Unitary group

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

We shall only mention the fact, that a unitary representation of the inhomogeneous proper Lorentz group is exhibited in this Hilbert space through the following identification of the generators of the... 

It can actually be shown that the factor to can be chosen to be l.s Furthermore, for restricted Lorentz transformations Via, A) must be unitary since every element of the group a, A can be written as the product of elements that are the same... 

We have noted that the unitary operators U(a,A) define a representation of the inhomogeneous group. If we denote by P and AT, the (hermitian) generators for infinitesimal translations and Lorentz transformations respectively, then... 

Structure of Nonunitary Groups.—Consider the group G, which contains both unitary and anti-unitary operators. These operators will be denoted by u and a respectively. Further it is convenient to write the anti-unitary operators as a = v8 where v is unitary and 0 is anti-unitary. No loss of generality results from our identification of 8 with the operation of time reversal. It can be shown 5 that the product of two unitary operators is unitary, the product of two anti-unitary operators is also unitary, and the product of an anti-unitary operator and a unitary operator is anti-unitary. Consequently nonunitary groups contain equal numbers of unitary and anti-unitary operators, and... 

Representation theory for nonunitary groups.—Before proceeding we should consider what is meant by a unitary and an anti-unitary operator.5 -6 If the hamiltonian of a system commutes with the operators u and a of the group 0, and T and O are state functions of the system, u is unitary if... 

D(l)(u) but B(1)(a) = to 2D<0(a) such that a common arbitrary phase factor remains in the co-representation matrices D(0(a) for the anti-unitary elements of nonunitary groups. 

In order to obtain explicit forms for the co-representation matrices D1 of the nonunitary group G, in terms of the representation matrices A (u) of the unitary subgroup H, it is necessary to know how the set of functions atransforms under the operations of H. Let a0 = iffa, then... 

We will not be concerned further with the explicit forms of the co-representation matrices. Instead we need ask only to which of the three cases a specific representation A (u) of the group H belongs when H is considered as a subgroup of O. The co-representation matrices can be written down immediately once this is known. The irreducible representations of H can be obtained by standard means since H is unitary. It, therefore, remains to obtain a method by which one can decide between the three cases given the group 0 and an irreducible representation of H.9 In order to do this we need the fact that the matrices / and A (u) may be assumed to be unitary,6 and that the A((u) matrices satisfy the usual orthogonality relation... 

Lorentz group, inhomogeneous proper, unitary representation in Hilbert space, 497... 

A representation of the Lie group will be unitary if the operators R(t) are unitary ... 

Thus unitary operators for the group are associated with anti-Hermitian operators for the Lie algebra. Replacing P — iP, gives P = P ... 

The set of all such transformations constitutes the group U(2) which is isomorphic to the group of all unitary matrices of order 2. It is a 4 parameter, continuous, connected, compact, Lie group. The subgroup of U(2) which contains all the unitary matrices of order two with determinant +1, is the set of matrices whose general element is... 

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

This result amounts to a 1 to 2 homomorphic mapping of the unitary group SU(2) onto the rotation group. From (28) it follows that the two unitary matrices... 

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