , ψ = h¯ δ 1. Therefore, it can be seen that the { α β} αβ one representing the bosonic field and the other representing fermionic ladder operators bˆ and bˆ† satisfy the following an- the fermionic field. Leaving this aside for the moment, to de- ticommutation relations: rive the supersymmetric Hamiltonian, let us define the bosonic valuable qˆ and the bosonic momentum pˆ, respectively, as fol- bˆ, bˆ† = 1, bˆ, bˆ = bˆ†, bˆ † = 0 . (13) lows: { } { } { } It is possible using Eq. (10) to write the bosonic position and ∂ momentum variables in terms of the bosonic ladder operators Á qˆ = q Á2, pˆ = ih¯ 2 . (4) − ∂q as follows: On the other hand, the fermionic variables ψˆ1 and ψˆ2 and the † 2ω h¯ † α aˆ + aˆ = qˆ qˆ = (aˆ + aˆ), fermionic momenta π , respectively, have to be defined as r h¯ ⇒ r 2ω 2 h¯ ω h h aˆ aˆ† = ipˆ pˆ = i (aˆ† aˆ) . ˆ ¯ ˆ ¯ ψ1 = σ1, ψ2 = σ2, − r h¯ ω ⇒ r 2 − r 2 r 2 (14) α ∂L i αβ π = = δ ψβ, α, β 1, 2 . (5) ∂ψ˙α − 2 ∈{ } Similarly, using Eq. (12), we can write the fermionic variables ψ1 and ψ2 in terms of the fermionic ladder operators as follows: Hence, using Eqs. (4, 5), we can rewrite the previous Hamilto- nian (3) in the following form: 2 h¯ bˆ† + bˆ = ψˆ ψˆ = (bˆ + bˆ†), h¯ 1 ⇒ 1 2 2 r r 1 2 d 2 2 1 Hˆ = h¯ + ω q Á2 + h¯ ω σ3 , (6) † 2 h¯ † 2 − dq2 2 bˆ bˆ = i ψˆ2 ψˆ2 = i (bˆ bˆ ) . − r h¯ ⇒ r 2 − (15) where Á is the 2 2 identity matrix and σ are the Pauli matri- 2 × j ces; that is, Actually, we derived Eqs. (14, 15) here; however, it will be used 1 0 0 1 0 i 1 0 later. At this stage, let us define the following two Hermitian Á = , σ = , σ = − , σ = . (7) 2 0 1 1 1 0 2 i 0 3 0 1 operators, the bosonic number operator Nˆ , and the fermionic − B number operator Nˆ F: We want to obtain solutions of the time-independent Schr¨odinger Ψ = Ψ 1 equation H E ; that is, Nˆ aˆ†aˆ = ω2qˆ2 h¯ ω + pˆ2 , B ≡ 2¯hω − 2 1 2 d 2 2 1 † i h¯ + ω q Á2 + h¯ ω σ3 = EΨ , (8) Nˆ bˆ bˆ = − ψˆ ψˆ . (16) 2 − dq2 2 F 1 2 ≡ h¯ with the wave function Ψ(x) constraint to satisfy appropriate Using Eq. (16), we can rewrite the Hamiltonian (6) in the oper- boundary conditions. ator formalism As we mentioned, the supersymmetric Hamiltonian can be 1 1 Hˆ = h¯ ω Nˆ + + h¯ ω Nˆ . (17) written as a summation of two terms representing the bosonic B 2 F − 2 and the fermionic components of the Hamiltonian This means that the energy eigenstates can be labeled with the H = HB(p, q)+ HF(ψ) . (9) eigenvalues of nB and nF. The action of the ladder operators, aˆ, aˆ†, bˆ and bˆ†, upon an To this end, we define the lowering (annihilation) and raising energy eigenstate nB, nF is listed as follows: (creation) operators for bosons aˆ† and aˆ, respectively, they are | i aˆ n , n = √n n 1, n , aˆ† n , n = √n + 1 n + 1, n , | B Fi B| B − Fi | B Fi B | B Fi 1 † 1 bˆ n , n = √n n , n 1 , bˆ† n , n = √n + 1 n , n + 1 .
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