Two-time physics, Carroll symmetry and Jordan algebras

Bulgarian Journal of Ph ysics v ol. XX issue X (20XX) 1 – 6 T w o-time ph ysics, Carr oll symmetry and Jor dan algebras A. Kamenshchik 1 , A. Marrani 2 F . Muscolino 3 1 Dipartimento di Fisica e Astronomia “ A. Righi”, Univ ersit ` a di Bologna and INFN, via Irnerio 46, 40126 Bologna, Italy 2 School of Physics, Engineering and Computer Science, Univ ersity of Hertford- shire, AL10 9AB Hatfield, UK 3 Dipartimento di Matematica e Applicazioni, Univ ersit ` a di Milano Bicocca, via Roberto Cozzi 55, 20125 Milano, Italy Receiv ed Day Month Y ear (Insert date of submission) Abstract. W e describe Carroll particles with nonzero ener gy (i.e., par - ticles that remain at rest) within the frame work of two-time (2T) physics dev eloped by Bars and collaborators. In a spacetime with one additional time and one additional space dimension, one can gauge the phase-space symmetry that exchanges generalized coordinates with their conjugate momenta, thereby unifying the description of apparently different one- time systems. W e de v elop both classical and quantum descriptions of the Carroll particle arising from 2T physics, and explore links between the extended phase space of 2T physics and Freudenthal triple systems constructed over a semisimple cubic Jordan algebra (the Lorentzian spin factor). K E Y W O R D S : Carroll symmetry , Lorentzian spin factors, Freudenthal triple systems, two-time physics. 1 Introduction Theories with extra spatial dimensions hav e become quite traditional since the times when they were put forw ard in the famous works by Kaluza and Klein. Theories with more than one time dimension look much less intuitiv e and plau- sible. Ne v ertheless, an impressive series of papers dev oted to the so called two- time (2T) physics was produced by I. Bars and his co-authors since 1996 (see the book [ 1 ] and references therein). Classical and quantum physics of simple sys- tems such as non-relati vistic particle, massiv e and massless relati vistic particles, harmonic oscillator, hydrogen-like atoms can be described in the framework of 2T -physics from a unifying point of vie w . The language of 2T physics is also 1310–0157 © 2005 Heron Press Ltd. 1 T wo-time physics, Carroll symmetry and Jordan algebras well adapted to field theories and to gravity . Ho wev er , relations between the 2T physics and the Carroll symmetry [ 2 – 4 ] had not been e xplored before. This was done in our paper [ 5 ]. From the point of vie w of the 2T Physics, usual physical systems living in a one-time world represent projections from the spacetime with one additional temporal dimension one additional spatial dimension. These additional dimen- sions are introduced to construct a ne w gauge theory , based on the localization of the phase-space symmetry described by the symplectic group S p (2 , R ) . In- troduction of the gauge fields implies the presence of three first-class constraints X · X = 0 , P · P = 0 , X · P = 0 , where X and P are coordinates and momenta in the e xtended 2 + d -dimensional spacetime. The usual ph ysics in 1 + ( d − 1) - dimensional spacetime is obtained by means of a gauge-fixing procedure. Dif- ferent gauge-fixings (or, in other words, dif ferent parametrizations of the phase space coordinates X and P in terms of usual coordinates x and momenta p ) give different e xpressions for the 1 + ( d − 1) -dimensional Hamiltonians and time parameters. The Carroll Lie algebra and the Carroll group are obtained from the contraction of the Poincar ´ e group by putting the velocity of light equal to zero [ 2 , 3 ]. The Carrol particle with non-zero ener gy should be always in rest. The Carroll parti- cle with zero energy is always moving. These two cases are disconnected. This is because the Carroll boosts K i = x i ∂ ∂ t commute with the Hamiltonian and do not change the ener gy , in contrast to the Lorentzian and Galilean boosts. In Section 2 we shall consider the classical Carroll particle in rest from the point of view of the 2T physics. In the third section we discuss its quantization. The fourth section is de v oted to relations between 2T physics and Jordan algebras, extensi vely treated in [ 11 ]. The last section contains concluding remarks. 2 Carroll par ticle in tw o-time spacetime: classical theory The action for the Carroll particle can be represented as [ 4 ] S = − Z dτ  ˙ tE − ˙ x · p − λ ( E − E 0 )  , (1) where τ is the proper time, t is the physical time, E represents the classical Hamiltonian and x i and p i are the space coordinates and the momenta, for i = 1 , . . . , d − 1 . The quantity E 0  = 0 represents the rest energy of the Carroll particle and λ plays the role of a Lagrange multiplier . W e would like to obtain this action from the 2T action S = Z dτ ( P A ˙ X A − A ij Q ij , where A ij are the gauge fields, playing the role of Lagrange multipliers while the constraints are Q 11 = X · X , Q 12 = X · P , Q 22 = P · P . Let us introduce 2 A. Kamenshchik, A. Marrani, F . Muscolino the light-cone coordinates X + = 1 2  X 1 ′ + X 0 ′  , X − = 1 2  X 1 ′ − X 0 ′  . Choosing the two-time coordinates and momenta as in [ 5 ] X + = E 0 t, X − = x i p i E 0 + t E 0  E − E 0 + p i p i 2  , X 0 = p x i x i , X i = x i + tp i , P + = E 0 , P − = 1 E 0  E − E 0 + p i p i 2  , P 0 = 0 , P i = p i , we obtain the action ( 1 ), provided E = E 0 , which is obtained by solving the constraints Q ij = 0 . 3 Carroll par ticle in tw o-time spacetime: quantum theor y The commutation relation for the position and momentum operators in the stan- dard d − 1 -dimensional space are [ x i , p j ] = i δ ij . Upon quantization, operator ordering becomes an issue. All the operators should be Hermitian, b ut this requirement is not suf ficient. W e have to resort to the cov ariant quantization in the 2T spacetime. The quantum generators L M N of S O (2 , d ) must satisfy the Lie algebra under commutators. The generators of the Lorentz group S O (2 , d ) , L M N which become operators should constitute the Lie algebra with respect to the commutators. This requirement also does not de- fine the ordering in the quantum generators in a unique way and one should use also the properties of the Casimir operators of the unitary representations of both the groups S O (2 , d ) and 1 S p (2 , R ) . The constraints play the role of the genera- tors of the symmetry with respect to the S p (2 , R ) group. They should be applied to the acceptable quantum states of the system according to the prescription of the Dirac quantization of systems with first-class constraints Q | Ψ ⟩ = 0 . 1 W e will henceforth use the physicists’ notation of symplectic groups : namely , S p (2 , R ) is the split real form of the Lie group whose algebra is (in the usual Cartan’ s notation) c 1 , when considered over the comple x numbers. 3 T wo-time physics, Carroll symmetry and Jordan algebras The same should be valid also for the Casimir operators. In paper [ 6 ] this tech- nique was implemented to reproduce the quantization scheme and the spectrum for the hydrogen-like atom. If we manage to fix the ordering in the generators of S O (2 , d ) group at τ = 0 , then the same ordering will be conserv ed. Our parametrization of the v ariables X M , P M at τ = 0 coincides with that used for the description of the hydrogen atom in [ 6 ] provided we ha ve already put E = E 0 . It is amazing because these physical systems are quite different and their actions are also different. W e can use this fact to quantize our Carroll particle at rest. It does not mean that we shall obtain the discrete spectrum. The combination of the squared mo- mentum and the inv erse radius is not connected with the Hamiltonian. The mo- mentum is not connected with the velocity (which is equal to zero). What is the role of the momentum? It enters into the commutation relations and, hence, the Heisenberg inequality of uncertainties ∆ x i · ∆ p j ≥ 1 2 δ ij is valid. In contrast to the standard non-relativistic quantum mechanics, we can choose the quantum states with a dispersion of the coordinate ∆ x as small as we wish, because the growth of the dispersion of the momentum ∆ p is not impor- tant. Thus, a particle can be localized with an arbitrary high precision. 4 T wo-time physics and Jor dan algebras The choice of the parametrizations of the coordinates and momenta in the 2T spacetime looks as some kind of the craftsman w ork. It would be interesting to find a general algebraic structure behind the 2T spacetime. Perhaps, some rather abstract constructions connected with Jordan algebras can play this role (see [ 11 ] for further details). Jordan algebras [ 7 , 8 ] were in vented with an intention to create a new mathematical apparatus for quantum theory . They did not fulfill these expectations, but instead ha ve found a lot of other applications in both mathematics and physics [ 9 ]. A (real) Jordan algebra ( J , ◦ ) is a v ector space defined ov er a ground field F (in our case R ) equipped with a bilinear product ◦ which is commutativ e and power-associati ve, satisfying X ◦ Y = Y ◦ X, X 2 ◦ ( X ◦ Y ) = X ◦ ( X 2 ◦ Y ) , ∀ X , Y ∈ J . A cubic Jordan algebra is endo wed with a cubic form N : J → R , such that N ( λX ) = λ 3 N ( X ) , ∀ λ ∈ R , X ∈ J . Besides, an element, named base point c ∈ J exists, satisfying N ( c ) = 1 . The definition of the cubic norm permits to construct a cubic Jordan algebra, in which all the properties of the Jordan algebra are determined by the cubic form itself. W e are interested in a special 4 A. Kamenshchik, A. Marrani, F . Muscolino class of the cubic Jordan algebras, called pseudo-Euclidean spin-factors. Ha ving a ( m + n ) -dimensional pseudo-Euclidean spacetime Γ m,n we can construct a cubic Jordan algebra J = R ⊕ Γ m,n with the cubic norm N 3 ( X = ξ ⊕ γ ) := ξ γ a γ b η ab . it is important that to transform a Mink owski spacetime Γ 1 ,d − 1 into a cubic Jordan algebra, we should add to this spacetime an additional spatial dimension. Starting from a cubic Jordan algebra J , we can construct a Freudenthal triple system [ 10 ], which is defined as the vector space F ( J ) := R ⊕ R ⊕ J ⊕ J . An element x ∈ F ( J ) can formally be written as a “ 2 × 2 matrix” : x =  x X Y y  , x, y ∈ R , X , Y ∈ J . A Freudenthal triple system is endo wed with a non-degenerate symplectic bilin- ear form, a quartic in variant, and a trilinear triple product. It is important to note that, when constructing a Freudenthal triple system from the pseudo-Euclidean spin factor , one algebraically doubles degrees of freedom from configuration to phase space. R ⊕ ( R ⊕ Γ 1 ,d − 1 ) | {z } coordinates ⊕ R ⊕ ( R ⊕ Γ 1 ,d − 1 ) | {z } momenta . Besides, we can try to use the in variants obtained from the Freudenthal triple system structure to classify different g auge-fixings in the two-time world. 5 Conclusions W e have found such a parametrization of the phase space variables in two-time spacetime, which permits to describe a Carroll particle in rest in the one-time spacetime. In quantum theory we ha ve seen an amusing correspondence be- tween our parametrization and that used for the description and quantization of the hydrogen atom. The case of the always moving particle (Carroll tachyon) is more complicated and it is under study . The most interesting line of research is probably an “algebraic” one: we hope to push forward the analysis of the rela- tions between Jordan algebras and Freudenthal triple systems and 2T physics to understand better how one can find dif ferent one-time w orlds hidden inside of the two-time spacetimes. Ackno wledg ements A.K. is grateful to the organizers of the conference QTS-13 in Y ere van, 2025 for the opportunity to giv e a talk. 5 T wo-time physics, Carroll symmetry and Jordan algebras References [1] I. Bars, J. T erning (2010) “ Extra Dimensions in Space and T ime ”. Springer , New Y ork. [2] J.-M. L ´ evy-Leblond (1965) Une nouvelle limite non-relativiste du groupe de Poincar ´ e. Ann. Inst. Henri P oincar ´ e A 3 , 1-12 [3] N. D. Sen Gupta (1966) On an Analogue of the Galilei Group. Nuovo Cimento 44 , 512-517 [4] J. de Boer et al (2022) Carroll Symmetry , Dark Energy and Inflation. F r ont. Phys. 10 , 810405 [5] A. Kamenshchik, F . Muscolino (2024) Looking for Carroll particles in two time spacetime. Phys. Re v . D 109 2, 025005 [6] I. Bars (1998) Conformal symmetry and duality between free particle, H - atom and harmonic oscillator . Phys. Rev . D 58 , 066006 [7] P . Jordan (1933) ¨ Uber die multiplikation quanten-mechanischer grossen. Zschr . f. Phys. 80 , 285 [8] P . Jordan, J. von Neumann, E.P . Wigner (1934) On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35 , 1, 29-64. [9] K. McCrimmon (2004) “ A T aste of J or dan Algebras ”. Springer-V erlag New Y ork Inc., New Y ork. [10] H. Freudenthal (1963) Lie groups in the foundations of geometry . Adv . Math. 1 , 145 [11] A. Kamenshchik, A. Marrani, F . Muscolino (2026) T wo Times for Freudenthal. e-Print: 2603.13067 [hep-th] 6