Interference and Oscillation in Nambu Quantum Mechanics
aa r X i v : . [ h e p - ph ] D ec Interference and Oscillation in Nambu Quantum Mechanics
Djordje Minic, ∗ Tatsu Takeuchi, † and Chia Hsiung Tze ‡ Center for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg VA 24061, USA
Nambu Quantum Mechanics, proposed in Phys. Lett. B536, 305 (2002), is a deformation ofcanonical Quantum Mechanics in which only the time-evolution of the “phases” of energy eigenstatesis modified. We discuss the effect this theory will have on oscillation phenomena, and place a boundon the deformation parameters utilizing the data on the atmospheric neutrino mixing angle θ . Introduction:
Quantum Mechanics (QM) is one of themost important and successful frameworks of modernphysics. The language of QM is essential for particle, nu-clear, atomic, condensed matter, and statistical physicsas well as chemistry, and it has lead to the current “sec-ond quantum revolution” in quantum information scienceand technology [1]. Nevertheless, the full understandingof the foundations and origins of QM is still an activearea of intense discussion and research [2–4]. It has beenargued that canonical QM should be replaced by a morefundamental or generalized framework, either in the con-text of quantum gravity and cosmology [5, 6], or in therealm of quantum measurement [7], or in the domain ofmacroscopic quantum systems [8].A deeper understanding of canonical QM could be ob-tained by comparing its predictions to those of its pos-sible generalizations, and confronting both with experi-ment. It would allow us to probe the robustness of theoriginal tenet or axiom that was relaxed to generalize thetheory, thereby identify the theoretical bedrock on whichQM rests. (See e.g. [9] and references therein).Various proposals for alternative QM theories can befound in the literature. The field/division algebra overwhich the state space is constructed has been modifiedfrom C to R [10], H [11], O [12–14], and the finite fields F q [15–21]. Non-linear corrections to the Schr¨odinger equa-tion have been considered by Weinberg [22, 23]. Otherideas address issues in quantum measurement and quan-tum gravity [5–7].In this letter, we look at Nambu QM introduced in[24] by Minic and Tze, and one of its consequences. Thework was inspired by a profound and far-reaching paperby Nambu [25]. The starting point of the approach isthe geometric formulation of QM [26] in which the timeevolution of pure quantum states is described as a “clas-sical” area preserving Hamiltonian flow within the state“phase” space. For a single energy eigenstate, this is justthe evolution of its phase, the real and imaginary partsof which constitute the “phase” space, with the Hamilto-nian being that of a harmonic oscillator. Nambu’s ideain [25] was to extend the classical equation of motion˙ F = −{ H, F } , where { A, B } = ε ij ∂A∂q i ∂B∂q j (1) is the Poisson bracket, to ˙ F = −{ H , H , F } , where { A, B, C } = ε ijk ∂A∂q i ∂B∂q j ∂C∂q k , (2) i.e. the Nambu bracket. In Poisson dynamics time evolu-tion is generated by the one conserved quantity H , whileNambu dynamics requires two: H and H , and the gen-erated flow is volume preserving. An application of theNambu equation is the asymmetric top in which the evo-lution of its angular momentum ~L can be generated bythe Nambu bracket with the energy E and total angularmomentum L / H and H . Theproposal of [24] was to enlarge the “phase” space of eachenergy eigenstate from two dimensions to three, and as-sume that the “classical motion” of the “phase” was gov-erned by Nambu asymmetric top dynamics instead ofthat of a Poisson harmonic oscillator. Note that this de-formation of canonical QM is particularly attractive sinceit is minimalistic: it only deforms the time-evolution ofthe “phase” of energy eigenstates and everything else iskept fixed. Furthermore, this deformation can be contin-uously turned on and off.In the following, we first review canonical QM in thetwo-component real vector notation and the treatmentof oscillations in that language. Then, we formulateNambu QM as the three-component real vector exten-sion to canonical QM and derive an explicit formula foroscillations in this context. This formula can be probedexperimentally, most promisingly in neutrino oscillations. Canonical QM:
Let | n i denote the n -th eigenstate ofthe Hamiltonian ˆ H with eigenvalue E n = ~ ω n :ˆ H | n i = E n | n i . (3)A generic state | ψ ( t ) i can be expanded in terms of | n i as | ψ ( t ) i = X n | n i h n | ψ ( t ) i | {z } ψ n ( t ) = X n ψ n ( t ) | n i , (4)where the coefficients ψ n ( t ) evolve in time as ψ n ( t ) = N n e − iω n ( t − t n ) . (5)Here, we take N n to be real and positive, and t n is theboundary time at which ψ n ( t ) is phaseless. The complexvalued ψ n ( t ) can also be expressed as a two-componentreal vector as ~ψ n ( t ) ≡ (cid:20) Re ψ n ( t )Im ψ n ( t ) (cid:21) = N n (cid:20) cos ω n ( t − t n ) − sin ω n ( t − t n ) (cid:21) . (6)The inner product between two states | ψ i and | φ i in thistwo-component real vector notation is h ψ | φ i = X n ψ ∗ n φ n = X n ( ~ψ n · ~φ n ) | {z } g ( ψ, φ ) + i X n ( ~ψ n × ~φ n ) | {z } ε ( ψ, φ ) . (7)Note that g ( ψ, φ ) and ε ( ψ, φ ) depend only on the mag-nitudes of, and the relative angles between the ( ~ψ n , ~φ n )pairs. They are invariant under 2D rotations. The abso-lute value of the inner product squared is then |h ψ | φ i| = g ( ψ, φ ) + ε ( ψ, φ ) . (8)Now, consider two energy eigenstates | i and | i andtwo flavor eigenstates | α i and | β i which are related by (cid:20) | α i| β i (cid:21) = (cid:20) c θ s θ − s θ c θ (cid:21) (cid:20) | i| i (cid:21) , (9)where s θ = sin θ , c θ = cos θ . In vector notation, we have ~α = c θ ~n , ~α = s θ ~n ,~β = − s θ ~n , ~β = c θ ~n , (10)where ~n represents a phaseless state: ~n = (cid:20) (cid:21) . (11)Let | ψ (0) i = | α i , that is ~ψ (0) = ~α = c θ ~n , ~ψ (0) = ~α = s θ ~n . (12)At a later time, these will have evolved into ~ψ ( t ) = c θ ~n ( t ) , ~ψ ( t ) = s θ ~n ( t ) , (13)where ~n ( t ) = (cid:20) c − s (cid:21) , ~n ( t ) = (cid:20) c − s (cid:21) , (14)with s i = sin ω i t and c i = cos ω i t . To find the sur-vival probability P ( α → α ) of flavor α , and the transi-tion probability P ( α → β ) to flavor β , we need h α | ψ ( t ) i and h β | ψ ( t ) i . The symmetric and antisymmetric parts ofthese inner products are g ( α, ψ ( t )) = ~α · ~ψ ( t ) + ~α · ~ψ ( t ) = c θ c + s θ c , (15) ε ( α, ψ ( t )) = ~α × ~ψ ( t ) + ~α × ~ψ ( t ) = − c θ s − s θ s ,g ( β, ψ ( t )) = ~β · ~ψ ( t ) + ~β · ~ψ ( t ) = − s θ c θ c + s θ c θ c , ε ( β, ψ ( t )) = ~β × ~ψ ( t ) + ~β × ~ψ ( t ) = s θ c θ s − s θ c θ s , and the survival and transition probabilities will be P ( α → α ) = |h α | ψ ( t ) i| = g ( α, ψ ( t )) + ε ( α, ψ ( t )) = 1 − P ( α → β ) ,P ( α → β ) = |h β | ψ ( t ) i| = g ( β, ψ ( t )) + ε ( β, ψ ( t )) = sin θ sin (cid:20) ( ω − ω ) t (cid:21) . (16)Making the relativistic replacement ω i t → ( ω i t − k i L ) natural units −−−−−−−−→ ( E i t − p i L ) , (17)and assuming that the energies of the two states are com-mon, E = E = E ≫ m i , we have( Et − p i L ) ≈ E ( t − L ) + m i E L , (18)leading to the identification( ω − ω ) t → δm E L ≡ ∆ . (19)This gives us the familiar neutrino oscillation formula P ( α → β ) = sin θ sin ∆ . (20) Nambu QM:
The deformation of canonical QM which isdetailed in Ref. [24], i.e.
Nambu QM, can be summarizedas follows. Extend the two-component real vector ~ψ n introduced above to a three-component real vector ~ Ψ n : ~ψ n → ~ Ψ n . (21)In the two-component case, the components evolved asEq. (6). For the three-component extension, it is assumedthat ~ Ψ n ( t ) = N n c ξ cn(Ω n ( t − t n ) , k ) − κc ξ sn(Ω n ( t − t n ) , k ) − s ξ dn(Ω n ( t − t n ) , k ) , (22)where cn( u, k ), sn( u, k ), and dn( u, k ) are Jacobi’s el-liptical functions [27], and s ξ = sin ξ , c ξ = cos ξ , and κ = p k tan ξ . The period of cn( u, k ) and sn( u, k )in u is 4 K , where K = K ( k ) is the complete ellipticalintegral of the first kind [28], and Ω n = (2 K/π ) ω n . Thetwo parameters k and ξ are the deformation parameters,and when they are both set to zero, the time evolution ofthe first two components of ~ Ψ n reduce to that of the twocomponents of ~ψ n , while the third component of ~ Ψ n van-ishes. In principle, we can make the deformation param-eters k and ξ depend on n , but for the sake of simplicity,we keep them common to all n .Note that the time evolution assumed in Eq. (22) isthat of the angular momentum vector ~L of a free asym-metric top in its co-rotating frame [29]. Though theequations that govern this motion are non-linear (ormore precisely multi-linear ), the presence of the two con-served quantities of energy E and angular momentum L renders the motion solvable, norm-preserving, and pe-riodic. ~L evolves along the intersection of the sphere L = constant and the ellipsoid E = constant. Due tothe norm preserving nature of Eq. (22), this time evo-lution is unitary . However, the time evolution operatorcannot be expressed as a matrix as in canonical QM (ex-cept when k = 0) due to the evolution being non-linear.In essence, the “phase” of the state evolves periodicallyon S instead of on S .The symmetric and antisymmetric parts of the innerproduct between two states are extended to g (Ψ , Φ) = X n ( ~ Ψ n · ~ Φ n ) ,~ε (Ψ , Φ) = X n ( ~ Ψ n × ~ Φ n ) , (23)where the dot and cross products are now defined inthree dimensions. Consequently, ~ε has three compo-nents, which in the k = ξ = 0 limit reduces to X n ( ~ Ψ n × ~ Φ n ) k = ξ =0 −−−−→ P n ( ~ψ n × ~φ n ) . (24)The square of the absolute value of h Ψ | Φ i is extended to |h Ψ | Φ i| = g (Ψ , Φ) + ~ε (Ψ , Φ) · ~ε (Ψ , Φ) , (25)which is invariant under 3D rotations of the “phase”space. This expression allows us to make predictionsbased on Nambu QM. Since the deformation is in thetime-evolution of the “phase” of each energy eigenstate,we can expect deviations from canonical QM to occurin phenomena that involve the evolution of interferenceterms.Consequently, let us look at oscillation in Nambu QM.We consider flavor eigenstates | α i and | β i to be super-positions of energy eigenstates | i and | i as in Eq. (9).The three-component vector notation of | α i and | β i areformally the same as Eq. (10), except with ~n replaced by the three component object ~n = c ξ − s ξ . (26)This corresponds to a “zero phase” state. To clarify thatwe are working in the three-component formalism, wewill replace the label α with A , and β with B in thefollowing.Let | Ψ(0) i = | A i , that is: ~ Ψ (0) = c θ ~n , ~ Ψ (0) = s θ ~n . (27)At a later time t , these will evolve to ~ Ψ ( t ) = c θ ~n ( t ) , ~ Ψ ( t ) = s θ ~n ( t ) , (28)where ~n i ( t ) = c ξ cn i − κc ξ sn i − s ξ dn i , (29)with sn i = sn(Ω i t, k ), cn i = cn(Ω i t, k ), dn i = dn(Ω i t, k ).The symmetric parts of h A | Ψ( t ) i and h B | Ψ( t ) i are g ( A, Ψ( t )) = ~A · ~ Ψ ( t ) + ~A · ~ Ψ ( t ) (30)= c θ (cid:16) c ξ cn + s ξ dn (cid:17) + s θ (cid:16) c ξ cn + s ξ dn (cid:17) ,g ( B, Ψ( t )) = ~B · ~ Ψ ( t ) + ~B · ~ Ψ ( t )= − s θ c θ (cid:16) c ξ cn + s ξ dn (cid:17) + s θ c θ (cid:16) c ξ cn + s ξ dn (cid:17) , while the antisymmetric parts are ~ε ( A, Ψ( t )) = ~A × ~ Ψ ( t ) + ~A × ~ Ψ ( t ) (31)= c θ − κs ξ c ξ sn s ξ c ξ (cid:0) dn − cn (cid:1) − κc ξ sn + s θ − κs ξ c ξ sn s ξ c ξ (cid:0) dn − cn (cid:1) − κc ξ sn ,~ε ( B, Ψ( t )) = ~B × ~ Ψ ( t ) + ~B × ~ Ψ ( t )= − s θ c θ − κs ξ c ξ sn s ξ c ξ (cid:0) dn − cn (cid:1) − κc ξ sn + s θ c θ − κs ξ c ξ sn s ξ c ξ (cid:0) dn − cn (cid:1) − κc ξ sn . From these expressions, we find the survival and transi-tion probabilities to be P ( A → A ) = g ( A, Ψ( t )) + ~ε ( A, Ψ( t )) · ~ε ( A, Ψ( t )) = 1 − P ( A → B ) ,P ( A → B ) = g ( B, Ψ( t )) + ~ε ( B, Ψ( t )) · ~ε ( B, Ψ( t ))= sin θ " − (cid:8) c ξ (cid:0) sn sn + cn cn (cid:1) + s ξ (cid:0) k sn sn + dn dn (cid:1)(cid:9) . (32)For the ease of comparison with Eq. (20), we expand theJacobi functions in powers of k [30–32] :sn(Ω t, k ) = (cid:18) k (cid:19) sin( ωt ) + k
16 sin(3 ωt ) + · · · , cn(Ω t, k ) = (cid:18) − k (cid:19) cos( ωt ) + k
16 cos(3 ωt ) + · · · , dn(Ω t, k ) = (cid:18) − k (cid:19) + k ωt ) + · · · , (33)from which we find to order k sn sn + cn cn = cos ∆ − k sin ∆ ,k sn sn + dn dn = 1 − k (cid:0) (cid:1) sin ∆ , (34)where ∆ = ( ω − ω ) t and Σ = ( ω + ω ) t . Averagingover time makes the cos Σ terms vanish. Therefore, P ( A → B ) = (cid:18) c ξ + s ξ k (cid:19) sin θ sin ∆ . (35)Note that 0 ≤ k <
1. Thus, the effect of the Nambudeformation is an overall suppression factor compared tothe undeformed canonical case, Eq. (20). This is themain result of this letter.
Discussion:
In this letter, we have considered NambuQM, a deformation of canonical QM in which the “phase”space of energy eigenstates is enlarged from 2D to 3D,and the “phase” dynamics is deformed from that of a har-monic oscillator to that of an asymmetric top. This de-formation maintains the Born rule, i.e. the conservationof norm, which is embedded in the “classical” dynamicsof the “phase”. The invariance of physical predictionson 2D rotations of the “phase” space is modified to thatunder 3D rotations, a feature responsible for the projec-tivity of the state space. (Note that we cannot associatea “phase” shift with a constant shift of Ω t in Eq. (22).)This invariance can, in principle, be gauged in the fieldtheoretic version of Nambu QM, but since the symmetryis SO (3), it could lead to non-abelian features thoughonly the “phase” of a single field will be gauged. The S geometry of the “phase” space, as opposed to the canon-ical S , also suggests that the path integral of NambuQM is not the usual integral of e iS .We have investigated the effect of the Nambu QM de-formation on oscillation phenomena, relevant e.g. forneutrinos and neutral mesons [33], and have derived anexplicit formula, Eq. (35), involving the deformation pa-rameters k and ξ . Given that the sin θ term can-not increase beyond one, the suppression factor cannotalways be absorbed into θ . For instance, the currentbound on the atmospheric mixing angle gives sin θ > . , . , .
952 respectively at 1, 2, and 3 σ for nor- mal ordering [34]. This indicates s ξ (cid:18) − k (cid:19) < .
027 (1 σ ) , .
037 (2 σ ) , .
048 (3 σ ) , (36)though the value of θ itself is not yet precisely known.Future improvements in the determination of θ at Ice-Cube [35], JUNO [36], and DUNE [37] could improveupon this bound.Note that oscillation is but one possible phenomenonthat could be affected by Nambu QM. There may bemany others involving interference and the resulting cor-relations given that the “phase” vectors are assumed tomove in a very particular way on S . They could shednew light on various issues in quantum foundations andin entanglement, and call for a thorough investigation.Apart from these phenomenological considerations, wewould like to highlight the fact that the original paper ofNambu [25] has inspired very many works on the math-ematical and foundational nature of the Nambu bracket,Eq. (2), and its related structures [38–40], and on thequantization of those structures and their relevance instring theory (see [41–52] and references therein). Morerecently, such a structure was discovered [53–56] in thecontext of a new formulation of non-perturbative stringtheory and quantum gravity based on quantum space-time [57–64].We also note that analogies with the asymmetric topare ubiquitous in various classical and quantum physi-cal systems [29]. This has particularly been the case inphenomenological particle physics. What we have un-covered here in Nambu QM relates closely to, and for-mally extends in a new direction the top like Hamilto-nians used in dynamical models of neutrino oscillations[65, 66]. They belong to the family of integrable quantumspin Gaudin models of wider applications in condensedmatter physics. The time oscillatory features we havededuced in this letter along with the above mentionedconnection further suggests the construction and phe-nomenological testing of a family of dynamical, integrable SO (3) Nambu top models of, say, neutrinos oscillationswith, not a trigonometric but a novel Jacobian elliptictime evolutions with two periods – presumably with oneperiod being much, much smaller than the other. Theywould add a new prediction for neutrino oscillations inour quest to see theoretically and experimentally beyondthe Standard Model. Acknowlegments:
We thank P. Huber and R. Pestesfor helpful discussions. DM and TT are supported in partby the DOE (DE-SC0020262). DM is also supported bythe Julian Schwinger Foundation, and TT by the NSF(PHY-1413031).We dedicate this letter to the memory of Prof. YoichiroNambu, a great physicist and a wonderful human being,on the occasion of his upcoming centennial in 2021. ∗ [email protected] † [email protected] ‡ [email protected][1] I. H. Deutsch, PRX Quantum , 020101 (2020).[2] J. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press).[3] Y. Aharonov and D. Rohrlich,
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