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In mathematics, a Clifford–Klein form is a double coset space Γ\G/H, where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G/H. A suitable discrete subgroup Γ may or may not exist, for a given G and H.
The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R p,q. The Clifford algebra on R p,q is denoted Cl p,q (R). A standard orthonormal basis {e i} for R p,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + = ,where the curly brackets {,} represent the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.
The Clifford gates do not form a universal set of quantum gates as some gates outside the Clifford group cannot be arbitrarily approximated with a finite set of operations. An example is the phase shift gate (historically known as the π / 8 {\displaystyle \pi /8} gate):
In mathematical physics, the Dirac algebra is the Clifford algebra, ().This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- 1 / 2 particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.
In Euclidean space the Dirac operator has the form = = where e 1, ..., e n is an orthonormal basis for R n, and R n is considered to be embedded in a complex Clifford algebra, Cl n (C) so that e j 2 = −1.
Clifford's theorem states that for an effective special divisor D, one has: 2 ( ℓ ( D ) − 1 ) ≤ d {\displaystyle 2(\ell (D)-1)\leq d} , and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.