aa r X i v : . [ qu a n t - ph ] A ug On the symmetry definitions for the constraint dynamical systems
Alexei M. Frolov ∗ Department of Applied MathematicsUniversity of Western Ontario, London, Ontario N6H 5B7, Canada (Dated: April 3, 2018)
Abstract
The problem of proper symmetry definition for constraint dynamical systems with Hamiltoni-ans is considered. Finally, we choose a definition of symmetry which agrees with the analogousdefinition used for the non-constraint dynamical systems with Hamiltonians. Our symmetry defi-nition allows one to consider the whole spectrum of the Hamiltonian without splitting it into a fewdifferent parts. ∗ E–mail address: [email protected] H . The corresponding Schr¨odinger equation for such a system takes the formˆ H Ψ = E Ψ , or ˆ H Ψ = ı ¯ h ∂∂t Ψ (1)where Ψ is the wave function, while ¯ h is the reduced Planck constant (¯ h = h π ) and ı is theimaginary units. The parameter E in Eq.(1) is the total energy of the quantum system.The first equation in Eq.(1) is the time-independent Schr¨odinger equation, while the secondequation is the time-dependent Schr¨odinger equation. In reality, many quantum systemshave additional symmetry, e.g., geometrical, dynamical and/or ‘hidden’ symmetry. Briefly,such a symmetry means invariance of the Schr¨odinger equation for some (closed) group ofphysical, finite transformations. The ‘conditions of invariance’ can be re-written in terms ofinfinitesimal (or contact) transformations which form a closed algebraic structure in terms ofcommutation relations between each pair of the corresponding generators. In general, eachgroup of such transformations has finite number of generators of infinitesimal transforma-tions which are represented as self-adjoint operators. These generators are designated belowby the notation A i , where i = 1 , . . . , N . They form a closed structure which is called thealgebra Lie of infinitesimal transformations. The corresponding group of the finite (or physi-cal) transformations is uniformly reconstructed, if its algebra Lie is known. The commutatorbetween each pair of generators of the Lie algebra plays a fundamental role in the wholetheory of symmetry. It was shown by Sophus Lie (see [1] and references therein) that sucha commutator is always written in the form [ A i , A j ] = P k C kij A k , where C kij are the groupconstants, or structural constants of the corresponding Lie algebra. As follows directly fromthe definition of the commutator the group constants C kij must be antisymmetric upon theboth i and j indexes, i.e. C kij = − C kji . From here one finds that, e.g., [ A i , A j ] = − [ A j , A i ]and [ A i , A i ] = 0. All fundamental facts about groups, their Lie algebras, etc can be found,e.g., in [2], [3]. Here we do not want to repeat these definitions and discuss propertieswhich follow from such definitions. Instead, we want to consider some possible definitionsof physical symmetry.One of the first definitions of symmetry in Quantum Mechanics was formulated in termsof the generators A i of the Lie algebra as the set of conditions [ A i , ˆ H ] = 0, where i = 1 , . . . , N and N is the total number of generators in the Lie algebra. However, in the middle of 1960’ssuch a definition was found to be quite restrictive in actual applications. In particular, that2efinition was applicable only to the energy levels (or states) of the system which have thesame energy. It was not possible to move up and/or down along the spectra of states. Sincethen another extended symmetry definition has been proposed and applied. The extendeddefinition can be formulated in the two different (but equivalent) forms: (a) [ A i , ˆ H ]Ψ = 0for i = 1 , . . . , N , and (b) A i ˆ H Ψ = ˆ HA i Ψ for i = 1 , . . . , N , where Ψ is the solution of theSchr¨odinger equation, i.e. ( ˆ H − E )Ψ = 0. The first definition means that all generators of theLie algebra commute with the Hamiltonian on solutions of the Schr¨odinger equation. Thesecond definition means that the generators A i of the Lie algebra transforms one solution ofthe Schr¨odinger equation into another solution of the same Schr¨odinger equation, i.e. if Ψis a solution of the Schr¨odinger equation, then the functions Φ i = A i Ψ for i = 1 , . . . , N arealso its solutions. Such a definition represents the ‘dynamical’ symmetry, i.e. the symmetrywhich is only important for the actual (dynamical) motion of the system, or motion whichagrees with the dynamics of quantum system. An obvious difference with the old-fashioneddefinition of symmetry is clear. Below, we shall use only the dynamical symmetry definition.Now, we need to make another step forward and discuss a few possible definitions ofsymmetry for constrained Hamiltonian systems, i.e. for quantum systems which have Hamil-tonians and a number of constraints. In this study, we assume that all constraints are thefirst class constraints. An important example of the constrained Hamiltonian systems is thefree electromagnetic field. At the end of 1920’s the quantization of the free electromagneticfield was a serious problem, since it was clear that the two gauge conditions ∂φ∂t = 0 and div A = 0 cannot be imposed on the components of the four-potential ( φ, A ) of this field.In reality, it did lead to very serious contradictions in the whole quantization procedure forthe four-vector ( φ, A ). Fermi [4] proposed an effective approach which allows one to solveall such troubles at once. Fermi [4] assumed that the conditions ∂φ∂t = 0 and div A = 0for the components of 4-potential must be replaced by the corresponding conditions for thefield wave function Ψ, i.e. (cid:16) ∂φ∂t (cid:17) | Ψ i = 0 and ( div A ) | Ψ i = 0, where the notation | Ψ i stands for the wave function Ψ, i.e. Ψ = | Ψ i . Moreover, only such wave functions Ψ mustbe considered at the following steps of the procedure. Dirac immediately realized that weare dealing with the new Hamiltonian mechanics, which leads to the new type of motionin the Hamiltonian systems with constraints. Later such systems were called the constraintdynamical systems. In reality, it took almost 20 years for Dirac to develop the closed theoryof the Hamiltonian systems with constraints [5]. Below we discuss only a restricted version3f this theory (analysis of more general cases can be found, e.g., in [7]).The total Hamiltonain ˆ H tot of an arbitrary quantum system with constraints is repre-sented as the sum of its dynamical part ˆ H d and constraint part ˆ H c , i.e. ˆ H tot = ˆ H d + ˆ H c , whereˆ H c Ψ = 0. Let us assume that we are dealing with the system which has N p primary con-straints ˆ p i , N s secondary constraints ˆ s j and N t tertiary constraints ˆ t k , where N p ≥ N s ≥ N t .The constraint part of the Hamiltonian ˆ H tot is represented as a linear function of the primary,secondary and tertiary constraints, i.e.ˆ H tot = ˆ H d + ˆ H c = ˆ H d + N p X i =1 v i ˆ p i + N s X j =1 u j ˆ s j + N t X k =1 w k ˆ t k (2)According to the definition of the primary, secondary and tertiary constraints [6] we canwrite [ˆ p i , ˆ H d ] = N s X l =1 a il ˆ s l + N t X m =1 b im ˆ t m , [ˆ s j , ˆ H d ] = N t X l =1 c jq ˆ t q , [ˆ t k , ˆ H d ] = 0 (3)where some of the numerical coefficients a il , b im and c jq can be equal zero identically. Notethat the numerical coefficients a il , b im , c jq in Eq.(3) and v i , u j , w k in Eq.(2) are the fielddepended values, while the group constants C kij defined above cannot depend upon thesevalues, i.e. they are truly constants. It follows directly from Eq.(2) and definitions of theconstraints that ( ˆ H tot − ˆ H d )Ψ = 0. Therefore, we can write( ˆ H tot − E )Ψ = ( ˆ H d − E )Ψ = 0 (4)At this point we need to propose an accurate and workable definition of the physical sym-metry which can be applied for constraint dynamical systems with Hamiltonians. First of all,it is clear that the dynamical part of the Hamiltonian H d must commute with all generators A i of the contact Lie algebra on solutions of the Schr¨odinger equation, i.e. [ A i , H d ]Ψ = 0, or A i H d Ψ = H d A i Ψ. In other words, if Ψ is the solution of the Schr¨odinger equation with theHamiltonian H d , then A i Ψ (for i = 1 , . . . , N ) are also solutions of the same equation. Briefly,this means that if ( H d − E )Ψ = 0, then we also have ( H d − E )( A i Ψ) = 0 for i = 1 , . . . , N .The second part of this definition must contain information which allows one to determinethe commutation relations between generators of the Lie algebra and operators which rep-resent constraints. Formally, we can write for the primary constraints A α ˆ p i Ψ = ˆ p i ( A α Ψ).Since ˆ p i Ψ = 0, then we have ˆ p i ( A α Ψ) = 0, i.e. we have new primary constraints whichare defined on the set of functions φ α = A α Ψ, where α = 1 , . . . , N p . Analogously, for the4econdary and tertiary constraints, i.e. A α ˆ s j Ψ = ˆ s j ( A α Ψ) = 0 and A α ˆ t j Ψ = ˆ t j ( A α Ψ) = 0.It is important to note that the total numbers of primary, secondary and tertiary constraintscannot increase during applications of the symmetry operators. Furthermore, applicationof the symmetry operation does not mix constraints, i.e. after application of the symmetryoperations all primary constraints are represented as the linear combinations of the primaryconstraints only. The same statement is true for all secondary and tertiary constraints. Afteran extensive analysis it became clear that alternative definitions of symmetry which allowto mix constraints lead to some fundamental changes in the dynamics of quantum system.Therefore, such definitions cannot be accepted.It should be mentioned that there are some additional relations between constraintsknown for the dynamical systems. For instance, in our paper [8] on the free gravity fieldsit was shown that this system has four primary ˆ p i and four secondary ˆ s i constraints (notertiary constraints have been found). It was shown in [8] that the Poisson between thefour primary constraints and four secondary constraints equals to the product of the metrictensor g µν (with the additional coefficient − and secondary constraint with two temporalindexes (or (00) constraint) (for more details, see [8]). Very likely, any non-zero Poissonbracket between different constraints for one dynamical systems must be represented as alinear combination of other constraints and operator ( H d − E ). The coefficients of such linearcombination are some filed-dependent functions, i.e. they are not constants. In general, thisstatement has never been proved. However, in those cases when the wave functions Ψ of oursystem are normalized, i.e. have unit norm, the proof of this statement is straightforward.For the goals of our study it is important to note that such additional relations betweenconstraints of the system may complicate applications of the symmetry definition developedabove. [1] S. Lie and F. Engel, Theory of Transformationgruppen , (Taubner, Leipzig, vol.1 (1888), vol.2(1890), vol.3 (1893)) [in German].[2] L.S. Pontryagin,
Continuous Gruops , (Science, Moscow (1974)) [in Russian].[3] L.P. Eisenhart,
Continuous Gruops of Transformations , (Dover Publ., New York (2003)).[4] E. Fermi, Rev. Mod. Phys. , 131 (1931).
5] P.A.M. Dirac, Can. J. Math. , 147 (1950).[6] P.A.M. Dirac, Lectures on Quantum Mechanics , (Dover Publications Inc., New York, 2001).[7] D.M. Gitman and I.V. Tyutin,
Quantization of Fields with Constraints (Springer, Berlin, 1990).[8] A.M. Frolov, N. Kiriushcheva and S.V. Kuzmin, Gravitation , 314 - 323 (2011) (see alsoarXiv:0809.1198 [hep-th])., 314 - 323 (2011) (see alsoarXiv:0809.1198 [hep-th]).