Response to the Comment by G. Emch on Projective Group Representations in Quaternionic Hilbert Space

We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embeddings, and leads to a detailed theory of quaternionic, as well as complex, projective group representations. I. PRELIMINARIES NOT INVOLVING GROUP STRUCTURE Before turning to a discussion of what is an appropriate definition of a quaternionic projective group representation, we first address several issues that do not involve the notion of a group of symmetries. We follow throughout the Dirac notation used in our recent book [1], in which linear operators in Hilbert space act on ket states from the left and on bra states from the right, as in O|f〉 and 〈f |O, while quaternionic scalars in Hilbert space act on ket states from the right and on bra states from the left, as in |f〉ω and ω〈f |. 1 We begin by recalling the statement (see Sec. 2.3 of Ref. [1]) of the quaternionic extension of Wigner’s theorem, which gives the Hilbert space representation of an individual symmetry in quantum mechanics. Physical states in quaternionic quantum mechanics are in one-to-one correspondence with unit rays of the form |f〉 = {|f〉ω}, with |f〉 a unit normalized Hilbert space vector and ω a quaternionic phase of unit magnitude. A symmetry operation S is a mapping of the unit rays |f〉 onto images |f 〉, which preserves all transition probabilities, S|f〉 = |f 〉 |〈f |g〉| = |〈f |g〉|. (1) Wigner’s theorem, as extended to quaternionic Hilbert space, asserts that by an appropriate S-dependent choice of ray representatives for the states, the mapping S can always be represented (in Hilbert spaces of dimension greater than 2) by a unitary transformation US on the state vectors, so that |f 〉 = US |f〉 . (2) Conversely, any unitary transformation of the form of Eq. (2) clearly implies the preservation of transition probabilities, as in Eq. (1). When only one symmetry transformation is involved, the issue of projective representations does not enter, since Wigner’s theorem asserts that this transformation can be given a unitary representation on appropriate ray representative states in Hilbert space. The issue of projective representations arises only when we are dealing with two (or more) symmetry transformations, in which case the ray representative choices which reduce the first symmetry transformation to unitary form may not be compatible with the ray representative choices which reduces a second symmetry transformation to unitary form. Thus we disagree with Emch’s statement, in the semifinal paragraph of his Comment, that Wigner’s theorem (which he notes is a form of the first fundamental theorem of projective geometry) may be dependent on the definition adopted for quaternionic projective group representations. In the first section of his Comment, Emch proves a Proposition stating that if an operator O commutes with all of the projectors |f〉〈f | of a quaternionic Hilbert space of dimension 2 2 or greater, then O must be a real multiple of the unit operator 1 in Hilbert space. When O is further restricted to be a unitary operator (as obtained from a symmetry transformation via the Wigner theorem), the real multiple is further restricted to be ±1. Since we will refer to this result in the next section, let us give an alternative proof, based on the spectral representation of a general unitary operator U in quaternionic Hilbert space,