Small magnetic charges and monopoles in non-associative quantum mechanics

Weak magnetic monopoles with a continuum of charges less than the minimum implied by Dirac's quantization condition may be possible in non-associative quantum mechanics. If a weakly magnetically charged proton in a hydrogen atom perturbs the standard energy spectrum only slightly, magnetic charges could have escaped detection. Testing this hypothesis requires entirely new methods to compute energy spectra in non-associative quantum mechanics. Such methods are presented here, and evaluated for upper bounds on the magnetic charge of elementary particles.

In 1931, Dirac [1] showed that magnetic monopoles with charge g can be consistently described by wave functions provided the quantization condition eg = N holds with half-integer N . Since the elementary electric charge e (or, rather, the fine structure constant) is small, the elementary magnetic charge is large. Therefore, there are strict limits on the possible magnetic charge of, say, a proton in a hydrogen nucleus because the strong magnetic charge would significantly alter the energy spectrum [2].
The aim of this letter is to point out and analyze the fact that Dirac's argument relies on properties of wave functions in a Hilbert space, and therefore implicitly assumes that quantum mechanics is associative. If the assumption of associativity is dropped, there is no Hilbertspace representation of the algebra of observables (which by necessicity would always be associative), but quantum mechanics may still be meaningful [3][4][5][6]. Indeed, the existence of consistent non-associative algebras for magnetic charge densities has recently been demonstrated [7][8][9][10][11]. Non-associative quantum mechanics can therefore be defined by replacing the operator product of observables with an abstract product, such thatâ 1 (â 2â3 ) = (â 1â2 )â 3 in general. States are defined as expectation-value functionals that assign complex numbers â to algebra elementsâ, subject to certain consistency conditions which make sure that uncertainty relations are respected. No wave functions appear in this formalism, and there is no analog of "single-valuedness" used crucially by Dirac. Without wave functions, Dirac's argument therefore loses its footing. Magnetic monopoles are then possible with small charges much less than the smallest non-zero value, g 0 = 1 2 /e, allowed by Dirac. It is conceivable that a small magnetic charge of the proton could have escaped detection in precision spectroscopy such as [12].
Here, we show that even a small magnetic charge of the nucleus would significantly shift the ground-state energy of a hydrogen atom. To the best of our knowledge, this is the first time that properties of energy spectra have been computed in non-associative quantum mechanics. We provide new methods to compute spectra in an algebraic manner, which may also be useful in other contexts.
Harmonic oscillator: We first demonstrate the new methods in an application to the harmonic oscillator in standard, associative quantum mechanics. We have two distinguished observablesq andp with [q,p] = i , and the quantum HamiltonianĤ = 1 2 (p 2 /m + mω 2q2 ). An eigenstate |ψ E ofĤ with eigenvalue E obeys the equationĤ|ψ E = E|ψ E , which implies for the expectation value · E taken in |ψ E , whereâ can be any polynomial inq andp. We will first show that (27), which amounts to infinitely many equations given the freedom of choosingâ, allows one to compute the spectrum ofĤ even if the eigenstates |ψ E are not known. In [13,14], it has been shown how observables can be computed using algebraic relations between moments of a state. The methods used here are closely related to these papers but provide a new application to energy spectra. In this way, we will set up a method to compute eigenvalues without using wave functions or boundary conditions. The same method can then be applied to the Coulomb problem in non-associative quantum mechanics. The demonstration is based on recurrence with respect to the degree of the polynomialâ inq andp. The groundstate energy can be obtained by elementary calculations as follows: First,â =1 (the identity operator) gives E = 1 2 ( p 2 E /m + mω 2 q 2 E ). Forâ not the identity, it is useful to refer to the equation with the complex conjugatez of a complex number z. In particular, [q,Ĥ] E = i p E /m = 0 from a =q and [p,Ĥ] E = −i mω 2 q E = 0 fromâ = p. From quadratic monomials, we obtain [q 2 ,Ĥ] E = 1 2 i qp +pq E /m = 0 and [qp,Ĥ] E = i ( p 2 E /m − mω 2 q 2 E ) = 0. Therefore, any eigenstate has fluctuations obeying ∆ E p = mω∆ E q, and zero covariance 0 = C E qp = 1 2 qp +pq E − q E p E . From the condition forâ =1, (∆ E q) 2 = E/(mω 2 ) and (∆ E p) 2 = mE.
So far, we have computed moments of a bound state in terms of its energy value E. We obtain a restriction on E by making sure that the fluctuations we derived obey the uncertainty relation: and therefore E ≥ 1 2 ω. In order to evaluate all the conditions imposed on eigenstates by (27), we follow [15,16] and introduce the operatorsT m,n := (q mpn ) Weyl where m and n are non-negative integers, and the subscript indicates that the product is taken in the totally symmetric ordering. The Hamiltonian is a linear combinationĤ = 1 2 (T 2,0 /m + mω 2T 0,2 ) ofT 2,0 andT 0,2 , and therefore (27) contains products of the formT m,nTm ′ ,n ′ . Using the basic commutation relation ofq andp, such products can always be rewritten as sums over individualT m ′′ ,n ′′ of order m + n + m ′ + n ′ or less, as derived explicitly in [17]. The condition (27) is therefore equivalent to a recurrence relation for T m,n E which is shown and discussed in more detail in our supplementary material. (This material also uses an algebraic notion of states [18] and makes contact with effective constraints [19,20].) In addition to higher-order moments T m,n E of an eigenstate, we have higher-order uncertainty relations. They can be obtained just like Heisenberg's version, by applying the textbook derivation to integer powers ofq andp or their products instead of justq andp. A systematic procedure to organize these higher-order, or generalized, uncertainty relations has been given in [15,16]. For our purposes, a subset of these relations is sufficient, which can be constructed as follows: We defineξ J as the 2J-dimensional column vector consisting of allT m,0 and T m−1,1 up to order m = 2J, where J is an integer or halfinteger. According to the generalized uncertainty principle, the matrix M J = ξ Jξ † J is positive semi-definite for all J, where the expectation value is taken element by element. For J = 1/2, we have Heisenberg's uncertainty principle because a positive semi-definite matrix has a non-negative determinant.
As outlined in the supplementary material, positive semi-definiteness of M J can be reduced to the conditions for all integer n ≥ 1, where α k = (2k − 1)/2 are the odd half-integer multiples. Considered as functions of E for all n, these expressions have nodes at ωα k up to some maximum k that depends on the particular value of n. Between nodes, the functions are non-zero and alternate in sign. Moreover, sending n to n + 1 causes the signs at fixed E to alternate. This behavior combined with the non-negativity of (4) implies that the only allowable values for E occur at the nodes. We can exclude negative values of E because we have already shown that E ≥ 1 2 ω. Thus, the only possible values for E are such that E/ ω = 1 2 , 3 2 , 5 2 , . . . in agreement with the well-known eigenvalues of the harmonic oscillator.
Moreover, the arguments just given show that, for each eigenvalue E n = (n − 1 2 ) ω, there is a generalized uncertainty relation which restricts higher-order moments and is saturated by the corresponding excited state with energy E n . This result generalizes the well-known statement that the ground state of the harmonic oscillator saturates Heisenberg's uncertainty relation. Also note that our derivation, based on expectation values, still applies if the state used is mixed, given by a density matrix.
Since we obtain the usual energy spectrum of the harmonic oscillator, it follows that mixed states do not to enlarge the spectrum.
As another consequence, we obtain the full energy spectrum of the harmonic oscillator from the unfamiliar condition (27) on energy eigenvalues. This result serves as a proof of concept of the new algebraic method introduced here, which we now apply to the Coulomb problem. We will then be ready to generalize the results to nonassociative quantum mechanics, where the usual methods of computing eigenvalues are not available.
Hydrogen: The hydrogen atom has the Hamiltonian is more useful. Closely related variables have been used, quite differently, to compute hydrogen spectra in deformation quantization [21][22][23]. These operators have linear commutation relations and there is a Casimir operator that commutes withr,P andQ. A direct calculation in terms of the position and momentum components in (5) shows thatK is equal to the total angular momentum squared. We should keep in mind that not all the distinguished observables are self-adjoint. We do haver † =r, butQ † =Q − 3i and As in our demonstration using the harmonic oscillator, we will be interested in expectation values of monomials inr,P andQ evaluated in eigenstates that obey (27). We have another useful relationship between certain expectation values given by the virial theorem: The procedure used for the harmonic oscillator does not directly apply to the Coulomb problem because the Hamiltonian is no longer quadratic, leading to highly coupled recurrence relations. We therefore reformulate the condition (27) in terms of a constraint linear inP andr, introducingĈ The condition on the spectrum ofĤ then takes the form âĈ E E = 0 for all polynomialsâ inr,r −1 ,P andQ. Unlike the Hamiltonian,Ĉ E is not self-adjoint. It is still useful to apply commutator identities as in (2), but with a non-self-adjointĈ E , there are additional terms: In an eigenstate such that âĈ E E = 0 and â †Ĉ E E = 0, Witĥ using (8), we have Forâ =Q, If we replaceQ 2 using the Casimir operatorK, and P E using Ĉ E E = 0, we have 0 = 3α + 4E r E − K ℓ r −1 E /m. The eigenvalues K ℓ = ℓ(ℓ + 1) 2 ofK follow from angular-momentum quantization, and r −1 E is related to E by (9). With these ingredients and similar calculations forâ =rQ, we obtain In order to determine the allowed eigenvalues E, as before, we have to impose uncertainty relations. We are interested here in the ground state, for which we can focus on the lowest-order uncertainty relations, computed for our non-canonical operatorsr,P andQ using the Cauchy-Schwarz inequality. There is only one non-trivial relation, with two covariances. Again using (13), we compute Inserting all the required moments and factorizing the resulting polynomial in E, (16) gives the condition It is saturated for all energy eigenvalues with maximal ℓ, for which Assuming the well-known degeneracy of the hydrogen spectrum, we obtain the full set of bound-state energies.
As in the example of the harmonic oscillator, every eigenstate saturates an uncertainty relation, in this case (16). Non-associative hydrogen: We are now in a position to derive our main result. In the presence of a magnetic central charge, we cannot use canonical momenta because they require a vector potential of the magnetic field B. Instead, we generate an algebra using kinematical electron momenta, quantizing p i = mẋ i . Their commutators are obtained by generalizing the case in which there is a vector potential A depending on x, and canonical mo-menta are π i = p i + eA i . Therefore, while [x j ,p k ] = i δ jk is unchanged. The final result depends only on B and therefore can be used to define the commutators [p j ,p k ] also if ∇· B = 0 in the presence of magnetic charges. A direct calculation shows that these commutators then no longer obey the Jacobi identity: Even a single point-like monopole cannot be excised, as in Dirac's construction, if we consider weak charges that do not obey the quantization condition. However, a nonassociative algebra generated by commutingx i and noncommutingp j , with standard commutators betweenx i andp j , is still meaningful [3,4]. Another direct calculation shows that the commutators of (r,Q,P ) remain unchanged provided that r × B = 0. This result, which relies on unexpected cancellations of the extra terms in commutators implied by (19), is crucial for the new application in this letter. In this case, B = g( r) r. For a static magnetic field, we have ∇ × B = 0, which implies that g(r) is spherically symmetric. A monopole density ∇ · B = 0 then requires that g(r) = Q m (r)/(4πr 3 ) with the magnetic charge Q m (r) = 4π ∇ · B(r)r 2 dr (21) enclosed in a sphere of radius r. For a single monopole at r = 0, g(r) = g is constant.
The virial theorem relies only on algebraic properties and remains valid. With monopole commutators for momentum components, however, the modified angular momentumˆ L ′ =ˆ L + egˆ r/r, notˆ L itself, satisfies the usual commutators of angular momentum [24,25]. The Casimir of the algebra generated by (r,Q,P ) is still equal toK =ˆ L 2 , but in terms of the modified angular momentum it has an extra term: For a single monopole at the center, the spectrum ofK has a simple shift compared with the standard spectrum ofL 2 , which is known to break the ℓ-degeneracy of the hydrogen spectrum [2]. Moreover, the allowed values of ℓ are restricted for non-zero g becauseK, by definition, is positive, and so must be its eigenvalues. Therefore, ℓ = 0 is not possible for g = 0, and larger ℓ may be ruled out as well for strong magnetic charges. We will focus now on the range of weak magnetic charges given by None of these values could be modeled by a Dirac monopole (they would not correspond to single-valued wave functions), but they can be considered if quantum mechanics is non-associative. Since the algebraic relations used to derive (17) are still applicable, we obtain conditions on the energy spectrum. The only difference is that the eigenvalues ofK are now given by K ℓ = ℓ(ℓ + 1) 2 − e 2 g 2 , which can be taken into account by replacing ℓ in (17) with For quantized magnetic charges, the corresponding eigenvalues for which the first parenthesis in (17) is zero are indeed included in the spectrum found in [2], but they no longer constitute the full spectrum. For weak magnetic charges, positivity ofK requires that the smallest possible ℓ is ℓ = 1/2, which we use for the ground state. The correspondingl is equal tõ and lies in the range 1 2 ( √ 3 − 1) <l < 1 2 . This range does not come close to the integer values 0 or 1 which would amount to standard hydrogen eigenvalues. Therefore, even for weak magnetic monopoles the energy spectrum of hydrogen is strongly modified. The ground-state energy is discontinuous in the central magnetic charge as a consequence of the positivity condition K ≥ 0, which is the reason why even a small magnetic charge is not a simple perturbation of the usual hydrogen spectrum.
This result would seem to rule out any non-zero magnetic charge of the proton. However, from a purely experimental perspective, the smallest eigenvalue of the total angular momentum, used in our evaluation of K ≥ 0, is zero only within some uncertainty. The angular momentum spectrum is very basic and hard to modify. For instance, the conservation law and its role played in parity considerations implies that, for a single component, it has the form of a ladder centered around zero. It is, however, conceivable that its values are washed out to within some δL 2 . To estimate this quantity, we are not restricted to hydrogen-like systems because all energy levels depend in some way on the eigenvalues ofL 2 . The best relative precision, of about 5 · 10 −19 , is obtained for spectral lines used in atomic clocks [26]. In SI units, a non-zero upper bound g ≤ 4πǫ 0 √ δL 2 c 2 e ≈ 4.7·10 −18 Am = 1.4·10 −9 g Dirac (26) then follows from K ≥ 0 and (22), where g Dirac is the smallest magnetic charge allowed by Dirac.
For the proton, this bound is not as strong as existing ones [27,28]. However, the bounds in [27,28] are obtained by limiting the total magnetic charge of a macroscopic object, adding the individual charges of all electrons or nucleons. Our bound is obtained directly for a single proton. Moreover, the magnetic charge of the muon is more difficult to bound [28]. Our bound, on the other hand, also applies to a muon as the nucleus of muonium, and to antimatter such as the antiproton in antihydrogen [29,30] or the positron in positronium [31].

Supplementary material
For an application to non-associative quantum mechanics, we present here details of the systems considered in the letter in algebraic notation.
Harmonic oscillator: The quantum harmonic oscillator is defined by an algebra generated by two distinguished observablesq andp with [q,p] = i, containing the HamiltonianĤ = 1 2 ω(p 2 +q 2 ). Compared with the usual expressions, we have applied a linear transformationq = mω/ q,p =p/ √ mω to simplify the original Hamiltonian 1 2 (p 2 /m + mω 2q2 ). The same transformation removes from the commutator.
The algebra is equipped with an adjointness relation such thatq † =q andp † =p, which impliesĤ † =Ĥ. A quantum state, in algebraic terms [18], is a positive linear map Ω from the algebra to the complex numbers, such that Ω(â †â ) ≥ 0 for all algebra elementsâ. The positivity condition imposes uncertainty relations, and it also implies that the state is normalized, Ω(1) = 1. If the algebra is represented on a Hilbert space, any wave function |ψ or density matrixρ defines such a map by Ω |ψ (â) = ψ|â|ψ or Ωρ(â) = tr(ρâ), respectively. The algebraic notion, however, can also be used in cases without a Hilbert space, as in non-associative quantum mechanics. As in the letter, we define an energy eigenstate Ω E ofĤ with eigenvalue E by the condition Ω E (â(Ĥ − E)) = 0 (27) for all algebra elementsâ, or all polynomials inq andp. Following [15,16], we define the algebra elementŝ T m,n := (q mpn ) Weyl where m and n are non-negative integers, and the subscript indicates that the product is taken in the totally symmetric ordering. Through the basic commutation relation, products of the form T m,nTm ′ ,n ′ can always be rewritten as sums over indi-vidualT m ′′ ,n ′′ of order m + n + m ′ + n ′ or less; see [17] for an explicit statement of the relevant reordering identity. For an eigenstate ofĤ with eigenvalue E, we have Ω E (T m,n (Ĥ − E)) = 0 for all m, n ≥ 0. Sincê H = 1 2 ω(T 2,0 +T 0,2 ), this equation provides recurrence relations for Ω E (T m,n ): and hold for all m, n ≥ 0. From the second relation, starting with m = 0 or n = 0, we see that the moments are zero unless both m and n are even. Only moments of the form Ω E (T 2m,2n ) are therefore non-zero. Since (28) is symmetric under switching m and n, Ω E (T 2m,2n ) is a function only of m + n. Starting with the initial condition Ω E (T 0,0 ) = 1 using the normalization condition of states, (28) implies where the a E ℓ are determined by the recurrence relation in a single independent variable, ℓ, with initial conditions a E 0 = 1 and a E 1 = E/ ω which follow from normalization and our leading-order results presented in the letter, respectively. Equations (30) and (31) determine all orders of moments in terms of E/ ω. It follows that a E ℓ is a polynomial in E/ ω of degree ℓ, with only even terms for ℓ even and only odd terms for ℓ odd.
So far, there is no restriction on E because we have not yet imposed the positivity condition on Ω E , or uncertainty relations. Positivity can efficiently be formulated using the (J + 1)(2J + 1)-dimensional column vector,ξ J , consisting of allT m,n up to order m + n = 2J, where J is an integer or half-integer. Again following [15,16], if a linear map Ω E is positive, the square matrix is positive semi-definite, where Ω E is applied element by element. It turns out that the positivity condition onM J is redundant as far as computing the spectrum ofĤ is concerned. The conditionM J ≥ 0 implies that M J ≥ 0, where M J is a matrix formed by deleting fromM J any number of rows and their corresponding columns. Equivalently, M J is the matrix formed by deleting entries from ξ J to form a new vectorξ J and then taking In particular, consider the matrix M J formed by takinĝ ξ J to contain only operators of the formT m,0 andT m−1,1 . The recurrence relation (31) implies relations between some of the components of M J , which can be exploited to bring this matrix to block diagonal form, (A 0 , . . . , A 2J ), with 2 × 2-matrices A n . Then, M J ≥ 0 implies that A n ≥ 0 for all n ≥ 0.
For a fixed n, this is a constraint involving moments up to order 2n, which can in turn be written in terms of E/ ω using (30) and (31). Evaluating the determinants of A n , det A n (E) = 1 4 n−1 n k=1 (E/ ω − α k )(E/ ω + α k ) ≥ 0 (35) with α k = (2k − 1)/2, we obtain the energy spectrum as shown in the letter.
Hydrogen: In the letter, we have shown the relevant details for the Coulomb problem. Here, we give an alternative demonstration of the commutator relationship used to compute moments of eigenstates. The derivation here illustrates how our method in this example is related to quantum constrained systems, which have been analyzed in terms of moments in [19,20]. These methods may be useful for further derivations.
If Ω E (âĈ E ) = 0, Ω E (âF E (ǫ)) = Ω E (â) withF E (ǫ) = exp(−iǫĈ E / ), for all real ǫ. The state Ω E is therefore invariant under the flow Ω E (Ô)[ǫ] = Ω E (F E (ǫ) †ÔF E (ǫ)) generated byĈ E , akin to the evolution generated by a HamiltonianĤ. However, sinceĈ E is not self-adjoint, F E (ǫ) is not unitary. Moreover, the infinitesimal flow dΩ E (Ô)[ǫ]/dǫ is not equal to the commutator [Ô,Ĉ E ], evaluated in Ω E and divided by i , but is given by Witĥ = Ω E (P ) 2m as evaluated in the letter. We have used angular-momentum eigenvalues in order to determine eigenvalues of the Casimir operatorK. In the non-associative model, in particular, we applied the standard eigenvalues to the modified angular momentum L ′ which obeys the same commutator algebra as the stan-dardˆ L. This step might be questioned becaue we do not have a Hilbert-space representation of a non-associative algebra, and therefore it is not obvious how to apply the standard derivation, for instance using ladder operatorsL ′ ± . However, a brief argument shows that the usual eigenvalues are still correct: An algebraic state ω, that is, a mapping from the algebra of observables to the complex numbers, induces a state on any unital subalgebra, such as the algebra spanned by the components ofˆ L ′ together with the identity operator. Since this subalgebra is associative even if the full algebra is non-associative, we can use the Gelfand-Naimark-Segal (GNS) construction [18] to induce a Hilbert-space representation. In this representation, the standard derivation of eigenvalues applies, such that the modified angular momentumˆ L ′ of a non-associative theory does not have modified eigenvalues.