Adv. Appl. Clifford Algebras (2018) 28:81 c  2018 Springer Nature Switzerland AG https://doi.org/10.1007/s00006-018-0894-3 Advances in Applied Clifford Algebras Quantum Symmetries: From Clifford and Hurwitz Algebras to M-Theory and Leech Lattices Sultan Catto ∗ , Yasemin G¨ urcan, Amish Khalfan and Levent Kurt Abstract. We explore some consequences of a theory of internal symme- tries for elementary particles based on exceptional quantum mechani- cal spaces based on Jordan algebra formulation that admit exceptional groups as gauge groups. 1. Preface We explore some consequences of a theory of internal symmetries for elemen- tary particles based on exceptional quantum mechanical spaces based on Jor- dan algebra formulation that admit exceptional groups as gauge groups. We then move onto showing the connection between the lattices of certain groups that arise in superstring theories and discrete elements of Jordan algebras as- sociated with the magic square. Such correspondence suggests a profound role that the division algebras and exceptional Jordan algebras are likely to play in superstring theories and their compactification. It also points to the exis- tence of a larger theory connected with a chiral 27-dimensional lattice that generalizes Conway’s lattice and unifies all known superstring theories. 2. Introduction There exist different formulations of quantum mechanics, such as those of Er- win Schrodinger, Werner Heisenberg, and Paul Dirac. Another formulation of quantum mechanics, which is often glossed over, is that due to Pascual This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie. Work supported in part by DOE contracts no. DE-AC-0276-ER 03074 and 03075; NSF Grant no. DMS-8917754. * Corresponding author. 0123456789().: V,-vol