QQuantum Mechanical Effects from Deformation Theory
A. Much
Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, GermanyInstitute for Theoretical Physics, University of Leipzig, 04009 Leipzig, Germany
October 30, 2018
Abstract
We consider deformations of quantum mechanical operators by using the novel con-struction tool of warped convolutions. The deformation enables us to obtain severalquantum mechanical effects where electromagnetic and gravitomagnetic fields play arole. Furthermore, a quantum plane can be defined by using the deformation tech-niques. This in turn gives an experimentally verifiable effect.
Contents
Deformation theory is an interesting subject of research, both from a mathematicaland a physical point of view. Nowadays, many fundamental theories are reconsideredas deformations of more subtle theories. A fundamental example of deformation the-ory in a physical context is the deformation of classical mechanics to quantum physics,where the deformation parameter in that case is Planck’s constant (cid:126) . In this contextthe Poincaré group can also be considered as a deformation of the Galilei group,where the parameter characterizing the deformation is given by the speed of light, i.e. /c . The opposite of a deformation in a group theoretical context is a contraction. Itis induced by taking the limit of the deformation parameter to zero. In the example a r X i v : . [ m a t h - ph ] F e b of the Poincaré group, this would mean that we take the limit /c → . This limitis often taken by physicists as a consistency check and rarely recognized as a contrac-tion. Another interesting example is the deformation of the Poincaré group to theAnti-de Sitter group by using the cosmological constant Λ as a deformation parameter.Thus, from a physical point of view, deformation theory enters the game bythe physical dimensionality of the deformation parameter. In this work, we emphasizethe importance of choosing the deformation constant , in order to obtain physicaleffects .One justified critique usually spoken out in the context of deformation theoryis that the rightful deformation is only guessed after the physical theory has beenformulated. Thus, to consider such deformations as fundamental, is often put intothe category of wishful thinking of theoretical physicists. Therefore, the main aimof the current work is to understand a variety of physical effects, in a quantummechanical context, by a deformation of the free theory. Furthermore, we propose aneffect coming from deformation considerations.The method that is used, in the current work, for deformation is known under the nameof warped convolutions, [GL07, BS, BLS11]. Usually, this method is used in the realmof quantum field theory to deform free quantum fields and to construct non-trivialinteracting fields which was done in [Ala, GL07, GL08, Lec12, LST13, MM11, Alb12].It was also used in quantum measurement theory [And13]. One of the majoradvantages of this method is its easy accessibility to a physical regimen.By using this novel tool in a quantum mechanical context, we recast manyfundamental physical effects involving electromagnetism. This is done by theadjustment of the deformation parameter. Moreover, we are able to producegravitomagnetic effects and interaction between magnetic and gravitomagnetic fieldsby this deformation procedure.This paper is organized as follows: In Section 2 we give a brief introduction ofthe method of warped convolutions and introduce the basic notations for deformationin a quantum mechanical context. The free Hamiltonian is deformed in Section 3.We are obliged to show that the warped convolutions formula, originally formulatedfor a subset of bounded operators, is well-defined in the case of the deformation ofunbounded operators. Section 4 is devoted to the emergence of physical effects fromthe deformation procedure. Since we constantly use warped convolutions we lay out the novel deformationprocedure in this section and present the most important definitions, lemmas andpropositions for the current paper. For proofs of the lemmas and propositions werefer the reader to the original works.We start by assuming the existence of a strongly continuous unitary group U that is a representation of the additive group R n , on some separable Hilbert space H . Let D be the dense domain of vectors in H which transform smoothly underthe adjoint action of U . Then, the warped convolutions for operators F ∈ C ∞ , where C ∞ is the *-algebra of smooth elements with respect to the adjoint action of U , aregiven by the following definition. Definition
Let B be a real skew-symmetric matrix on R n , let F ∈ C ∞ andlet E be the spectral resolution of the unitary operator U . Then, the correspondingwarped convolution F B of F is defined on the domain D according to F B := (cid:90) α Bx ( F ) dE ( x ) , (2.1)where α denotes the adjoint action of U given by α k ( F ) = U ( k ) F U ( k ) − .The restriction in the choice of operators is owed to the fact that the deformation isperformed with operator valued integrals. Furthermore, one can represent the warpedconvolution of A ∈ C ∞ by (cid:82) α Bx ( A ) dE ( x ) or (cid:82) dE ( x ) α Bx ( A ) , on the dense domain D ⊂ H of vectors smooth w.r.t. the action of U , in terms of strong limits (cid:90) α Bx ( A ) dE ( x )Φ = (2 π ) − n lim (cid:15) → (cid:90) (cid:90) d n x d n y χ ( (cid:15)x, (cid:15)y ) e − ixy U ( y ) α Bx ( A )Φ , where χ ∈ S ( R n × R n ) with χ (0 ,
0) = 1 . This representation makes calculationsand proofs concerning the existence of integrals easier. In this work we use bothrepresentations.In the following lemma we introduce the deformed product, also known as theRieffel product [Rie93] by using warped convolutions. The two deformations areinterrelated since warped convolutions supply isometric representations of Rieffel’sstrict deformations of C ∗ -dynamical systems with actions of R n . Lemma
Let B be a real skew-symmetric matrix on R n and let A, E ∈ C ∞ . Then A B E B Φ = ( A × B E ) B Φ , Φ ∈ D . where × B is known as the Rieffel product on C ∞ and is given by, ( A × B E )Φ = (2 π ) − n lim (cid:15) → (cid:90) (cid:90) d n x d n y χ ( (cid:15)x, (cid:15)y ) e − ixy α Bx ( A ) α y ( E )Φ . (2.2)Another proposition that seems a matter of technicality in the original work but hasgreat physical significance is the following. Proposition
Let B , B be skew symmetric matrices. Then ( A B ) B = A B + B , A ∈ C ∞ . (2.3)Next, we adopt Formula (2.1) to define the warped convolutions for an unboundedoperator, with a real vector-valued function of the coordinate operator. To apply thedefinition of warped convolutions, we need self-adjoint operators that commute alongtheir components. For this purpose let us give the following theorem, [RS75, TheoremVIII.6]. Theorem
Let Q ( . ) be an unbounded real vector-valued Borel function on R n and let the dense domain D Q be given as, D Q = { φ | ∞ (cid:90) −∞ | Q j ( x ) | d ( φ, P x φ ) < ∞ , j = 1 , . . . , n } , where { P x } are projection valued measures on H . Then, Q ( X ) defined on D Q is aself-adjoint operator. In this paper we consider unbounded real vector-valued functions of the coordinateoperator and therefore we give the following definition.
Definition
Let B be a real skew-symmetric matrix on R n and let χ ∈ S ( R n × R n ) with χ (0 ,
0) = 1 . Moreover, let Q ( X ) be given as in Theorem 2.1. Then, thewarped convolutions of an operator A with operator Q , denoted as A B, Q are defined,in the same manner as in [BLS11], namely A B, Q := (2 π ) − n lim (cid:15) → (cid:90) (cid:90) d n y d n k e − iy l k l χ ( (cid:15)y, (cid:15)k ) V ( k ) α By ( A ) . (2.4)The automorphisms α are implemented by the adjoint action of the strongly continuousunitary representation V ( y ) = e iy k Q k of R n given by α y ( A ) = V ( y ) A V ( y ) − , y ∈ R n . Since we deform unbounded operators we are obliged to prove that the deformationformula, given as an oscillatory integral, is well-defined. This is the subject of thenext section.
At first, we study deformations of the simplest Hamiltonian of quantum mechanics,that of a free particle. Further on we explore the physical consequences of the defor-mation and introduce to the reader how one can obtain a variety of physical effectsusing this method. For a deformation of the Hamiltonian we choose to work in thestandard realization of quantum mechanics, the so called
Schrödinger represen-tation , [BEH08, RS75, Tes01]. In this representation the pair of operators ( P i , X j ) ,satisfying the canonical commutation relations (CCR) [ X i , P k ] = − iη ik , (3.1)are represented as essentially self-adjoint operators on the dense domain S ( R n ) . Here X i and P k are the closures of x i and multiplication by i∂/∂x k on S ( R n ) respectively.In quantum mechanics the Hamiltonian of a free particle is given as follows H = − P j P j m . (3.2)This operator describes a non-relativistic and non-interacting particle. For thefollowing considerations, we restrict the deformation to three space dimensions. Thisrestriction is obvious due to its physical relevance. Let us start this section witha theorem concerning the domain of self-adjointness and the spectrum of the freeundeformed Hamiltonian H , [Tes01]. Theorem
The free Schrödinger operator H is self-adjoint on the domain D ( H ) given as D ( H ) = H ( R ) = { ϕ ∈ L ( R ) || P | ϕ ∈ L ( R ) } , and its spectrum is characterized by σ ( H ) = [0 , ∞ ) . Before proceeding with the deformation, one problem arises at this point of our work.The deformation formula given by warped convolutions is only well-defined in thestrong operator topology for a subset of bounded operators that are smooth w.r.t.the unitary representation U of R n . In view of the fact that we deal with unboundedoperators, we are obliged to investigate the validity of the deformation Formula (2.4)for H . For this purpose we need a dense domain E ⊆ S ( R ) that fulfills additionalrequirements. Lemma
Consider the self-adjoint operator Q ( X ) = X / | X | n , n ∈ R . (3.3) Then, for all n ∈ R there exists a dense domain E ⊆ S ( R ) such that (cid:107){ P j , [ Q , P j ] } Φ (cid:107) < ∞ , (cid:107) [ Q , P j ][ Q , P j ]Φ (cid:107) < ∞ , Φ ∈ E . (3.4) Proof.
From Theorem 2.1 it is follows that all operators of the form X / | X | n are self-adjoint on their respective domains. Further we show the existence of a dense domain,satisfying Inequalities (3.4). To simplify calculations let us give general formulas forthe commutators [ P j , | X | − n ] = i n X j | X | − ( n +2) , (3.5) [ P j , X k / | X | n ] = i (cid:0) η jk + n X k X j / | X | (cid:1) | X | − n . (3.6)Thus, for an arbitrary n ∈ R and Q = X / | X | n the anti-commutator in Inequality(3.4) is calculated as follows { P j , [ P j , Q k ] } = [ P j , [ P j , Q k ]] + 2[ P j , Q k ] P j = i [ P j , (cid:0) η jk + nX k X j / | X | (cid:1) | X | − n ] + 2 i (cid:0) η jk + nX k X j / | X | (cid:1) | X | − n P j = (cid:0) n − n (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) =: a ( n ) X k | X | − ( n +2) + 2 i (cid:0) η jk + n X k X j / | X | (cid:1) | X | − n P j , where in the last lines we used the CCR, Equations (3.5) and (3.6). The norm of theanti-commutator is given by (cid:107){ P j , [ Q , P j ] } Φ (cid:107) = (cid:107) e k (cid:16) a ( n ) X k | X | − ( n +2) + 2 i (cid:0) η jk + n X k X j / | X | (cid:1) | X | − n P j (cid:17) Φ (cid:107)≤ (cid:107) e k a ( n ) X k | X | − ( n +2) Φ (cid:107) + (cid:107) e k (cid:0) η jk + n X k X j / | X | (cid:1) | X | − n P j Φ (cid:107)≤ (cid:107) a ( n ) | X | − ( n +1) Φ (cid:107) + (cid:107) | X | − n P Φ (cid:107) + (cid:107) n | X | − ( n +1) X j P j Φ (cid:107) . The term in the second inequality in (3.4) is given by (cid:107) [ Q , P j ][ Q , P j ]Φ (cid:107) = (cid:107) (cid:0) η jl + n X l X j / | X | (cid:1) (cid:0) η jl + n X l X j / | X | (cid:1) | X | − n Φ (cid:107) = (cid:107) ( n − n + 3) | X | − n Φ (cid:107) . It is clear that if n ∈ R − Inequalities (3.4) are satisfied for vectors in the densedomain S ( R ) , since the expressions in the norm are positive polynomial functionsof the coordinate operator. For n ∈ R + we consider the domain E which denotes thelinear hull of the dense vectors [Thi81, Theorem 3.2.5] Φ( x ) = x k x k x k exp ( − | x | , k i = 0 , , , . . . . Since the dense domain E remains invariant under the action of positive functions ofthe coordinate and momentum operator (see proof of [Thi81, Theorem 3.2.5]), theremaining task is to show the finiteness of (cid:107)| X | ( λ − Φ (cid:107) = (cid:90) d x | x | λ − e −| x | , λ ∈ R − = π (cid:90) dφ π (cid:90) dϑ sin ϑ ∞ (cid:90) r λ e − r dr = 2 π ∞ (cid:90) −∞ x λ e − x dx. This integral exists for all λ and it is easily seen to be an analytic function in λ ,[GS68, Chapter 1, Section 3.6]. Note that we choose the polynomial functions ofcomponents of x to be equal to one. This choice is for the sake of argument, sincepositive polynomial functions improve the behavior of the integral.By using the former lemma, we show in the next proposition that the scalar productof the deformed free Hamiltonian, i.e. (cid:104) Ψ , ( H ) B, Q Φ (cid:105) = (2 π ) − lim ε → (cid:90) (cid:90) d y d k e − iy l k l χ ( εy, εk ) (cid:104) Ψ , V ( k ) α By ( H )Φ (cid:105) , is bounded for ∀ Ψ ∈ H and Φ ∈ E ⊆ S ( R ) . Proposition
Let Q ( X ) be a self-adjoint operator of the form Q ( X ) = X / | X | n , n ∈ R , and let ( H ) B, Q denote the deformed free Hamiltonian (see Formula (2.4)). Then, thescalar product (cid:104) Ψ , ( H ) B, Q Φ (cid:105) is bounded by a finite constant C B as follows, |(cid:104) Ψ , ( H ) B, Q Φ (cid:105)| ≤ C B (cid:107) Ψ (cid:107) , ∀ Ψ ∈ H , Φ ∈ E ⊆ S ( R ) . Therefore, the deformation formula for the unbounded operator H , given as an oscil-latory integral, is well-defined and the explicit result of the deformation is ( H ) B, Q Φ = − m (cid:0) P j + i ( BQ ) k [ Q k , P j ] (cid:1) (cid:0) P j + i ( BQ ) r [ Q r , P j ] (cid:1) Φ . (3.7) Proof.
To prove the boundedness of the scalar product (cid:104) Ψ , V ( k ) α By ( H )Φ (cid:105) , we firstderive the adjoint action of V ( By ) on H given by, α By ( H ) = − m V ( By ) P j P j V ( − By )= − m V ( By ) P j V ( − By ) V ( By ) P j V ( − By ) . To solve this expression we first calculate the adjoint action of V ( By ) on the momen-tum operator P j by using the Baker-Campbell-Hausdorff formula, V ( By ) P j V ( − By ) = P j + i ( By ) k [ Q k , P j ] (cid:124) (cid:123)(cid:122) (cid:125) =: − iX kj ( X ) + i By ) l ( By ) k [ Q l , [ Q k , P j ]] + ... (cid:124) (cid:123)(cid:122) (cid:125) =0 , (3.8) where in the last lines we used the CCR given in (3.1), and the commutativity of thecoordinate operator, i.e. [ X i , X j ] = 0 . Thus, the adjoint action w.r.t. V ( By ) on H is α By ( H ) = − m ( P j + ( By ) s X sj )( P j + ( By ) r X rj )= H − ( By ) s m (cid:0) P j X sj + X sj P j (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) =: N s − ( By ) r ( By ) s m X sj X rj (cid:124) (cid:123)(cid:122) (cid:125) =: R rs = H − ( By ) s N s − ( By ) r ( By ) s R rs . Moreover, without loss of generality, one can choose the skew-symmetric matrix B tohave the form B ij = ε ijk B j , where ε ijk is the three dimensional epsilon-tensor. Then,we are able to derive the following inequality, | ( By ) i e i | ≤ √ | B || y | . (3.9)This is easily seen by using Cauchy-Schwarz and the inequality | a | − | b | ≤ | a | + | b | . | ( By ) i e i | = ( − B ij y j B is y s )= − ε ijk B k y j ε isr B r y s = − (cid:0) δ sj δ rk − δ rj δ sk (cid:1) B r B k y j y s = (cid:0) B r y r B k y k − B r B r y j y j (cid:1) ≤ | B | | y | Thus, by using the adjoint action of V ( By ) on the free Hamiltonian and for B ij = ε ijk B k we have the following inequality |(cid:104) Ψ , V ( k ) α By ( H )Φ (cid:105)| ≤ (cid:107) Ψ (cid:107) (cid:107) ( H − ( By ) s N s − ( By ) r ( By ) s R rs ) Φ (cid:107)≤ (cid:107) Ψ (cid:107) (cid:18) (cid:107) H Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C +2 | y | | B |√ (cid:107) N Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C + | y | | B | (cid:107) R Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C (cid:19) ≤ C B (cid:107) Ψ (cid:107) (1 + | y | ) . (3.10)A finite constant C B obeying the inequality exists, since C , C and C are finite for Φ ∈ E , where E is a dense set of vectors specified in Lemma 3.1. Therefore, the scalarproduct is polynomially bounded to the second order in y , i.e. |(cid:104) Ψ , ( H ) B, Q Φ (cid:105)| C B (cid:107) Ψ (cid:107) ≤ (2 π ) − lim ε → (cid:90) (cid:90) d y d k e − iy l k l χ ( εy, εk ) (1 + | y | ) = (2 π ) − lim ε → (cid:18)(cid:90) d y lim ε → (cid:18)(cid:90) d ke − ik r y r χ ( ε k ) (cid:19) χ ( ε y ) (1 + | y | ) (cid:19) = lim ε → (cid:18)(cid:90) d y δ ( y ) χ ( ε y ) (1 + | y | ) (cid:19) = 1 . Here we used the fact that the oscillatory integral does not depend on thecut-off function chosen. As in [Rie93], we chose χ ( εk, εy ) = χ ( ε k ) χ ( ε y ) with χ l ∈ S ( R × R ) and χ l (0 ,
0) = 1 , l = 1 , , and obtained the delta distribution δ ( y − q ) in the limit ε → , [Hör04, Section 7.8, Equation 7.8.5]. Since the formerinequality proves the convergence of the oscillatory integral, we conclude that thedeformation of the unbounded operator is well-defined. Next, we turn to the actual result of the deformation. To simplify calculationsit is easier to work in the spectral measure representation (see Equation 2.1). Thiscan be done, since the two representations, one in terms of the spectral measure andthe other as the limit of oscillatory integrals, are equal and we have proven that thedeformation is well-defined. ( H ) B, Q Φ = (cid:90) dE ( y ) α By ( H ) Φ= − m (cid:90) dE ( y ) (cid:0) ( P j + i ( By ) s [ Q s , P j ]) (cid:0) P j + i ( By ) r [ Q r , P j ] (cid:1)(cid:1) Φ= − m ( P j + i ( BQ ) s [ Q s , P j ]) (cid:0) P j + i ( BQ ) r [ Q r , P j ] (cid:1) Φ . The essential point of the proposition is that the deformation with the coordinateoperator amounts to a non-constant shift in the momentum. In physics this isusually referred to as minimal substitution . Such a minimal substitution is inQM based on Galilei invariance and then implemented accordingly by an externalelectromagnetic field (see [JM67]). In our approach we obtain such a substitution bydeformation. The connection between deformation and an external electromagneticfield is explored in the next sections.For the next proposition we deform the momentum operator. Since the mo-mentum operator is unbounded, we are as before obliged to show that deformationFormula (2.4) is given as a well-defined oscillatory integral.
Proposition
Let Q ( X ) be a self-adjoint operator of the form Q ( X ) = X / | X | n , n ∈ R , and let P B, Q denote the deformed momentum operator (see Formula 2.4). Then, thescalar product (cid:104) Ψ , P B, Q Φ (cid:105) is bounded by a finite constant C D as follows, |(cid:104) Ψ , P B, Q Φ (cid:105)| ≤ C D (cid:107) Ψ (cid:107) , ∀ Ψ ∈ H , Φ ∈ E ⊆ S ( R ) . Therefore, the deformation of the unbounded momentum operator, given as an oscilla-tory integral, is well-defined. Moreover, the explicit result of the deformation is givenas P jB, Q Φ = (cid:0) P j + i ( BQ ) k [ Q k , P j ] (cid:1) Φ . (3.11) Proof.
As in the proof of the former proposition we show that |(cid:104) Ψ , V ( k ) α By ( P )Φ (cid:105)| ,is polynomially bounded. To do so, we use the adjoint action of the unitary opera-tor V ( By ) on the momentum operator (see Equation (3.8)) and the Cauchy-Schwarzinequality, |(cid:104) Ψ , V ( k ) α By ( P )Φ (cid:105)| ≤ (cid:107) Ψ (cid:107) (cid:13)(cid:13)(cid:0) P + i ( By ) j [ Q j , P ] (cid:1) Φ (cid:13)(cid:13) ≤ (cid:107) Ψ (cid:107) (cid:107) P Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C + | y | √ | B | (cid:107) [ Q , P ]Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C ≤ C D (cid:107) Ψ (cid:107) (1 + | y | ) , where we used Inequality (3.9) and the fact that a finite constant C D obeying theinequality exists, since C and C are finite for Φ ∈ E (see Lemma 3.1). Therefore, the whole expression is polynomially bounded to first order in y , i.e. |(cid:104) Ψ , P B, Q Φ (cid:105)| C D (cid:107) Ψ (cid:107) ≤ (2 π ) − lim ε → (cid:90) (cid:90) d y d k e − iy l k l χ ( εy, εk ) (1 + | y | )= (2 π ) − lim ε → (cid:18)(cid:90) d y lim ε → (cid:18)(cid:90) d ke − ik r y r χ ( ε k ) (cid:19) χ ( ε y ) (1 + | y | ) (cid:19) = lim ε → (cid:18)(cid:90) d y δ ( y ) χ ( ε y ) (1 + | y | ) (cid:19) = 1 . As before, we argue that due to the convergence of the integral the deformation ofthe momentum operator for all Ψ ∈ H and Φ ∈ E is well-defined.Next, we turn to the actual result of the deformation and again for simplicitywe use the spectral measure for deformation, P jB, Q Ψ = (cid:90) dE ( y ) α By (cid:0) P j (cid:1) Ψ= (cid:90) dE ( y ) (cid:0) P j + i ( By ) s [ Q s , P j ] (cid:1) Ψ= (cid:0) P j + i ( BQ ) s [ Q s , P j ] (cid:1) Ψ . Since the deformed Hamiltonian could be defined as the scalar product of the deformedmomentum operators, we need to investigate the possible outcome. The investigationof the arbitrariness in the definition of the deformed free Hamiltonian is subject of thefollowing theorem.
Theorem
The scalar product of the deformed momentum vectors is equal tothe deformed free Hamiltonian (see Equation 3.7), i.e. ( H ) B, Q Ψ = − m P B, Q j P jB, Q Ψ , Ψ ∈ E ⊆ S ( R ) . Proof.
For the proof we calculate the Rieffel product, defined with the operator-valuedvector Q ( X ) , of the deformed momentum vectors, i.e. ( P k × B, Q P j ) Ψ = (2 π ) − lim ε → (cid:90) (cid:90) d x d y χ ( (cid:15)x, (cid:15)y ) e − ixy α Bx ( P k ) α y ( P j )Ψ= (cid:0) P k P j − iB ls ∂ k Q l ∂ j Q s (cid:1) Ψ , where in the last lines we used the CCR and the fact that the only terms that do notvanish are those of equal odd order in x and y , (see proof of Lemma 5.3 in [Alb12]).Now by summing over all components we obtain (cid:0) P k × B, Q P k (cid:1) Ψ = P k P k − i B ls ∂ k Q l ∂ k Q s (cid:124) (cid:123)(cid:122) (cid:125) =0 Ψ = P k P k Ψ , where we used the skew-symmetry of B and commutativity of the coordinate operator.Thus, by using the last equation and Lemma 2.1 the following equality is given ( H ) B, Q Ψ = − m (cid:0) P k P k (cid:1) B, Q Ψ= − m (cid:0) P k × B, Q P k (cid:1) B, Q Ψ= − m P B, Q k P kB, Q Ψ , Ψ ∈ S ( R ) . This is an important result resolving the question of arbitrariness of the deformation.Moreover, it is a group theoretical circumstance, since the deformation of a freeHamiltonian can be understood as the deformation of generators of the centralextended Galilei (CEG) group (for CEG see for example [Bal98]). The deformationwith the coordinate operator leaves all generators of the group invariant except forthe momentum and the Hamiltonian. Since, the Hamiltonian is a function of themomentum it follows from the former proposition that the deformation respects thestructure of the group. This fact is owed to the deformed product. Also note thatthe deformed momentum operator does not commute along its components.As already mentioned in Section 2, for some arguments we deform the coordi-nate operator by using the momentum operator. Before doing so, we show in the nextproposition that the deformation formula is well-defined even though the coordinateoperator is unbounded.
Proposition
The scalar product (cid:104) Ψ , X θ, P Φ (cid:105) is bounded by a finite constant C E as follows, |(cid:104) Ψ , X θ, P Φ (cid:105)| ≤ C E (cid:107) Ψ (cid:107) , ∀ Ψ ∈ H , Φ ∈ S ( R ) . Therefore, the deformation of the unbounded coordinate operator, given as an oscilla-tory integral, is well-defined. Moreover, the explicit result of the deformation is givenas X jθ, P Ψ = (cid:0) X j − ( θP ) j (cid:1) Ψ , Ψ ∈ S ( R ) . (3.12) Proof.
Similar to the proofs of the former propositions we show that the scalar product |(cid:104) Ψ , V ( k ) α θy ( X )Φ (cid:105)| is polynomially bounded, |(cid:104) Ψ , V ( k ) α θy ( X )Φ (cid:105)| ≤ (cid:107) Ψ (cid:107) (cid:13)(cid:13)(cid:0) X + i ( θy ) j [ P j , X ] (cid:1) Φ (cid:13)(cid:13) ≤ (cid:107) Ψ (cid:107) (cid:107) X Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C + | y | √ | θ | (cid:107) Φ (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) =: C ≤ C E (cid:107) Ψ (cid:107) (1 + | y | ) , where in the last lines we used Inequality (3.9), and the fact that a finite constant C E obeying the inequality exists, since C and C are finite for Φ ∈ S ( R ) . Therefore,the whole expression is polynomially bounded to the first order in y , i.e. |(cid:104) Ψ , X θ, P Φ (cid:105)| ≤ C E (cid:107) Ψ (cid:107) , where we used the same arguments made in Proposition 3.2.Next, we turn to the result of the deformation and again for simplicity we usethe spectral measure for the deformation, X jθ, P Ψ = (cid:90) dE ( y ) α θy (cid:0) X j (cid:1) Ψ= (cid:90) dE ( y ) (cid:0) X j + i ( θy ) s [ P s , X j ] (cid:1) Ψ= (cid:0) X j − ( θP ) j (cid:1) Ψ . After this rather more technical part we turn in the next section to the physicalimplications of the deformation technique.
One of the most important aspects of the interplay between mathematics and physicslies in the physical dimensionality of the physical constants. The main motivationof this work is the search for the physical meaning of the deformation parameter.Quantum mechanical deformations give us a variety of interesting answers and theyare presented in this section.
An example of a dynamical system interacting with a magnetic field in a quantummechanical setting, is given by the Landau effect. It is also an important example of theappearance of quantum space in a physical context. The Landau effect describes thedynamics of a system of non-relativistic electrons confined to a plane , for examplethe y − z plane ( (cid:126)A = (0 , y, z ) ), in the presence of a homogeneous magnetic field (cid:126)B = B (1 , , . In the symmetric gauge the Hamiltonian of the Landau effect isgiven by, [Eza08, Equation 9.2.1] H L = − m ( P i + eA i ) (cid:0) P i + eA i (cid:1) , where the gauge field is given as A i = − ε ijk B k X j . (4.1)Next, we show that the deformed Hamiltonian ( H ) B, X reproduces the Landau modelafter setting the parameters of the deformation matrix equal to a constant withphysical dimension. This is the result of the following lemma. Lemma
Let the deformation matrix B ij be given as, B ij = − ( e/ ε ijk B k , where B k characterizes a constant homogeneous magnetic field and e is the electriccharge. Then, the deformed free Hamiltonian ( H ) B, X becomes the Hamiltonian H L of the Landau problem, i.e. ( H ) B, X Ψ = H L Ψ , Ψ ∈ E . Proof.
For the proof we consider the free deformed Hamiltonian ( H ) B, X , given inEquation (3.7), with Q j = X j ( H ) B, X Ψ = − m (cid:0) P j + i ( BX ) k [ X k , P j ] (cid:1) (cid:0) P j + i ( BQ ) r [ X r , P j ] (cid:1) Ψ= − m ( P j + B jk X k )( P j + B jr X r )Ψ , Ψ ∈ E , (4.2) .2 Zeemaneffect 12 where in the last line we used the CCR. By setting the deformation matrix equal to B ij = − ( e/ ε ijk B k , where B k is a homogeneous magnetic field in the x -direction( B k = B (1 , , ), we obtain the Landau quantization.This is an interesting result. We started with the free Hamiltonian and deformedit with warped convolutions using the coordinate operator. By simply taking thedeformation parameters of the matrix B ij to be equal to certain physical quantitieswe obtain the Landau problem. Therefore, the quantization with the coordinateoperator is physically of great importance. Note that our model is formulated in ageneral manner, and just for the specific choice of the deformation parameters weobtained the Landau effect.A remark is in order about the current result. It is well-known that the non-commutative coordinates of the Landau quantization can be generated by minimallyshifting the ordinary coordinate operator by a skew-symmetric matrix times themomentum operator. This rather ad hoc but remarkably insightful result is well-known as the Bopp-shift. In the context of deformation theory, we were able togive a systematic derivation of the Landau quantization, rather than postulating adhoc a substitution. This derivation can be further applied to a variety of quantummechanical effects involving gauge fields. The Hamiltonian of the hydrogen atom is given as follows, [Thi81, Equation 4.1.1] H A = − P j P j m + e | X | . By solving the stationary Schrödinger equation H A ψ = Eψ one obtains the energyspectrum of a hydrogen atom, the so called Balmer series , [Str02]. In the presenceof a constant magnetic field, an interesting physical effect occurs to the spectral linesof the hydrogen atom. The spectral lines split into further spectral lines dependingon the presence of a homogeneous magnetic field B k . This phenomenon is called the Zeemaneffect and the Hamiltonian of this effect is given as follows, [Thi81, Equation4.2.1] H AZ = − m ( P j − ( e/ ε jik B k X i )( P j − ( e/ ε jnl B l X n ) + e | X | . (4.3)We recognized in the last section that the deformation with the coordinate operatorinduces a gauge field. Due to this lesson we preform a deformation on the Hamiltonianof the hydrogen atom to obtain the Hamiltonian of the Zeemanneffect. Lemma
Let the deformation matrix B ij be given as, B ij = − ( e/ ε ijk B k , where B k characterizes a constant homogeneous magnetic field. Then, the deformedHamiltonian of the hydrogen atom, denoted by ( H A ) B, X , becomes the Hamiltonian ofthe Zeemaneffect H AZ , i.e. ( H A ) B, X Ψ = H AZ Ψ , Ψ ∈ E . Proof.
Due to the fact that the coordinate operator commutes with itself the onlypart of the Hamiltonian H A which is affected is the free part and therefore we obtain ( H A ) B, X Ψ = (cid:18) − m ( P j + B jk X k )( P j + B jr X r ) + e | X | (cid:19) Ψ , (4.4) .3 Aharonov-Bohm effect 13 the Hamiltonian of the Zeemaneffect for a homogeneous magnetic field in the x -direction, i.e. ( H A ) B, X = H AZ .As in the case of Landau quantization, the deformation parameter plays the role ofthe magnetic field which leads to this wonderful physical effect. In the last sections we recognized the consequence of a deformation with the coordinateoperator. Warped convolutions with the coordinate operator induce a gauge field .Now since we work in a quantum mechanical setting we want to reproduce otherphysical effects where magnetic fields play a significant role. One of the most strikingones is the
Aharanov-Bohm (AB) effect . It takes place in a system in which thegauge field influences the dynamics of a charged particle even in regions where themagnetic field vanishes, [Ber00, Eza08]. The gauge field of the magnetic AB effect,for a homogeneous magnetic field in x -direction, takes the following form A i = φ M π ( X + X ) ε ijk e k X j , (4.5)where φ M is the magnetic flux and e k is the unit vector in x -direction. Moreover,from quantum mechanical considerations it follows that the interference pattern isthe same for two values of fluxes φ and φ if only if e ( φ − φ ) = 2 πn, n ∈ Z . (4.6)In this section we take the free Hamiltonian and deform it with a vector-valued functionof the coordinate operator. As before, after setting the deformation parameter equalto a physical constant, namely that of a magnetic flux, we obtain the AB effect. Proposition
Let the deformation matrix B ij and the operator Q j ( X ) begiven as B ij = − e φ M π ε ijk e k , Q j ( X ) := X j / ( − (cid:88) s =2 X s X s ) / , (4.7) where φ M characterizes the magnetic flux. Then, the deformed Hamiltonian ( H ) B, Q ,is equal to Hamiltonian of the Aharonov-Bohm, i.e. ( H ) B, Q Ψ = 12 m ( P − e A ) Ψ , where A is the gauge field of the Aharanov-Bohm effect (see Equation 4.5). Further-more, if the deformation parameters of the matrices B and B fulfill Equation (4.6),the physical systems described by the Hamiltonians H B ,F and H B ,F have the sameinterference pattern.Proof. For the deformation of H we use Proposition 3.1, with Q j ( X ) as given inEquation (4.7), ( H ) B, Q Ψ = − m (cid:0) P j + i ( BQ ) k [ Q k , P j ] (cid:1) (cid:0) P j + i ( BQ ) r [ Q r , P j ] (cid:1) Ψ= − m (cid:32) P j + ( BX ) j / ( − (cid:88) s =2 X s X s ) (cid:33) (cid:32) P j + ( BX ) j / ( − (cid:88) r =2 X r X r ) (cid:33) Ψ , where in the last lines we used the skew-symmetry of B and commutator relation 3.6.Thus, by setting the deformation matrix B ij = − ( e φ M / π ) ε ijk e k , the gauge field A i ( x ) induced by deformation is the gauge field of the AB effect for a homogeneousmagnetic field in x -direction. .4 Physical Moyal-Weyl plane 14 This is an interesting result. We were able to induce the AB-gauge field by deformingthe free Hamiltonian with a vector-valued function of the coordinate operator. Inthis case the deformation parameter corresponds to the magnetic flux rather, as inthe previous cases, to the magnetic field.There are two ways to interpret these results. The first one lies in understandingdeformation, in the case of QM, as the rightful minimal substitution. Thus theprocedure sheds new light on quantum mechanical effects involving magnetic fields.The fields can be understood as the outcome of a deformation with vector-valuedfunctions of the coordinate operator. The other way of understanding the result isthe following. The coupling of an external magnetic field in QM is well understoodand studied for various physical applications and models. Deformation on theother hand is a mathematical tool, rather than a procedure that generates physicaleffects. Hence, in these examples deformation of a QM system can be understoodas the coupling of an external field. Thus, if the deformation goes hand in handwith Moyal-type spaces one sees in these examples that Moyal spaces correspond toordinary spaces in the presence of an external field. By having this observation inmind it does not seem far fetched that certain deformations of spacetime correspondto gravitation. Let us describe in the next sections how Moyal-Weyl spaces arise inthis context.
To describe the circular motion of an electron in the lowest Landau level we define theso called guiding center coordinates Q , [Eza08, Sza04] Q i := X i + 12 ( B − ) ik P k , with matrix B ij = − ( e/ ε ijk B k . Note that the inverse corresponds to the non-degenerate sub-matrix of B ij . By using the CCR it becomes apparent that the guidingcenter coordinates span a three dimensional Moyal-Weyl plane, i.e. [ Q i , Q j ] = i ( B − ) ij . (4.8)Thus, the Landaueffect is an example of a physical noncommutative space. Now canwe generate these noncommuting coordinates by the deformation procedure warpedconvolutions? Yes we can! Lemma
The deformed coordinate operators X jθ, P given as (see Equation 3.12) X jθ, P = X j − θ jr P r , (4.9) satisfy the commutation relations of the Moyal-Weyl plane R − θ , [ X iθ, P , X jθ, P ] = − iθ ij . (4.10) Moreover, let − θ ij be ( B − ) ij then the deformed coordinate operators X iθ, P are equalto the guiding center coordinates given in Equation (4.8).Proof. The commutator of the deformed coordinate operator is calculated by using thecanonical commutation relations and the skew-symmetry of the deformation matrix θ jk . [ X jθ, P , X kθ, P ] = [ X j − θ jr P r , X k − θ kl P l ]= − θ kl [ X j , P l ] + θ jl [ X k , P l ]= − iθ jk . .5 Gravitomagnetism in QM 15 Lemma 4.3 gives a well defined path to obtain an effective quantum plane by thedeformation using warped convolutions. As we showed, the lemma follows from wellunderstood physical models and ideas, which are in circulation in condensed matterfield theory, for quite some time. In the example of the Landau problem one definesguiding center coordinates, which satisfy the commutator relations of the Moyal-WeylPlane. The reader is cautioned to notice that the effective quantum plane obtained bythe Landau problem is not merely an abstract construct but has the precise meaning,that the space coordinates can not be measured simultaneously. A more precise math-ematical way to obtain this Moyal-Weyl plane is introduced in this work. We obtainthe Landau problem by deforming the Hamiltonian of a free non-relativistic particlewith the coordinate operator and by setting the deformation parameter equal to amagnetic field. Furthermore, we show that the noncommuting coordinates referredto as the guiding coordinates are obtained by deforming the coordinate operator, us-ing the momentum operator. In our opinion, this method can be further used in thequantum field theoretical (QFT) approach to define an effective quantum plane.
The emergence of gravitomagnetism in QM from deformation theory is one of thecenterpieces of this work. Before we prove the emergence of these effects let us in-troduce some basic notations. We consider slowly varying weak gravitational fieldswith energy momentum tensor of ordinary matter (dust-like). In this description themetric can be written as g µν = η µν + h µν , where h is the small perturbation from the flat spacetime and the energy momentumtensor can be written as T µν = ρu µ u ν , where u is the -velocity and ρ the scalar density. For slowly varying fields, thelinearized Einstein field equations can be described by Maxwell-like field equationsgiven by ∆ φ = 4 πGρ, ∆ h j = − πGρv j , φ − ∇ · h j = 0 , with the definitions of the potentials φ := h / , h j := h j , where we used the Lorentz condition and the fact that the fields considered are slowlyvarying, i.e. ¨ φ , ˙ h k and ¨ h k can be neglected, (see for example [AC10], [Wei72]). Analo-gously to the electromagnetic case the gravitoelectric field g and the gravitomagneticfield Ω are both defined by the potentials as g = − ∇ φ, Ω = ∇ × h . There are a few important examples that can be considered in the context gravito-magnetism. One of them is example of the vector potential h inside a hollow spinningsphere with radius r hs and spin ω that is given by, [Wei72, Eq. 9.4.35] h ( x ) = x × Ω , (4.11)where Ω = 2
M G/r hs is the constant gravitomagnetic field inside the hollow sphere.Now in [AC10] the authors derived the non-relativistic Schrödinger equationfor a particle that is minimally coupled to an external electromagnetic andgravitoelectromagnetic field. The equation is given by [AC10, Eq. 5.1], .5 Gravitomagnetism in QM 16 H GEM
Ψ = − m ( P − eA ) i ( P − eA ) i Ψ − h i ( P − eA ) i Ψ , (4.12)where we set the potential V, φ equal to zero and neglected the term of secondderivative in Ψ . This can be done since the term is just a relativistic correction whichfor slowly moving bodies can be neglected.By using the former definitions and results we are able to reproduce the caseof a constant gravitomagnetic field by deformation. Lemma
Let the deformation matrix B ij be given as B ij = m ε ijk Ω k , (4.13) where Ω k = (2 GM/r hs ) ω k is a constant gravitomagnetic field for a hollow spinningsphere. Then, the deformed free Hamiltonian ( H ) B, X , becomes the Hamiltonian ofa quantum mechanical particle minimally coupled to a constant gravitomagnetic field,i.e. ( H ) B, X Ψ( x ) = − m ( P j + m h j )( P j + m h j )Ψ= H Ψ( x ) − h j P j Ψ( x ) + O ( h ) , Ψ ∈ E , where the vector h j = ε jkl x k Ω l represents the gravitomagnetic vector potential for ahollow spinning sphere (see Equation (4.11)).Proof. The free deformed Hamiltonian ( H ) B, X is given by ( H ) B, X Ψ = − m ( P j + B jk X k )( P j + B jr X r )Ψ= − m ( P j + m h j )( P j + m h j )Ψ= H Ψ( x ) − h j P j Ψ( x ) + O ( h ) , Ψ ∈ E , where in the last line we set the deformation matrix B ij = m ε ijk (2 GM/r hs ) ω k andwe neglected second order terms since we work in the linear approximation.This an important result, since this means that we obtain gravitational effects froma well-defined deformation procedure by simply adjusting the deformation constantsaccordingly. Thus, gravitomagnetism can be understood as the outcome of a defor-mation procedure. Moreover, the physical constant used as deformation parameterin the gravitomagnetic case is the gravitational constant G . Since by setting thegravitational constant to zero, i.e. neglecting gravitational effects, the deformedHamiltonian describing gravitomagnetic effects becomes the free Hamiltonian.Next we use Proposition 2.1 of the deformation technique to obtain the elec-tromagnetic and gravitomagnetic coupling. Proposition
Let the deformation matrix B ij be given as B ij = m ε ijk Ω k , where Ω k = (2 GM/r hs ) ω k is the constant gravitomagnetic field for a hollow spinningsphere and let the deformation matrix B ij be given as B ij = − ( e/ ε ijk B k , .6 Lense-Thirring Precession 17 where B k is a homogeneous magnetic field.Then, the deformed free Hamiltonian (cid:0) ( H ) B , X (cid:1) B , X becomes the Hamiltonian H GEM (see Equation 4.12) of a quantum mechanical particle minimally coupled to aconstant external magnetic and gravitomagnetic field, i.e. ( H ) B + B , X Ψ = H GEM Ψ , Ψ ∈ E . Proof.
First of all by the virtue of Proposition 2.1 the deformed Hamiltonian satisfies (cid:0) ( H ) B , X (cid:1) B , X Ψ = ( H ) B + B , X Ψ . Next we consider the free deformed Hamiltonian ( H ) B + B , X , (see Equation (4.2)). ( H ) B + B , X Ψ = − m (cid:0) P + (cid:0) ( B + B ) X (cid:1)(cid:1) j (cid:0) P + (cid:0) ( B + B ) X (cid:1)(cid:1) j Ψ= − m ( P j − eA j + mh j )( P j − eA j + mh j )Ψ= − m ( P − eA ) j ( P − eA ) j Ψ − h j ( P − eA ) j Ψ + O ( h ) , where we set the deformation matrix B ij = mε ijk (2 GM/r hs ) ω k and B ij = − ( e/ ε ijk B k and h j is the gravitomagnetic vector potential given in Equation (4.11)for a hollow spinning sphere and − A j the magnetic vector potential given in theLandau quantization, (see Equation (4.1)).From this result it becomes clear that in a quantum mechanical setting one can ob-tain electromagnetic and gravitomagnetic effects by a deformation procedure. In theframework of deformation these effects simply correspond to certain deformation pa-rameters that in turn are given by physical constants. One should also note that weobtained the Hamiltonian of a quantum mechanical system that is coupled to an ex-ternal magnetic and gravitomagnetic field by deformation, rather than by advancedcalculations and considerations as done in [AC10]. Another important gravitomagnetic effect is known under the name of Lense–Thirringprecession. The effect is a general-relativistic correction to the precession of a gyro-scope outside a massive stationary spinning sphere. The vector potential for suchgravitomagnetic field is given as h = − (2 GI/r ) x × ω , (4.14)where I is the moment of inertia of the sphere and r = | x | the radius.As for the constant gravitomagnetic field, we are also able to produce the vec-tor potential of the Lense–Thirring effect. Proposition
Let the deformation matrix B ij and the operator Q j be given as B ij = m ε ijk Ω k , Q j ( X ) = X j / | X | / . (4.15) where Ω k = (2 GI ) ω k and I is the moment of inertia of a spinning sphere. Then,the deformed free Hamiltonian ( H ) B, Q , becomes the Hamiltonian of a quantum me-chanical particle minimally coupled to the gravitomagnetic field of the Lense-Thirring .6 Lense-Thirring Precession 18 effect, i.e. ( H ) B, Q Ψ = − m (cid:0) P j + ( BX ) j / | X | (cid:1) (cid:0) P j + ( BX ) j / | X | (cid:1) Ψ , Ψ ∈ E = H Ψ − h j P j Ψ + O ( h ) , where the vector potential induced by deformation is the gauge field of the Lense-Thirring effect, i.e. h j = m ε jkl (2 GI ) ω l X k / | X | .Proof. To prove this proposition we use the spectral measure representation. Thedeformation of H is then given as follows, ( H ) B, Q Ψ = − m ( P j + ( BQ ) s [ Q s , P j ])( P j + ( BQ ) r [ Q r , P j ])Ψ= − m (cid:0) P j + ( BX ) j / | X | (cid:1) (cid:0) P j + ( BX ) j / | X | (cid:1) Ψ= − m (cid:18) P j + m h j (cid:19)(cid:18) P j + m h j (cid:19) Ψ= H Ψ − h j P j Ψ + O ( h ) , where in the last lines we used the skew-symmetry of B and the commutator relation . .Next, we use the deformation technique to obtain the electromagnetic and gravito-magnetic coupling in the case of the Lense-Thirring effect. The effects emerge bya double deformation where once we use the coordinate operator and after that theoperator-valued vector Q j ( X ) . Note that the order of the deformation is irrelevant,since the two operators commute. Remark . The proof that the deformation with two different operators is well-defined, is equivalent to proving Proposition 3.1, where one replaces the free Hamil-tonian in Inequality (3.10) with ( H ) B , X . It then follows that for Φ ∈ S ( R ) , theexpression (cid:107) ( H ) B , X Φ (cid:107) is finite. Proposition
Let the deformation matrix B ij be given as B ij = m ε ijk Ω k , (4.16) where Ω k = (2 GI ) ω k and let the deformation matrix B ij be given as B ij = − ( e/ ε ijk B k , (4.17) where B k is a homogeneous magnetic field. Moreover, let the operator Q j ( X ) be givenby Q j ( X ) = X j / | X | / . Then, the deformed free Hamiltonian (( H ) B , X ) B , Q becomes the Hamiltonian H GEM of a quantum mechanical particle minimally coupled to a constant external magneticand the gravitomagnetic field of the Lense-Thirring effect, i.e. (( H ) B , X ) B , Q Ψ = H GEM Ψ , Ψ ∈ E . Proof.
The free deformed Hamiltonian ( H ) B , X is given by, (see Equation (4.2)). ( H ) B , X Ψ = − m (cid:0) P + (cid:0) B X (cid:1)(cid:1) j (cid:0) P + (cid:0) B X (cid:1)(cid:1) j Ψ . Due to the commutativity of the coordinate operators, deformations with Q j ( X ) do not influence the gauge field ( B X ) and vice versa, i.e. (( H ) B , X ) B , Q = .7 Gravitomagnetic Zeemaneffect 19 (( H ) B , Q ) B , X . Thus, after choosing the deformation parameters as stated in Equa-tions (4.16) and (4.17) we obtain for the deformed free Hamiltonian, (( H ) B , Q ) B , X Ψ = − m ( P − eA + m h ) j ( P − eA + m h ) j Ψ= − m ( P − eA ) j ( P − eA ) j Ψ − h j ( P − eA ) j Ψ + O ( h ) , where h j is the gravitomagnetic gauge field of the Lense-Thirring effect (see Equation(4.14)) and − A j the magnetic vector potential given in the Landau quantization, (seeEquation (4.1)). Similar to the magnetic case, where the Zeemaneffect emerged by deforming the hydro-gen Hamiltonian with the same deformation matrix used in the Landau quantization,we precede in the gravitomagnetic case. Thus, we are able to predict a gravitomagneticZeemaneffect by deforming the hydrogen atom and using the constant gravitomagneticdeformation matrix.
Lemma
Let the deformation matrix B ij be given as, B ij = m ε ijk Ω k , where Ω k = (2 GM/r hs ) ω k is the constant gravitomagnetic field for a hollow spinningsphere. Then, the deformed Hamiltonian of the hydrogen atom, denoted by ( H A ) B , X ,becomes the Hamiltonian of the gravitomagnetic Zeemaneffect, i.e. ( H A ) B , X Ψ = − m ( P j + m ε jkl Ω l X k )( P j + m ε jrs Ω s X r )Ψ + e | X | Ψ , Ψ ∈ E = − m ( P j + m h j )( P j + m h j )Ψ + e | X | Ψ= H Ψ − h j P j Ψ + e | X | Ψ + O ( h ) . Proof.
The only difference to the proof of Lemma 4.2 consists in the choice of thedeformation matrix, i.e. the proof is equivalent.Analogously to the magnetic case, the presence of a constant gravitomagnetic fieldwill split the spectral lines of the hydrogen atom. In this case the splitting depends onthe strength of the gravitomagnetic field. This phenomenon is the gravitomagneticZeemaneffect , [Mas00]. Note that the linear approximation works just fine, since thequadratic terms of the gauge field are already neglected in the magnetic Zeemaneffect,[Str02].In the next proposition we couple the two constant forces by a double defor-mation.
Proposition
Let the deformation matrix B ij be given as B ij = m ε ijk Ω k , where Ω k = (2 GM/r hs ) ω k is the constant gravitomagnetic field for a hollow spinningsphere and let the deformation matrix B ij be given as B ij = − ( e/ ε ijk B k , .8 Arbitrary static gauge field 20 where B k is a homogeneous magnetic field.Then, the deformed Hamiltonian of the hydrogen atom, (cid:0) ( H A ) B , X (cid:1) B , X be-comes the Hamiltonian of the Zeemaneffect generated by a a constant externalmagnetic and gravitomagnetic field, i.e. ( H A ) B + B , X Ψ = H GEM Ψ , Ψ ∈ E . Proof.
Since the deformation with the coordinate operator commutes with the po-tential term of the hydrogen atom, the proof is analog to the proof of Proposition4.4.
By only assuming the principle of Galilei-invariance the author in [Jau64] succeededin deriving the minimally coupled Hamiltonian plus a potential. Thus by demandingthat our deformed Hamiltonian respects the Galilei-invariance, we have to add apotential. This is justified since we showed that the deformation of a free Hamiltonianinduces electromagnetism and gravitomagnetism. Moreover, in [Jau64] it was shownthat the gauge field and the potential can only depend on the coordinates. Therefore,our deformation covers the whole range of abelian gauge fields, since we can inducesuch fields by choosing functions of the coordinate operator to obtain an arbitrarygauge field. This fact is used in the next sections to induce a variety of physical effects.In the next proposition we show the importance of adding a potential to theHamiltonian and for this purpose we need the four-momentum given as P µ = ( H , P i ) = (cid:0) − P k P k / (2 m ) + g φ ( X ) , P i (cid:1) , where φ ( X ) is the electromagnetic potential φ E or the gravitoelectromagnetic potential − φ G . Moreover, g is a coupling constant given by e in the electromagnetic case andby − m in the gravitoelectromagnetic case. Proposition
Let the gauge field induced by deformation of the Hamiltonian ( H ) B, Q (see Proposition 3.1) be defined as − g A r ( X ) := ( BQ ( X )) k ∂ r Q k ( X ) , (4.18) where A is the electromagnetic or the gravitoelectromagnetic vector potential. Then,the commutator of the deformed momentum vectors gives the spatial part of the fieldstrength tensor F ij , [ P B, Q i ,P B, Q j ] = − ig F ij ( X ) . (4.19) Furthermore, the commutator of the deformed Hamiltonian with the deformed momen-tum gives the Lorentz force F Lj , i.e. [ P B, Q ,P B, Q j ] = − g [ φ ( X ) , P j ] − i gm F jk ( X ) P kB, Q = iF Lj . (4.20) Moreover, by using the Jacobi identities for the commutator relations betweenthe deformed momentum and Hamiltonian operators the homogeneous Maxwell-equations emerge.Proof.
According to Proposition 3.1, the deformation of H by an operator Q is givenas follows ( H ) B, Q Ψ = − m (cid:18) P j + i ( BQ ) s [ Q s , P j ] (cid:19)(cid:18) P j + i ( BQ ) r [ Q r , P j ] (cid:19) Ψ + g φ ( X )Ψ= 12 m (cid:18) P − g A ( X ) (cid:19) Ψ + g φ ( X )Ψ , .8 Arbitrary static gauge field 21 where we used the fact that the potential can only be a function of the coordinateoperator and thus remains invariant under deformation. Since in the former propo-sitions we identified deformations induced by the coordinate operator with the gaugefield − g A , the induced term in this deformation can be identified with a generalstatic gauge field. Next, we calculate Commutator (4.19). The deformed momentumoperator is given in Proposition 3.11 as P jB, Q Ψ = ( P j + i ( BQ ) s [ Q s , P j ])Ψ= ( P j − g A j ( X ))Ψ , where in the last lines we identified the gauge field by Equation (4.18). Now by usingthe commutator relations [ X i , X j ] = 0 and the fact that the commutator [ Q s , P j ] isagain only a function of the coordinate operator, we obtain the following commutatorrelations for the deformed momentum operator [ P kB, Q , P lB, Q ] = − g [ P k , A l ( X )] − k ↔ l = − ig F kl ( X ) . Now we can also rewrite the deformed Hamiltonian ( H ) B, Q as ( H ) B, Q = − (1 / m ) (cid:16) P jB, Q P B, Q j (cid:17) + g φ ( X ) . This form of the deformed Hamiltonian simplifies the calculation of Commutator(4.20). [( H ) B, Q , P kB, Q ] = − (1 / m ) [ (cid:16) P jB, Q P B, Q j (cid:17) , P kB, Q ] + g [ φ ( X ) , P k ]= i gm P B, Q j F jk ( X ) − ig ∂ k φ ( X )= ig (cid:16) E k ( X ) − ε kjl V B, Q j B l ( X ) (cid:17) , In the last lines we used the commutator relation [ P B, Q j , F jk ( X )] = 0 and theHeisenberg-equation to identify the velocity operator with the deformed momentum.Moreover, the fields E and B are the electromagnetic or the gravitoelectromagneticfields, depending on the considered case. It is not surprising that the Lorentz forceis obtained by calculating the commutator between the deformed Hamiltonian andthe momentum, since it gives the equations of motion for the deformed system.This in turn is properly identified with a particle coupled to an electromagnetic orgravitoelectromagnetic force.Next, we use the Jacobi identities for the commutators of the deformed mo-mentum operator P µ to obtain the homogeneous Maxwell-equations. From the Jacobiidentity for the spatial part we have [ P B, Q k , [ P B, Q i ,P B, Q j ]] + cyclic = − ig [ P B, Q k ,F ij ( X )] + cyclic = g ∂ k F ij ( X ) + cyclic = 0 . The last equation is the relativistic expression for the spatial part of the homogeneousMaxwell-equations. To obtain the homogeneous Maxwell-equations involving F j = E j we look at the other Jacobi identity of the deformed momentum, i.e. [ P B, Q , [ P B, Q i ,P B, Q j ]] + [ P B, Q i , [ P B, Q j ,P B, Q ]] − i ↔ j = 0 . (4.21) .9 Gravitomagnetic Moyal-Weyl plane 22 Let us take a look at the first term, [ P B, Q , [ P B, Q i ,P B, Q j ]] = − ig [ P B, Q ,F ij ( X )]= − g ∂ F ij ( X ) − gm P B, Q k ∂ k F ij ( X ) , where we used the Heisenberg-equation. The other two terms in Equation (4.21) give − i [ P B, Q i , F Lj ] − i ↔ j = − g ( ∂ i E j ( X ) − ∂ j E i ( X )) − gm P kB, Q ( ∂ i F jk ( X ) − ∂ j F ik ( X )) . By summing the two terms and using the spatial part of the homogeneous Maxwell-equations we obtain [ P B, Q , [ P B, Q i ,P B, Q j ]] + cyclic = − g ( ∂ i E j ( X ) − ∂ j E i ( X )) − g ∂ F ij ( X ) . Remark . In [Dys90] the author recollected an argument given by Feynman aboutthe most general force, compatible with the naive commutation relations [ X i , X j ] = 0 , m [ X i , ˙ X j ] = iδ ij . It turns out that by assuming a noncommutative structure for the velocity operators,one obtains the electromagnetic force. Furthermore, by using the Jacobi identitiesthe Maxwell equations follow. Therefore, in a sense the deformations with warpedconvolutions reproduce the result of Feynman in a more sophisticated language, andmoreover, the technique gives concrete representations of the operators that generatethe electromagnetic and gravitoelectromagnetic fields. Thus, by the virtue of thedeformation technique we have a deeper understanding of the surprising result ofFeynman.A crucial point is implied in the last proposition. The linear field equations of generalrelativity or the Maxwell equations emerge from a well-defined deformation procedure.The emergence is owed to the Jacobi identities. This observation gives an insightfulhint how to receive substantial field equations from a deformation procedure.
By using the deformations techniques we are able to understand how to generatethe Landau quantization. Thus, by using the same procedures and by setting thedeformation parameter equal to a constant gravitomagnetic field (see Lemma 4.4)we also obtain a Landau quantization in the gravitomagnetic case.Analogously to the Landau quantization in the magnetic case, we can solvethe eigenvalue equation of the deformed Hamiltonian ( H ) B,X with deformationmatrix B ij = mε ijk Ω e k , i.e. with a constant homogeneous gravitomagnetic field in x -direction. The eigenvalue problem can be solved by diagonalizing the Hamiltonianand we obtain the following energy eigenvalues E B,n = p m + (cid:18) n + 12 (cid:19) ω B , p ∈ R , n ∈ N , where the frequency of this harmonic oscillator is given by ω B = 2Ω . Therefore,quantum mechanical particles in a constant gravitomagnetic field can only occupyorbits with discrete energy values. This effect is the gravitomagnetic analogue of theLandau quantization in the magnetic case. In Section 4.4 we identified the noncommutative coordinates of the Landau-quantization with the deformed coordinate operator. In the same manner we canobtain a physical Moyal-Weyl plane for the constant gravitomagnetic case. For thispurpose, we set the deformation matrix of X θ, P equal to the inverse of a constantgravitomagnetic field times the coupling constant m . The deformed coordinateoperators X B − , P are then equal to the guiding center coordinates of an electron inthe lowest Landau level and are given by X B − , P i = X i + 12 ( B − ) ik P k , with commutator relations [ X B − , P i , X B − , P j ] = i ( B − ) ij , and deformation matrix B ij = mε ijk Ω k . From the commutator relations we have thefollowing uncertainty relations (∆ X B − , P )(∆ X B − , P ) ≥ (cid:126) / ( m Ω) . These uncertainty relations have the precise meaning that coordinates of an electroncan not be measured more accurately than the area π (cid:126) / ( m Ω) . This is a physicaleffect that we predicted from a deformation procedure and may be experimentallyverified. We obtained a variety of physical effects, in a QM context, containing electromag-netism and gravitomagnetism. These effects were generated by the deformationprocedure warped convolutions. Thus, in this sense those two fundamental forcescan be understood as deformations of free theories. The fundamental deformationparameters , for those forces, are given by the elementary electric charge e and bythe gravitational constant G . Therefore, not only (cid:126) and c can be used to deform theclassical case (Galilei group) but also e and G play the role of deformation parametersresponsible for electromagnetism and gravitomagnetism.The deformation also shed a new light on the dynamics of a quantum me-chanical particle in the presence of electromagnetic and gravitomagnetic forces.Namely, it gives a systematic derivation of a non-relativistic Hamiltonian in thepresence of electromagnetic and gravitomagnetic effects.Another striking implication of the deformation procedure is the deduction ofa physical Moyal-Weyl plane. This plane is generated from the gravitomagnetic fieldtimes the mass and thus the strength of noncommutativity of the coordinates, inthe lowest Landau level, is given by the inverse of the constant m Ω / (cid:126) . This effectwas purely deduced from deformation and could be one of the first effects that istheoretically predicted by deformation and verified experimentally. This would be amajor physical breakthrough for deformation theory.To obtain electric and gravitoelectric fields in the framework of deformation,we would have to extend the definition of warped convolutions. For example, it isnot possible to obtain the Stark effect from deformation of the free Hamiltonian withwarped convolutions. The reason herein lies in the fact, that the deformation leavesthe spectrum of the operator invariant. Since the free Hamiltonian has a positivespectrum and the Hamiltonian of the Stark effect has the whole real line as spectrum, EFERENCES 24 a deformation respecting spectrum conditions can not reproduce such an effect.Another line of work involves the extension of warped convolutions to a non-abelian setting. If this succeeds, we would be able to reproduce the weak and stronginteraction as deformations. In this context, it seems intuitive to lift such deforma-tions to a QFT setting. Thus, recasting the fundamental forces as deformations andmost likely simplifying calculations involving interactions.These and many other lines of work, in deformation theory, are to this dateopen and provide a broad, interesting and exciting area of research.
Acknowledgments
I would like to thank Prof. K. Sibold for insightful remarks. I am particularly gratefulfor the assistance given by Dr. G. Lechner. Moreover, I thank S. Alazzawi, A. Ander-sson, S. Pottel, J. Zschoche and Hyun Seok Yang for many discussions. I would liketo express my great appreciation to Dr. Z. Much for an extensive proofreading.
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