Classical analog of extended phase space SUSY and its breaking
aa r X i v : . [ m a t h - ph ] J un Classical analog of extended phase space SUSY and its breaking
G Ter-Kazarian ∗ Byurakan Astrophysical Observatory, Byurakan 378433, Aragatsotn District, Armenia
We derive the classical analog of the extended phase space quantum mechanics of the particle withodd degrees of freedom which gives rise to (N=2)-realization of supersymmetry (SUSY) algebra. Bymeans of an iterative procedure, we find the approximate groundstate solutions to the extendedSchr¨odinger-like equation and use these solutions further to calculate the parameters which measurethe breaking of extended SUSY such as the groundstate energy. Consequently, we calculate a morepractical measure for the SUSY breaking which is the expectation value of an auxiliary field. Weanalyze non-perturbative mechanism for extended phase space SUSY breaking in the instantonpicture 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 generatingalgebra.
Keywords: Extended phase space, SUSY quantum mechanics, Path-integrals, SUSY breaking, Instantonpicture, Spectrum generating algebra
I. INTRODUCTION
An interesting question of keeping the symmetry between canonical coordinates and momenta in the process ofquantization 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 ofcanonical 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 actionsminimum along the actual trajectories in ( p, q ) − , rather than in q − space. This leads to the phase space formulationof quantum mechanics. Consequently this formalism was developed further in [3] by addressing the extended phasespace stochastic quantization of Hamiltonian systems with first class holonomic constraints. This in a natural wayresults in the Faddeev-Popov conventional path-integral measure for gauge systems. Continuing along this line in thepresent article we address the classical analog of the extended phase space (N=2)-SUSY quantum mechanics [4] ofthe particles which have both bosonic and fermionic degrees of freedom, i.e., the quantum field theory in (0 + 1)-dimensions in ( q, p ) − space, exhibiting supersymmetry (for conventional SUSY quantum mechanics see [5]-[12]). Weanalyze in detail the non-perturbative mechanism for supersymmetry breaking in the instanton picture ([13]). Thispaper has been organized as follows. In the first part (Sects. 2, 3), we derive the classical analog of the extendedphase space SUSY quantum mechanics and obtain the integrals of motion. Consequently, we describe the extendedphase space (N=2)-SUSY algebra. In the second part (Sects. 4 - 5), by means of an iterative scheme, first, we findthe approximate groundstate solutions to the extended Schr¨odinger-like equation, and then calculate the parameterswhich measure the breaking of extended SUSY such as the groundstate energy. We calculate a more practical measurefor the SUSY breaking, in particular in field theories which is the expectation value of an auxiliary field. We analyzenon-perturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that thishas resulted from tunneling between the classical vacua of the theory. The section 6 deals with the independentgroup theoretical methods with nonlinear extensions of Lie algebras from the perspective of extended phase spaceSUSY quantum mechanics and, further, shows how it can be useful for spectrum generating algebra. The concludingremarks are given in section 7. Unless otherwise stated we take the geometrized units ( ~ = c = 1). Also, an implicitsummation on repeated indices are assumed throughout this paper. II. THE INTEGRALS OF MOTION
For the benefit of those not familiar with the framework of extended phase space quantization, enough details aregiven below to make the rest of the paper understandable. The interested reader is invited to consult the originalpapers [1, 2] for further details. In the framework of the proposed formalism, the extended Lagrangian can be written ∗ Electronic address: gago˙[email protected] as L ext ( p, q, ˙ p, ˙ q ) = − ˙ q i p i − q i ˙ p i + L q + L p , (1)where a dynamical system with N degrees of freedom described by the 2N independent coordinates q = ( q , . . . q N )and momenta p = ( p , . . . 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 respectto 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 q i = ∂ L ext ∂ ˙ q i = ∂ L q ∂ ˙ q i − p i , π p i = ∂ L ext ∂ ˙ p i = ∂ L p ∂ ˙ p i − q i . One may now definesan extended Hamiltonian H ext ( p, q, π p , π q ) = π q i ˙ q i + π p i ˙ p i − L ext ( p, q, ˙ p, ˙ q ) = H ( p + π q , q ) − H ( p, q + π p ) , (2)where H ( p, q ) = p i ˙ q i − L q = q i ˙ p 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 mechanicsthis choice picks up a pure state ( actual path ). Otherwise, one is dealing with a mixed state ( virtual path ). Onemay, however, envisage that the full machinery of the conventional quantum mechanical dynamics is extendible tothe extended dynamics as alluded to above. Here p and q will be considered as independent c-number operators onthe integrable complex function χ ( q, p ). One of the key assumptions of extended phase space quantization [1, 2] is thedifferential operators and commutation brackets for π p and π q borrowed from the conventional quantum mechanics: π q i = − i ∂∂ q i , [ π q i , q j ] = − iδ ij , π p i = − i ∂∂ p i , [ π p i , p j ] = − iδ ij . (3)Note also the following [ p i , q j ] = [ p i , p j ] = [ q i , q j ] = (cid:2) π p i , π q j (cid:3) = (cid:2) π p i , π p j (cid:3) = (cid:2) π q i , π q j (cid:3) = 0 . (4)By the virtue of Eq. (3) and Eq. (4), 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 Schr¨odinger-like equation: i ∂∂ t χ = H ext χ. (5)Solutions of Eq. (5) are χ ( q, p, t ) = χ r ( q, p, t ) e − ipq = a αβ ψ α ( q, t ) φ ∗ β ( p, t ) e − i pq , (6)where a = a † , positive definite , tr a = 1 , and ψ α and φ ∗ α are solutions of the conventional Schr¨odinger equationin q − and p − representations, respectively. They are mutually Fourier transforms. Note that the α and β are not, ingeneral, eigenindices. The normalizable χ ( R χ dp dq = tr ( a ) = 1) is a physically acceptable solution. The exponentialfactor is a consequence of the total time derivative, − d ( qp ) /dt, in Eq. (1) which can be eliminated. Actually, it iseasily verified that ( p + π q ) χ r ( q, p, t ) = ( π q χ r ( q, p, t )) e − ipq , (7)and so on. Substitution of Eq. (7) in Eq. (5) gives i ∂∂ t χ ( q, p, t ) = H ext χ r ( q, p, t ) , (8)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 particlewith 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. Theclassical extended Lagrangian Eq. (1) can be written L ext ( p, q, ˙ p, ˙ q ) = − ˙ q p − q ˙ p + 12 ˙ q − F ( q ) + 12 ˙ p − G ( p ) − R ( q, p ) N + 12 ψ α ˙ ψ α , (9)provided by N = ψ ψ = − i ψ + ψ − . Here F ( q ) : R → R , G ( p ) : R → R and R ( q, p ) : R → R are arbitrarypiecewise 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 Eq. (9) are { q, π q } = 1 , { p, π p } = 1 , { ψ α , ψ β } = δ αβ , { ψ + , ψ − } = 1 , ψ ± = 0 , ψ ± = 1 √ ψ ± iψ ) . (10)The extended Hamiltonian H ext Eq. (2) reads H ext ( p, q, π p , π q ) = 12 ( p + π q ) + F ( q ) −
12 ( q + π p ) − G ( p ) + R ( q, p ) N, (11)which, according to Eq. (7), reduces to H ext ( p, q, π p , π q ) = 12 π q + F ( q ) − π p − G ( p ) + R ( q, p ) N. (12)The Hamiltonian Eq. (12) yields the following equations of motion:˙ q = π q , ˙ p = π p , ˙ π q = − F ′ q ( q ) − R ′ q ( q, p ) N, ˙ π p = − G ′ p ( p ) + R ′ p ( q, p ) N, ˙ ψ ± = ± iR ( q, p ) ψ ± . (13)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 ± ( t ) exp (cid:20) ± i Z tt R ( q ( τ ) , p ( τ )) dτ (cid:21) . (14)Hence the odd quantities θ ± = θ ± ( t ) exp (cid:20) ∓ i Z tt R ( q ( τ ) , p ( τ )) dτ (cid:21) (15)are nonlocal in time integrals of motion. In trivial case R = 0 , we have ˙ ψ ± = 0 , and θ ± = θ ± . Suppose the systemhas even complex conjugate quantities B q,p ± , ( B q,p + ) ∗ = B q,p − , whose evolution looks up to the term proportionalto N like the evolution of odd variables in Eq. (13). Then local odd integrals of motion could be constructed in theform Q q,p ± = B q,p ∓ ψ ± . (16)Let us introduce the oscillator-like bosonic variables B q,p ± in q − and p -representations B q ∓ : L ( R ) → L ( R ) , B q ∓ = [ p + π q ± iW ( q )] , B p ∓ : L ( R ) → L ( R ) , B p ∓ = [ q + π p ± iV ( p )] . (17)In the expressions (17), W ( q ) : R → R and V ( p ) : R → R are the piecewise continuously differentiable functionscalled SUSY potentials. In particular case if R ( q, p ) = R q ( q ) − R p ( p ), for the evolution of B q,p ± we obtain˙ B q ∓ = (cid:2) − (cid:0) F ′ q + R ′ q N (cid:1) ± iW ′ q ( q ) ( p + π q ) (cid:3) , ˙ B p ∓ = (cid:2) − (cid:0) G ′ p + R ′ p N (cid:1) ± iV ′ p ( p ) ( q + π p ) (cid:3) . (18)Consequently,˙ Q q ± = ± i (cid:2)(cid:0) W ′ q − R ′ q ± (cid:1) ± i (cid:0) F ′ q − W W ′ q (cid:1) ψ ∓ (cid:3) , ˙ Q p ± = ± i (cid:2)(cid:0) V ′ p − R ′ p ± (cid:1) ± i (cid:0) G ′ p − V V ′ p (cid:1) ψ ∓ (cid:3) . (19)This shows that either ˙ Q q ± = 0 or ˙ Q p ± = 0 when W ′ q ( q ) = R ′ q ± ( q ) and F ′ q = ( W ) ′ q or V ′ p ( p ) = R ′ p ± ( p ) and G ′ p = ( V ) ′ p , respectively. Therefore, when the functions R q,p and F ( q ) , G ( p ) are related as R ′ q ± ( q ) = W ′ q ( q ) , F q = ( W ) + C q , R ′ p ± ( p ) = V ′ p ( p ) , F p = ( V ) + C p , (20)where C q,p are constants, then odd quantities Q q,p ± are integrals of motion in addition to H ext and N. According toEq. (2) and Eq. (12), let us present H ext in the form H ext = H q − H p , where H q = 12 π q + F ( q ) + R q N, H p = 12 π p + G ( p ) + R p N. (21)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 { Q q,p + , Q q,p − } = − i ( H 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 , (22)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 givenby the extended Lagrangian L ext ( p, q, ˙ p, ˙ q ) = 12 π q − W ( q ) + 12 π p − V ( p ) + ψ ψ ( W ′ q + V ′ p ) + 12 ψ α ˙ ψ α . (23)We conclude that the classical system Eq. (9) is characterized by the presence of two additional local in time oddintegrals of motion Eq. (16) being supersymmetry generators. Along the actual trajectories in q − space, the Eq. (23)reproduces the results obtained in [6]. III. THE PATH INTEGRAL FORMULATION
In the matrix formulation of extended phase space (N=2)-SUSY quantum mechanics, the ˆ ψ ± will be two realfermionic creation and annihilation nilpotent operators describing the fermionic variables. The ˆ ψ ± , having anticom-muting c-number eigenvalues, implyˆ ψ ± = r (cid:16) ˆ ψ ± i ˆ ψ (cid:17) , n ˆ ψ α , ˆ ψ β o = δ αβ , n ˆ ψ + , ˆ ψ − o = 1 , ˆ ψ ± = 0 . (24)They can be represented by finite dimensional matrices ˆ ψ ± = σ ± , where σ ± = σ ± σ are the usual raising andlowering operators for the eigenvalues of σ which is the diagonal Pauli matrix. The fermionic operator ˆ f readsˆ f : C → C , ˆ f = 12 h ˆ ψ + , ˆ ψ − i , which commutes with the H ext and is diagonal in this representation with conservedeigenvalues ± . Due to it the wave functions become two-component objects: χ ( q, p ) = (cid:18) χ +1 / ( q, p ) χ − / ( q, p ) (cid:19) = (cid:18) χ ( q, p ) χ ( q, p ) (cid:19) = (cid:18) ψ ( q ) φ ( p ) ψ ( q ) φ ( p ) (cid:19) , (25)where the states ψ , ( q ) , φ , ( p ) correspond to fermionic quantum number f = ±
12 , respectively, in q − and p − spaces.They belong to Hilbert space H = H ⊗ C = (cid:2) L ( R ) ⊗ L ( R ) (cid:3) ⊗ C . Hence the Hamiltonian H ext of extended phasespace (N=2)-SUSY quantum mechanical system becomes a 2 × H ext = (cid:18) H + H − (cid:19) = 12 (cid:16) ˆ π q + W (ˆ q ) + iW ′ q (ˆ q ) h ˆ ψ , ˆ ψ i(cid:17) − (cid:16) ˆ π p + V (ˆ p ) + iV ′ q (ˆ p ) h ˆ ψ , ˆ ψ i(cid:17) . (26)To infer the extended Hamiltonian Eq. (26) equivalently one may start from the c-number extended Lagrangian ofextended phase space quantum field theory in (0 + 1)-dimensions in q − and p − spaces: L ext ( p, q, ˙ p, ˙ q ) = − ˙ q p − q ˙ p + 12 "(cid:18) dqdt (cid:19) − W ( q ) + f W ′ q ( q ) + 12 "(cid:18) dpdt (cid:19) − V ( p ) + f V ′ p ( p ) . (27)In dealing with abstract space of eigenstates of the conjugate operator ˆ ψ ± which have anticommuting c-numbereigenvalues, suppose | − > is the normalized zero-eigenstate of ˆ q and ˆ ψ − :ˆ q | − > = 0 , ˆ ψ − | − > = 0 . (28)The state | > is defined by | > = ˆ ψ + | − > = 0 , (29)then ˆ ψ + | > = 0 , ˆ ψ − | > = | − > . Taking into account that ˆ ψ †± = ˆ ψ ∓ , we get < ∓ | ˆ ψ ± = 0 , < ∓ | ˆ ψ ∓ = < ± | . (30)Now we may introduce the notation α, β, . . . for the anticommuting eigenvalues of ˆ ψ ± . Consistency requires: α ˆ ψ ± = − ˆ ψ ± α, α | ± > = ±| ± > α. (31)The eigenstates of ˆ q, ˆ ψ − can be constructed as | qα − > = e − iq ˆ p − α ˆ ψ + | − >, (32)and thus, ˆ q | qα − > = q | qα − >, ˆ ψ − | qα − > = α | qα − > . (33)Then, the ˆ π q and ˆ ψ + eigenstates are obtained by Fourier transformation: | qβ + > = − Z dα e αβ | qα − >, | π q α ± > = − Z dq e iqπ q | qα ± >, | pβ + > = − Z dα e αβ | pα − >, | π p α ± > = − Z dp e ipπ p | pα ± >, (34)which gives ˆ π q | π q α ± > = π q | π q α ± >, ˆ ψ + | ( q, π q ) β + > = β | ( q, π q ) β + >, ˆ π p | π p α ± > = π p | π p α ± >, ˆ ψ + | ( p, π p ) β + > = β | ( p, π p ) β + > . (35)The following completeness relations hold: − Z dα dq | qα ± >< ∓ α ∗ q | = 1 , − Z dα dπ q π | π q α ± >< ∓ α ∗ π q | = 1 , − Z dα dp | pα ± >< ∓ α ∗ p | = 1 , − Z dα dπ p π | π p α ± >< ∓ α ∗ π p | = 1 . (36)The time evolution of the state | t > is now given χ − ( qα pβ t ) = − Z dα ′ dq ′ dβ ′ dp ′ K ( qα pβ t | q ′ α ′ p ′ β ′ t ′ ) . (37)The kernel reads K ( qα pβ t | q ′ α ′ p ′ β ′ t ′ ) = < + qα ∗ pβ ∗ | e − iH ext ( t − t ′ ) | q ′ α ′ p ′ β ′ >, (38)which can be evaluated by the path integral. Actually, an alternative approach to describe the state space anddynamics of the extended phase space quantum system is by the path integral [3], which reads K ff ′ ( qpt | q ′ p ′ t ′ ) = < qpf | e − iH ext ( t − t ′ ) | q ′ p ′ f ′ > . (39)In the path integral Eq. (39) the individual states are characterized by the energy and the fermionic quantum number f. With the Hamiltonian H ext , the path integral Eq. (39) is diagonal: K ff ′ ( qpt | q ′ p ′ t ′ ) = K ff ′ ( qt | q ′ t ′ ) K ff ′ ( pt | p ′ t ′ ) = δ ff ′ Z qq ′ D q Z pp ′ D p exp (cid:18) i Z tt ′ L ext ( p, q, ˙ p, ˙ q ) dt (cid:19) . (40)Knowing the path integral Eq. (40), it is sufficient to specify the initial wave function χ f ( q ′ , p ′ , t ′ ) to obtain all possibleinformation about the system at any later time t, by χ f ( q, p, t ) = X f ′ Z dq ′ dp ′ K ff ′ ( qpt | q ′ p ′ t ′ ) χ f ′ ( q ′ , p ′ , t ′ ) , (41)with χ ± / ( q, p, t ) = χ , ( q, p, t ) (Eq. (25)). In terms of anticommuting c-number operators ζ and η defining ψ = r (cid:18) η + ζi ( η − ζ ) (cid:19) , the path integral Eq. (40) becomes K ( qαpβt | q ′ α ′ p ′ β ′ t ′ ) = Z q,α,p,βq ′ ,α ′ ,p ′ ,β ′ D q D p D ζ D η exp (cid:18) i Z tt ′ L ext ( p, q, ˙ p, ˙ q ) dt (cid:19) . (42)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. IV. SOLUTION OF THE EXTENDED SCHR ¨ODINGER EQUATION WITH SMALL ENERGYEIGENVALUE ε Adopting the technique developed in [13], first, we use the iterative scheme to find the approximate groundstatesolutions to the extended Schr¨odinger-like equation H ext χ ( q, p ) = ( H q − H p ) χ ( q, p ) = ε χ ( q, p ) , (43)with energy ε . We will then use these solutions to calculate the parameters which measure the breaking of extendedSUSY such as the groundstate energy. The approximation, which went into the derivation of solutions of Eq. (43)meets our interest that the groundstate energy ε is supposedly small. As we mentioned above the solutions fornon-zero ε come in pairs of the form χ ↑ ( q, p ) = (cid:18) χ ( q, p )0 (cid:19) or χ ↓ ( q, p ) = (cid:18) χ ( q, p ) (cid:19) , (44)related by supersymmetry, where χ , ( q, p ) = ψ , ( q ) φ , ( p ) . The state space of the system is defined by all thenormalizable solutions of Eq. (43) and the individual states are characterized by the energies ε q and ε q and thefermionic 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 isnormalizable, and they cannot represent the groundstate of the system. The Eq. (43) yields the following relationsbetween energy eigenstates with fermionic quantum number ± : (cid:20)(cid:18) ∂∂q + W q ( q ) (cid:19) − (cid:18) ∂∂p + V p ( p ) (cid:19)(cid:21) ψ ( q ) φ ( p ) = p ε q ψ ( q ) φ ( p ) − p ε p ψ ( q ) φ ( p ) , (45)and (cid:20)(cid:18) − ∂∂q + W q ( q ) (cid:19) − (cid:18) − ∂∂p + V p ( p ) (cid:19)(cid:21) ψ ( q ) φ ( p ) = p ε q ψ ( q ) φ ( p ) − p ε p ψ ( q ) φ ( p ) , (46)where ε = ε q − ε p , ε q and ε p are the eigenvalues of H q and H p , respectively. The technique now is to devise aniterative approximation scheme to solve Eq. (45) and Eq. (46) by taking a trial wave function for χ ( q, p ) , substitutethis into the first equation (45) and integrate it to obtain an approximation for χ ( q, p ) . This can be used as anansatz in the second equation Eq. (46) to find an improved solution for χ ( q, p ) , etc. As it was shown in [13], theprocedure converges for well-behaved potentials with a judicious choice of initial trial function. If the W q and V p areodd, then ψ ( − q ) = ψ ( q ) , φ ( − p ) = φ ( p ) , (47)since they satisfy the same eigenvalue equation. It is straightforward then, for example, to obtain (cid:20)(cid:18) ∂∂q + W q ( q ) (cid:19) − (cid:18) ∂∂p + V p ( p ) (cid:19)(cid:21) ψ ( q ) φ ( p ) = p ε q ψ ( − q ) φ ( p ) − p ε p ψ ( q ) φ ( − p ) . (48)The independent nature of q and p gives the freedom of taking q = 0 , p = 0 which yield an expression for energies: p ε q = W (0) + ψ ′ (0) / ψ (0) , p ε p = V (0) + φ ′ (0) / φ (0) . (49)Suppose 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 , ( q ) = δ ( q − q ± ) , φ (0)1 , ( p ) = δ ( p − p ± ) . (50)After one iteration, we obtain χ (1)1 ( q, p ) = 1 N q N p θ ( q − q − ) θ ( p − p − ) e − R q dq ′ W ( q ′ )+ R p dp ′ V ( p ′ ) ,χ (1)2 ( q, p ) = 1 N q N p θ ( q + − q ) θ ( p + − p ) e R q dq ′ W ( q ′ ) − R p dp ′ V ( p ′ ) , (51)where N q and N p are the normalization factors. The next approximation leads to χ (2)1 ( q, p ) = 1 N ′ e − R q dq ′ W ( q ′ )+ R p dp ′ V ( p ′ ) Z ∞ max ( − q,q − ) e − R q ′ dq ′′ W ( q ′′ ) dq ′ Z ∞ max ( − p,p − ) e R p ′ dp ′′ V ( p ′′ ) dp ′ ,χ (2)2 ( q, p ) = 1 N ′ e R q dq ′ W ( q ′ ) − R p dp ′ V ( p ′ ) Z min ( − q,q + ) ∞ e R q ′ dq ′′ W ( q ′′ ) dq ′ Z min ( − p,p + ) ∞ e − R p ′ dp ′′ V ( p ′′ ) dp ′ . (52)It can be easily verified that to this level of precision Eq. (51) is self-consistent solution. Actually, for example, for q ′ > q − the exponential e − R q ′ W ( q ′′ ) dq ′′ will peak sharply around q + and may be approximated by a δ − function cδ ( q + − q ′ ); similarly e R q ′ W ( q ′′ ) dq ′′ we may replace approximately by cδ ( q − − q ′ ) for q ′ < q + . The same argumentshold for the p-space. With these approximations equations (52) reduce to Eq. (51). The normalization constant N ′ is N ′ = Z ∞ q − dq e − R q W ( q ′ ) dq ′ Z ∞ p − dp e − R p ′ V ( p ′ ) dp ′ ! / . (53)The energy expectation value ε = ( χ , H ext χ ) (54)gives the same result as that obtained for odd potentials by means of equations (49) and (52). Assuming theexponentials e − R q − W ( q ) dq and e − R p − V ( p ) dp to be small, which is correct to the same approximations underlyingEq. (54), 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 = ~ W ′ ( q + )2 π e − W/ ~ − ~ V ′ ( p + )2 π e − V/ ~ , (55)which gives direct evidence for the SUSY breaking in the extended phase space quantum mechanical system. Herewe have reinstated ~ , to show the order of adopted approximation, and its non-perturbative nature. We also denoted∆ W = Z q − q + W ( q ) dq, ∆ V = Z p − p + V ( p ) dp. (56)However, a more practical measure for the SUSY breaking, in particular, in field theories is the expectation value ofan auxiliary field, which can be replaced by its equation of motion right from the start: < F > = ( χ ↑ , i { Q + , σ − } χ ↑ ) . (57)Taking into account the relation Q + χ ↑ = 0 , with Q + commuting with H ext , which means that the intermediate statemust have the same energy as χ , the Eq. (57) can be written in terms of a complete set of states as < F > = i ( χ ↑ , Q + χ ↓ ) ( χ ↓ , σ − , χ ↑ ) . (58)According to Eq. (54) we have ε = < H ext > = ( χ ↑ , H ext χ ↑ ) = 12 ( χ ↑ , Q + χ ↓ ) ( χ ↓ , Q − χ ↑ ) = ε q − ε p = < H q > − < H p > =( ψ ↑ , H q ψ ↑ ) − ( φ ↑ , H p φ ↑ ) = 12 ( ψ ↑ , Q q + ψ ↓ ) ( ψ ↓ , Q q − ψ ↑ ) −
12 ( φ ↑ , Q p + φ ↓ ) ( φ ↓ , Q p − φ ↑ ) , (59)where χ ↑↓ = ψ ↑↓ φ ↑↓ , and ψ ↑ = (cid:18) ψ (cid:19) , ψ ↓ = (cid:18) ψ (cid:19) , φ ↑ = (cid:18) φ (cid:19) , φ ↓ = (cid:18) φ (cid:19) . From SUSY algebra itfollows immediately that1 √ ε Q − χ ↑ = χ ↓ = ˆ ψ − χ ↑ , √ ε q Q q − ψ ↑ = ψ ↓ = ˆ ψ − ψ ↑ , √ ε p Q p − φ ↑ = φ ↓ = ˆ ψ − φ ↑ . (60)By virtue of Eq. (60), the Eq. (59) reads √ ε ( χ ↑ , Q + χ ↓ ) (cid:16) χ ↓ , ˆ ψ − χ ↑ (cid:17) = √ ε q ( ψ ↑ , Q q + ψ ↓ ) (cid:16) ψ ↓ , ˆ ψ − ψ ↑ (cid:17) − √ ε p ( φ ↑ , Q p + φ ↓ ) (cid:16) φ ↓ , ˆ ψ − φ ↑ (cid:17) , (61)Using the matrix representations of Q q + , Q p + and ˆ ψ − and the wave functions Eq. (52), one gets ([13]) (cid:16) ψ ↓ , ˆ ψ − ψ ↑ (cid:17) = r W ′ ( q + ) π e − ∆ W ∆ q, (cid:16) φ ↓ , ˆ ψ − φ ↑ (cid:17) = r V ′ ( p + ) π e − ∆ V ∆ p, ( ψ ↑ , Q q + ψ ↓ ) = i r W ′ ( q − ) π e − ∆ W , ( φ ↑ , Q p + φ ↓ ) = i r V ′ ( p − ) π e − ∆ V , (62)where ∆ q = q + − q − and ∆ p = p + − p − . Hence( χ ↑ , Q + χ ↓ ) (cid:16) χ ↓ , ˆ ψ − χ ↑ (cid:17) = 2 i √ ε "(cid:16) ε q ε (cid:17) √ ε q r W ′ ( q − ) π ∆ q − (cid:16) ε p ε (cid:17) √ ε p r V ′ ( p − ) π ∆ p , (63)and < F > = − √ ε "(cid:16) ε q ε (cid:17) r W ′ ( q + ) π e − W ∆ q − (cid:16) ε p ε (cid:17) r V ′ ( p ) π e − V ∆ p . (64)Along the actual trajectories in q − space, the Eq. (64) reproduces the results obtained in [13]. V. AN EXTENDED SUSY BREAKING IN THE INSTANTON PICTURE
In this subsection our goal is to show that the expressions Eq. (63) and Eq. (64) can be obtained in the path integralformulation of the theory by calculating the matrix elements, i.e., the effect of tunneling between two classical vacuaby using a one-instanton background. That is, the matrix elements of ˆ ψ ± , Q q ± and Q p ± can be calculated in thebackground of the classical solution ˙ q c = − W c and ˙ p c = − V c . In doing this we re-write the matrix element Eq. (61)in terms of eigenstates of the conjugate operator ˆ ψ ± : √ ε h < +0 q + p + | e − iH ext ( T − t ) Q + e − iH ext ( T + t ) | q − p − − >< − q − p − | e − iH ext ( T − t ) ˆ ψ − e − iH ext ( T + t ) | q + p + > i T →− i ∞ = √ ε q h < +0 q + | e − iH q ( T − t ) Q q + e − iH q ( T + t ) | q − − >< − q − | e − iH q ( T − t ) ˆ ψ − | e − iH q ( T + t ) q + > i T →− i ∞ −√ ε p h < +0 p + | e − iH p ( T − t ) Q p + e − iH p ( T + t ) | p − − >< − p − | ˆ ψ − | p + > i T →− i ∞ , (65)in the limit T → − i ∞ . This reduces to √ ε < +0 q + p + | Q + | q − p − − >< − q − p − | ˆ ψ − | q + p + > = √ ε q < +0 q + | Q q + | q − − >< − q − | ˆ ψ − | q + > −√ ε p < +0 p + | Q p + | p − − >< − p − | ˆ ψ − | p + >, (66)which, in turn, can be presented by path integrals defined in terms of anticommuting c-number operators ζ and η withEuclidean actions of the instantons in q − and p − spaces, respectively. Following [13], these functional integrals includean integration over instanton time τ which is due to the problem of zero modes of the bilinear terms in Euclideanactions. This arises from time-transformation of instantons, and SUSY transformations on them, respectively. Theexistence of zero modes gives rise to non-gaussian behaviour of the functional integral. Due to it the matrix elementsabove do not receive any contributions from either no-instanton or anti-instanton configurations. The zero modeproblem is solved by introducing a collective coordinate τ replacing the bosonic zero mode [14]. Whereas, thefuncional integrals depend only on the difference τ − τ . Note also that multi-instanton configurations could contributein 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 p ε q and p ε 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 , , . . . , N , one can put conditions on the SUSY potentials Z ∞ W ( q ′ ) dq ′ → ∞ at q → ±∞ for ψ +0 , Z ∞ W ( q ′ ) dq ′ → −∞ at q → ±∞ for ψ − , (67)and similar for V ( p ), that the extrema are well separated: R q ν +1 q ν W ( q ′ ) dq ′ ≫
1, and R q µ +1 p µ V ( p ′ ) dp ′ ≫
1. Around eachof the classical minima q ν and p µ of the potentials W ( q ) and V ( p ), respectively, one can approximate the theoryby a suppersymmetric harmonic oscillator. Then there are N ground states which have zero energy. These states aredescribed by upper or lower component of the wave function, depending on whether ν and µ are odd or even. Withthis provision the functional integrals are calculated in [13], which allow us consequently to write: < +0 q + | Q q + | q − − > = i r W ′ c ( q + ) π e − ∆ W c , < − q − | ˆ ψ − | q + > = r W ′ c ( q + ) π e − ∆ W c ∆ q c , (68)etc. Inserting this in Eq. (66), we arrive at the Eq. (63) < +0 q + p + | Q + | q − p − − >< − q − p − | ˆ ψ − | q + p + > =2 i √ ε "(cid:16) ε q ε (cid:17) √ ε q r W ′ c ( q + ) π ∆ q c − (cid:16) ε p ε (cid:17) √ ε p r V ′ c ( p + ) π ∆ p c , (69)and, thus, of Eq. (64) as its inevitable corollary. This proves that the extended SUSY breaking has resulted fromtunneling between the classical vacua of the theory. The corrections to this picture are due to higher order terms andquantum tunneling effects. VI. SPECTRUM GENERATING ALGEBRA
An extended Hamiltonian H ext Eq. (26) can be treated as a set of two ordinary two-dimensional partner Hamilto-nians [4] H ± = 12 (cid:2) π q − π p + U ± ( q, p ) (cid:3) , (70)provided by partner potentials U ± ( q, p ) = U q ± ( q ) − U p ± ( p ) , U q ± ( q ) = W ( q, a q ) ∓ W ′ q ( q, a q ) , U p ± ( p ) = V ( p, a p ) ∓ V ′ p ( p, a p ) . (71)A subset of the SUSY potentials for which the Schr¨odinger-like equations are exactly solvable share an integrabilityconditions of shape-invariance [15]: U + ( a , q, p ) = U − ( a , q, p ) + R ( a ) , a = f ( a ) , (72)where a and a are a set of parameters that specify phase-space-independent properties of the potentials, and thereminder R ( a ) is independent of ( q, p ) . A. Algebraic properties of shape invariance
Using the standard technique, we may construct a series of Hamiltonians H N , N = 0 , , , . . . ,H N = 12 " π q − π p + U − ( a N , q, p ) + N X k =1 R ( a k ) , (73)0where a N = f ( N ) ( a ) ( N is the number of iterations). From Eqs. (71) and (72) we obtain then N = n + m couplednonlinear differential equations which are the two recurrence relations of Riccati-type differential equations: W n +1 ( q ) + W ′ q ( n +1) ( q ) = W n ( q ) − W ′ qn ( q ) − µ n , V m +1 ( p ) + V ′ p ( m +1) ( p ) = V m ( p ) − V ′ pm ( p ) − ν m , (74)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 thatfor unbroken SUSY, the eigenstates of the potentials U q,p , respectively, are E ( − ) q = 0 , E ( − ) qn = n − X i =0 µ i , E ( − ) p = 0 , E ( − ) pm = m − X j =0 ν j , (75)that is, the ground states are at zero energies, characteristic of unbroken supersymmetry. The differential equa-tions (74) can be investigated to find exactly solvable potentials. The shape invariance condition Eq. (72) can beexpressed in terms of bosonic operators as B + ( x, a ) B − ( x, a ) − B − ( x, a ) B + ( x, a ) = R ( a ) = µ − ν ,B æ+ (æ , a æ0 ) B æ − (æ , a æ0 ) − B æ − (æ , a æ1 ) B æ+ (æ , a æ1 ) = ( µ or ν ) , (76)where x ( q, p )-is the coordinate in ( q, p ) − space, æ denotes concisely either q − or p − representations (no summationon æ is assumed). To classify algebras associated with the shape invariance, following [16] we introduce an auxiliaryvariables φ ( φ q , φ p ) and define the following creation and annihilation operators: J + = e ikφ B + ( x, χ ( i∂ φ )) , J − = B − ( x, χ ( i∂ φ )) e − ikφ , (77)where k ( k q , k p ) are an arbitrary real constants and χ ( χ q , χ p ) are an arbitrary real functions. Consequently, thecreation and annihilation operators in ( q, p ) − spaces can be written as J æ+ = e ik æ φ æ B æ+ (æ , χ æ ( i∂ φ æ )) , J æ − = B æ − (æ , χ æ ( i∂ φ æ )) e − ik æ φ æ . (78)The operators B ± ( x, χ ( i∂ φ )) and B æ ± (æ , χ æ ( i∂ φ æ )) are the generalization of Eqs. (76), where a → χ ( i∂ φ ) and a æ0 → χ æ ( i∂ φ æ ) . One can easily prove the following relations: e ikφ B + ( x, χ ( i∂ φ )) = B + ( x, χ ( i∂ φ + k )) e ikφ , B − ( x, χ ( i∂ φ )) e − ikφ = e − ikφ B − ( x, χ ( i∂ φ + k )) , (79)and that e ik æ φ æ B æ+ (æ , χ æ ( i∂ φ æ )) = B æ+ (æ , χ æ ( i∂ φ æ + k æ )) e ik æ φ æ ,B æ − (æ , χ æ ( i∂ φ æ )) e − ik æ φ æ = e − ik æ φ æ B æ − (æ , χ æ ( i∂ φ æ + k æ )) . (80)If we choose a function χ ( i∂ φ ) such that χ ( i∂ φ + k ) = f [ χ ( i∂ φ )] , then we have identified a → χ ( i∂ φ ) , and a = f ( a ) → f [ χ ( i∂ φ )] = χ ( i∂ φ + k ) . Similar relations can be obtained for the æ − representations. From Eq. (76) weobtain then B + ( x, χ ( i∂ φ )) B − ( x, χ ( i∂ φ )) − B − ( x, χ ( i∂ φ + k )) B + ( x, χ ( i∂ φ + k )) = R [ χ ( i∂ φ ))] ,B æ+ (æ , χ æ ( i∂ φ æ )) B æ − (æ , χ ( i∂ φ æ )) − B æ − (æ , χ æ ( i∂ φ æ + k æ )) B æ+ (æ , χ æ ( i∂ φ æ + k æ )) = R æ [ χ æ ( i∂ φ æ ))] . (81)Introducing the operators J = − ik ∂ φ and J æ3 = − ik æ ∂ φ æ , and combining Eqs. (78) and (81), we may arrive at adeformed Lie algebras: [ J + , J − ] = [ J q + , J q − ] − [ J p + , J p − ] = ξ ( J ) = ξ q ( J q ) − ξ p ( J p ) , [ J , J ± ] = ± J ± , [ J æ+ , J æ − ] = ξ æ ( J æ3 ) , [ J æ3 , J æ ± ] = ± J æ ± , (82)where ξ ( J ) ≡ R [ χ ( i∂ φ )] and ξ æ ( J æ3 ) ≡ R æ [ χ æ ( i∂ φ æ )] define the deformations. Different χ functions in Eq. (81)define different reparametrizations corresponding to several models. For example:1. The translational models ( a = a + k ) correspond to χ ( z ) = z. If R is a linear function of J the algebra becomesSO(2.1) or SO(3). Similar in many respects prediction is made in somewhat different method by Balantekin [17].2. The scaling models ( a = e k a ) correspond to χ ( z ) = e z , etc.1 B. The unitary representations of the deformed Lie algebra
In order to find the energy spectrum of the partner SUSY Hamiltonians2 H − ( x, χ ( i∂ φ )) = B − ( x, χ ( i∂ φ + k )) B + ( x, χ ( i∂ φ + k )) , H æ − (æ , χ æ ( i∂ φ æ )) = B æ − (æ , χ æ ( i∂ φ æ + k æ )) B æ+ (æ , χ æ ( i∂ φ æ + k æ )) , (83)one must construct the unitary representations of deformed Lie algebra defined by Eq. (82) [16, 18]. Using thestandard technique, one defines up to additive constants the functions g ( J ) and g æ ( J æ3 ): ξ ( J ) = g ( J ) − g ( J − , ξ æ ( J æ3 ) = g æ ( J æ3 ) − g æ ( J æ3 − . (84)The Casimirs of this algebra can be written as C = J − J + + g ( J ) and accordingly C æ2 = J æ − J æ+ + g æ ( J æ3 ) . In a basisin which J and C are diagonal, J − and J + are lowering and raising operators (the same holds for æ-representations).Operating on an arbitrary state | h > they yield J | h > = h | h >, J − | h > = a ( h ) | h − >, J + | h > = a ∗ ( h + 1) | h + 1 >, (85)where | a ( h ) | − | a ( h + 1) | = g ( h ) − g ( h − . (86)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 therepresentation of the algebra associated with a characteristic model, consequently we obtain the complete spectrum ofthe system. For example, without ever referring to underlying differential equation, we may obtain analytic expressionsfor the entire energy spectrum of extended Hamiltonian with Self-Similar potential. A scaling change of parametersis given as a = Qa , a æ1 = Q æ a æ0 , at the simple choice R ( a ) = − r a , where r is a constant. That is, ξ ( J ) ≡ − r exp( − kJ ) = ξ q ( J q ) − ξ p ( J p ) = − r q exp( − k q J q ) + r p exp( − k p J p ) , (87)which yields [ J + , J − ] = ξ ( J ) = − r exp( − kJ ) , [ J , J ± ] = ± J ± . (88)This is a deformation of the standard SO (2 .
1) Lie algebra, therefore, one gets g ( h ) = r e k − e − kh = − r − Q Q − h , Q = e k ,g æ ( h æ ) = r æ1 e k æ − e − k æ h æ = − r æ1 − Q æ Q − h æ æ , Q æ = e k æ . (89)For scaling problems [16] one has 0 < q < , which leads to k <
0. The unitary representation of this algebra formonotonically decreasing profile of the function g ( h ), are infinite dimensional. Let the lowest weight state of the J be h min , then a ( h min ) = 0 . One can choose the coefficients a ( h ) to be real. From Eq. (86), for an arbitrary h = h min + n, n = 0 , , , . . . , we obtain a ( h ) = g ( h − n − − g ( h −
1) = r Q n − Q − Q − h . (90)The spectrum of the extended Hamiltonian H − ( x, a ) reads H − | h > = a ( h ) | h > = r Q n − Q − Q − h | h >, (91)with the eigenenergies E n ( h ) = r α ( h ) Q n − Q − , α ( h ) ≡ Q − h . (92)2Similar expressions can be obtained for the H æ − and eigenenergies E qn and E pn ( n ≡ ( n , n ) , n , = 0 , , , . . . ),as H q − | h q > = a q ( h q ) | h q > = r q Q n q − Q q − Q − h q q | h q >,H p − | h p > = a p ( h p ) | h p > = r p Q n p − Q p − Q − h p p | h p >, (93)and that E qn ( h q ) = r q α q ( h q ) Q n q − Q q − , α q ( h q ) ≡ Q − h q q ,E pn ( h p ) = r p α p ( h p ) Q n p − Q p − , α p ( h p ) ≡ Q − h p p . (94)Hence E n ( h ) = E qn ( h q ) − E pn ( h p ) . (95) VII. CONCLUSIONS
We addressed the classical analog of the extended phase space quantum mechanics of particle which have bothbosonic and fermionic degrees of freedom, i.e., 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 findthe approximate groundstate solutions to the extended Schr¨odinger-like equation and calculate the parameters whichmeasure the breaking of extended SUSY such as the groundstate energy. The approximation, which went into thederivation of solutions of Eq. (43) meets our interest that the groundstate energy ε is supposedly small. This givesdirect 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 non-perturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this hasresulted from tunneling between the classical vacua of the theory. Finally, we present an analysis on the independentgroup theoretical methods with nonlinear extensions of lie algebras from the extended phase space SUSY quantummechanics. Using the factorization procedure we explore the algebraic property of shape invariance and spectrumgenerating algebra. Most of these Hamiltonians posses this feature and hence are solvable by an independent grouptheoretical method. We construct the unitary representations of the deformed Lie algebra. Acknowledgments
I would like to thank Y. Sobouti for drawing my attention to the extended phase space formulation of quantummechanics.
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