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Journal of Generalized Lie Theory and Applications
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Hidden structures in quantum mechanics


Department of Physics and Microelectronics Engineering, Kyrgyz-Russian Slavic University Bishkek, Kievskaya Str. 44, 720021, Kyrgyz Republic

*Corresponding Author:
Department of Physics and Microelectronics Engineering,
Kyrgyz-Russian Slavic University Bishkek,
Kievskaya Str. 44, 720021, Kyrgyz Republic
E-mail: [email protected]

Received date: September 21, 2008 Accepted Date: November 29, 2008

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It is shown that some operators in quantum mechanics have hidden structures that are unobservable in principle. These structures are based on a supersymmetric decomposition of the momentum operator, and a nonassociative decomposition of the spin operator.


A majority of physicists these days believes that quantum mechanics does not have a hidden structure: Experiments have shown that a vast class of hidden variable theories is incompatible with observations. In essence, these theories assume existence of some hidden variables behind quantum mechanics, which could be measured in principle. In this paper we would like to show that in quantum mechanics there exist hidden structures based on a supersymmetric decomposition of the momentum operator, and a nonassociative decomposition of the spin operator. The constituents of this nonassociative decomposition are inaccessible to the experiment, because nonassociative parts of operators are unobservable.

In Ref. [1] the attempt is made to introduce a nonassociative structure in quantum chromodynamics. As a consequence, not all states in the corresponding octonionic Hilbert space will be observable, because the propositional calculus of observable states as developed by Birkho and von Neumann [2] can only have realizations as projective geometries corresponding to Hilbert spaces over associative composition algebras, whereas octonions are nonassociative. An observable subspace arises in the following way: Within Fock space there will be states that are observable (longitudinal, in the notation of Ref. [1]), which are the linear combinations of u0 and equation Conversely, the states in transversal direction (spanned by ui and equation) are unobservables (equationare split octonions).

A hidden structure in supersymmetric quantum mechanics is found in Ref. [3]. There, the Hamiltonian in supersymmetric quantum mechanics is decomposed as a bilinear combination of operators built from octonions, a nonassociative generalization of real numbers.

In some sense, the Maxwell and Dirac equations have hidden nonassociative structures as well. In Ref's [4] and [5] it is shown that: (a) classical Maxwell equations can be written as the single continuity equation in the algebra of split octonions, and (b) the algebra of split octonions suces to formulate a system of di erential equations equivalent to the standard Dirac equation.

In this paper we present hidden structures in traditional quantum mechanics. These structures are based on a supersymmetric decomposition of the momentum operator, and a nonassociative decomposition of the spin operator. We would like to emphasize that a "hidden nonassociative structure" presented here is not the same as a "hidden variable theory", because the nonassociative constituents of a hidden structure can not be measured in principle, in contrast to hidden variables which can be measured in principle.

Througout the paper we use notions from the textbooks for nonassociative algebras [8,9], the physical applications of nonassociative algebras in physics can be found in [10] and [11,12].

A supersymmetric decomposition of the momentum operator

The Poincare algebra is de ned with the generators Mμv Pμ and the following commutator relations


where equation is the Minkowski metric, Pμ are generators of the translation group, and Mμv = xμPv − xvPμ are generators of the Lorentz group.

The simplest supersymmetric algebra is de ned as follows




having indicesequation are the Pauli matrices. The relation (2.2) can be inverted as follows:


Equation (2.3) can be interpreted as a \square root" of the quantum mechanical momentum operator Pμ. It allows us to bring forward the question: Is it possible to decompose other operators in quantum mechanics in a similar manner, for example, spin equation and angular momentum Mμv?

The undotted indices are raised with


The dotted indices are lowered with


A nonassociative decomposition of the spin operator

Let us consider the split-octonion numbers, designated as qi (i = 1; 2; .,.,.,7). Table 1 represents the chosen multiplication rules for the qi.


Table 1: The split-octonion multiplication table

The split-octonions have the following commutators and associators

[qi+3; qj+3] = −2εijkqk (3.1)

[qi; qj ] = 2εijkqk (3.2)

(qi+3; qj+3; qk+3) = (qi+3qj+3) qk+3 − qi+3 (qj+3qk+3) = 2εijkq7

here i; j; k = 1; 2; 3. The commutator (3.2) shows that qi; i = 1; 2; 3 form a subalgabra. This subalgebra is called quaternion algebra H; qi are quaternions.

The commutator (3.2) can be rewritten in the form


which is similar to the commutator relationship for spin operators equation (i are the Pauli matrices). It allows us to say that nonrelativistic spin operators have a hidden nonassociative structure (3.1). The multiplication table 1 shows that the nonrelativistic spin operator can be decomposed as the product of two nonassociative numbers


In order to see it more concisely, let us represent the split-octonions via the Zorn vector matrices


where a; b are real numbers and equation are 3-vectors, with the product de ned as


Here, equation denote the usual scalar and vector products.

If the basis vectors of 3D Euclidean space are equation then we can rewrite the split-octonions as matrices


here i = 1; 2; 3: Thus, the nonrelativistic spin operators have two representations: The first one is as Pauli matrices equation with the usual matrix product, and the second one is as Zorn matrices equation with nonassociative product (3.4). In the second case, the spin is decomposed into a product (3.3) of two unobservables qj+3; qk+3.

Quantum mechanical applications

In the supersymmetric approach, the operators Pμ with commutator relations (2.1) are generators of the Poincare group. We interpret these as quantum mechanical operators, and consider nonrelativistic quantum mechanics. Taking the decomposition (2.3) and (3.3) into account, we have


We o er the following interpretation of Eq's (4.1): Quantum mechanics has hidden supersym- metric and nonassociative structures, which can be expressed through decomposition of classical momentum and spin operators, into bilinear combinations of some operators that are either supersymmetric or nonassociative.

Probably, such nonassociative hidden structure can not be found experimentally in principle, because the nonassociative parts q5;6;7 generate unobservables (for details of unobservability, see Ref. [6]).

Discussion and conclusions

We have shown that some quantum mechanical operators can be decomposed into supersymmetric and nonassociative constituents. The following questions outline further investigation in this direction:

1. Does a 4D generalization of relations (4.1) exist?

2. Do Qa;Q_a have dynamical equations?

3. Is a similar nonassociative decomposition of quantum eld theory possible?

The rst question can be formulated mathematically as follows: Find a nonassociative algebra R, with commutators and associators


where equation Also, ask whether a linear representation for equation exists. This question arises, because supersymmetric operators equation have a linear representation


These operators are generators of translation, in a superspace with coordinates zM = (xμ; θa; θa), where θa; θa are Grassmanian numbers obeying


The second question from above is important, because for any classical quantum mechanical operator L we can write the Hamilton equation


where H is a Hamiltonian. For nonassociative parts of operators, however, there is an obstacle for such equation: Because H is generated from a product of two or more constituents, their nonassociativity demands to de ne the order of brackets in the product of HL and LH.

In Ref. [10] the question is considered: What is the most general nonassociative algebra A which is compatible with Eq. (5.1). Let us consider the consistency condition

equationwhich requires validity of

[H; xy] = x [H; y] + [H; x] y: (5.2)

The validity of Eq. (5.2) is not obvious for a general algebra. There exists the following

Theorem (Myung [7]). The necessary and suficient condition for


is that A is flexible and Lie-admissible, i.e.


Finally, a few notes about the third question. In Ref. [1] the idea is o ered that by quantization of strongly interacting elds (in particular in quantum chromodynamics), nonassociative properties of quantum eld operators may arise. In [1] it is proposed that a quark spinor field ψ can be presented as a bilinear combination of usual spinor fields ψi and nonassociative numbers (split octonions) qi. Both ideas, in Ref. [1] and here, are qualitatively similar: Quantum operators can be decomposed into nonassociative constituents.

An unsolved problem exists in quantum chromodynamics: the con nement. The challenging property is that a quark-antiquark pair cannot be separated. Physically speaking, this means that a single quark cannot be observed. Mathematically, the problem is that we do not know the algebra yet, which models eld operators for strongly interacting elds (gluons for quantum chromodynamics). One can suppose that the unobservability of quarks can be connected with a non-associative structure of the algebra of gluon operators.


I am grateful to the Alexander von Humboldt Foundation for nancial support, to V. Mukhanov for invitation to Universitat Munchen for research, and to J. Koplinger for fruitful discussion.


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