only as tensors but also as spinors. If one adds to this a classification by parity, these can be extended, for example, into notions of scalars () and pseudoscalars () which are rotationally invariant. vectors () and axial vectors (also called pseudovectors) () which both transform as vectors under rotation. One can define reflections such as which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing x-, y-, and z-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used. Parity forms the abelian group due to the relation . All Abelian groups have only one-dimensional irreducible representations. For , there are two irreducible representations: one is even under parity, , the other is odd, . These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase. Representations of O(3) An alternative way to write the above classification of scalars, pseudoscalars, vectors and pseudovectors is in terms of the representation space that each object transforms in. This can be given in terms of the group homomorphism which defines the representation. For a matrix scalars: , the trivial representation pseudoscalars: vectors: , the fundamental representation pseudovectors: When the representation is restricted to , scalars and pseudoscalars transform identically, as do vectors and pseudovectors. Classical mechanics Newton's equation of motion (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity. However, angular momentum is an axial vector, In classical electrodynamics, the charge density is a scalar, the electric field, , and current are vectors, but the magnetic field, is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector. Effect of spatial inversion on some variables of classical physics The two major divisions of classical physical variables have either even or odd parity. The way into which particular variables