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.

## REFERENCES

*[Proceedings of the Seventh Rochester Conference on High‐Energy Nuclear Physics 1957*(Interscience Publishers, Inc., New York, 1957), p. IX‐26].

*negation*$A\u2192\u223cA$ and the basic relation of

*implication*$A\u2282B.$ 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. $A\u2229B$ (

*A*and

*B*) and a l.u.b. $A\u222aB$ (

*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 $A\u2282B$ then $|A|<|B|;$ if $A\u2282B$ and $|A|\u2009=\u2009|B|$ then $A\u2009=\u2009B;$ $|O|\u2009=\u20090;$ and $|\u223cA|\u2009=\u2009|I|\u2212|A|.$ 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 $|A|\u2009=\u2009|B|\u2009=\u20091$ then there exists a

*C*with $|C|\u2009=\u20091$ such that $A\u222aB\u2009=\u2009B\u222aC\u2009=\u2009C\u222aA.$ 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 $A\u2229(B\u222aC)\u2009=\u2009(A\u2229B)\u222a(A\u2229C)$ 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.

*q*possesses an inverse $q\u22121.$ In Hamilton’s notation a quaternion is regarded as the sum of a “scalar” (

*real*) part and a “vector” (

*imaginary*) part: $q\u2009=\u2009q0+q\u22c5i.$ 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 $q\u2192qQ$ on the quaternions that is

*involutory*$(qQQ\u2009=\u2009q),$ Hermitian definite ($qQq$ is real, and vanishes only when $q\u2009=\u20090$), and

*anti‐automorphic*$(pQqQ\u2009=\u2009(qp)Q),$ and it is called the

*quaternion conjugate*(𝒬 conjugate): $ikQ\u2009=\u2009\u2212ik.$ On the other hand, the

*automorphisms*of the quaternions are all of the form $q\u2192aqa\u22121.$ (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

*left multiplication*$q\u2192aq,$ 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 $qQ\u2009=\u2009c0*\u2212i2c1$ where

*Q*denotes the quaternion, and the star the ordinary complex, conjugate. The “scalar product” of two quaternions $p,q,$ $(p,q)\u2009=\u2009pQq,$ then becomes (with $p\u2009=\u2009b0+i2b1$) $pQq\u2009=\u2009(b0*c0+b1*c1)+i2(b0c1\u2212b1c0).$ 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.

*General Analysis*(American Philosophical Society, Philadelphia, Pennsylvania, 1935);

*U*on the propositional calculus of $FQM$ is a mapping of

*propositions*to

*propositions*, $U:A\u2192A\u2032\u2009=\u2009AU,$ 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.

*I*and the complex conjugate $C:a\u2192a*;$ 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 $U0\u2009=\u20091$ is linear, all $Ut$ are necessarily linear. For $F\u2009=\u2009Q,$ the automorphisms of ℱ are the conjugations $a\u2192aq\u2009=\u2009qaq\u22121;$ any colinear operator

*T*can be expressed in terms of an associated (nonunique) linear operator

*L*and a quaternion

*q*according to

*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