Constraints on a Generalization of Geometric Quantum Mechanics from Neutrino and $B^0$-$\overline{B^0}$ Oscillations

Nambu Quantum Mechanics, proposed in Phys. Lett. B536, 305 (2002), is a deformation of canonical Quantum Mechanics in which the manifold over which the"phase"of an energy eigenstate time evolves is modified. This generalization affects oscillation and interference phenomena through the introduction of two deformation parameters that quantify the extent of deviation from canonical Quantum Mechanics. In this paper, we constrain these parameters utilizing atmospheric neutrino oscillation data, and $B^0$-$\overline{B^0}$ oscillation data from Belle. Surprisingly, the bound from atmospheric neutrinos is stronger than the bound from Belle. Various features of Nambu Quantum Mechanics are also discussed.


I. INTRODUCTION
In a previous letter [1], we looked at Nambu Quantum Mechanics proposed in [2] by Minic and Tze, which was inspired by an idea of Nambu [3] 1 , and argued that it could lead to observable consequences in oscillation phenomena.In this paper, we provide details of the derivation, expound on the formalism, and place further bounds on the deformation parameters introduced therein.
The objective of this work is to obtain a deeper understanding of canonical Quantum Mechanics (QM).In particular, we would like to demystify what its foundational principles are.To this end, we attempt to generalize QM, compare how the predictions of theory will change with respect to those of canonical QM as a consequence of the generalization, and confront both with experiment.This approach will allow us to probe how robust the original tenet or axiom that was relaxed to generalize QM is, and thereby identify the bedrock on which canonical QM rests.
Generalization or modification of canonical QM has been attempted in several distinct directions.One route has been the replacement of the field over which the state space is defined from complex numbers to other fields or division algebras, i.e. real numbers, quaternions, or octonions, thereby informing us how crucial the complex structure of Hilbert space is to the foundations of QM.For a formulation of QM on real vector spaces, see [8].Quaternionic QM is an active field of research and a well-developed formalism can be found in the literature, particularly in the works of Adler [9,10].For a discussion of QM based on the non-associative octonionic algebra see, for instance, [11].Formulation of QM utilizing vector spaces over finite fields is also possible: See [12][13][14][15][16] for discrete models of QM over Galois fields.
Another direction for generalizing canonical QM is relaxing the linearity of the Schrödinger equation.Suggestions for testing the linearity of canonical QM exist in the literature, see for instance [17] and [18] by Weinberg.However, this formulation allows for superluminal communication [19].In general, the addition of any non-linear terms to the equations of canonical QM is strongly constrained by experiments.For example, logarithmic additions to the Schrödinger equation [20] are tightly constrained by neutron interferometry [21].
Yet another research program that generalizes canonical QM is the study of generalized probabilistic theories [22][23][24][25][26].These theories contain canonical QM as a special case and share its non-classical features.In general, this research program includes the study of a broad class of probabilistic theories and the identification of their common operational features.One can then hope to single out the mathematical structure of canonical QM as a special case resulting from some physically motivated axioms, attempted, for instance, by Hardy [22].An example of such a theory that is relevant to our work is Sorkin's hierarchical classification of probabilistic theories according to the order of interference they exhibit [27,28].In this scheme, both classical probability theory and quantum theory can be seen as special cases of more general probabilistic theories exhibiting interference of orders higher than that of canonical QM.Namely, classical theory is that subset in which there is no interference and canonical QM is characterized by the absence of any interference higher than second order.
Nambu QM generalizes canonical QM via the geometric formulation of QM [29][30][31][32][33][34].The circular S 1 "phase" space of the energy eigenstates in geometric QM is extended to spherical S 2 , with the canonical S 1 forming the equator of the Nambu S 2 .The periodic evolution of the energy eigenstate "phase" is deformed from a simple circular motion along the equator to a more general periodic motion on S 2 .This generalization of the "phase" space necessarily has consequences on the interference between energy eigenstates.
The organization of this paper is as follows: In Section II, we review the basics of the geometric formulation of canonical QM [29] and introduce concepts and notation convenient for describing the Nambu QM extension and contrasting it against canonical QM.Section III contains the presentation of Nambu QM as a generalization of geometric QM, which can be viewed as a continuous deformation of canonical QM with two deformation parameters.How this generalization impacts interference and oscillation phenomena is also discussed.Section IV compares the formalism developed to atmospheric neutrino data to place bounds on the deformation parameters, detailing how the results in [1] were obtained.In Section V, we contrast how B 0 -B 0 oscillations are treated in both canonical and Nambu QM, and use B 0 -B 0 oscillation data from Belle [35] to constrain the deformation parameters of Nambu QM.
The somewhat surprising result is that the bounds on the deformation parameters from atmospheric neutrinos are stronger than those from Belle.This is due to the atmospheric neutrino data strongly preferring maximal mixing.
Section VI provides a summary of the results, a discussion on the pros and cons of the Nambu approach, and lists possible future directions of research.The Appendix includes information that did not fit well into the main text: In Appendix A we review some properties of Poisson brackets and contrast them with (ternary) Nambu brackets.The properties of the asymmetric top are also discussed.Appendix B lists properties of the Jacobi elliptical functions relevant for this work.

II. CANONICAL QUANTUM MECHANICS
A. The Geometric Formulation of Canonical Quantum Mechanics Due to the rigid and robust structure of canonical QM, any extension or deformation of its mathematical framework necessarily distorts or discards some of its cherished assumptions and/or principles.To see which of these properties are compromised or maintained in the generalization proposed in this paper, it is worthwhile to start by listing said properties in a way that would facilitate the comparison of canonical QM and the Nambu extension.
We maintain the following properties of canonical QM: 1. Any quantum state ψ of the system under consideration is described as a superposition of energy eigenstates: Here, we assume for the sake of simplicity that the energy eigenstates are discrete and can be labelled by an integer n.In the following, we will always work in this basis and no other.We do not consider any change of basis.
2. In canonical QM, each coefficient ψ n in the above expansion is an element of the complex number field C, which can be written as ψ n = A n e iθn where A n ∈ R is the amplitude and θ n ∈ R is the phase.Since e iθ is periodic in θ with period 2π, the phase can be restricted to θ ∈ R/(2πZ) = [0, 2π).We do not restrict the amplitude to non-negative reals for latter convenience.
We will generalize this to "numbers" which will maintain the property that they can be written as where the x-ponent x(θ) is a map from S 1 to another manifold which is periodic in θ with period 2π: The product of two such "numbers" is defined to be: where AB is just the usual product between real numbers A and B. It is clear that the x-ponent x(θ) is a representation of U (1) ∼ = SO(2).However, no sum of the "numbers" will be defined, i.e. the "numbers" will not comprise a field or division algebra.This means that our state space will not retain the full vector space structure we have in canonical QM.However, it turns out that this will not be a problem if we work solely in the fixed energy eigenstate basis.
3. The phase θ n in the coefficient ψ n of the n-th energy eigenstate time-evolves as where ℏω n is identified as the energy of the energy eigenstate n , and t n is the time at which θ n = 0. We characterize energy eigenstates in this fashion without introducing a Hamiltonian or Schrödinger equation.
When considering unstable states, we allow the amplitude A n to be time dependent as We will not add an imaginary part to the energy eigenvalue, since it will not lead to exponential decay of the coefficient in the Nambu extension.
4. There exists an inner product ⟨ϕ|ψ⟩ between states ϕ and ψ such that the energy eigenstates are orthonormal, ⟨n|m⟩ = δ nm , and We assume that all stable states are normalized, i.e. ⟨ψ|ψ⟩ = 1 for all ψ . 2.The Born rule: When the system is in state ψ , the probability of measuring it to be in state ϕ is given by We also describe the x-ponent and inner product of canonical QM as follows: 6.In canonical QM, the x-ponent x(θ) is e iθ .This can also be represented as a point on a unit circle, which we denote: The coefficient ψ n is represented by a point on a circle of radius A n : Thus, with an abuse of notation we can write Each ⃗ ψ n will time-evolve clockwise on a circle of radius A n with constant angular velocity ω n .

The inner product between two states
is defined as where It is straightforward to show that this definition agrees with the usual one using complex numbers.Note that and Let us write that is, C n is the product of the magnitudes of ⃗ ϕ n and ⃗ ψ n , and ζ n is the angle between the two.Then where P (n) represents the probability that the nth energy eigenstate contributes to |⟨ϕ|ψ⟩| 2 , while I 2 (n, m) is the pairwise interference between the nth and mth eigenstates a la Sorkin [27].Note that the interference A phase shift of each energy eigenstate keeps the angle ζ n defined in Eq. ( 17) invariant since it corresponds to the rotation by angle α n of the nth S 1 phase manifold.Therefore, the inner product ⟨ϕ|ψ⟩ is invariant under such phase shifts.Consequently, a global phase shift of all superpositions of the energy eigenstates by a common angle α, leaves the inner product between states invariant.This property will only be partially maintained in our extension.
In the case of canonical QM, one can have different phase shifts on each state in which case the symmetric and antisymmetric parts of the inner product transform as leading to the invariance of |⟨ψ|ϕ⟩| 2 .Thus, an arbitrary change of phase does not lead to any physical consequences, so states that only differ by a phase are identified as representing the same physical state.This leads to the projective vector space nature of the canonical QM state space.This property will be lost in our extension.
In the geometric formulation of canonical QM [29][30][31][32][33][34], the time evolution of the coefficient of the nth energy eigenstate is viewed as the classical evolution of a classical harmonic oscillator along the circle q 2 n + p 2 n = A 2 n in phase space.For a system with N energy eigenstates, the system is described by the evolution of a point on a 2N -dimensional Kähler manifold with the symmetric and antisymmetric parts of the inner product respectively providing the bilinear and symplectic forms which are connected by where the complex structure J acts on the coefficients ⃗ ψ n of ψ as Note that J 2 = −1.Our Nambu extension can be viewed as a generalization of canonical QM via this geometric formulation.

B. Oscillation
Before we describe our generalized QM, let us first demonstrate that the above description of canonical QM allows us to derive the usual neutrino oscillation formulae.
Let α and β be the flavor eigenstates, which are defined as superpositions of the energy eigenstates 1 and 2 via where c θ = cos θ, s θ = sin θ.In our phase vector notation, we have where represents a phaseless state.Let the state ψ at time t = 0 be equal to α : that is At a later time, these coefficients will have evolved into We find Therefore, the survival and transition probabilities will be These are the usual oscillation equations.To render the expression relativistic, we make the replacement If the energies are common, Therefore, the replacement is which precisely recovers the usual neutrino oscillation relations [36].
), the complete elliptical integral of the first kind.Shown is the graph for 4K, which is the period of sn(u, k) and cn(u, k) in u.Note that K → ∞ as k 2 → 1.

A. Deformation of Quantum Mechanics into a theory of Asymmetric Tops
The deformation of canonical QM discussed in Refs.[1,2] can be summarized as follows: The x-ponent, which is a map from S 1 to S 1 in canonical QM, is deformed to a map from S 1 to S 2 as: where cn(u, k), sn(u, k), dn(u, k), with 0 ≤ k < 1, are Jacobi's elliptical functions [37] 3 , and c ξ = cos ξ, s ξ = sin ξ, where 0 ≤ ξ ≤ π/2 is the second deformation parameter.The reason for the minus sign on the third component is explained later.Note that Jacobi's elliptical functions satisfy the relations so ⃗ X(θ|k, ξ) has unit norm.The periods of cn(u, k) and sn(u, k) for real u is 4K while that of dn(u, k) is 2K, where K = K(k 2 ) is the complete elliptical integral of the first kind [37].See FIGS. 1 and 2. In the above map, the arguments of the elliptical functions are rescaled so that the period in θ is 2π for cn and sn, and π for dn.For latter convenience, we introduce the notation to refer to these functions with their arguments rescaled.Due to the period of Dn(θ) being π instead of 2π, a phase shift by π does not change the sign of ⃗ X(θ|k, ξ): except when ξ = 0 in which case the third component of ⃗ X(θ|k, 0) is missing.To allow for a sign flip, we let the magnitudes of the coefficients be negative as well as positive.
The dependence of the trajectory of ⃗ X(θ|k, ξ) on S 2 on the deformation parameters k and ξ is shown FIG. 3.For instance, when k = 0 the trajectory reduces to namely, the path of constant latitude −ξ on S 2 .When ξ = 0 this will be the equator, whereas when ξ = π/2 this will be the south pole.Setting ξ = 0 for a non-zero k would lead to which traces the equator of S 2 but with a non-canonical dependence on θ.
The coefficients of the nth energy eigenstate will time-evolve as where t n sets the initial condition for each n.Note that when ξ = 0 the trajectory of ⃗ X(−ωt|k, 0) will be along the equator of S 2 , but it will not time-evolve with constant angular velocity ω except for the canonical QM case k = 0.
The trajectory of ⃗ Ψ n (t) on S 2 is that of the angular momentum of a free asymmetric top in the frame fixed to itself.There, the angular momentum and energy are both fixed so the angular momentum vector will evolve along the intersection of the two surfaces A 2 =constant and E =constant.k = 0 corresponds to the symmetric top with I 1 = I 2 .However, the dynamics we chose in Eq. ( 43) is not that of the asymmetric top.Indeed, when the intersection of the two surfaces is the equator, ξ = 0, the angular momentum of the asymmetric (symmetric at the equator) top stays fixed and does not time evolve.This must be the case since the ξ = 0 limit must correspond to the same physical configuration for the top regardless of the value of k.See Appendix A 3 for details.Our dynamics here is chosen so that k = 0 and ξ = 0 recovers canonical QM.The only thing that is maintained is the direction of evolution along the trajectory, by the choice of the sign of the third component.
Note that this is but one way to generate a closed path on S 2 .Other deformations of the x-ponent are possible, each leading to different generalizations of canonical QM.However, we will not dwell on this question further in this paper.

B. Inner Product
We extend the inner product of two states to the quaternion where The dot and cross products are now defined in three dimensions.This inner product results if we map the 3D phase vector ⃗ Ψ to a purely imaginary quaternion and define the inner product to be Note that and If we write where where γ nm is the angle between ⃗ C n and ⃗ C m .Unlike the canonical QM case, cf.Eq. ( 18), the interference I 2 (n, m) will depend on the sum ζ n + ζ m as well as the difference The angle ζ n defined in Eq. ( 52) is invariant under 3D rotations of the nth S 2 phase manifold associated with the nth energy eigenstate, generalizing the canonical case which was invariant under 2D rotations of the S 1 phase manifold.Note, however, that in general, a phase shift of an energy eigenstate does not correspond to a 3D rotation of S 2 , except for the symmetric top case k n = 0, in which case this phase shift becomes a rotation of angle α around the north-south axis of S 2 .Thus, to maintain the invariance of the angles ζ n , and consequently that of the inner product ⟨Φ|Ψ⟩ under phase shifts of the individual energy eigenstates, we must choose k n = 0 for all n.Even then, we do not have the freedom to shift the phases of different states, i.e. different linear combinations of the energy eigenstates, independently since |⟨Φ|Ψ⟩| 2 will not remain invariant.So the projective nature of the state space is lost.This difference between the inner products in canonical QM and Nambu QM has consequences in oscillation phenomena.We demonstrate this using the calculation of neutrino oscillation probabilities next, allowing k n ̸ = 0 to see their effect.

C. Neutrino Oscillations in Nambu QM
As an example of oscillation phenomena, we calculate the two-flavor neutrino oscillation probabilities.For simplicity, we take the deformation parameters k n and ξ n to be the same for all energy eigenstates.
Let the flavor eigenstates A and B be given by superpositions of mass eigenstates 1 and 2 as Then, where represents a "phaseless" state.Let Ψ(0) = A , that is: At a later time t, we have where Here, we use the shorthand Then, the symmetric parts of ⟨A|Ψ(t)⟩ and ⟨B|Ψ(t)⟩ are while the antisymmetric parts of ⟨A|Ψ(t)⟩ and ⟨B|Ψ(t)⟩ are Therefore, the survival and transition probabilities are given by Note that the conservation of probability is manifest: Let us define so that we can write Utilizing the addition and subtraction theorems given in Appendix B, Eq. (B6), we can rewrite the combinations (Sn 1 Sn 2 + Cn 1 Cn 2 ) and (k 2 Sn 1 Sn 2 + Dn 1 Dn 2 ) as functions of ωt and δω t, and we find To render this expression relativistic and appropriate for neutrino oscillations, we make the replacement where we assume E ≫ m i .Note that Thus, for F(ωt, ∆) to depend only on ∆, we must set k = 0, in which case we have Other than the factor of c 2 ξ , this is the same as the expression for canonical QM.However, due to the c 2 ξ factor, oscillations vanish in the limit ξ → π 2 .Note that when k = 0, ξ = π/2, the phase vectors of all energy eigenstates will be pointing to the south pole of the S 2 phase space, and do not evolve with time.The angles between arbitrary pairs of phase vectors will always be zero, and consequently, the asymmetric part of the inner product, ⃗ ε(Φ, Ψ), will be identically zero.It is tempting to think of this as the "classical" limit of the theory, in which all oscillation phenomena vanish.However, pairwise interferences do not vanish altogether in this limit since they continue to exist in the g(Φ, Ψ) 2 contribution to |⟨Φ|Ψ⟩| 2 .Indeed, these interferences are necessary to maintain unitarity.Due to this, we will call the k = 0, ξ = π/2 case "pseudo-classical."FIG. 4. The graph of F(ωt, δω t) when 2ω/δω = (ω1 + ω2)/(ω1 − ω2) = 100, for the cases k 2 = 0.8, 0.9, 0.99, 0.999 and ξ = 0, π 6 , π 4 , π 3 .Graphs for the same value of k 2 are arranged in columns while those for the same value of ξ are arranged in rows.Red: graph of F(δω t) which averages over the rapid oscillations of F(ωt, δω t).Blue dashed: graph of Eq. ( 75).Note that the red graph is well approximated by the blue, especially for small ξ, until k 2 closely approaches 1.In the limit k 2 → 1, the locally averaged graph (red line) converges to the saw-tooth function indicated with the black dotted line on the graphs in the rightmost column (k 2 = 0.999).
For k ̸ = 0 the dependence on ωt remains.However, when E ≫ m i we have ω ≫ δω, and in the expression for F(ωt, δω t) given in Eq. (69), Sn 2 (ωt, k) will oscillate very rapidly compared to Sn 2 (δω t/2, k).We expect such rapid oscillations to be unobservable, and that F(ωt, ∆) can be replaced by the function F(∆) where We demonstrate this in FIG. 4 which shows graphs of F(ωt, δω t) for several choices of k 2 and ξ for the case We can see that F(ωt, δω t) oscillates rapidly around the more slowly varying function F(δω t) shown in red.Using the expansion of the Jacobi elliptic functions in k 2 given in Appendix B, we can show that F(δω t) can be approximated by This graph is also shown on FIG. 4 in dashed blue, and we see that this approximation breaks down only when k 2 is very close to 1.
IV. BOUND FROM ATMOSPHERIC NEUTRINOS FIG. 5. k 2 -ξ contour plot showing the 1σ (blue), 2σ (orange), and 3σ (green) bounds from atmospheric neutrino data and Eq.(75), assuming Normal Ordering.ξ is in units of π.The contours are not continued to the right of k 2 = 0.9 where the validity of Eq. (75) becomes problematic as k 2 → 1.We expect a more thorough analysis to lead to the contours dropping down to the k 2 axis near k 2 = 1.
The expression we obtained in Eq. ( 75) is just the usual neutrino oscillation formula scaled by the constant 2 .If we consider atmospheric neutrino oscillations, this factor cannot always be absorbed into sin 2 (2θ 23 ) ≤ 1, so we can interpret the bounds on the quantity sin So the bounds for both the NO and IO cases are more or less the same.The NO numbers translate to For the k = 0 case, this translates to ξ π < 0.053 (1σ) , 0.062 (2σ) , 0.070 (3σ) .
For non-zero k, see FIG. 5 for the likelihood contours on the k 2 -ξ plane.In FIG. 5, the contours are not extended all the way to k 2 = 1 since Eq. ( 75) is unreliable in that region.They can be expected to turn downward and reach the k 2 axis before reaching k 2 = 1 since the oscillation function F(∆) will start to deviate from the sinusoidal as k 2 → 1 as shown in FIG. 4. A complete oscillation analysis would be necessary to work out the bounds on ξ and k 2 in this region.We do not perform this analysis in this paper due to reasons that are given in the discussion session.Instead, we look at B 0 -B 0 oscillation next.
V. B 0 -B 0 OSCILLATION Another oscillation phenomenon that has been extensively studied in experiments is that of time-dependent CP asymmetry in meson-antimeson oscillations [39,40] for which a wealth of data exist from Belle and BABAR.In the following, we place bounds on k and ξ utilizing the B 0 -B 0 oscillation data from Belle.

A. Canonical QM Case
Consider the oscillation of B 0 into B 0 and vice versa before they decay into a common physical state f .Recall that in the usual canonical QM formulation, the CP eigenstates are ( B 0 ± B 0 )/ √ 2, whereas the mass eigenstates are [41] B 0 where |p| = |q| = 1/ √ 2, and p and q have opposite phase: To leading order in the Standard Model, the CP violating phase ϕ is given by [41] ⃗ x(2ϕ To facilitate the application of our Nambu extension later, we work in the mass eigenstate basis, in which the B 0 and B 0 states are expressed as superpositions of the mass eigenstates B 0 L and B 0 H : In the rest frame of the particle, the mass eigenstates B 0 L and B 0 H evolve as so the B 0 superposition evolves as We assume that Γ L = Γ H , and drop the common decay factor from this point on.Then, we find where The survival and transition probabilities evolve with time as with similar relations for P (B 0 → B 0 ) and P (B 0 → B 0 ).Consider the asymmetry where ⟨f |B 0 (t)⟩ and ⟨f |B 0 (t)⟩ are the amplitudes of the states B 0 (t) and B 0 (t) decaying into f respectively.Let that is Then, and Similarly, and we find Note that It is straightforward to show that the above definitions of S and C agree with the usual expressions [41] involving We refrain from this rewriting, however, since the analogs of A f and Āf are ill defined in Nambu QM.

B. Nambu QM Extension
Let us now apply the Nambu extension to the above formalism.We assume that the B 0 and B 0 states are given by the superpositions where Here, we suppress the dependence of ⃗ X on the deformation parameters k and ξ to simplify our notation.In the rest frame of the particle, the mass eigenstates B 0 L and B 0 H evolve as We assume Γ L = Γ H as in the canonical case and suppress this common decay factor from this point on.The state B 0 (0) = B 0 evolves to Therefore, where we have used the shorthand The transition probability is given by where the function F was defined in Eq. ( 69), and m = (m L + m H )/2 and ∆m = m H − m L .The survival probably is also found to be Note these expressions are the same as the neutrino oscillation case we considered earlier, Eq. ( 64), with mixing angle set to π/4.The CP violating phase ϕ only appears as a shift of the first argument of F.
The mass difference between B 0 L and B 0 H is [42,43] whereas the average mass is [42,43] m = 5279 ± 0.12 MeV . Therefore, and we conclude that Sn 2 (ϕ + mt) oscillates very rapidly compared to Sn 2 (∆m t) and will average out in F(ϕ + mt, ∆m t), just as in the neutrino oscillation case.So the transition probability is well approximated by where the rightmost expression is valid when k 2 is not too close to one.
Let us now consider the asymmetry A(t) defined in Eq. (89).We assume Then and we find where Θ 1 and Θ 2 are, respectively, the angles between the pairs of unit vectors The sign of sin Θ 1 has been chosen so that sin Θ We note that cos Θ 1 and sin Θ 1 can take on any value in the range [−1, 1], whereas the range of cos Θ 2 is limited to Since Θ 1 and Θ 2 are both functions of λ and η, their values are correlated and not all possible values of Θ 1 and Θ 2 in their respective ranges can be simultaneously realized.To see this, we show in FIG.6 the contour plots of cos Θ 1 , sin Θ 1 , and cos Θ 2 on the (λ, η) plane for a single period in each direction.k 2 is taken to be 0.5 in these plots, but the basic features will remain the same for any value of k 2 ∈ (0, 1).For instance, the maxima of cos Θ 1 , sin Θ 1 , and cos Θ 2 will be along the red lines, and their minima will be along/at the white lines/dots for any value of k 2 .Note that cos Θ 1 = 1 necessarily implies cos Θ 2 = 1, since both functions are at their maxima of 1 along the η = λ line.However, we have cos Θ 2 = 1 along the η = −λ ± π lines as well, which intersect the contours for all possible values of cos Θ 1 and sin Θ 1 .Also, we have cos Θ 1 = −1, sin Θ 1 = 0 along the lines η = λ ± π, which intersect the contours for all possible value of cos Θ 2 .Therefore, any value of cos Θ 1 and sin Θ 1 can be realized when cos Θ 2 = 1, and any value of cos Θ 2 can be realized when cos Θ 1 = −1, sin Θ 1 = 0.However, an arbitrary contour for cos Θ 1 does not necessarily cross all the contours for cos Θ 2 , and vice versa, e.g. the cos Θ 1 = 1 contour, so not all value pairs can be realized.
Here we use the fact that m ≫ ∆m to replace the F(ϕ + mt, ∆m t), G(ϕ + mt, ∆m t), H(ϕ + mt, ∆m t), and I(ϕ + mt, ∆m t) functions with those that are averaged over a period of the first argument.It is straightforward to see that Graphs of the function G(∆m t), where is compared to that of G(mt, ∆m t) in FIG.7 for several values of k 2 , where the mass ratio was taken to be 2m/∆m = 100.Using the k 2 -expansions of the Jacobi elliptic functions given in Appendix B, we can demonstrate that The graph of sin(∆m t) is also shown in FIG.7 and we can see that the approximation is very good until k 2 exceeds 0.9 or so.Thus, we arrive at the expression for A(t) in Nambu QM: For k = 0, this simplifies to If we further set ξ = 0, we recover the canonical QM expression Eq. ( 95).On the other hand, if we set k = 0, ξ = π/2, the limit in which oscillations vanish, the expression becomes that is, A(t) will be constant as expected.Note that since A L and A H can be either positive or negative in Nambu QM, this constant can be any number in the range In this pseudo-classical limit, the B 0 state stays a B 0 state, and the B 0 state stays a B 0 state.The ratio of their decay rates to f is which can be any non-negative number.
In FIG. 8, we illustrate the dependence of A(t) on ξ and k.There, the other parameters are fixed to for which the k = ξ = 0 case is A(t) = sin(∆m t), whereas the k = 0, ξ = π/2 case is A(t) = −1.The deviation of the curve from sinusoidal is small until k 2 exceeds 0.9 or so.Increasing ξ toward π/2 when k = 0 suppresses the oscillation amplitude and shifts the graph toward the flat no-oscillation limit.

Fit to Belle Data
Let us now use actual Belle data to constrain the deformation parameters k and ξ.We use the data from [35].We first consider the k = 0 case and look at how the fit of Eq. (121) to the Belle data constrains the parameter ξ.
In interpreting the experimental data, one should take into account the miss-tag factors w B 0 and w B 0 that respectively characterizes the probability of incorrectly tagging the B 0 as B 0 , and vice versa.The probability amplitudes are modified to and A(t) to where D = 1−2⟨w⟩ is the dilution factor, ⟨w⟩ = (w B 0 +w B 0 )/2 the average miss-tag probability, and ∆w = w B 0 −w B 0 .The dilution factor D simply suppresses the overall amplitude ) is already restricted to have magnitude smaller than one, the presence of D does not add any new constraints.The difference in miss-tag probabilities, ∆w, leads to an offset of A(t) in the vertical direction.This is known to be negligibly small for analyses involving a low number of events, and indeed we find that its preferred value when included in our fits is approximately zero.Therefore, we set ∆w = 0 in the following.
Firstly, we fit Eq. ( 121) to the Belle data allowing 2A L A H /(A 2 L + A 2 H ), (η − λ), and ξ to float.The mass difference ∆m is set to the current world average, Eq. ( 106), from the Review of Particle Properties [42].The minimum of the χ 2 is found to be 15.06 for 24 − 3 degrees of freedom at that is, canonical QM is preferred over a non-zero ξ.The best-fit curve is shown against the data of [35] in FIG. 10 as the canonical QM fit.The likelihood function for ξ is shown in FIG. 9.It has the characteristic feature that it suddenly drops to zero around ξ/π ∼ 0.2.This drop is caused by the amplitude | ≤ 1 being unable to compensate for the flattening of the oscillation curve as ξ is increased beyond this point.This is similar to the atmospheric neutrino analysis in which sin 2 2θ ≤ 1 was unable to compensate for an increase in ξ, and the resulting suppression of c 2 ξ .From this likelihood curve, we find the upper bounds on ξ to be ξ π < 0.098 (1σ, 68%), 0.168 (2σ, 95%), 0.192 (3σ, 99.7%) .
Note that these are weaker than the atmospheric neutrino bounds given in Eq. (79).Next, we fit Eq. ( 120) to the Belle data to constrain both ξ and k.The fit parameters are 2A L A H /(A 2 L + A 2 H ), λ, η, k 2 , and ξ.The mass difference ∆m is set to Eq. ( 106).The minimum of the χ 2 is found to be 14.32 for 24 − 5 = 19 degrees of freedom when Θ 2 (λ, η) is unconstrained due to the best fit value of ξ being zero, leading to degeneracies on the (λ, η) plane.The best fit curve is shown in FIG. 10 against the data and the canonical QM best fit.The likelihood contours on the k 2 -ξ plane are shown in FIG.11.Note that the best fit point is at ξ = 0 and k 2 = 0.957, exceptionally close to the maximum allowed value of k 2 .This is due to the fact that the data prefers an oscillation which slightly deviates from the sinusoidal, as can be seen in FIG. 10, while Jacobi's elliptical functions show such deviations in a noticeable way only as k 2 approaches 1.This leads to k 2 being only very weakly constrained.The 3σ contour on FIG.11 traces a precipice in the likelihood function along which it plummets to zero, just as in the k = 0 case, cf.FIG. 9.This is again due to the constraint

VI. SUMMARY AND DISCUSSION
In this paper, we have reviewed how Nambu QM [2] extends canonical QM via its geometrical formulation [29][30][31][32][33][34].All states are expressed as superpositions of energy eigenstates with coefficients that have magnitudes and "phases."The extension involves generalizing the concept of the "phase" to a map from S 1 (a circle) to S 2 (a sphere).The 2N dimensional Kähler manifold of geometric QM for N energy eigenstates is generalized to a 3N dimensional manifold.Canonical QM corresponds to the "phase" being a map from S 1 to the equator of S 2 .
Deviations of the map from the equator is described by two deformation parameters ξ and k.The trajectory of the map is assumed to follow the path of the angular momentum of an asymmetric top, the motion of which can be described by the triple Nambu bracket [3] with two conserved quantities, one of which maintains the "phase" vector on the surface of S 2 .The dependence of the 3D "phase" vector on the phase parameter is described by Jacobi's elliptical functions cn(u, k), sn(u, k), and dn(u, k), where k is the eccentricity of the ellipse from which these function are defined.
Under this deformation, the coefficients of the energy eigenstates are no longer elements of a field or division algebra since the addition of coefficients is ill defined.The inner product defined for the extension is also not invariant under generic phase shifts of the coefficients.Therefore, the projective vector space structure of the state space of canonical QM is lost.Since the coefficients do not add, there is no path-integral formulation of the Nambu extension in which such a coefficient is associated with each path, and the paths interfere with each other via superposition.Nevertheless, the energy eigenstates with those coefficients do interfere allowing for oscillation phenomena, which deform those of canonical QM.
We have used atmospheric neutrino [38] and Belle [35] data to constrain the deformation parameters ξ and k, as shown in FIGS. 5 and 11.Of the two deformation parameters ξ is the better constrained, the 3σ upper bounds for the k = 0 case being ∼ 0.07π and ∼ 0.2π for atmospheric neutrinos and Belle, respectively, cf.Eq. ( 79) and (129).This is due to the parameter ξ suppressing oscillations as it is increased from 0 to π/2, while the overall amplitude (sin 2 2θ in the case of atmospheric neutrinos and 2A L A H /(A 2 L + A 2 H ) in the case of B 0 -B 0 oscillations, which are both bounded by 1) is unable to compensate for it.The somewhat surprising result is that the bound from atmospheric neutrinos is stronger than the Belle bound.
The bound on k 2 in contrast is very weak.This is due to k 2 causing significant deviations of the oscillation functions from the sinusoidal only when k 2 is very close to one.Indeed, the best fit value to the Belle data was k 2 = 0.957, Eq. ( 130), due to the data preferring a slight deviation of the oscillation curve from the sinusoidal.See FIG. 10.However, this was based on a simplified oscillation formula which resulted due to the sum of the eigenfrequencies of the interfering energy eigenstates being much larger that the difference, allowing us to average out the fast oscillations.For situations in which the sum and difference of the frequencies are much closer to each other, there will be significant deviations from sinusoidal oscillations at the frequency difference, and we expect the bound on k 2 to be much stronger.Furthermore, the k = 0 case is preferred theoretically, since it will allow for phase changes of the energy eigenstates without affecting physical probabilities, justifying the somewhat arbitrary phase choices in Eqs. ( 55) and (98).
In addition to the loss of the projective vector space structure of the state space, Nambu QM has other shortcomings as well.First, unstable states cannot be made to decay by giving them complex energy eigenvalues.This is due to the doubly periodic nature of the Jacobi elliptic functions which are periodic in the imaginary direction as well as the real.Thus, we must let the amplitudes of the coefficients decay exponentially by hand.Second, since the Nambu extension relies on a fixed energy-eigenstate basis, it is not clear how it should be applied to situations in which the energy eigenstates and their respective eigenvalues are themselves time-dependent.For instance, when considering matter effects on neutrino oscillations, the eigenstates of the neutrino Hamiltonian change depending on the background electron density.This is the reason why we did not analyze neutrino oscillations directly in this paper.
Despite these drawbacks, Nambu QM has many interesting features beyond the deformation of oscillations that can be constrained by experiment.First, when k = 0 but ξ ̸ = 0, the inner product becomes invariant under a global phase shift of all states by a common phase.Thus, if we gauge the theory it will become a U (1) gauge theory in which all states carry a common U (1) charge.This may provide a new way to quantize U (1) charges without the need to embed the theory into a non-Abelian gauge theory.
Second, the parameter choice ξ = π/2, k = 0 leads to a model which is pseudo-classical, i.e. all oscillations vanish in this limit (though interferences do not).Thus Nambu QM provides a framework in which one can interpolate between quantum and classical-like behavior continuously by tuning the deformation parameter ξ.This property may allow us to study how the Leggett-Garg inequalities [44][45][46][47][48] set in as pseudo-classicality is approached.
We note that the loss of addition of the "numbers" that constitute the state coefficients may be what is necessary to render a quantum-like theory classical (or pseudo-classical).Indeed, in [15] it has been shown that in the q → 1 limit of F q QM [12][13][14]16], which is a quantum mechanical model constructed over the finite Galois field F q , the theory behaves "classically," while F q becomes the so-called "field with one element" F 1 , or F un [49], for which the elements multiply but do not add.
Third, it may be possible to construct a model with Sorkin's triple-path interference [27,50,51] using the Nambu approach.In both canonical and Nambu QM, interference between different energy eigenstates is pairwise due to the Born Rule.They result from the cross terms that appear in the squares of the symmetric and anti-symmetric parts of the inner product, namely: However, in the Nambu extension ( ⃗ Φ n × ⃗ Ψ n ) is a 3D vector, allowing us to construct the 3D volume which could be interpreted as the interference of the nth, mth, and kth energy eigenstates.If such a term can be incorporated into a unitary model, we would have an extension of canonical QM with triple-path interference.Note that the existence of such a model is crucial in experimentally probing the validity of the Born rule [50,52] as an alternative theory against which to compare canonical QM.
Furthermore, though we only consider an extension of the "phase" to a map from S 1 to S 2 , the Nambu bracket [3] exists for any number of dimensions and it can be used to extend the "phase" to a map from S 1 to S n−1 , n = 4, 5, • • • .In those higher dimensional extensions, the phase vector will be n-dimensional, and n-dimensional hypervolumes can be constructed, which could correspond to n-path interferences.Therefore, the possibility exists for a ladder of models, the n-dimensional case possessing Sorkin's multi-path interference up to the nth order.However, if one insists on the top-like behavior of the phase vector, then one may need to jump to 7D [53][54][55][56] in which the model will have connections to octonions [57].
If we do not restrict our attention to models in which the phase vector is top-like, then even in the 3D case we can use "phase" maps from S 1 to S 2 that are different from what was used in this paper.For instance, we could let the phase vector of each energy eigenstate follow different great circles on S 2 .Each choice of "phase" map will lead to a different phenomenology from what was discussed.
The possibility of multi-path interferences also suggests the relevance of Nambu QM to the quantization of gravity.Indeed, in the new approach to quantum gravity dubbed "gravitization of quantum theory" [58,59], the phenomena of triple and higher order interference are naturally expected to occur [51].The reason for this stems from the fact that "gravitization of quantum theory" by definition implies a dynamical geometry of quantum theory, and thus the generalization of the Born rule, which is tied to the canonical geometry of complex projective spaces.This canonical geometry is maximally symmetric and via the Born rule it precludes any intrinsic triple and higher order interference.However, given the non-linear and non-polynomial nature of gravitational interaction, it is expected that "gravitization of quantum theory" will lead to triple and higher interference of arbitrary order.
Other avenues worth exploring include the quantum brachistochrone problem, which is usually stated as follows: Find the Hamiltonian that takes one from a given initial state to a given final state in the least amount of time, under the constraint that the difference between the largest and the smallest eigenvalues of the Hamiltonian is fixed [60][61][62][63].Since we have not introduced a Hamiltonian operator in Nambu QM, and also since we can expect transition speeds to depend on the sums of energy eigenvalues as well as the differences when k ̸ = 0, the problem must be restated somewhat to: Find the energy spectrum that takes one from a given initial state to a given final state in the least amount of time, under the constraint that the largest and the smallest energy eigenvalues are fixed.In particular, how does the answer depend on k and ξ and what happens in the pseudo-classical limit k = 0, ξ → π/2?Answering this will provide us with a deeper understanding of the dynamics of Nambu QM, and perhaps illuminate what happens in the classical limit of a quantum theory.
Classical Nambu theory fits into the general Hamiltonian framework of the principle of least action δS = 0, where the action of classical Nambu dynamics is the integral of a 2-form involving 2 Hamiltonians [4] (as opposed to the integral of a 1-form involving one Hamiltonian, valid in canonical classical theory).In general, one has integrals over n-forms, with n-Hamiltonians.The corresponding quantum theory can be formulated as an ℏ deformation of the classical action principle, by invoking the Schwinger variational principle δS• = iℏδ•, where in the usual case one acts (δ) on a complex wavefunction, or alternatively the real and imaginary parts of the wave function, respectively.The real and imaginary parts of the wavefunctions are dual to each other in this formulation that gives the canonical Schrödinger evolution.In quantum Nambu theory, the action δ is on one of the components of a three-component (vector-like) wavefunction, where the dual to each component is given by the cyclic product of the other two components.Obviously this can be generalized to n-components.In this case the variation of the action δS, in general, involves n Hamiltonians of the Nambu dynamics.This general formulation leads in principle to a new view of how probabilities are calculated in Nambu quantum theory.From this point of view, Nambu quantum theory fits into a more general understanding of quantum mechanics as quantum measure theory [27], with general probabilities related to triple [50] and higher order interference phenomena [51].
In this context another curious question is what happens when we discretize the "phase" space variables of Nambu QM.This question is worth exploring especially in connection to Spekkens' toy model [64] and its generalization, dubbed Quasi-quantization [65], in which one considers classical physical systems and imposes a limitation on the amount of knowledge that can be possessed by an observer about those systems.The theory derived from those assumptions is able to qualitatively reproduce various phenomena that are ordinarily considered characteristic of quantum theory.For instance, the version of such theory in which the phase space variables p and q are restricted to take on values in F 2 can reproduce a large subset of phenomena associated with quantum spins.Among the phenomena reproduced therein, the ones that are of most interest to us are coherent superposition and interference, owing to our goal of modeling triple-path interference.As such, our future work will explore Quasi-quantization of theories with three "phase" space variables in general and their discrete version in particular.However, we need to have a better understanding of the third "phase" space variable before attempting this study.This is because the analog of epistemic restriction in quantum theory is Heisenberg's uncertainty principle and trying to define its three-variable version deserves careful consideration.
There remain numerous avenues to be pursued in further development and understanding of Nambu QM.We intend to address these possibilities in our forthcoming publications.
Since both H 1 and H 2 are conserved, the ellipsoid H 1 = constant and the sphere H 2 = constant must have an intersection in the 3D (q 1 , q 2 , q 3 ) space for a solution to exist.This requires the shortest axis of the ellipsoid to be shorter than the radius of the sphere while the longest axis of the ellipsoid is longer than the radius of the sphere.We assume in which case the two intersections are respectively in the northern and southern hemispheres, and in the limit I 1 = I 2 → H 2 /H 1 the intersections come together at the equator.When this condition is satisfied, the solution to the equation of motion is given by the Jacobi elliptic functions, namely: q 1 (t) = N 1 cn(Ωt, k) , q 2 (t) = −N 2 sn(Ωt, k) , q 3 (t) = −N 3 dn(Ωt, k) , (A19) where N 1 , N 2 , and N 3 are all positive, and The deformation parameter ξ introduced in Eq. ( 37) is given by where and k ′ = √ 1 − k 2 .We have q n+1/2 1 − q 2n+1 sin((2n + 1)v) , q n+1/2 1 + q 2n+1 cos((2n + 1)v) , q n 1 + q 2n cos(2nv) , where 2π/K and q can also be expanded as

FIG. 3 .
FIG. 3. Phase trajectories on S 2 in Nambu QM for positive amplitudes.Only the southern hemisphere is shown.Trajectories in the northern hemisphere occur when the amplitude is negative.The colored lines in each figure indicate ξ = 0 (Black), ξ = π/8 (Blue), ξ = π/4 (Orange), ξ = 3π/8 (Green), and ξ = π/2 (Red).When ξ = 0, the trajectory always follows the equator regardless of the value of k.However, the phase time-evolves with uniform angular velocity along the equator only when k = 0.When k = 0 and ξ = π/2, the phase is stationary at the south pole.

2 FIG. 6 .
FIG.6.Contour plots of (a) cos Θ1, (b) sin Θ1, and (c) cos Θ2 for the case k 2 = 0.5 on the (λ, η) plane for a single period in each direction.The thick red lines on the graphs indicate the points at which each function is at their maxima, while the white lines on the cos Θ1 graph and the while dots on the cos Θ2 graph indicate points at which they are at their respective minima.

FIG. 9 .
FIG. 9.The likelihood function for ξ for the k = 0 case.The vertical axis is in units so that the area underneath the curve is one.The sudden drop at ξ/π ∼ 0.2 is due to the amplitude |2ALAH /(A 2 L + A 2 H )| ≤ 1 being unable to compensate for the flattening of the oscillation curve as ξ is increased, cf.FIG. 8.The dashed line indicates how the likelihood function would have looked without the amplitude constraint.The vertical dotted lines indicate the locations of the 1σ (68%), 2σ (95%), and 3σ (99.7%) bounds.