Werner Heisenberg the lattice theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the application of which has, , led to the discovery of the
Heisenberg group "n"-dimensional systems, and most generally, to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg matrices is given by: Since the multiplication is not commutative, the group is non-abelian. The neutral element of the Heisenberg group is the identity matrix, and inverses are given by The group is a subgroup of the 2-dimensional affine group Aff(2): formula_2 acting on formula_6 corresponds to the affine transform formula_7. There are several prominent examples of the three-dimensional case. If , are real numbers (in the ring R) then one has the continuous Heisenberg group H(R). It is a nilpotent