A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The role played by the new imaginaries is studied. The principal conceptual difficulty concerns the theory of composite systems where the ordinary tensor product fails due to noncommutativity. It is shown that the natural resolution of this difficulty introduces new degrees of freedom similar to isospin and hypercharge. The problem of the Schrödinger equation, ``which i should appear?'' is studied and a generalization of Stone's theorem is used to resolve this problem.

von Neumann
Ann. Math.
The first suggestion of quaternion quantum mechanics appears in a footnote of this paper. C. N. Yang has also pointed out the interest of this possibility [Proceedings of the Seventh Rochester Conference on High‐Energy Nuclear Physics 1957 (Interscience Publishers, Inc., New York, 1957), p. IX‐26].
We can present the propositional calculus of general quantum mechanics as follows, if we consider finite‐dimensional Hilbert spaces only, thus excluding systems with continuous variables except as limiting cases. The elements A,B,C,⋯ (which may indifferently be regarded as representative ensembles, propositions about a physical system, or “operational” rules for testing the truth of statements) are subject to the basic operation of negationA→∼A and the basic relation of implicationA⊂B. In addition, unlike the classical propositional calculus, the propositions make up (are the points of) a topologicai space. The axioms are: 1. The axioms for a complemented lattice. Implication is reflexive, transitive, and antisymmetric (reversible only for equals). Any A,B, possess both a g.l.b. A∩B (A and B) and a l.u.b. A∪B (A or B) with respect to implication. There exists an over‐all g.l.b. O and l.u.b. I. Negation is an involutory anti‐automorphism of the lattice. 2. Axioms of cardinality. To each proposition A may be associated a non‐negative integer |A| such that if A⊂B then |A|<|B|; if A⊂B and |A| = |B| then A = B;|O| = 0; and |∼A| = |I|−|A|. We assume, without loss of generality, that the least of the positive values assumed by this integer is normalized to be 1; otherwise any positive integral multiple of A would also satisfy these requirements. 3. Axiom of superposition. If |A| = |B| = 1 then there exists a C with |C| = 1 such that A∪B = B∪C = C∪A. 4. Axiom of continuity. |A| is a continuous function of A. Evidently Axioms 1 and 2 are valid for the propositional calculus of a classical system with a finite number of states. Axiom 3 is the very essence of quantum logic; the C whose existence it asserts is a superposition of A and B in the quantum sense. Were the classical distributive law of logic A∩(B∪C) = (A∩B)∪(A∩C) adjoined to Axiom 1, Axiom 3 would be inconsistent. Even without continuity (Axiom 4) it would follow from Axioms 1–3 that the propositions correspond one‐to‐one to substances of some vector space over a skew field. Since every subspace of the vector space is utilized in this one‐to‐one realization, there is no room for superselection principles in what we have called general quantum mechanics; but if Axiom 3 is simply dropped we find a “supersum” (direct sum with superselection rules between addends) of systems for each of which Axiom 3 is satisfied. Thus it is not necessary to go beyond the quaternions until Axioms 1, 2, or 4 are weakened. We have consigned this matter to inferior print and omitted much mathematical beauty; it concerns mostly how one arrives at 𝒬 quantum mechanics, and in the final analysis it is more important to know where a theoretical path leads than how one fell upon it.
We give here the algebra of quaternions. Every quaternion can be written in the form
q = q0+q1i1+q2i2+q3i3
, where the four coefficients qk are real. The multiplication of quaternions is associative, distributive, and obeys
ik2 = −1 k = 1,2,3
i1i2i3 = −1
. In the last equation, the anticyclic order of factors might have been taken. Every quaternion q possesses an inverse q−1. In Hamilton’s notation a quaternion is regarded as the sum of a “scalar” (real) part and a “vector” (imaginary) part: q = q0+qi. The quaternions that commute with all other quaternions are just the reals. The quaternions that commute with a given nonreal quaternion form a subset isomorphic to the complex numbers. There exists an operation q→qQ on the quaternions that is involutory(qQQ = q), Hermitian definite (qQq is real, and vanishes only when q = 0), and anti‐automorphic(pQqQ = (qp)Q), and it is called the quaternion conjugate (𝒬 conjugate): ikQ = −ik. On the other hand, the automorphisms of the quaternions are all of the form q→aqa−1. (The quaternion a associated with a particular automorphism is not uniquely defined by this equation; by requiring that the norm of a, meaning aQa, be unity, the ambiguity is reduced to an extremely important matter of sign.) It is sometimes convenient to represent quaternions by pairs of complex numbers (c0,c1) according to
q = c0+i2c1
, where c0,c1 commute with i3, and are therefore essentially complex numbers. Treating these pairs as vectors in a two‐dimensional complex vector space C2, we find that every linear transformation of 𝒬 is represented by a linear transformation of C2, that is by a 2×2 complex matrix. In particular the left multiplicationq→aq, by a fixed quaternion a, is represented by a matrix aij, the symplectic representation of a, The symplectic representations of left multiplication by i1,i2,i3 are just the Pauli spin operators (times i). But the symplectic representation of right multiplications by quaternions are sums of linear and antilinear operators. Computation yields that qQ = c0*−i2c1 where Q denotes the quaternion, and the star the ordinary complex, conjugate. The “scalar product” of two quaternions p,q,(p,q) = pQq, then becomes (with p = b0+i2b1) pQq = (b0*c0+b1*c1)+i2(b0c1−b1c0). We separated the quaternion with respect to i2 and identified i3 with the complex i. But, of course, we could have used any pair of anticommuting units as well.
Usually it is the scalar product that is taken as fundamental, but except with the Dirac notation this leads to a doubling of symbols, and to an ambiguity about which factor is the linear one, which the antilinear. We shall find the Dirac notation extremely convenient for 𝒬 quantum mechanics, since it manages automatically certain rules of order that are not important in 𝒞 quantum mechanics. Equivalent definitions of H(Q) are given by von Neumann and Birkhoff, (reference 3);
E. H. Moore, General Analysis (American Philosophical Society, Philadelphia, Pennsylvania, 1935);
Z. Math.
Again two approaches present themselves, the “synthetic” and the “analytic”; just as the definition of “general quantum mechanics” in footnote 2 is the “synthetic” version of the “analytic” one given in the text of this section for FQM. Again we relegate the “synthetic” formulation to a footnote: An automorphism U on the propositional calculus of FQM is a mapping of propositions to propositions, U:A→A′ = AU, that possesses an inverse and preserves the operation of negation and the relation of implication. It is then a theorem that every such mapping is effected by a mapping of vectors of the kind to be cailed co‐unitary above. Likewise any mapping that preserves implication is represented by a colinear vector transformation.
Why unitary and not simply co‐unitary? Since after all the essential requirements from the point of view of the propositional calculus are that implication and negation of propositions (linear dependence and orthogonality of vectors) be preserved by the passage of time, and this is a property of the co‐unitary operators. The reason the unitary operators are sufficient varies slightly for the three cases F = R,C,Q: For F = R, the only automorphism of ℱ is the identity a→a; all colinear operators are linear, and all co‐unitary operators are unitary. For F = C, the automorphisms of ℱ are the identity I and the complex conjugate C:a→a*; all colinear operators are either linear, or if not, are called antilinear, the two classes being disconnected. Since a one‐parameter group Ut is connected and U0 = 1 is linear, all Ut are necessarily linear. For F = Q, the automorphisms of ℱ are the conjugations a→aq = qaq−1; any colinear operator T can be expressed in terms of an associated (nonunique) linear operator L and a quaternion q according to
TΨ = LΨq (*)
. Now we see that the linear operators (q real) are continuously connected to the other colinear operators (q not real). Thus the continuity argument does not work here. On the other hand the colinear T and the associated linear operator L of (*) define the same correspondence of propositions to propositions (subspaces to subspaces). Therefore, for 𝒬 quantum mechanics every such correspondence, being representable by a colinear operator, is representable by a linear operator. By choosing the q in (*) to be of unit norm, it is readily seen, L is determined up to sign and is unitary if T is co‐unitary. We thus obtain a unitary function of time Ut obeying
UtUt′ = ±Ut+t′
. By continuity, it is always possible to redefine Ut so that the upper sign is chosen.
Many of the properties of a system of three anticommuting, anti‐Hermitian unitary operators on H(Q) like i1,i2,i3 have been discussed by Teichmüller, reference 4.
F. Gürsey (private communication).
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