Classical Analog of Extended Phase Space SUSY and Its Breaking

Abstract

We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom which gives rise to (𝑁=2)-realization of supersymmetry (SUSY) algebra. By means of an iterative procedure, we find the approximate ground state solutions to the extended Schrödinger-like equation and use these solutions further to calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. Consequently, we calculate a more practical measure for the SUSY breaking which is the expectation value of an auxiliary field. We analyze nonperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra.

1. Introduction

An interesting question of keeping the symmetry between canonical coordinates and momenta in the process of quantization deserves an investigation. From its historical development, this aspect of statistical quantum mechanics, unfortunately, has attracted little attention. However, much use has been made of the technique of ordering of canonical coordinates (𝑞) and momenta (𝑝) in quantum mechanics [1, 2]. It was observed that the concept of an extended Lagrangian ℒ(𝑝,𝑞,𝑝,̇𝑞)̇ in phase space allows a subsequent extension of Hamilton’s principle to actions minimum along the actual trajectories in (𝑝,𝑞)-, rather than in 𝑞-space. This leads to the phase space formulation of quantum mechanics. Consequently this formalism was developed further in [3] by addressing the extended phase space stochastic quantization of Hamiltonian systems with first class holonomic constraints. This in a natural way results in the Faddeev-Popov conventional path-integral measure for gauge systems. Continuing along this line in the present paper we address the classical analog of the extended phase space (𝑁=2)-SUSY quantum mechanics [4] of the particles which have both bosonic and fermionic degrees of freedom, that is, the quantum field theory in (0+1)-dimensions in (𝑞,𝑝)-space, exhibiting supersymmetry (for conventional SUSY quantum mechanics see [5–12]). We analyze in detail the non-perturbative mechanism for supersymmetry breaking in the instanton picture ([13]). This paper has been organized as follows. In the first part (Sections 2 and 3), we derive the classical analog of the extended phase space SUSY quantum mechanics and obtain the integrals of motion. Consequently, we describe the extended phase space (𝑁=2)-SUSY algebra. In the second part (Sections 4 and 5), by means of an iterative scheme, first, we find the approximate ground state solutions to the extended Schrödinger-like equation and then calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. We calculate a more practical measure for the SUSY breaking, in particular in field theories, which is the expectation value of an auxiliary field. We analyze nonsperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Section 6 deals with the independent group theoretical methods with nonlinear extensions of Lie algebras from the perspective of extended phase space SUSY quantum mechanics and, further, shows how it can be useful for spectrum generating algebra. The concluding remarks are given in Section 7. Unless otherwise stated we take the geometrized units (ℏ=𝑐=1). Also, an implicit summation on repeated indices is assumed throughout this paper.

2. The Integrals of Motion

For the benefit of those not familiar with the framework of extended phase space quantization, enough details are given in the following to make the rest of the paper understandable. The interested reader is invited to consult the original papers [1, 2] for further details. In the framework of the proposed formalism, the extended Lagrangian can be written as ℒ_ext𝑝,𝑞,𝑝,̇𝑞̇=−𝑞̇_𝑖𝑝_𝑖−𝑞_𝑖𝑝̇_𝑖+ℒ^𝑞+ℒ^𝑝, (1) where a dynamical system with 𝑁 degrees of freedom described by the 2𝑁 independent coordinates 𝑞=(𝑞₁,…𝑞_𝑁) and momenta 𝑝=(𝑝₁,…𝑝_𝑁) which are not, in general, canonical pairs. A Lagrangian ℒ^𝑞(𝑞,𝑞)̇ is given in 𝑞-representation and the corresponding ℒ^𝑝(𝑝,𝑝)̇ in 𝑝-representation. A dot will indicate differentiation with respect to 𝑡. The independent nature of 𝑝 and 𝑞 gives the freedom of introducing a second set of canonical momenta for both 𝑝 and 𝑞 through the extended Lagrangian 𝜋_𝑞𝑖=𝜕ℒ_ext𝜕𝑞̇_𝑖=𝜕ℒ^𝑞𝜕𝑞̇_𝑖−𝑝_𝑖,𝜋_𝑝𝑖=𝜕ℒ_ext𝜕𝑝̇_𝑖=𝜕ℒ^𝑝𝜕𝑝̇_𝑖−𝑞_𝑖. (2) One may now define an extended Hamiltonian 𝐻_ext𝑝,𝑞,𝜋_𝑝,𝜋_𝑞=𝜋_𝑞𝑖𝑞̇_𝑖+𝜋_𝑝𝑖𝑝̇_𝑖−ℒ_ext𝑝,𝑞,𝑝,̇𝑞̇=𝐻𝑝+𝜋_𝑞,𝑞−𝐻𝑝,𝑞+𝜋_𝑝, (3) where 𝐻(𝑝,𝑞)=𝑝_𝑖𝑞̇_𝑖−ℒ^𝑞=𝑞_𝑖𝑝̇_𝑖−ℒ^𝑝 is the conventional Hamiltonian of the system. In particular, vanishing of 𝜋_𝑞/or 𝜋_𝑝 is the condition for 𝑝 and 𝑞 to constitute a canonical pair. In the language of statistical quantum mechanics this choice picks up a pure state (actual path). Otherwise, one is dealing with a mixed state (virtual path). One may, however, envisage that the full machinery of the conventional quantum mechanical dynamics is extendible to the extended dynamics as alluded to the above. Here 𝑝 and 𝑞 will be considered as independent 𝑐-number operators on the integrable complex function 𝜒(𝑞,𝑝). One of the key assumptions of extended phase space quantization [1, 2] is the differential operators and commutation brackets for 𝜋_𝑝 and 𝜋_𝑞 borrowed from the conventional quantum mechanics as follows: 𝜋_𝑞𝑖𝜕=−𝑖𝜕𝑞_𝑖,𝜋_𝑞𝑖,𝑞_𝑗=−𝑖𝛿_𝑖𝑗,𝜋_𝑝𝑖𝜕=−𝑖𝜕𝑝_𝑖,𝜋_𝑝𝑖,𝑝_𝑗=−𝑖𝛿_𝑖𝑗. (4) Note also the following: 𝑝_𝑖,𝑞_𝑗=𝑝_𝑖,𝑝_𝑗=𝑞_𝑖,𝑞_𝑗=𝜋_𝑝𝑖,𝜋_𝑞𝑗=𝜋_𝑝𝑖,𝜋_𝑝𝑗=𝜋_𝑞𝑖,𝜋_𝑞𝑗=0. (5) By the virtue of (4) and (5), 𝐻_ext is now an operator on 𝜒. Along the trajectories in (𝑝,𝑞)-space, however, it produces the state functions, 𝜒(𝑝,𝑞,𝑡), via the following Schrödinger-like equation: 𝑖𝜕𝜕𝑡𝜒=𝐻_ext𝜒. (6) Solutions of (6) are 𝜒𝑞,𝑝,𝑡=𝜒_𝑟𝑞,𝑝,𝑡𝑒^{−𝑖𝑝𝑞}=𝑎_𝛼𝛽𝜓_𝛼𝑞,𝑡𝜙^∗_𝛽𝑝,𝑡𝑒^{−𝑖𝑝𝑞}, (7) where 𝑎=𝑎^†, positive definite, tr𝑎=1, and 𝜓_𝛼 and 𝜙^∗_𝛼 are solutions of the conventional Schrödinger equation in 𝑞- and 𝑝-representations, respectively. They are mutually Fourier transforms. Note that the 𝛼 and 𝛽 are not, in general, eigenindices. The normalizable 𝜒 (∫𝜒𝑑𝑝𝑑𝑞=tr(𝑎)=1) is a physically acceptable solution. The exponential factor is a consequence of the total time derivative, −𝑑(𝑞𝑝)/𝑑𝑡, in (1) which can be eliminated. Actually, it is easily verified that 𝑝+𝜋_𝑞𝜒_𝑟𝑞,𝑝,𝑡=𝜋_𝑞𝜒_𝑟𝑞,𝑝,𝑡𝑒^{−𝑖𝑝𝑞} (8) and so on. Substitution of (8) in (6) gives 𝑖𝜕𝜕𝑡𝜒𝑞,𝑝,𝑡=𝐻_ext𝜒_𝑟𝑞,𝑝,𝑡, (9) provided by the reduced Hamiltonian, 𝐻_ext. From now on we replace 𝐻_ext by 𝐻_ext and 𝜒_𝑟(𝑞,𝑝,𝑡) by 𝜒_𝑟(𝑞,𝑝,𝑡), respectively, and retain former notational conventions.

It is certainly desirable to derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom directly from what may be taken as the first principle. Therefore, following [10–12], let us consider a nonrelativistic particle of unit mass with two (𝛼=1,2) odd (Grassmann) degrees of freedom. The classical extended Lagrangian equation (1) can be written ℒ_exṫ𝑞̇̇̇1𝑝,𝑞,𝑝,=−𝑞𝑝−𝑞𝑝+2𝑞̇²𝑞+1−𝐹2𝑝̇²𝑝1−𝐺−𝑅𝑞,𝑝𝑁+2𝜓_𝛼𝜓̇_𝛼, (10) provided by 𝑁=𝜓₁𝜓₂=−𝑖𝜓₊𝜓₋. Here 𝐹(𝑞)∶ℛ→ℛ,  𝐺(𝑝)∶ℛ→ℛ, and 𝑅(𝑞,𝑝)∶ℛ→ℛ are arbitrary piecewise continuously differentiable functions given over the 1-dimensional Euclidean space ℛ. The 𝜓_𝛼 are two odd (Grassmann) degrees of freedom. The nontrivial Poisson-Dirac brackets of the system (10) are 𝑞,𝜋_𝑞=1,𝑝,𝜋_𝑝=1,𝜓_𝛼,𝜓_𝛽=𝛿_𝛼𝛽,𝜓₊,𝜓₋=1,𝜓²_±=0,𝜓_±=1√2𝜓₁±𝑖𝜓₂. (11) The extended Hamiltonian 𝐻_ext (3) reads 𝐻_ext𝑝,𝑞,𝜋_𝑝,𝜋_𝑞=12𝑝+𝜋_𝑞²+𝐹²𝑞−12𝑞+𝜋_𝑝²−𝐺²𝑝+𝑅𝑞,𝑝𝑁, (12) which, according to (8), reduces to 𝐻_ext𝑝,𝑞,𝜋_𝑝,𝜋_𝑞=12𝜋²_𝑞+𝐹²𝑞−12𝜋²_𝑝−𝐺²𝑝+𝑅𝑞,𝑝𝑁. (13) The Hamiltonian equation (13) yields the following equations of motion: 𝑞̇=𝜋_𝑞,𝑝̇=𝜋_𝑝,𝜋̇_𝑞=−𝐹^′_𝑞𝑞−𝑅^′_𝑞𝜋̇𝑞,𝑝𝑁,_𝑝=−𝐺^′_𝑝𝑝+𝑅^′_𝑝𝑞,𝑝𝑁,𝜓̇_±=±𝑖𝑅𝑞,𝑝𝜓_±. (14) A prime will indicate differentiation with respect either to 𝑞 or 𝑝. Thus, 𝑁 is the integral of motion additional to 𝐻_ext. Along the trajectories 𝑞(𝑡) and 𝑝(𝑡) in (𝑝,𝑞)-spaces, the solution to equations of motion for odd variables is 𝜓_±(𝑡)=𝜓_±𝑡₀exp±𝑖^𝑡_𝑡0𝑅𝑞(𝜏),𝑝(𝜏)𝑑𝜏. (15) Hence the odd quantities 𝜃_±=𝜃_±(𝑡)exp∓𝑖^𝑡_𝑡0𝑅𝑞(𝜏),𝑝(𝜏)𝑑𝜏 (16) are nonlocal in time integrals of motion. In trivial case 𝑅=0, we have 𝜓̇_±=0 and 𝜃_±=𝜃_±. Suppose that the system has even complex conjugate quantities 𝐵_𝑞,𝑝±,(𝐵_𝑞,𝑝+)^∗=𝐵_𝑞,𝑝−, whose evolution looks up to the term proportional to 𝑁 like the evolution of odd variables in (14). Then local odd integrals of motion could be constructed in the form 𝑄_𝑞,𝑝±=𝐵_𝑞,𝑝∓𝜓_±. (17) Let us introduce the oscillator-like bosonic variables 𝐵_𝑞,𝑝± in 𝑞- and 𝑝-representations as follows: 𝐵_𝑞∓∶ℒ²(ℛ)⟶ℒ²(ℛ),𝐵_𝑞∓=𝑝+𝜋_𝑞𝑞,𝐵±𝑖𝑊_𝑝∓∶ℒ²(ℛ)⟶ℒ²(ℛ),𝐵_𝑝∓=𝑞+𝜋_𝑝±𝑖𝑉𝑝. (18) In the expressions (18), 𝑊(𝑞)∶ℛ→ℛ and 𝑉(𝑝)∶ℛ→ℛ are the piecewise continuously differentiable functions called SUSY potentials. In particular case if 𝑅(𝑞,𝑝)=𝑅_𝑞(𝑞)−𝑅_𝑝(𝑝), for the evolution of 𝐵_𝑞,𝑝± we obtain 𝐵̇_𝑞∓=−𝐹^′_𝑞+𝑅^′_𝑞𝑁±𝑖𝑊^′_𝑞𝑞𝑝+𝜋_𝑞,𝐵̇_𝑝∓=−𝐺^′_𝑝+𝑅^′_𝑝𝑁±𝑖𝑉^′_𝑝𝑝𝑞+𝜋_𝑝. (19) Consequently, 𝑄̇_𝑞±=±𝑖𝑊^′_𝑞−𝑅^′_𝑞±±𝑖𝐹^′_𝑞−𝑊𝑊^′_𝑞𝜓_∓,𝑄̇_𝑝±=±𝑖𝑉^′_𝑝−𝑅^′_𝑝±±𝑖𝐺^′_𝑝−𝑉𝑉^′_𝑝𝜓_∓. (20) This shows that either 𝑄̇_𝑞±=0 or 𝑄̇_𝑝±=0 when 𝑊^′_𝑞(𝑞)=𝑅^′_𝑞±(𝑞) and 𝐹^′_𝑞=(1/2)(𝑊²)^′_𝑞 or 𝑉^′_𝑝(𝑝)=𝑅^′_𝑝±(𝑝) and 𝐺^′_𝑝=(1/2)(𝑉²)^′_𝑝, respectively. Therefore, when the functions 𝑅_𝑞,𝑝 and 𝐹(𝑞), 𝐺(𝑝) are related as 𝑅^′_𝑞±𝑞=𝑊^′_𝑞𝑞,𝐹_𝑞=12𝑊²+𝐶_𝑞,𝑅^′_𝑝±𝑝=𝑉^′_𝑝𝑝,𝐹_𝑝=12𝑉²+𝐶_𝑝, (21) where 𝐶_𝑞,𝑝 are constants, then odd quantities 𝑄_𝑞,𝑝± are integrals of motion in addition to 𝐻_ext and 𝑁. According to (3) and (13), let us present 𝐻_ext in the form 𝐻_ext=𝐻_𝑞−𝐻_𝑝, where 𝐻_𝑞=12𝜋²_𝑞+𝐹²𝑞+𝑅_𝑞𝐻𝑁,_𝑝=12𝜋²_𝑝+𝐺²𝑝+𝑅_𝑝𝑁. (22) Then, 𝑄_𝑞,𝑝± and 𝑁 together with the 𝐻_𝑞 and 𝐻_𝑝 form the classical analog of the extended phase space SUSY algebra as follows: 𝑄_𝑞,𝑝+,𝑄_𝑞,𝑝−=−𝑖𝐻_𝑞,𝑝−𝐶_𝑞,𝑝,𝐻_𝑞,𝑝,𝑄_𝑞,𝑝±=𝑄_𝑞,𝑝±,𝑄_𝑞,𝑝±=0,𝑁,𝑄_𝑞,𝑝±=±𝑖𝑄_𝑞,𝑝±,𝑁,𝐻_𝑞,𝑝=0, (23) with constants 𝐶_𝑞,𝑝 playing a role of a central charges in (𝑞,𝑝)-spaces; 𝑁 is classical analog of the grading operator. Putting 𝐶_𝑞=𝐶_𝑝=0, we arrive at the classical analog of the extended phase space SUSY quantum mechanics given by the extended Lagrangian ℒ_exṫ𝑞̇=1𝑝,𝑞,𝑝,2𝜋²_𝑞−12𝑊²𝑞+12𝜋²_𝑝−12𝑉²𝑝+𝜓₁𝜓₂𝑊^′_𝑞+𝑉^′_𝑝+12𝜓_𝛼𝜓̇_𝛼. (24) We conclude that the classical system (10) is characterized by the presence of two additional local in time odd integrals of motion equation (17) being supersymmetry generators. Along the actual trajectories in 𝑞-space, (24) reproduces the results obtained in [6].

3. The Path Integral Formulation

In the matrix formulation of extended phase space (𝑁=2)-SUSY quantum mechanics, the 𝜓_± will be two real fermionic creation and annihilation nilpotent operators describing the fermionic variables. The 𝜓_±, having anticommuting 𝑐-number eigenvalues, imply 𝜓_±=12𝜓₁±𝑖𝜓₂,𝜓_𝛼,𝜓_𝛽=𝛿_𝛼𝛽,𝜓₊,𝜓₋=1,𝜓²_±=0. (25) They can be represented by finite dimensional matrices 𝜓_±=𝜎^±, where 𝜎^±=(𝜎₁±𝜎₂)/2 are the usual raising and lowering operators for the eigenvalues of 𝜎₃ which is the diagonal Pauli matrix. The fermionic operator 𝑓 reads 𝑓∶𝒞²→𝒞²,𝑓=(1/2)[𝜓₊,𝜓₋], which commutes with the 𝐻_ext and is diagonal in this representation with conserved eigenvalues ±1/2. Due to it the wave functions become two-component objects as follows: 𝜒=𝜒𝑞,𝑝^+1/2𝜒𝑞,𝑝_−1/2=𝜒𝑞,𝑝𝜒¹𝑞,𝑝₂=𝜓𝑞,𝑝¹𝑞𝜙₁𝑝𝜓₂𝑞𝜙₂𝑝, (26) where the states 𝜓_1,2(𝑞), 𝜙_1,2(𝑝) correspond to fermionic quantum number 𝑓=±1/2, respectively, in 𝑞- and 𝑝-spaces. They belong to Hilbert space ℋ=ℋ₀⊗𝒞²=[ℒ²(ℛ)⊗ℒ²(ℛ)]⊗𝒞². Hence the Hamiltonian 𝐻_ext of extended phase space (𝑁=2)-SUSY quantum mechanical system becomes a 2×2 matrix as follows: 𝐻_ext=𝐻₊00𝐻₋=12𝜋²_𝑞+𝑊²𝑞+𝑖𝑊^′_𝑞𝑞𝜓₁,𝜓₂−12𝜋²_𝑝+𝑉²𝑝+𝑖𝑉^′_𝑞𝑝𝜓₁,𝜓₂. (27) To infer the extended Hamiltonian equation (27) equivalently one may start from the 𝑐-number extended Lagrangian of extended phase space quantum field theory in (0+1)-dimensions in 𝑞- and 𝑝-spaces as follows: ℒ_exṫ𝑞̇̇̇1𝑝,𝑞,𝑝,=−𝑞𝑝−𝑞𝑝+2𝑑𝑞𝑑𝑡²−𝑊²𝑞+𝑓𝑊^′_𝑞𝑞+12𝑑𝑝𝑑𝑡²−𝑉²𝑝+𝑓𝑉^′_𝑝𝑝. (28) In dealing with abstract space of eigenstates of the conjugate operator 𝜓_± which have anticommuting 𝑐-number eigenvalues, suppose that |00−⟩ is the normalized zero-eigenstate of 𝑞 and 𝜓₋ as follows: 𝑞|00−⟩=0,𝜓₋|00−⟩=0. (29) The state |00+⟩ is defined by |00+⟩=𝜓₊|00−⟩=0, (30) then 𝜓₊|00+⟩=0,𝜓₋|00+⟩=|00−⟩. Taking into account that 𝜓^†_±=𝜓_∓, we get ⟨∓00|𝜓_±=0,⟨∓00|𝜓_∓=⟨±00|. (31) Now we may introduce the notation 𝛼,𝛽,… for the anticommuting eigenvalues of 𝜓_±. Consistency requires 𝛼𝜓_±=−𝜓_±𝛼,𝛼|00±⟩=±|00±⟩𝛼. (32) The eigenstates of 𝑞, 𝜓₋ can be constructed as 𝑞𝛼−=𝑒^−𝑖𝑞𝑝_−𝛼𝜓₊|00−⟩, (33) and thus 𝑞𝑞𝛼−=𝑞𝑞𝛼−,𝜓₋𝑞𝛼−=𝛼𝑞𝛼−. (34) Then, the 𝜋_𝑞 and 𝜓₊ eigenstates are obtained by Fourier transformation as follows: 𝑞𝛽+=−𝑑𝛼𝑒^𝛼𝛽,𝜋𝑞𝛼−_𝑞𝛼±=−𝑑𝑞𝑒^{𝑖𝑞𝜋𝑞}𝑞𝛼±,𝑝𝛽+=−𝑑𝛼𝑒^𝛼𝛽,𝜋𝑝𝛼−_𝑝𝛼±=−𝑑𝑝𝑒^{𝑖𝑝𝜋𝑝}𝑝𝛼±, (35) which gives 𝜋_𝑞𝜋_𝑞𝛼±=𝜋_𝑞𝜋_𝑞,𝜓𝛼±₊𝑞,𝜋_𝑞𝛽+=𝛽𝑞,𝜋_𝑞,𝜋𝛽+_𝑝𝜋_𝑝𝛼±=𝜋_𝑝𝜋_𝑝,𝜓𝛼±₊𝑝,𝜋_𝑝𝛽+=𝛽𝑝,𝜋_𝑝𝛽+. (36) The following completeness relations hold: −𝑑𝛼𝑑𝑞𝑞𝛼±∓𝛼^∗𝑞−=1,𝑑𝛼𝑑𝜋_𝑞𝜋2𝜋^𝑞𝛼±∓𝛼^∗𝜋_𝑞=1,−𝑑𝛼𝑑𝑝𝑝𝛼±∓𝛼^∗𝑝−=1,𝑑𝛼𝑑𝜋_𝑝𝜋2𝜋^𝑝𝛼±∓𝛼^∗𝜋_𝑝=1. (37) The time evolution of the state |𝑡⟩ is now given as follows: 𝜒₋𝑞𝛼𝑝𝛽𝑡=−𝑑𝛼^′𝑑𝑞^′𝑑𝛽^′𝑑𝑝^′𝐾𝑞𝛼𝑝𝛽𝑡∣𝑞^′𝛼^′𝑝^′𝛽^′𝑡^′. (38) The kernel reads 𝐾𝑞𝛼𝑝𝛽𝑡∣𝑞^′𝛼^′𝑝^′𝛽^′𝑡^′=+𝑞𝛼^∗𝑝𝛽^∗𝑒^{−𝑖𝐻ext(𝑡−𝑡′)}𝑞^′𝛼^′𝑝^′𝛽^′, (39) which can be evaluated by the path integral. Actually, an alternative approach to describe the state space and dynamics of the extended phase space quantum system is by the path integral [3], which reads 𝒦_𝑓𝑓′𝑞𝑝𝑡∣𝑞^′𝑝^′𝑡^′=𝑞𝑝𝑓𝑒^{−𝑖𝐻ext(𝑡−𝑡′)}𝑞^′𝑝^′𝑓^′. (40) In the path integral equation (40) the individual states are characterized by the energy and the fermionic quantum number 𝑓. With the Hamiltonian 𝐻_ext, the path integral equation (40) is diagonal as follows: 𝒦_𝑓𝑓′𝑞𝑝𝑡∣𝑞^′𝑝^′𝑡^′=𝒦_𝑓𝑓′𝑞𝑡∣𝑞^′𝑡^′𝒦_𝑓𝑓′𝑝𝑡∣𝑝^′𝑡^′=𝛿_𝑓𝑓′^𝑞_𝑞′𝒟𝑞^𝑝_𝑝′𝒟𝑝exp𝑖^𝑡_𝑡′ℒ_ext𝑝,𝑞,𝑝,̇𝑞̇𝑑𝑡. (41) Knowing the path integral equation (41), it is sufficient to specify the initial wave function 𝜒_𝑓(𝑞^′,𝑝^′,𝑡^′) to obtain all possible information about the system at any later time 𝑡, by 𝜒_𝑓𝑞,𝑝,𝑡=_𝑓′𝑑𝑞^′𝑑𝑝^′𝒦_𝑓𝑓′𝑞𝑝𝑡∣𝑞^′𝑝^′𝑡^′𝜒_𝑓′𝑞^′,𝑝^′,𝑡^′, (42) with 𝜒_±1/2(𝑞,𝑝,𝑡)=𝜒_1,2(𝑞,𝑝,𝑡) (26). In terms of anticommuting 𝑐-number operators 𝜁 and 𝜂 defining 𝜓=1/2^𝜂+𝜁_{𝑖(𝜂−𝜁)}, the path integral equation (41) becomes 𝒦𝑞𝛼𝑝𝛽𝑡∣𝑞^′𝛼^′𝑝^′𝛽^′𝑡^′=_𝑞^{𝑞,𝛼,𝑝,𝛽}_{′,𝛼′,𝑝′,𝛽′}𝒟𝑞𝒟𝑝𝒟𝜁𝒟𝜂×exp𝑖^𝑡_𝑡′ℒ_ext𝑝,𝑞,𝑝,̇𝑞̇𝑑𝑡. (43) The functional integral is taken over all trajectories from 𝑞^′,𝛼^′ to 𝑞,𝛼 and 𝑝^′,𝛽^′ to 𝑝,𝛽 between the times 𝑡^′ and 𝑡.

4. Solution of the Extended Schrödinger Equation with Small Energy Eigenvalue 𝜀

Adopting the technique developed in [13], first, we use the iterative scheme to find the approximate ground state solutions to the extended Schrödinger-like equation 𝐻_ext𝜒𝑞,𝑝=𝐻_𝑞−𝐻_𝑝𝜒𝑞,𝑝=𝜀𝜒𝑞,𝑝, (44) with energy 𝜀. We will then use these solutions to calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. The approximation, which went into the derivation of solutions of (44), meets our interest that the ground state energy 𝜀 is supposedly small. As we mentioned above the solutions for nonzero 𝜀 come in pairs of the form 𝜒_↑=𝜒𝑞,𝑝¹0𝑞,𝑝or𝜒_↓=0𝜒𝑞,𝑝₂𝑞,𝑝, (45) related by supersymmetry, where 𝜒_1,2(𝑞,𝑝)=𝜓_1,2(𝑞)𝜙_1,2(𝑝). The state space of the system is defined by all the normalizable solutions of (44) and the individual states are characterized by the energies 𝜀_𝑞 and the fermionic quantum number 𝑓. One of these solutions is acceptable only if 𝑊(𝑞) and 𝑉(𝑝) become infinite at both 𝑞→±∞ and 𝑝→±∞, respectively, with the same sign. If this condition is not satisfied, neither of the solutions is normalizable, nor they can represent the ground state of the system. Equation (44) yields the following relations between energy eigenstates with fermionic quantum number ±1/2: 𝜕𝜕𝑞+𝑊_𝑞𝑞−𝜕𝜕𝑝+𝑉_𝑝𝑝𝜓₁𝑞𝜙₁𝑝=2𝜀_𝑞𝜓₂𝑞𝜙₁𝑝−2𝜀_𝑝𝜓₁𝑞𝜙₂𝑝,−𝜕𝜕𝑞+𝑊_𝑞𝑞−−𝜕𝜕𝑝+𝑉_𝑝𝑝𝜓₂𝑞𝜙₂𝑝=2𝜀_𝑞𝜓₁𝑞𝜙₂𝑝−2𝜀_𝑝𝜓₂𝑞𝜙₁𝑝, (46)(47) where 𝜀=𝜀_𝑞−𝜀_𝑝 and 𝜀_𝑞 and 𝜀_𝑝 are the eigenvalues of 𝐻_𝑞 and 𝐻_𝑝, respectively. The technique now is to devise an iterative approximation scheme to solve (46) and (47) by taking a trial wave function for 𝜒₂(𝑞,𝑝), substitute this into the first equation (46), and integrate it to obtain an approximation for 𝜒₁(𝑞,𝑝). This can be used as an ansatz in the second equation (47) to find an improved solution for 𝜒₂(𝑞,𝑝) and so forth. As it was shown in [13], the procedure converges for well-behaved potentials with a judicious choice of initial trial function. If the 𝑊_𝑞 and 𝑉_𝑝 are odd, then 𝜓₁−𝑞=𝜓₂𝑞,𝜙₁−𝑝=𝜙₂𝑝, (48) since they satisfy the same eigenvalue equation. It is straightforward then, for example, to obtain 𝜕𝜕𝑞+𝑊_𝑞𝑞−𝜕𝜕𝑝+𝑉_𝑝𝑝𝜓₁𝑞𝜙₁𝑝=2𝜀_𝑞𝜓₁−𝑞𝜙₁𝑝−2𝜀_𝑝𝜓₁𝑞𝜙₁−𝑝. (49) The independent nature of 𝑞 and 𝑝 gives the freedom of taking 𝑞=0,𝑝=0 which yield an expression for energies as follows: 2𝜀_𝑞(0)+𝜓=𝑊^′1(0)𝜓₁(0),2𝜀_𝑝(0)+𝜙=𝑉^′1(0)𝜙₁(0). (50) Suppose that the potentials 𝑊_𝑞(𝑞) and 𝑉_𝑝(𝑝) have a maximum, at 𝑞₋ and 𝑝₋, and minimum, at 𝑞₊ and 𝑝₊, respectively. For the simplicity sake we choose the trial wave functions as 𝜓⁽⁰⁾_1,2𝑞=𝛿𝑞−𝑞_±,𝜙⁽⁰⁾_1,2𝑝=𝛿𝑝−𝑝_±. (51) After one iteration, we obtain 𝜒₁⁽¹⁾=1𝑞,𝑝𝑁_𝑞𝑁_𝑝𝜃𝑞−𝑞₋𝜃𝑝−𝑝₋×𝑒^{−∫𝑞0𝑑𝑞′𝑊(𝑞′)+∫𝑝0𝑑𝑝′𝑉(𝑝′)},𝜒₂⁽¹⁾=1𝑞,𝑝𝑁_𝑞𝑁_𝑝𝜃𝑞₊−𝑞𝜃𝑝₊−𝑝×𝑒^{∫𝑞0𝑑𝑞′𝑊(𝑞′)−∫𝑝0𝑑𝑝′𝑉(𝑝′)}, (52) where 𝑁_𝑞 and 𝑁_𝑝 are the normalization factors. The next approximation leads to 𝜒₁⁽²⁾=1𝑞,𝑝𝑁^′𝑒^{−∫𝑞0𝑑𝑞′𝑊(𝑞′)+∫𝑝0𝑑𝑝′𝑉(𝑝′)}×^∞_{max−𝑞,𝑞−}𝑒^{−2∫𝑞′0𝑑𝑞′′𝑊(𝑞′′)}𝑑𝑞^′×^∞_{max−𝑝,𝑝−}𝑒^{2∫𝑝′0𝑑𝑝′′𝑉(𝑝′′)}𝑑𝑝^′,𝜒₂⁽²⁾=1𝑞,𝑝𝑁^′𝑒^{∫𝑞0𝑑𝑞′𝑊(𝑞′)−∫𝑝0𝑑𝑝′𝑉(𝑝′)}×^{min(−𝑞,𝑞+)}_∞𝑒^{2∫𝑞′0𝑑𝑞′′𝑊(𝑞′′)}𝑑𝑞^′×^{min(−𝑝,𝑝+)}_∞𝑒^{−2∫𝑝′0𝑑𝑝′′𝑉(𝑝′′)}𝑑𝑝^′. (53) It can be easily verified that to this level of precision (52) is self-consistent solution. Actually, for example, for 𝑞^′>𝑞₋ the exponential 𝑒^{−2∫𝑞′0𝑊(𝑞′′)𝑑𝑞′′} will peak sharply around 𝑞₊ and may be approximated by a 𝛿-function 𝑐𝛿(𝑞₊−𝑞^′); similarly 𝑒^{2∫𝑞′0𝑊(𝑞′′)𝑑𝑞′′} may be replaced approximately by 𝑐𝛿(𝑞₋−𝑞^′) for 𝑞^′<𝑞₊. The same arguments hold for the 𝑝-space. With these approximations equations (53) reduce to (52). The normalization constant 𝑁^′ is 𝑁^′=^∞_𝑞−𝑑𝑞𝑒^{−2∫𝑞0𝑊(𝑞′)𝑑𝑞′∞}_𝑝−𝑑𝑝𝑒^{−2∫𝑝′0𝑉(𝑝′)𝑑𝑝′3/2}. (54) The energy expectation value 𝜀=𝜒₁,𝐻_ext𝜒₁ (55) gives the same result as that obtained for odd potentials by means of (50) and (53). Assuming the exponentials 𝑒^{−2∫0𝑞−𝑊(𝑞)𝑑𝑞} and 𝑒^{−2∫0𝑝−𝑉(𝑝)𝑑𝑝} to be small, which is correct to the same approximations underlying (55), the difference is negligible and the integrals in both cases may be replaced by Gaussians around 𝑞₊ and 𝑝₊, respectively. Hence, it is straightforward to obtain 𝜀=ℏ𝑊^′𝑞₊𝑒2𝜋^{−2Δ𝑊/ℏ}−ℏ𝑉^′𝑝₊𝑒2𝜋^{−2Δ𝑉/ℏ}, (56) which gives direct evidence for the SUSY breaking in the extended phase space quantum mechanical system. Here we have reinstated ℏ, to show the order of adopted approximation, and its non-perturbative nature. We also denoted Δ𝑊=^𝑞−_𝑞+𝑊𝑞𝑑𝑞,Δ𝑉=^𝑝−_𝑝+𝑉𝑝𝑑𝑝. (57) However, a more practical measure for the SUSY breaking, in particular, in field theories is the expectation value of an auxiliary field, which can be replaced by its equation of motion right from the start as follows: ⟨𝐹⟩=𝜒_↑,𝑖𝑄₊,𝜎₋𝜒_↑. (58) Taking into account the relation 𝑄₊𝜒_↑=0, with 𝑄₊ commuting with 𝐻_ext, which means that the intermediate state must have the same energy as 𝜒, (58) can be written in terms of a complete set of states as ⟨𝐹⟩=𝑖𝜒_↑,𝑄₊𝜒_↓𝜒_↓,𝜎₋,𝜒_↑. (59) According to (55), we have 𝜀=𝐻_ext=𝜒_↑,𝐻_ext𝜒_↑=12𝜒_↑,𝑄₊𝜒_↓𝜒_↓,𝑄₋𝜒_↑=𝜀_𝑞−𝜀_𝑝=𝐻_𝑞−𝐻_𝑝=𝜓_↑,𝐻_𝑞𝜓_↑−𝜙_↑,𝐻_𝑝𝜙_↑=12𝜓_↑,𝑄_𝑞+𝜓_↓𝜓_↓,𝑄_𝑞−𝜓_↑−12𝜙_↑,𝑄_𝑝+𝜙_↓𝜙_↓,𝑄_𝑝−𝜙_↑, (60) where 𝜒_↑↓=𝜓_↑↓𝜙_↑↓, 𝜓_↑=^𝜓1₀, 𝜓_↓=⁰_𝜓2, 𝜙_↑=^𝜙1₀, and 𝜙_↓=⁰_𝜙2. From SUSY algebra it follows immediately that 1√𝜀𝑄₋𝜒_↑=𝜒_↓=𝜓₋𝜒_↑,1𝜀_𝑞𝑄_𝑞−𝜓_↑=𝜓_↓=𝜓₋𝜓_↑,1𝜀_𝑝𝑄_𝑝−𝜙_↑=𝜙_↓=𝜓₋𝜙_↑. (61)

By virtue of (61), (60) reads √𝜀𝜒_↑,𝑄₊𝜒_↓𝜒_↓,𝜓₋𝜒_↑=𝜀_𝑞𝜓_↑,𝑄_𝑞+𝜓_↓𝜓_↓,𝜓₋𝜓_↑−𝜀_𝑝𝜙_↑,𝑄_𝑝+𝜙_↓𝜙_↓,𝜓₋𝜙_↑. (62) Using the matrix representations of 𝑄_𝑞+,𝑄_𝑝+, and 𝜓₋ and the wave functions equation (53), one gets [13] 𝜓_↓,𝜓₋𝜓_↑=𝑊^′𝑞₊𝜋𝑒^−Δ𝑊𝜙Δ𝑞,_↓,𝜓₋𝜙_↑=𝑉^′𝑝₊𝜋𝑒^−Δ𝑉𝜓Δ𝑝,_↑,𝑄_𝑞+𝜓_↓=𝑖𝑊^′𝑞₋𝜋𝑒^−Δ𝑊,𝜙_↑,𝑄_𝑝+𝜙_↓=𝑖𝑉^′𝑝₋𝜋𝑒^−Δ𝑉, (63) where Δ𝑞=𝑞₊−𝑞₋ and Δ𝑝=𝑝₊−𝑝₋. Hence 𝜒_↑,𝑄₊𝜒_↓𝜒_↓,𝜓₋𝜒_↑=2𝑖√𝜀𝜀_𝑞𝜀√𝜀_𝑞𝑊^′𝑞₋𝜋−𝜀Δ𝑞_𝑝𝜀√𝜀_𝑝𝑉^′𝑝₋𝜋,⟨𝐹⟩√Δ𝑝=−2𝜀𝜀_𝑞𝜀𝑊^′𝑞₊𝜋𝑒^−2Δ𝑊−𝜀Δ𝑞_𝑝𝜀𝑉^′𝑝𝜋𝑒^−2Δ𝑉Δ𝑝. (64)(65) Along the actual trajectories in 𝑞-space, (65) reproduces the results obtained in [13].

5. An Extended SUSY Breaking in the Instanton Picture

In this Section our goal is to show that the expressions equations (64) and (65) can be obtained in the path integral formulation of the theory by calculating the matrix elements, that is, the effect of tunneling between two classical vacua by using a one-instanton background. That is, the matrix elements of 𝜓_±,𝑄_𝑞±, and 𝑄_𝑝± can be calculated in the background of the classical solution 𝑞̇_𝑐=−𝑊_𝑐 and 𝑝̇_𝑐=−𝑉_𝑐. In doing this we rewrite the matrix element equation (62) in terms of eigenstates of the conjugate operator 𝜓_± as follows: √𝜀+0𝑞₊𝑝₊𝑒^{−𝑖𝐻ext(𝑇−𝑡)}𝑄₊𝑒^{−𝑖𝐻ext(𝑇+𝑡)}𝑞₋𝑝₋0−−0𝑞₋𝑝₋×𝑒^{−𝑖𝐻ext(𝑇−𝑡)}𝜓₋𝑒^{−𝑖𝐻ext(𝑇+𝑡)}𝑞₊𝑝₊0+_{𝑇→−𝑖∞}=𝜀_𝑞+0𝑞₊𝑒^{−𝑖𝐻𝑞(𝑇−𝑡)}𝑄_𝑞+𝑒^{−𝑖𝐻𝑞(𝑇+𝑡)}𝑞₋0−−0𝑞₋×𝑒^{−𝑖𝐻𝑞(𝑇−𝑡)}𝜓₋𝑒^{−𝑖𝐻𝑞(𝑇+𝑡)}𝑞₊0+_{𝑇→−𝑖∞}−𝜀_𝑝+0𝑝₊𝑒^{−𝑖𝐻𝑝(𝑇−𝑡)}𝑄_𝑝+𝑒^{−𝑖𝐻𝑝(𝑇+𝑡)}×𝑝₋0−−0𝑝₋𝜓₋𝑝₊0+_{𝑇→−𝑖∞}, (66) in the limit 𝑇→−𝑖∞. This reduces to √𝜀+0𝑞₊𝑝₊𝑄₊𝑞₋𝑝₋0−−0𝑞₋𝑝₋𝜓₋𝑞₊𝑝₊=0+𝜀_𝑞+0𝑞₊𝑄_𝑞+𝑞₋0−−0𝑞₋𝜓₋𝑞₊−0+𝜀_𝑝+0𝑝₊𝑄_𝑝+𝑝₋0−−0𝑝₋𝜓₋𝑝₊0+, (67) which, in turn, can be presented by path integrals defined in terms of anticommuting 𝑐-number operators 𝜁 and 𝜂 with Euclidean actions of the instantons in 𝑞- and 𝑝-spaces, respectively. Following [13], these functional integrals include an integration over instanton time 𝜏₀ which is due to the problem of zero modes of the bilinear terms in Euclidean actions. This arises from time-transformation of instantons and SUSY transformations on them, respectively. The existence of zero modes gives rise to non-Gaussian behaviour of the functional integral. Due to it the matrix elements above do not receive any contributions from either no-instanton or anti-instanton configurations. The zero mode problem is solved by introducing a collective coordinate 𝜏₀ replacing the bosonic zero mode [14]. Whereas; the functional integrals depend only on the difference 𝜏−𝜏₀. Note also that multi-instanton configurations could contribute in principle, provided they have not more than one normalizable fermionic zero mode. But as it was shown in [13], their contribution is clearly smaller with respect to 2𝜀_𝑞 and 2𝜀_𝑝. In the case when the SUSY potentials in 𝑞- and 𝑝-spaces have more than two extrema 𝑞_𝜈 and 𝑝_𝜇, 𝜈,𝜇=1,2,…,𝑁, one can put conditions on the SUSY potentials as follows ^∞₀𝑊𝑞^′𝑑𝑞^′⟶∞at𝑞⟶±∞for𝜓⁺₀,_∞0𝑊𝑞^′𝑑𝑞^′⟶−∞at𝑞⟶±∞for𝜓⁻₀, (68) and similar to 𝑉(𝑝), that the extrema are well separated as follows: ∫^𝑞𝜈_𝑞⁺¹_𝜈𝑊(𝑞^′)𝑑𝑞^′≫1, ∫^𝑞𝜇_𝑝⁺¹_𝜇𝑉(𝑝^′)𝑑𝑝^′≫1. Around each of the classical minima 𝑞_𝜈 and 𝑝_𝜇 of the potentials 𝑊²(𝑞) and 𝑉²(𝑝), respectively, one can approximate the theory by a supersymmetric harmonic oscillator. Then there are 𝑁 ground states which have zero energy. These states are described by upper or lower component of the wave function, depending on whether 𝜈 and 𝜇 are odd or even. With this provision the functional integrals are calculated in [13], which allow us consequently to write +0𝑞₊𝑄_𝑞+𝑞₋0−=𝑖𝑊^′_𝑐𝑞₊𝜋𝑒^{−Δ𝑊𝑐},−0𝑞₋𝜓₋𝑞₊0+=𝑊^′_𝑐𝑞₊𝜋𝑒^{−Δ𝑊𝑐}Δ𝑞_𝑐, (69) and so forth. Inserting this in (67), we arrive at (64) +0𝑞₊𝑝₊𝑄₊𝑞₋𝑝₋0−−0𝑞₋𝑝₋𝜓₋𝑞₊𝑝₊√0+=2𝑖𝜀𝜀_𝑞𝜀𝜀_𝑞𝑊^′_𝑐𝑞₊𝜋Δ𝑞_𝑐−𝜀_𝑝𝜀𝜀_𝑝𝑉^′_𝑐𝑝₊𝜋Δ𝑝_𝑐, (70) and, thus, of (65) as its inevitable corollary. This proves that the extended SUSY breaking has resulted from tunneling between the classical vacua of the theory. The corrections to this picture are due to higher-order terms and quantum-tunneling effects.

6. Spectrum Generating Algebra

An extended Hamiltonian 𝐻_ext equation (27) can be treated as a set of two ordinary two-dimensional partner Hamiltonians [4] as follows: 𝐻_±=12𝜋²_𝑞−𝜋²_𝑝+𝑈_±𝑞,𝑝, (71) provided by partner potentials 𝑈_±𝑞,𝑝=𝑈_𝑞±𝑞−𝑈_𝑝±𝑝,𝑈_𝑞±𝑞=𝑊²𝑞,𝑎_𝑞∓𝑊^′_𝑞𝑞,𝑎_𝑞,𝑈_𝑝±𝑝=𝑉²𝑝,𝑎_𝑝∓𝑉^′_𝑝𝑝,𝑎_𝑝. (72) A subset of the SUSY potentials for which the Schrödinger-like equations are exactly solvable share an integrability conditions of shape-invariance [15] as follows: 𝑈₊𝑎₀,𝑞,𝑝=𝑈₋𝑎₁,𝑞,𝑝+𝑅𝑎₀,𝑎₁=𝑓𝑎₀, (73) where 𝑎₀ and 𝑎₁ are a set of parameters that specify phase-space-independent properties of the potentials, and the reminder 𝑅(𝑎₀) is independent of (𝑞,𝑝).

6.1. Algebraic Properties of Shape Invariance

Using the standard technique, we may construct a series of Hamiltonians 𝐻_𝑁,𝑁=0,1,2,…, 𝐻_𝑁=12𝜋²_𝑞−𝜋²_𝑝+𝑈₋𝑎_𝑁,𝑞,𝑝+^𝑁_𝑘=1𝑅𝑎_𝑘, (74) where 𝑎_𝑁=𝑓^(𝑁)(𝑎) (𝑁 is the number of iterations). From (72) and (73) we obtain then 𝑁=𝑛+𝑚 coupled nonlinear differential equations which are the two recurrence relations of Riccati-type differential equations as follows: 𝑊²_𝑛+1𝑞+𝑊^′_𝑞()𝑛+1𝑞=𝑊²_𝑛𝑞−𝑊^′_𝑞𝑛𝑞−𝜇_𝑛,𝑉²_𝑚+1𝑝+𝑉^′_𝑝(𝑚+1)𝑝=𝑉²_𝑚𝑝−𝑉^′_𝑝𝑚𝑝−𝜈_𝑚, (75) where we denote 𝑊_𝑛(𝑞)≡𝑊(𝑞,𝑎_𝑞𝑛), 𝑉_𝑚(𝑝)≡𝑉(𝑝,𝑎_𝑝𝑚), 𝜇_𝑛≡𝑅_𝑞(𝑎_𝑞𝑛), and 𝜈_𝑚≡𝑅_𝑝(𝑎_𝑝𝑚). Here we admit that for unbroken SUSY, the eigenstates of the potentials 𝑈_𝑞,𝑝, respectively, are 𝐸⁽⁻⁾_𝑞0=0,𝐸⁽⁻⁾_𝑞𝑛=^𝑛−1_𝑖=0𝜇_𝑖,𝐸⁽⁻⁾_𝑝0=0,𝐸⁽⁻⁾_𝑝𝑚=^𝑚−1_𝑗=0𝜈_𝑗, (76) that is, the ground states are at zero energies, characteristic of unbroken supersymmetry. The differential equations (75) can be investigated to find exactly solvable potentials. The shape invariance condition equation (73) can be expressed in terms of bosonic operators as 𝐵₊𝑥,𝑎₀𝐵₋𝑥,𝑎₀−𝐵₋𝑥,𝑎₁𝐵₊𝑥,𝑎₁=𝑅𝑎₀=𝜇₀−𝜈₀,𝐵_æ+æ,𝑎_æ0𝐵_æ−æ,𝑎_æ0−𝐵_æ−æ,𝑎_æ1𝐵_æ+æ,𝑎_æ1=𝜇₀or𝜈₀, (77) where 𝑥(𝑞,𝑝)- is the coordinate in (𝑞,𝑝)-space æ denotes concisely either 𝑞- or 𝑝-representations (no summation on æ is assumed). To classify algebras associated with the shape invariance, following [16], we introduce an auxiliary variables 𝜙(𝜙_𝑞,𝜙_𝑝) and define the following creation and annihilation operators: 𝐽₊=𝑒^𝑖𝑘𝜙𝐵₊𝑥,𝜒𝑖𝜕_𝜙,𝐽₋=𝐵₋𝑥,𝜒𝑖𝜕_𝜙𝑒^{−𝑖𝑘𝜙}, (78) where 𝑘(𝑘_𝑞,𝑘_𝑝) are an arbitrary real constants and 𝜒(𝜒_𝑞,𝜒_𝑝) are an arbitrary real functions. Consequently, the creation and annihilation operators in (𝑞,𝑝)-spaces can be written as 𝐽_æ+=𝑒^{𝑖𝑘æ𝜙æ}𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ,𝐽_æ−=𝐵_æ−æ,𝜒_æ𝑖𝜕_𝜙æ𝑒^{−𝑖𝑘æ𝜙æ}. (79) The operators 𝐵_±(𝑥,𝜒(𝑖𝜕_𝜙)) and 𝐵_æ±(æ,𝜒_æ(𝑖𝜕_𝜙æ)) are the generalization of (77), where 𝑎₀→𝜒(𝑖𝜕_𝜙) and 𝑎_æ0→𝜒_æ(𝑖𝜕_𝜙æ). One can easily prove the following relations: 𝑒^𝑖𝑘𝜙𝐵₊𝑥,𝜒𝑖𝜕_𝜙=𝐵₊𝑥,𝜒𝑖𝜕_𝜙+𝑘𝑒^𝑖𝑘𝜙,𝐵₋𝑥,𝜒𝑖𝜕_𝜙𝑒^{−𝑖𝑘𝜙}=𝑒^{−𝑖𝑘𝜙}𝐵₋𝑥,𝜒𝑖𝜕_𝜙+𝑘, (80) and that 𝑒^{𝑖𝑘æ𝜙æ}𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ=𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ𝑒^{𝑖𝑘æ𝜙æ},𝐵_æ−æ,𝜒_æ𝑖𝜕_𝜙æ𝑒^{−𝑖𝑘æ𝜙æ}=𝑒^{−𝑖𝑘æ𝜙æ}𝐵_æ−æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ. (81) If we choose a function 𝜒(𝑖𝜕_𝜙) such that 𝜒(𝑖𝜕_𝜙+𝑘)=𝑓[𝜒(𝑖𝜕_𝜙)], then we have identified 𝑎₀→𝜒(𝑖𝜕_𝜙), and 𝑎₁=𝑓(𝑎₀)→𝑓[𝜒(𝑖𝜕_𝜙)]=𝜒(𝑖𝜕_𝜙+𝑘). Similar relations can be obtained for the æ-representations. From (77) we obtain then 𝐵₊𝑥,𝜒𝑖𝜕_𝜙𝐵₋𝑥,𝜒𝑖𝜕_𝜙−𝐵₋𝑥,𝜒𝑖𝜕_𝜙+𝑘𝐵₊𝑥,𝜒𝑖𝜕_𝜙+𝑘=𝑅𝜒𝑖𝜕_𝜙,𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ𝐵_æ−æ,𝜒𝑖𝜕_𝜙æ−𝐵_æ−æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ=𝑅_æ𝜒_æ𝑖𝜕_𝜙æ. (82) Introducing the operators 𝐽₃=−(𝑖/𝑘)𝜕_𝜙 and 𝐽_æ3=−(𝑖/𝑘_æ)𝜕_𝜙æ and combining (79) and (82), we may arrive at a deformed Lie algebras as follows: 𝐽₊,𝐽₋=𝐽_𝑞+,𝐽_𝑞−−𝐽_𝑝+,𝐽_𝑝−=𝜉𝐽₃=𝜉_𝑞𝐽_𝑞3−𝜉_𝑝𝐽_𝑝3,𝐽₃,𝐽_±=±𝐽_±,𝐽_æ+,𝐽_æ−=𝜉_æ𝐽_æ3,𝐽_æ3,𝐽_æ±=±𝐽_æ±, (83) where 𝜉(𝐽₃)≡𝑅[𝜒(𝑖𝜕_𝜙)] and 𝜉_æ(𝐽_æ3)≡𝑅_æ[𝜒_æ(𝑖𝜕_𝜙æ)] define the deformations. Different 𝜒 functions in (82) define different reparametrizations corresponding to several models. For example, (1)the translational models (𝑎₁=𝑎₀+𝑘) correspond to 𝜒(𝑧)=𝑧. If 𝑅 is a linear function of 𝐽₃, the algebra becomes SO(2.1) or SO(3). Similar in many respects prediction is made in somewhat different method by Balantekin [17];(2)the scaling models (𝑎₁=𝑒^𝑘𝑎₀) correspond to 𝜒(𝑧)=𝑒^𝑧 and so forth.

6.2. The Unitary Representations of the Deformed Lie Algebra

In order to find the energy spectrum of the partner SUSY Hamiltonians 2𝐻₋𝑥,𝜒𝑖𝜕_𝜙=𝐵₋𝑥,𝜒𝑖𝜕_𝜙+𝑘×𝐵₊𝑥,𝜒𝑖𝜕_𝜙+𝑘,2𝐻_æ−æ,𝜒_æ𝑖𝜕_𝜙æ=𝐵_æ−æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ×𝐵_æ+æ,𝜒_æ𝑖𝜕_𝜙æ+𝑘_æ, (84) one must construct the unitary representations of deformed Lie algebra defined by (83) [16, 18]. Using the standard technique, one defines up to additive constants the functions 𝑔(𝐽₃) and 𝑔_æ(𝐽_æ3) as follows: 𝜉𝐽₃=𝑔𝐽₃−𝑔𝐽₃,𝜉−1_æ𝐽_æ3=𝑔_æ𝐽_æ3−𝑔_æ𝐽_æ3−1. (85) The Casimirs of this algebra can be written as 𝐶₂=𝐽₋𝐽₊+𝑔(𝐽₃) and accordingly 𝐶_æ2=𝐽_æ−𝐽_æ++𝑔_æ(𝐽_æ3). In a basis in which 𝐽₃ and 𝐶₂ are diagonal, 𝐽₋ and 𝐽₊ are lowering and raising operators (the same holds for æ-representations). Operating on an arbitrary state ∣ℎ>, they yield 𝐽₃|ℎ⟩=ℎ|ℎ⟩,𝐽₋|ℎ⟩(ℎ⟩,𝐽=𝑎)|ℎ−1₊|ℎ⟩=𝑎^∗(ℎ+1)|ℎ+1⟩, (86) where |𝑎(ℎ)|²−|𝑎(ℎ+1)|²=𝑔(ℎ)−𝑔(ℎ−1). (87) Similar arguments can be used for the operators 𝐽_(𝑝,𝑞)3, 𝐶_æ2, and 𝐽_æ±, which yield similar relations for the states ℎ_æ. The profile of 𝑔(ℎ) (and, thus, of 𝑔_æ(ℎ_æ)) determines the dimension of the unitary representation. Having the representation of the algebra associated with a characteristic model, consequently we obtain the complete spectrum of the system. For example, without ever referring to underlying differential equation, we may obtain analytic expressions for the entire energy spectrum of extended Hamiltonian with self-similar potential. A scaling change of parameters is given as 𝑎₁=𝑄𝑎₀, 𝑎_æ1=𝑄_æ𝑎_æ0, at the simple choice 𝑅(𝑎₀)=−𝑟₁𝑎₀, where 𝑟₁ is a constant. That is, 𝜉𝐽₃≡−𝑟₁exp−𝑘𝐽₃=𝜉_𝑞𝐽_𝑞3−𝜉_𝑝𝐽_𝑝3=−𝑟_𝑞1exp−𝑘_𝑞𝐽_𝑞3+𝑟_𝑝1exp−𝑘_𝑝𝐽_𝑝3, (88) which yields 𝐽₊,𝐽₋=𝜉𝐽₃=−𝑟₁exp−𝑘𝐽₃,𝐽₃,𝐽_±=±𝐽_±. (89) This is a deformation of the standard SO(2.1) Lie algebra, and therefore one gets 𝑔(ℎ)=𝑟₁𝑒^𝑘𝑒−1^−𝑘ℎ𝑟=−¹𝑄1−𝑄^−ℎ,𝑄=𝑒^𝑘,𝑔_æℎ_æ=𝑟_æ1𝑒^𝑘æ𝑒−1^{−𝑘æℎæ}𝑟=−^æ11−𝑄_æ𝑄^−ℎæ_æ,𝑄_æ=𝑒^𝑘æ. (90) For scaling problems [16] one has 0<𝑞<1, which leads to 𝑘<0. The unitary representation of this algebra for monotonically decreasing profile of the function 𝑔(ℎ), is infinite dimensional. Let the lowest weight state of the 𝐽₃ be ℎ_min, and then 𝑎(ℎ_min)=0. One can choose the coefficients 𝑎(ℎ) to be real. From (87), for an arbitrary ℎ=ℎ_min+𝑛, 𝑛=0,1,2,…, we obtain 𝑎(ℎ)²=𝑔(ℎ−𝑛−1)−𝑔(ℎ−1)=𝑟₁𝑄^𝑛−1𝑄𝑄−1^1−ℎ. (91) The spectrum of the extended Hamiltonian 𝐻₋(𝑥,𝑎₁) reads 𝐻₋|ℎ⟩=𝑎(ℎ)²|ℎ⟩=𝑟₁𝑄^𝑛−1𝑄𝑄−1^1−ℎ|ℎ⟩, (92) with the eigenenergies 𝐸_𝑛(ℎ)=𝑟₁𝛼(ℎ)𝑄^𝑛−1(ℎ)𝑄−1,𝛼≡𝑄^1−ℎ. (93) Similar expressions can be obtained for the 𝐻_æ− and eigenenergies 𝐸_𝑞𝑛1 and 𝐸_𝑝𝑛2 (𝑛≡(𝑛₁,𝑛₂),𝑛_1,2=0,1,2,…) as 𝐻_𝑞−ℎ_𝑞=𝑎_𝑞ℎ_𝑞²ℎ_𝑞=𝑟_𝑞1𝑄^𝑛1_𝑞−1𝑄_𝑞𝑄−1^{1−ℎ𝑞𝑞}ℎ_𝑞,𝐻_𝑝−ℎ_𝑝=𝑎_𝑝ℎ_𝑝²ℎ_𝑝=𝑟_𝑝1𝑄^𝑛2_𝑝−1𝑄_𝑝𝑄−1^{1−ℎ𝑝𝑝}ℎ_𝑝, (94) and that 𝐸_𝑞𝑛1ℎ_𝑞=𝑟_𝑞1𝛼_𝑞ℎ_𝑞𝑄^𝑛1_𝑞−1𝑄_𝑞,𝛼−1_𝑞ℎ_𝑞≡𝑄^1−ℎ𝑞_𝑞,𝐸_𝑝𝑛2ℎ_𝑝=𝑟_𝑝1𝛼_𝑝ℎ_𝑝𝑄^𝑛2_𝑝−1𝑄_𝑝,𝛼−1_𝑝ℎ_𝑝≡𝑄^1−ℎ𝑝_𝑝. (95) Hence 𝐸_𝑛(ℎ)=𝐸_𝑞𝑛1ℎ_𝑞−𝐸_𝑝𝑛2ℎ_𝑝. (96)

7. Conclusions

We addressed the classical analog of the extended phase space quantum mechanics of particle which have both bosonic and fermionic degrees of freedom, that is, the particle with odd degrees of freedom which gives rise to (𝑁=2)-realization of the supersymmetry algebra. We obtain the integrals of motion. We use the iterative scheme to find the approximate ground state solutions to the extended Schrödinger-like equation and calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. The approximation, which went into the derivation of solutions of (44), meets our interest that the ground state energy 𝜀 is supposedly small. This gives direct evidence for the SUSY breaking. However, we calculate a more practical measure for the SUSY breaking, in particular in field theories which is the expectation value of an auxiliary field. We analyze in detail the nonperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Finally, we present an analysis on the independent group theoretical methods with nonlinear extensions of Lie algebras from the extended phase space SUSY quantum mechanics. Using the factorization procedure we explore the algebraic property of shape invariance and spectrum generating algebra. Most of these Hamiltonians possess this feature and hence are solvable by an independent group theoretical method. We construct the unitary representations of the deformed Lie algebra.

Acknowledgments

The author would like to thank Y. Sobouti for drawing his attention to the extended phase space formulation of quantum mechanics. The knowledgable comments from the anonymous referees are much appreciated.