$\mathbb{Z}_2\times \mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations

Prog. Theor. Exp. Phys. Z2 × Z2-graded Lie symmetries of the Lévy-Leblond equations N. Aizawa 2 Z. Kuznetsova 1 H. Tanaka 2 F. Toppan 0 Subject Index 0 CBPF , Rua Dr. Xavier Sigaud 150, Urca, cep 22290-180, Rio de Janeiro (RJ) , Brazil 1 UFABC , Av. dos Estados 5001, Bangu, cep 09210-580, Santo André (SP) , Brazil 2 Department of Physical Science, Graduate School of Science, Osaka Prefecture University , Nakamozu Campus, Sakai, Osaka 599-8531 Japan ................................................................................................................... The first-order differential Lévy-Leblond equations (LLEs) are the non-relativistic analogs of the Dirac equation, being square roots of (1+d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLEs possess a natural supersymmetry. In previous works it was shown that non-supersymmetric partial differential equations (notably the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural Z2-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric partial differential equation, a natural Z2 × Z2-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the (1 + 1)-dimensional Lévy-Leblond equations, both in the free case and for the harmonic potential. In the free case a Z2 × Z2-graded Lie superalgebra, realized by firstand second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the Z2- and Z2 × Z2-graded extensions are no longer allowed. The construction of the Z2 × Z2-graded Lie symmetry of the (1 + 2)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra. ................................................................................................................... Introduction I = X = Y = A = the split-quaternions, see Appendix A, can be represented as e0 = I , e1 = Y , e2 = X , e3 = A, e0 = II , e1 = AI , e2 = XA, e3 = YA. Some comments are in order: 2 × 2 : Cl(2, 1), 4 × 4 : Cl(3, 2), 8 × 8 : Cl(4, 3), Cl(0, 3), Cl(5, 0), Cl(1, 4), Cl(0, 7), and so on. Clifford algebras can be used to introduce supersymmetric quantum mechanics (SQM). In its simplest version (the one-dimensional, N = 2 supersymmetry, with x as a space coordinate) the two supersymmetry operators Q1, Q2 need to be block anti-diagonal, Hermitian, and complex-structure preserving first-order real differential operators. Moreover, they have to satisfy the N = 2 SQM algebra [H , Qi] = 0, where H is the Hamiltonian. The irreducible representation (in real counting) requires 4 × 4 matrices. By setting the complex structure to be J = IA, the most general solution, for an arbitrary function f (x) (the prepotential), can be expressed as Q1 = AI ∂x + YIf (x), anti-diagonal and anticommuting with the fermion number operator, so that it mutually exchanges bosons into fermions. The minimal matrix size to accommodate a Lévy-Leblond square root of the 1 + 1 heat equation is 2. Indeed, we can set (x, t) = 0, 1 1 = 2 (A + Y )∂t + 2 (A − Y )λ + X ∂x → heat,free = II (λ∂t − ∂x ) = I4(λ∂t − ∂x2). 2 2 2 2 heat = II (λ∂t − ∂x + f (x)2) + XXfx(x). where “·” denotes the ordinary matrix multiplication. Its square gives Sch can be expressed as 2 2 Sch = −AII λ∂t + III (−∂x + f (x)2) + IXXfx(x), the complex structure being defined by J = AII . It is straightforward to systematically construct, following this scheme and with the tools in [31], generalized Lévy-Leblond operators in (1 + d) dimensions. On symmetries of matrix partial differential equations (x) = 0 ⇒ =0 = 0. [ , Z ] = { , Z } = where Z (x) is an n × n matrix-valued function of the x space(-time) coordinates. If there is no ambiguity [certain operators, like the identity I, satisfy both Eqs. (3.2) and (3.3)] throughout the text we denote with s the symmetry operators satisfying Eq. (3.2) and with s the symmetry operators satisfying Eq. (3.3). In our applications we consider, initially, first-order differential symmetry operators. Secondorder differential symmetry operators are also constructed by taking suitable anticommutators of the previous operators. It is important to note that, if 1, 2 are two operators satisfying Eq. (3.2), then by construction the identity [ , [ 1, 2]] = 4. Symmetries of the Lévy-Leblond square root of the free heat and Schrödinger equations γ±γ∓ = 21 (I ± γ3), γ3γ± = ±γ± = −γ±γ3. They define the complex structure J given by (J 2 = −II = −I). = γ+∂t + γ−λ + γ3∂x. 2 = I · (λ∂t + ∂x2). { , } = By construction, γ±, γ3 defined in Eq. (4.5) commute with J ([J , γ±] = [J , γ3] = 0). The Lévy-Leblond operator of the free heat equation is introduced (both in the 2 × 2 and in the 4 × 4 representations) as Its square is Eq. (4.8) is given by H = I∂t, D = − (...truncated)