Unitary operations

This formula can be used directly, but to compare with more standard perturbation theory, we can use, for instance, the unitary operator (Kato 1980, p.99)... 

In this equation we transform both tf> and tft by a unitary operator U, which represents a rotation whose matrix is R (R acts on c), obtaining... 

Equivalently, this unitary operator S can be defined by the relation... 

If the hamiltonians H(0) and H0(0) are such that there exist no bound states, and the states F )+ are properly normalized, l(+) is a unitary operator. Furthermore, it has the property that... 

The representation of these commutation rules is again fixed by the requirement that there exist no-particle states 0>out and 0>ln. The -matrix is defined as the unitary operator which relates the in and out fields ... 

Such a unitary operator 8 must exist since tfiia and rout (and s inil and form equivalent representation of the commutation rules (11-60) and (11-65). Explicitly it can be computed as follows... 

In the Heisenberg-type description the existence of such a unitary or anti-unitary operator U is inferred from the fact that the set of observable Q and Q satisfy the same commutation rules. 

Let us next adopt the Schrodinger-type description. The statement that quantum electrodynamics is invariant under space inversion can now be translated into the statement that there exists a unitary operator U(it) such that... 

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... 

The transformation is induced by the unitary operator U(a,l) with the properly that... 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

If we now define the unitary operator Ue in accordance with (11-298) and (11-299) above, then... 

Assume that there exists a unitary operator U(it) which maps the Heisenberg operator Q(t) at time t into the operator (—<). Assume further that this mapping has the property of leaving the hamiltonian invariant, i.e., that U(it)SU(it)" 1 = H. Consider then the equation satisfied by the transformed operator... 

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... 

That is, unitary operators are linear and anti-unitary operators are antilinear. 

The time reversal operator may now be determined.6 The simplest anti-unitary operation is the transition to the complex conjugate. This operator K is clearly antilinear and anti-unitary. 

Furthermore, any antilinear operator can be written as the product of a unitary operator and the operator K. Specifically, we can write the time reversal operator as 6 = UK, and our problem is now that of... 

Case (c).- a0< produces a set of functions , which is independent of the set la, which forms a basis for the irreducible representation Ay(u) of H, which is inequivalent to A (u), but which has the same dimension. Df corresponds to two inequivalent irreducible representations of H, A (u), and A (u), such that in this case the anti-unitary operators cause A (u) and A (u) to become degenerate. 

In this case the anti-unitary operator a0 may be chosen to be (E a)8. Consider how this operator transforms the basis functions. 

Now, consider the case of spinless particles not subject to external electronic and magnetic fields. We may now choose the unitary operator U as the unit operator, that is, T = K. For the coordinate and momentum operators, one then obtains... 

Transformations that take one orthonormal set of basis vectors into another orthonormal set are called unitary transformations, the operators associated with them are called unitary operators. This definition preserves the norms and scalar products of vectors in Ln. The transformation (4) is in fact a set of linear equations... 

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

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

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