Quaternionic quantum theory admits universal dynamics only for two-level systems

We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schr\"odinger equation is the only possible choice, up to a shift of the global phase.


I. INTRODUCTION
Our understanding of quantum theory has significantly improved by investigating alternatives to quantum theory and analyzing how these alternatives would or would not be at variance with observations or expectations on the structure of a physical theory. Recently, these investigations are mostly based on the formalism of generalized probabilistic theories, where the fundamental objects are the convex sets of states and measurements. Different sets of assumptions have been found which are sufficient to single out quantum theory as the only possible theory [1][2][3][4][5][6][7] . A special role in these set of assumptions plays the analysis of the dynamics of such generalizations of quantum theory, see, for example, Ref. 8. Specifically, in quantum mechanics (and also in classical mechanics) there is an intimate relation between the Hamiltonian H as the energy observable and the generator of time shifts − i H as it occurs in the Schrödinger equation.
Maybe the most notable early alternatives to quantum theory that have been studied in great detail are real and quaternionic quantum mechanics. Those are based on the question, why the wave function in quantum theory is complex valued and whether it would also be possible to formulate quantum theory over different fields, in those cases using real valued or quaternionic valued wave functions. The two main concerns for the real and quaternionic case are the composition of systems via a tensor product and a suitable modification of the Schrödinger equitation. For real quantum theory, both topics lead to basically the same conclusion, namely that there must be a superselection rule [9][10][11][12] . In quaternionic quantum theory, the tensor product is complicated, at best [13][14][15][16] . However, the need for composing systems has also been questioned recently 17 . A consistent dynamics in quaternionic quantum theory has been formulated, however, also at the price of a superselection rule, where only a subspace of all self-adjoint operators can be used as the Hamiltonian of a system 18 . Nonetheless, Peres 19 suggested a possible experiment on the basis of noncommuting phases which could reveal the characteristics of quaternionic quantum theory. Corresponding experiments were realized using neutron interferometry 20 and using single photon interferometry [21][22][23] .
In this paper we use the standard quaternionic formulation 24 of states and observables, that is, states are normalized vectors and observables are self-adjoint matrices over the quaternions. We ask which dynamical evolution is admissible in this case. For canonical quantum theory, the Schrödinger equation implies that the state evolves according to a unitary group parametrized by time. Each such group is determined by the Hamiltonian of the system. We seek for a similar construction with the aim to derive a Schrödingertype equation for quaternionic quantum theory. In contrast to previous work 18, 24 , we are interested in the case where the set of Hamiltonians is unrestricted, that is, every self-adjoint operator must induce some dynamics. We find that this is only possible for one-level or two-level systems and that the corresponding Schrödinger-type equation is necessarily of the form where H is the Hamiltonian and A is a skew-adjoint operator which is independent of H. The term in the square brackets replaces here −iH from the canonical Schrödinger equation. We arrive at this result assuming that the term in the square brackets is an R-linear expression in H and that it commutes with H. The paper is organized as follows. In Section II we consider the case of canonical quantum theory. We review the connection between the Schrödinger equation, generators of time shifts, and Stone's theorem and develop then the axioms for the correspondence between the Hamiltonian and the generator of time shifts. In Theorem 2 we establish that these axioms are sufficient to reproduce the usual Schrödinger equation. In Section III we then turn to the quaternionic case. We first summarize quantum theory over the quaternions. In Theorem 5 we characterize the possible dynamics in quaternionic quantum theory and subsequently discuss alternatives to the axioms that lead to Eq. (1) before we conclude in Section IV.

II. TIME EVOLUTION IN CANONICAL QUANTUM THEORY
In quantum mechanics, the time evolution of a system is described by the Schrödinger equation, For a time-independent Hamiltonian this gives rise to the unitary time evolution operator which provides a solution of the Schrödinger equation via ψ(t 0 + t) = U S t ψ(t 0 ).

A. Unitary groups and Stone's theorem
There is a different way to obtain a time evolution operator U t of the same structure, without building on the Schrödinger equation. This is based on the assumptions that the transformation U t : ψ(t 0 ) → ψ(t 0 + t) is linear in ψ(t 0 ), preserves the norm of ψ(t 0 ), is independent of t 0 , and is continuous in t. More precisely, (U t ) t∈R must be a strongly continuous unitary group, that is, Condition (i) expresses linearity and isometry and U 0 = 1 is used to implement the identity ψ(t 0 ) = U 0 ψ(t 0 + 0). (We do not consider antiunitary transformations as allowed by Wigner's theorem.) Condition (ii) follows from the independence of t 0 and condition (iii) is a specification of the assumption of continuity. Strong continuity refers to the strong operator topology and reduces because of (i) and (ii) to lim t→0 U t ψ − ψ = 0 for all ψ. The fundamental representation theorem of strongly continuous one-parameter unitary groups is due to Stone (see, for example, Ref. 25). For every such group (U t ) t∈R there exists a unique self-adjoint operator G, such that holds. This result is valid for general Hilbert spaces, with subtleties occurring if t → U t ψ is not differentiable at t = 0 for some ψ. Clearly, in the finite-dimensional case this function is always differentiable.

B. Hamiltonians as generators of time shifts
From a physical perspective, the skew-adjoint operator −iG in Eq. (4) is responsible for time shifts and in this sense it is the generator of time shifts. The Schrödinger equation implies that the correspondence between the Hamiltonian and the generator of time shifts is obtained as −iG = − i H. But is this the only way to establish a correspondence between the generator of time shifts and the Hamiltonian and if not, how can we classify the different possibilities?
To answer this question, we write the Schrödinger equation in the forṁ yielding the time evolution operator U S t = e ΦS (H)t . Here Φ S (H) is the generator of time shifts associated to the Hamiltonian H, that is Φ S is a linear map from the self-adjoint matrices to the skew-adjoint matrices, We denote by H(C n ) [A(n, C)] the R-vector space of self-adjoint (skew-adjoint) complex n × n matrices. The skew-adjoint matrices obey A = −A † and together with the self-adjoint matrices, which satisfy H = H † , they span the set of all matrices, that is, The dimension of H(n, C) equals the dimension of A(n, C) and Φ S is a one-to-one mapping between these two vector spaces. This coincidence of dimensions is a peculiarity of the complex matrices. Over the reals as well as over the quaternions H, the dimensions differ, specifically, dim H(n, R) = dim A(n, R) + n and dim H(n, H) = dim A(n, H) − 2n.
In order to generalize Φ S , we consider an arbitrary relation ϕ ⊂ H(n, C) × A(n, C) between the Hamiltonians and the generators of time shifts. In literature, ϕ is sometimes called a dynamical correspondence 8,26,27 . We add several restrictions to this general case by requiring that ϕ is These assumptions have physical motivations and appeared earlier in literature, see, for example, Refs. 8 and 27. The relation ϕ should be R-homogeneous (DC1) to match the intuition that higher energies correspond in direct proportion to faster time evolutions and vice versa. The assumption of additivity (DC2) can be easily justified for the case where H and H ′ commute since in this case we have the addition of two Hamiltonians that differ only in spectrum. Then for any eigenstate of H and H ′ an argument similar to the motivation for assumption (DC1) can be applied. Noncommuting Hamiltonians most prominently appear in the form of interaction Hamiltonians, where H = H A +H B +µH I . In many situations the interaction strength µ is an external experimental parameter, for example, the strength of a magnetic field, and hence additivity is a reasonable assumption. This reasoning might be more difficult to apply, if the Hamiltonian does not emerge as an effective description, but is an unalterable property of the system. Finally, the relation ϕ should also be commuting (DC3), so that the Hamiltonian is invariant under time shifts. That is, the observable H should be a conserved quantity under its own time evolution, It is conceivable that the Hamiltonian alone does not determine the time evolution completely and that the map The set ϕ 1 (H) could also be empty, in which case H would not correspond to any dynamics, that is, H would be unphysical. Conversely, the same dynamics might arise from different Hamiltonians so that Also the set ϕ 2 (A) could be empty and hence the corresponding time evolution t → U t would be unphysical. In this paper, we focus on the first case and we only consider maps Φ : The latter condition is not a restriction, since we can always obtain ϕ from a family of maps The conditions (DC1)-(DC3) are equivalent to the condition that Φ is a commuting Rlinear map, that is, Φ is R-linear with HΦ(H) = Φ(H)H. We now demonstrate that such maps must have a very specific form. For this we use the following result by Brešar. Here, Z(A) denotes the center of A, that is, the elements of A that commute with all of A. Since Mat(n, C) is a simple unital ring with center C1, we can in principle apply Lemma 1 in order to characterize all maps Φ. However, we first need to extend the domain of Φ to be all of Mat(n, C). This is readily achieved by the canonical extension Φ c of Φ via where X = M 1 + iM 2 is the unique decomposition of X ∈ Mat(n, C) into its self-adjoint and skew-adjoint part and where λ ∈ R and B ∈ H(n, C).
Note that Φ(H) is covariant if and only if B is real multiple of 1. In quantum mechanics, we have λ = − 1 and B = 0. The value of λ constitutes a constant of nature, including a sign convention. The term involving B cannot be measured on a single system since it would only cause a global phase shift on the quantum state. In an interferometer-type setup even this phase shift would be accessible but is in contradiction to observation.

III. QUATERNIONIC QUANTUM THEORY
We have outlined the formalism for obtaining the Schrödinger equation for the familiar case of complex quantum mechanics. Our choices and assumptions have been made such that we can extend our considerations to construct a dynamical quantum theory over the quaternions. We start by summarizing a quaternionic version of quantum theory (see, for example, Ref. 24) and we then proceed by characterizing possible expressions for the Schrödinger equation.

A. The quaternions
The quaternions H are an extension of the real and complex numbers. They form an associative division ring where multiplication is noncommutative. Any quaternion q ∈ H can be written in the form where the coefficients a ℓ are real numbers and i, j, k are the quaternion units, which play a role similar to the complex unit i. The real part of q is Re(q) = a 1 and the imaginary part is the triple Im(q) = (a 2 , a 3 , a 4 ). The multiplication on H is commutative for the real numbers and otherwise determined by yielding ij = k = −ji, jk = i = −kj, ki = j = −ik. Similar to the complex numbers, conjugation is defined by yielding the rules (uv) * = v * u * , qq * = q * q, and (q * ) * = q. The modulus |q| = √ qq * induces the euclidean norm (a 1 , a 2 , a 3 , a 4 ) = |a 1 + a 2 i + a 3 j + a 4 k|. This way, the quaternions are a complete normed R-algebra.
We identify the complex numbers as a subset of the quaternions by identifying the complex unit i with the quaternion unit i. This allows us to write uniquely q = a + bj for a, b ∈ C. Similar to the representation of complex numbers as real 2 × 2 matrices, the quaternions can be represented as complex matrices, This map is a †-monomorphism of the corresponding real algebras, were the involution † reduces on the quaternions to the conjugation * .

B. Modules and matrices
Since quantum mechanics is formulated on the basis of complex Hilbert spaces, we use a similar structure over the quaternions, but taking into account the the noncommutativity of the quaternions. We consider here the n-fold direct product of quaternions, denoted by H n . It forms a free bimodule and possesses, apart from commutativity, most properties of a vector space. In particular, since it arises from a direct product, it can be equipped with the canonical basis (e (1) , e (2) , . . . , e (n) ).
For x, y ∈ H n with x = i x i e (i) and y similar we define the inner product to be of the canonical form, x, y = i x * i y i = y, x * , giving also rise to the norm x = x, x and turning H n into a Hilbert module. For scalar multiplication with α ∈ H we obtain the rules xα, y = α * x, y and x, yα = x, y α. This suggests that scalar multiplication in H n is preferably taken from the right, although, technically, H n is a bimodule.
We take linear maps M : H n → H m to be right-homogeneous, M (xα) = M (x)α which allows for a representation of M as an m × n matrix (M i,j ) i,j via M i,j = e (i) , M e (j) . Then x, M (yα) = i,j x * i M i,j y j α. We consider Mat(n, H) as an R-algebra, ignoring that it can be treated consistently as a left H-module. Where unambiguous, we use α ∈ H also as the linear map x → i αx i e (i) . The adjoint † of a linear map is defined as usual, x, M (y) = M † (x), y , and therefore (M † ) i,j = (M * j,i ) i,j . Since linear maps are wellrepresented by matrices, we mostly use the latter notion. Self-adjoint and skew-adjoint matrices are defined in the obvious way. For unitary matrices U ∈ Mat(n, H), we note that It is sometimes convenient to use the embedding Λ : Mat(n, H) → Mat(2n, C), This map is a †-monomorphism of the corresponding R-algebras. In particular, we have Λ[(rA + BC † )] = rΛ(A) + Λ(B)Λ(C) † for r ∈ R and A, B, C ∈ Mat(n, H). The map Λ −1 : Mat(2n, C) → Mat(n, H), is an R-linear left inverse of Λ, which, however, it is not preserving the algebraic properties of Mat(2n, C).

C. Stone's theorem
In order to study the dynamics in quaternionic quantum theory we proceed similar to the complex case by studying continuous unitary groups (U t ) t . Then, an analogous result to Stone's theorem holds. This theorem has been proved in Ref. 18. For completeness, we provide here a proof for finite dimensions.
Proof of Theorem 3. Since the embedding Λ is a mapping between finite-dimensional vector spaces, the family [Λ(U t )] t ⊂ Mat(2n, C) is also a continuous unitary group. By virtue of Stone's theorem there exists a skew-adjoint matrix B ∈ Mat(2n, C) such that Λ(U t ) = e Bt . The map W : t → Λ(U t ) is also differentiable, so that we can write for U : t → U t , The left hand side exists, proving that also U is differentiable. By letting A =U(0) and using U t+δ = U δ U t , we haveU It remains to show that A is skew-adjoint and satisfies U t = e At . Since Λ(U t ) is unitary, the identity Λ(A)Λ(U) = Λ(AU) = Λ(U) = d dt Λ(U) =Ẇ = BW = BΛ(U), allows us to conclude that Λ(A) = B. This implies that A is skew-adjoint, since B is. Finally, applying from the left gives us U t = e At . Uniqueness then follows immediately fromU(0) = A.
We mention that the smoothness of the map t → U t , which we show in the first part of the above proof, is a simple consequence of the fact that the unitary matrices form a Lie group. Indeed, for any Lie group G a continuous homomorphism R → G is necessarily smooth 30 . Also note that in contrast to the complex case, we consider here only the finite-dimensional case and therefore it is not necessary to use strong continuity.

D. Observables and generators of time shifts
We now head for the characterization of the dynamics in a quaternionic version of quantum theory. So far we have obtained a result about the structure of all possible dynamical evolutions. But for a dynamical evolution to be useful we need to specify a notion of states and observables. Here, we proceed in complete analogy to quantum mechanics, that is, states are represented by normalized vectors and observables by self-adjoint matrices 24 . The expectation value of an observable H for a system in state ψ is then given by H = ψ, Hψ . Clearly, the expectation value is always real and all states ψα are equivalent for all α ∈ H with |α| = 1. The spectral theorem for self-adjoint matrices can also be written as H = k h k Π k , with distinct eigenvalues h k ∈ R and self-adjoint projections Π k such that k Π k = 1. The expectation values of the projections correspond then to the probability p k for observing the eigenvalue h k , that is, p k = Π k . This way we recover a large bit of the structure and physical interpretation of quantum theory.
With the same arguments as in the complex case, we assume that the time evolution of a state is generated by a continuous unitary group (U t ) t and by virtue of Theorem 3 we have U t = e At . It is worth mentioning here a significant difference 24 to the complex case, which occurs if we add a global, time-dependent phase to a state, ψ(t) → ψ α (t) = ψ(t)α(t). We obtainψ α (t) = Aψ α (t) + ψ α (t)ϕ(t) where ϕ(t) = α * (t)α(t). Since ϕ(t) in general does not commute with ψ α , we cannot simply writeψ α (t) = (A + ϕ(t))ψ α (t), as it would be the case in the complex case.
In analogy to the complex case we are interested in the correspondence between observables and the generators of time shifts, in particular for the case where the observable is the Hamiltonian of the system. In the complex case, the multiplication with a purely imaginary number iλ is the right choice to establish this correspondence. The matrix A in Theorem 3 can written in the polar decomposition as 18 A = −XH, where X is unitary and skew-adjoint, H is self-adjoint and positive semidefinite, and [X, H] = 0 holds. It is conceivable to identify H in this decomposition with the Hamiltonian of the system while X is kept constant. This limits the possible set of Hamiltonians to those with [X, H] = 0 which basically reduces quaternionic quantum theory to complex quantum theory 18 .
Here, we are interested in the case where the Hamiltonian of the system can be any selfadjoint operator. The discussion in Section II B remains valid and leaves us with the task to characterize the commuting R-linear maps Φ : H(n, H) → A(n, H). However, we cannot proceed similar to above to obtain a result akin to Theorem 2. The main difficulty here is that it is not possible to use an extension of Φ as in Section II B, since such an extended map would be no longer commuting. We hence resort to a case-by-case study for different dimensions n.
For this it is useful to note that determining all admissible maps Φ can be reduced to finding the kernel of an R-linear map. Indeed, Φ is commuting if and only if the R-bilinear Here, sufficiency follows immediately for Y = X and necessity from [Φ(X + Y ), X + Y ] = 0. Thus, the set of commuting maps Φ is determined by the kernel of the R-linear map Φ → Q Φ . We perform this calculation with the help of a computer algebra system for n = 1, 2, 3 with the following results. For n = 1, all admissible maps are (obviously) given by Φ 1 (H) = αH for any α ∈ H with Re(α) = 0. For n = 2, all admissible maps are of the form Φ 2 (H) = AH + HA − tr(H)A, where A ∈ A(2, H) is arbitrary. For d = 3, only the trivial map Φ = 0 is commuting. This implies also that for d > 3 all commuting maps must be trivial, as it follows from the following contradiction. Proof. The real span of the matrices of the form (u i u * j ) i,j with u ∈ H n is H(n, H). Hence we can choose linearly independent vectors x, y, z ∈ H n such that y, Φ(X)z = 0 with Then there is an isometry τ : H 3 → H n such that x, y, z ∈ τ H 3 and π = τ τ † acts as identity on x, y, and z. Such an isometry can be constructed by means of the Gram-Schmidt procedure, yielding orthonormal vectors which are then used as columns of τ . We define the map Φ 3 as Φ 3 : H → τ † Φ(τ Hτ † )τ . By construction, this map is R-linear and maps self-adjoint matrices to skew-adjoint matrices. Since Φ is commuting, we have [Φ(τ Hτ † ), τ Hτ † ] = 0 for any H ∈ H(3, H) and by multiplication with τ † from the left and τ from the right, it follows that Φ 3 is also commuting. Finally, Φ is nontrivial for X ′ = τ † Xτ , y ′ = τ † y, and z ′ = τ † z in that we have y ′ , Φ 3 (X ′ )z ′ = πy, Φ(πXπ)πz = y, Φ(X)z = 0, where πXπ = X follows from πXπv = πx x, πv = x πx, v = Xv for all v ∈ H n .
In summary we have the following characterization.
where A ∈ A(n, H). Conversely, for n = 1, 2 every map of this form is commuting. For n > 2 every commuting R-linear map is trivial, Φ = 0.
For a two-level system, n = 2, the rank of Φ can be at most dim[H(n, H)] − 1 = 5 due to Φ(1) = 0, for any choice of A. The maximal rank is achieved, for example, for A being the diagonal matrix with diagonal (i, 0). A more intuitive choice for A might be − i 2 , which yields the quaternionic Schrödinger equation where H c = 1 2 (H + iHi * ) is the complex part of the matrix H. Hence the corresponding map Φ has an R-rank of only 3.
Theorem 5 has been obtained using the conditions (DC1)-(DC3). In contrast to the complex case, the resulting map Φ only obeys the covariance condition (DC4) if it is trivial. This can be seen by defining Covariance requires then Φ V = Φ for any unitary V . Without loss of generality, we can choose A = iD with D a real diagonal matrix and then the case V = j leads to Φ V = −Φ.
However, the no-go statement in Theorem 5 can be avoided by loosening our assumptions. For example, one can drop the assumption of additivity (DC2). Then a rather natural such candidate can be achieved as follows. We fix a spectral decomposition H = U † H D H U H for every H such that U rH is independent of r ∈ R. This allows us to define the R-homogeneous map Ψ : H(n, H) → A(n, H) as which can be easily seen to be commuting, but, in general, fails to be additive, due to Theorem 5. We can even satisfy the covariance condition (DC4) by requiring that U V HV † = V U H for all unitaries V . As mentioned before, another way to evade Theorem 5 is to limit the set of Hamiltonians to be purely complex 18 , [H, i] = 0. Then Φ S : H → − i H is admissible under (DC1)-(DC3) and we obtain the usual Schrödinger equation also in the quaternionic case.

IV. CONCLUSIONS
We studied the structure of universal dynamics in quantum theory using three main axioms (DC1)-(DC3). These axioms proved to be sufficient in order to recover the Schrödinger equation for the case of canonical quantum theory but when applied to quaternionic quantum theory they yield nontrivial dynamics only for dimension two (and one). For two-level systems, the resulting Schrödinger equation is not unique but can be modified by the choice of a skew-adjoint operator, see Eq. (1).
This makes quaternionic quantum theory for two-level systems exceptionally interesting. For higher dimensions, a possible conclusion from our analysis is to discard quaternionic quantum theory. However, it should be noted that the main reason for our no-go result is axiom (DC2), which requires that the sum of two Hamiltonian should correspond to the sum of two generators of time shifts. While this axiom is natural at least in canonical quantum theory it is an open question whether it is expendable nonetheless, and what this would imply for canonical quantum theory as well as quaternionic quantum theory.